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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p><span style="color: #3c3c3c; font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">Now that we have taken a look at linear regression</span>, in this section we will provide an overview of three common and very powerful analysis methods: logistic regression, decision trees and Cox proportional hazards models. These are the foundation for most data analysis conducted in clinical studies in addition to linear regression that was explained in the previous unit.</p>
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<h3>Learning objectives</h3>
<ol style="margin-top: 0pt; margin-bottom: 0pt;">
<li>Understanding the foundations of other common data analysis approaches for health data.</li>
<li>Present and interpret the results of these analysis types.</li>
</ol>
<p>As for previous sections of this course, some programming knowledge of R will be essential to understand the material displayed. </p>
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<h3>Credits</h3>
<p>Aldo Arévalo, Marie-Laure Charpignon, Mathew Samuel, Philips Samuel, Jesse D. Raffa, Marzyeh Ghassemi, Tristan Naumann, Mengling Feng and Douglas Hsu.</p>
<p><span style="color: rgb(49, 49, 49); font-family: "Open Sans", "Helvetica Neue", Helvetica, Arial, sans-serif;">Videos in this unit are presented by Jesse Raffa and Alistair Johnson.</span></p>
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<h2 class="hd hd-2 unit-title">Logistic Regression</h2>
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<p>Logistic regression allows us to extend the general framework of what we learned in the previous section on linear regression and apply these same principles to binary outcomes. Quite often, the variables we care most about with respect to health outcomes are discrete binary outcomes (e.g. dead/alive, cured/not cured). </p>
<p>Why not just use linear regression? As just one reason, probabilities need to be somewhere between zero and one, and there is nothing in linear regression to constrain the estimated probabilities to this interval, which is problematic as probabilities could be negative or greater than 1.</p>
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<h2 class="hd hd-2 unit-title">Logistic Regression: Contingency Tables</h2>
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<p dir="ltr" style="line-height: 1.38; margin-top: 0pt; margin-bottom: 0pt;">To understand the intuition behind what is happening, let us start by examining contingency tables.</p>
<p dir="ltr" style="line-height: 1.38; margin-top: 0pt; margin-bottom: 0pt;"></p>
<p dir="ltr" style="line-height: 1.38; margin-top: 0pt; margin-bottom: 0pt;"><span style="font-size: 11pt; font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">A contingency table cross-tabulates the outcome across two or more levels of a covariate. When you further stratify your outcome on a variable (feature, factor etc.) you can sometimes observe more dramatic differences within groups that suggest the variable may be important. Let us now consider the example of 28 day mortality distributed among individuals over and under the age of 55. </span></p>
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<p dir="ltr" style="line-height: 1.38; margin-top: 0pt; margin-bottom: 0pt;"><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">From the above table, you can see that 40 patients in the young group (≤55) died within 28 days, while 243 in the older group died. This corresponds to P(die|age≤55) = 40/(883+40) = 0.043 = 4.3 % and P(die|age > 55) = 0.284 or 28.4 %, where the “|” can be interpreted as “given” or “for those who have.” This difference is quite marked, and we know that age is an important factor in mortality, so this is not surprising.</span></p>
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<p dir="ltr" style="line-height: 1.38; margin-top: 0pt; margin-bottom: 0pt;"> </p>
<p dir="ltr" style="line-height: 1.38; margin-top: 0pt; margin-bottom: 0pt;"><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">It is also worth briefly reviewing the definition of odds: The odds of an event happening is the proportion that the event occurred divided by the proportion that the event did not occur. </span></p>
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<p class="MsoNormal" style="font-size: 16px;">Here, the odds of dying within 28 days in the young group is 0.043/(1-0.043) = 0.045, and in the older group is 0.284/(1-0.284) = 0.40. It is convenient to represent these two figures as a ratio, and the choice of what goes in the numerator and the denominator is somewhat arbitrary. In this case, we will choose to put the older group’s odds on the numerator and the younger in the denominator, and it’s important to make clear which group belongs in the numerator and the denominator in general. In this case, the odds ratio is 0.40/0.045 = 8.79, which indicates a very strong association between age and death, and means that the odds of dying in the older group is nearly 9 fold higher than when compared to the younger group.</p>
<p></p>
<p class="MsoNormal" style="font-size: 16px;">Whenever you perform calculations with odds, it is important to be clear in defining the group for comparison (i.e. what your reference group is). If this was a clinical trial with two drug arms and a placebo arm, it would not make sense to use a drug arm as the reference group; instead, it is important to use the placebo arm as the reference group for both drug arms to accurately understand the efficacy of the treatments. </p>
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<p>Though contingency tables are a good way of understanding the nature of your data, there are some limitations:</p>
<ul>
<li>What happens when the covariates of interest are continuous?</li>
<li>We could of course create categories from the covariate by establishing cut points, but we may still miss some important aspects of the relationship between the covariate and the outcome by not choosing the right cut points.</li>
<li>What happens when we know that a nuisance covariate is related to both the outcome and the covariate of interest?<br />This type of nuisance variable is called a confounder and occurs frequently in observational data, and although there are ways of accounting for confounding in contingency tables they become more difficult to use when there is more than one present.</li>
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<h2 class="hd hd-2 unit-title">Logistic Regression: Introduction</h2>
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<p>Logistic regression is one way to address both of these challenges. Logistic regression uses a link function known as the logit function to map probabilities in the interval [0,1] to a real number (-∞,∞).</p>
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<h3 dir="ltr" style="line-height: 1.38; margin-top: 0pt; margin-bottom: 0pt;">Code Example</h3>
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<h3>Coefficients</h3>
<p>As you can see, we get a coefficients table that is similar to the lm table we used earlier. Instead of a t-value, we get a z-value, but this can be interpreted similarly. The rightmost column is a p-value, for testing the null hypothesis β=0.</p>
<p>If you recall, the non-intercept coefficients are log-odds ratios, so testing if they are zero is equivalent to testing if the odds ratios are one. If an odds ratio is one the odds are equal in the numerator group and denominator group, indicating the probabilities of the outcome are equal in each group. So, assessing if the coefficients are zero will be an important aspect of doing this type of analysis.</p>
<p>Looking more closely at the coefficients. The intercept is −3.09 and the age.cat coefficient is 2.17. The coefficient for age.cat is the log odds ratio for the 2 x 2 table we previously did the analysis on. When we exponentiate 2.17, we get exp(2.17) = 8.79. This corresponds with the estimate using the 2 x 2 table.</p>
<p>For completeness, let’s look at the other coefficient, the intercept. If you recall, <img height="19" width="127" src="/assets/courseware/v1/b599513e6716804201760373a8b9753c/asset-v1:MITx+HST.953x+3T2020+type@asset+block/Screenshot_2020-11-29_at_13.02.26.png" alt="" />, so <span style="text-align: justify;">\(\beta_{0}\)</span><span style="font-size: 1em;"> is the log odds of the outcome in the younger group. Exponentiating again, exp(−3.09) = 0.045, and this corresponds with the previous analysis we did. Similarly, <img height="17" width="143" src="/assets/courseware/v1/9a6c9b6c926afef859317ab9141064fa/asset-v1:MITx+HST.953x+3T2020+type@asset+block/Screenshot_2020-11-29_at_12.59.31.png" alt="" /></span><span style="font-size: 1em;">, and the estimated odds of 28 day death in the older group is exp(−3.09 + 2.17) = 0.4, as was found above. </span>The case with multiple variables is similar to the previous example with linear regression, the key difference again being the need to exponentiate the coefficients for the sake of interpretation. </p>
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<h2 class="hd hd-2 unit-title">Logistic Regression: Question I</h2>
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<h2 class="hd hd-2 unit-title">Logistic Regression: Hypothesis Testing</h2>
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<h3>Hypothesis Testing</h3>
<p style="font-size: 16px;">Just as in the case of linear regression, there is a way to test hypotheses for logistic regression. It follows much of the same framework, with the null hypothesis being β = 0. If you recall, this is the log odds ratio, and testing if it is zero is equivalent to a test for the odds ratio being equal to one.</p>
<p style="font-size: 16px;">To examine this concept, the MIT Critical Data textbook <a href="https://www.springer.com/gp/book/9783319437408" target="[object Object]">Secondary Analysis of Electronic Health Records</a> page 232 walks through an example first fitting two competing models, a larger (alternative model), and a smaller (null model) and then using the ANOVA function to assess the difference. Using a <em>Chi-squared</em> test through the ANOVA function is a common way to test whether a categorical covariate should be retained in the model, as it can be difficult to assess using the z-value in the summary table, particularly when one is very statistically significant, and one is not.</p>
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<h3>Confidence Intervals</h3>
<p>Generating confidence intervals for either the log-odds ratios or the odds ratios are relatively straightforward through the use of the confint() function. </p>
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<h2 class="hd hd-2 unit-title">Logistic Regression: Question II</h2>
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<input type="checkbox" name="input_44d1e57d95834b32a8d49bcf7def6ed0_2_1[]" id="input_44d1e57d95834b32a8d49bcf7def6ed0_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="44d1e57d95834b32a8d49bcf7def6ed0_2_1-choice_0-label" for="input_44d1e57d95834b32a8d49bcf7def6ed0_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_44d1e57d95834b32a8d49bcf7def6ed0_2_1"> We are 95% confident that our confidence interval is the coefficient, with the standard error corresponding to how wrong we could be.
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<input type="checkbox" name="input_44d1e57d95834b32a8d49bcf7def6ed0_2_1[]" id="input_44d1e57d95834b32a8d49bcf7def6ed0_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="44d1e57d95834b32a8d49bcf7def6ed0_2_1-choice_1-label" for="input_44d1e57d95834b32a8d49bcf7def6ed0_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_44d1e57d95834b32a8d49bcf7def6ed0_2_1"> We are 95% confident that the interval contains the true value.
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<h2 class="hd hd-2 unit-title">Logistic Regression: Interpretation of Output</h2>
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Numerical Input
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<legend id="ee1b088afc8a4309a995d79377f5adba_2_1-legend" class="response-fieldset-legend field-group-hd">Using R and referencing the image above, what is the increase in the odds of the 28-day mortality for those &gt;55 years of age relative to the younger group, holding service units constant?</legend>
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<input type="radio" name="input_ee1b088afc8a4309a995d79377f5adba_2_1" id="input_ee1b088afc8a4309a995d79377f5adba_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ee1b088afc8a4309a995d79377f5adba_2_1-choice_1-label" for="input_ee1b088afc8a4309a995d79377f5adba_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ee1b088afc8a4309a995d79377f5adba_2_1"> 3.08
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<input type="radio" name="input_ee1b088afc8a4309a995d79377f5adba_2_1" id="input_ee1b088afc8a4309a995d79377f5adba_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="ee1b088afc8a4309a995d79377f5adba_2_1-choice_3-label" for="input_ee1b088afc8a4309a995d79377f5adba_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_ee1b088afc8a4309a995d79377f5adba_2_1"> 6.78
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<h2 class="hd hd-2 unit-title">Logistic Regression: Prediction</h2>
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<p>After fitting a final model, a common task is to generate subsequent predictions. Such a task may occur when doing a propensity score analysis or creating tools for clinical decision support. In the logistic regression setting, this involves attempting to estimate the probability of the outcome given the characteristics (covariates) of a patient. This quantity is often denoted P(outcome|X). We followed this by adding a pred column to our new data frame by using the predict function.</p>
<p>The predict function for logistic regression works similar to when we used it for linear regression, but this time we also specify type = ”response” which ensures the quantities computed are what we need, P(outcome|X). Outputting this new object shows our predicted probability of 28-day mortality for six hypothetical patients. Two in each of the service units where one is in the younger group and another in the older group. We see that our lowest prediction is for the youngest patients in the FICU, while the patients with the highest risk of 28-day mortality are the older group in the MICU, but the predicted probability is not all that much higher than the same age patients in the SICU. To do predictions on a different dataset, just replace the newdata argument with the other dataset. We could, for instance, pass newdata = dat and receive predictions for the dataset we built the model on. As was the case with linear regression, evaluating the predictive performance of our model on data used to build the model will generally be too optimistic as to how well it would perform in the real world. How to get a better sense of the accuracy of such models is covered in Chap. 17 of the <a href="https://www.springer.com/gp/book/9783319437408" target="[object Object]">textbook</a>.</p>
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<h2 class="hd hd-2 unit-title">Logistic Regression: Model Results</h2>
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<p dir="ltr" style="line-height: 1.38; margin-top: 0pt; margin-bottom: 0pt; margin-left: 6pt;">This section explains how results are often presented and how to interpret these models. The presentation generally follows what was previously described in a linear regression setting:</p>
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<p>Results should always be put into context.</p>
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<p>Reporting the results should always include some form of the coefficient estimate, a measure of uncertainty and likely a p-value.</p>
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<p>In medical and epidemiological journals, coefficients are usually exponentiated so that they are no longer on the log scale and reported as odds ratios.</p>
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<p>Frequently, multivariable analyses (analysis with more than one covariate) are distinguished from univariate analyses (one covariate) by denoting the estimated odds ratios as adjusted odds ratios (AOR).</p>
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<h2 class="hd hd-2 unit-title">Logistic Regression: Summary</h2>
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<p>Logistic regression is an extremely powerful tool for data analysis of health data, especially given the numerous binary and categorical outcomes.</p>
<p>As with linear regression, there are a number of concerns to validate such as outliers, missing data, colinearity, dependent/correlated outcomes, and additional modeling assumptions. It is important to ensure if the relationship between the log odds of the outcome is indeed linear with the covariate. In cases where the data has been divided into too many subgroups (or the study may be simply too small), you may encounter a level of a discrete variable where none (or very few) of one of the outcomes occurred (i.e. verify that there are a sufficient number of cases in stratified groups as extreme imbalance may cause issues).</p>
<p>Overall, logistic regression is a very valuable tool in modelling binary and categorical data. Although we did not cover the latter case, a similar framework is available for discrete data that is ordered or has more than one category (see ?multinom in the nnet package in R for details about multinomial logistic regression). This and other topics, such as assessing model fit and using logistic regression in more complicated study designs, are discussed in chapter 11 of the course <a href="https://www.springer.com/gp/book/9783319437408" target="[object Object]">textbook</a>.</p>
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<h2 class="hd hd-2 unit-title">Logistic Regression: Exercise</h2>
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<h3><span style="box-sizing: border-box;">Exercise</span></h3>
<p><span style="box-sizing: border-box; font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">Use the ICU study (icu.dta) from the aplore3 package in R. The ICU data set consists of a sample of 200 subjects who were part of a much larger study on the survival of patients following admission to an adult intensive care unit (ICU). The major goal of this study was to develop a logistic regression model to predict the probability of survival to hospital discharge of these patients. A number of publications have appeared which have focused on various facets of this problem.</span></p>
<pre class="r" style="-webkit-text-size-adjust: auto; box-sizing: border-box; overflow: auto; font-family: monospace; font-size: 13px; padding: 9.5px; margin-top: 0px; margin-bottom: 10px; line-height: 1.42857143; color: #333333; word-break: break-all; background-color: #f5f5f5; border: 1px solid #cccccc; border-top-left-radius: 4px; border-top-right-radius: 4px; border-bottom-right-radius: 4px; border-bottom-left-radius: 4px;"><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;"><code class="r" style="box-sizing: border-box; font-family: monospace; font-size: inherit; color: inherit; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px;"><span class="comment" style="box-sizing: border-box; color: #999988; font-style: italic;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="paren" style="box-sizing: border-box; color: #687687;"></span><span class="string" style="box-sizing: border-box; color: #dd1144;"></span><span class="paren" style="box-sizing: border-box; color: #687687;"></span></code></span></pre>
<p><span style="box-sizing: border-box; font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">a. Write down the equation for the logistic regression model of STA on AGE. Write down the equation for the logit transformation of this logistic regression model. What characteristic of the outcome variable STA leads us to consider the logistic regression model as opposed to the usual linear regression model to describe the relationship between STA and AGE?</span></p>
<p><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">The logistic regression model is:</span></p>
<p></p>
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padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.36em;"><span class="mn" id="MathJax-Span-338" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 8.4px;">1</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mi" id="MathJax-Span-339" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-340" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">)</span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px 0px 0px -2.378em; vertical-align: 0px; line-height: normal; clip: rect(3.039em, 1004.765em, 4.229em, -999.997em); top: -3.211em; left: 41.0234375px;"><span class="mrow" id="MathJax-Span-341" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mn" id="MathJax-Span-342" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">1</span><span class="mo" id="MathJax-Span-343" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;">+</span><span class="msubsup" id="MathJax-Span-344" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 3.098em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.42em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-345" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">e</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -4.283em; left: 0.42em;"><span class="texatom" id="MathJax-Span-346" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-347" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mo" id="MathJax-Span-348" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">(</span><span class="msubsup" id="MathJax-Span-349" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.658em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.336em, 1000.36em, 4.289em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-350" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">β<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.36em;"><span class="mn" id="MathJax-Span-351" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 8.4px;">0</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-352" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">+</span><span class="msubsup" id="MathJax-Span-353" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.658em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.336em, 1000.36em, 4.289em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-354" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">β<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.36em;"><span class="mn" id="MathJax-Span-355" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 8.4px;">1</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mi" id="MathJax-Span-356" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-357" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">)</span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(0.836em, 1004.884em, 1.253em, -999.997em); top: -1.307em; left: 0em;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border-width: 1.3px 0px 0px; border-top-style: solid; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; overflow: hidden; width: 4.884em; height: 0px;"></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 1.074em;"></span></span></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 2.562em;"></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border-width: 0px; border-left-style: solid; padding: 0px; margin: 0px; vertical-align: -0.996em; line-height: normal; overflow: hidden; width: 0px; height: 2.932em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation" style="box-sizing: border-box; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; position: static; padding: 0px; border: 0px; width: 201.703125px; display: inline; transition: none; margin: 0px; vertical-align: 0px; line-height: normal; height: 1px !important; overflow: hidden !important;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi></mi><mo stretchy="false"></mo><mi></mi><mo stretchy="false"></mo><mo></mo><mi></mi><mo stretchy="false"></mo><mi></mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false"></mo></mrow><mi></mi><mo stretchy="false"></mo><mo></mo><mfrac><msup><mi></mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false"></mo><msub><mi></mi><mn></mn></msub><mo></mo><msub><mi></mi><mn></mn></msub><mi></mi><mo stretchy="false"></mo></mrow></msup><mrow><mn></mn><mo></mo><msup><mi></mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false"></mo><msub><mi></mi><mn></mn></msub><mo></mo><msub><mi></mi><mn></mn></msub><mi></mi><mo stretchy="false"></mo></mrow></msup></mrow></mfrac></math></span></span></div>
<div class="MathJax_Display" style="box-sizing: border-box; text-align: center; margin: 1em 0em; position: relative; max-width: none; max-height: none; min-width: 0px; min-height: 0px; width: 345px;"><span class="MathJax" id="MathJax-Element-10-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>w</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>e</mi><mspace width="thinmathspace" /><mi>x</mi><mo>=</mo><mi>A</mi><mi>G</mi><mi>E</mi><mspace width="thinmathspace" /><mi>a</mi><mi>n</mi><mi>d</mi><mspace width="thinmathspace" /><mi>&#x03C0;</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mi>r</mi><mi>o</mi><mi>b</mi><mo stretchy="false">(</mo><mi>S</mi><mi>T</mi><mi>A</mi><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>A</mi><mi>G</mi><mi>E</mi><mo stretchy="false">)</mo></math>" role="presentation" style="box-sizing: border-box; display: inline; line-height: normal; font-size: 14px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;"><nobr aria-hidden="true" style="box-sizing: border-box; transition: none; border: 0px; padding: 0px; margin: 0px; max-width: 5000em; max-height: 5000em; min-width: 0px; min-height: 0px; vertical-align: 0px; line-height: normal;"><span class="math" id="MathJax-Span-358" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 23.813em;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 19.824em; height: 0px; font-size: 16.799999237060547px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.67em, 1019.765em, 2.92em, -999.997em); top: -2.557em; left: 0em;"><span class="mrow" id="MathJax-Span-359" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-360" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">w</span><span class="mi" id="MathJax-Span-361" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">h</span><span class="mi" id="MathJax-Span-362" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">e</span><span class="mi" id="MathJax-Span-363" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">r<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-364" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">e</span><span class="mspace" id="MathJax-Span-365" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="mi" id="MathJax-Span-366" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-367" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">=</span><span class="mi" id="MathJax-Span-368" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">A</span><span class="mi" id="MathJax-Span-369" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">G</span><span class="mi" id="MathJax-Span-370" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">E<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mspace" id="MathJax-Span-371" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="mi" id="MathJax-Span-372" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">a</span><span class="mi" id="MathJax-Span-373" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">n</span><span class="mi" id="MathJax-Span-374" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">d<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mspace" id="MathJax-Span-375" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="mi" id="MathJax-Span-376" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">π<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span class="mo" id="MathJax-Span-377" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="mi" id="MathJax-Span-378" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-379" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="mo" id="MathJax-Span-380" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">=</span><span class="mi" id="MathJax-Span-381" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">P</span><span class="mi" id="MathJax-Span-382" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">r<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-383" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">o</span><span class="mi" id="MathJax-Span-384" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">b</span><span class="mo" id="MathJax-Span-385" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="mi" id="MathJax-Span-386" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">S<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-387" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">T<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span class="mi" id="MathJax-Span-388" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">A</span><span class="mo" id="MathJax-Span-389" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">=</span><span class="mn" id="MathJax-Span-390" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">1</span><span class="texatom" id="MathJax-Span-391" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-392" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mo" id="MathJax-Span-393" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">|</span></span></span><span class="mi" id="MathJax-Span-394" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">A</span><span class="mi" id="MathJax-Span-395" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">G</span><span class="mi" id="MathJax-Span-396" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">E<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-397" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 2.562em;"></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border-width: 0px; border-left-style: solid; padding: 0px; margin: 0px; vertical-align: -0.282em; line-height: normal; overflow: hidden; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation" style="box-sizing: border-box; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; position: static; padding: 0px; border: 0px; width: 333.375px; display: inline; transition: none; margin: 0px; vertical-align: 0px; line-height: normal; height: 1px !important; overflow: hidden !important;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi></mi><mi></mi><mi></mi><mi></mi><mi></mi><mspace width="thinmathspace"></mspace><mi></mi><mo></mo><mi></mi><mi></mi><mi></mi><mspace width="thinmathspace"></mspace><mi></mi><mi></mi><mi></mi><mspace width="thinmathspace"></mspace><mi></mi><mo stretchy="false"></mo><mi></mi><mo stretchy="false"></mo><mo></mo><mi></mi><mi></mi><mi></mi><mi></mi><mo stretchy="false"></mo><mi></mi><mi></mi><mi></mi><mo></mo><mn></mn><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false"></mo></mrow><mi></mi><mi></mi><mi></mi><mo stretchy="false"></mo></math></span></span></div>
<p><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">With logit transformation:</span></p>
<p></p>
<div class="MathJax_Display" style="box-sizing: border-box; text-align: center; margin: 1em 0em; position: relative; max-width: none; max-height: none; min-width: 0px; min-height: 0px; width: 345px;"><span class="MathJax" id="MathJax-Element-11-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>l</mi><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac><mrow><mi>&#x03C0;</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x03C0;</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mfrac><mrow class="MJX-TeXAtom-ORD"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow><mo>=</mo><msub><mi>&#x03B2;</mi><mn>0</mn></msub><mo>+</mo><msub><mi>&#x03B2;</mi><mn>1</mn></msub><mi>x</mi></math>" role="presentation" style="box-sizing: border-box; display: inline; line-height: normal; font-size: 14px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;"><nobr aria-hidden="true" style="box-sizing: border-box; transition: none; border: 0px; padding: 0px; margin: 0px; max-width: 5000em; max-height: 5000em; min-width: 0px; min-height: 0px; vertical-align: 0px; line-height: normal;"><span class="math" id="MathJax-Span-398" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 13.872em;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 11.551em; height: 0px; font-size: 16.799999237060547px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(2.503em, 1011.551em, 5.301em, -999.997em); top: -4.164em; left: 0em;"><span class="mrow" id="MathJax-Span-399" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-400" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">l<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-401" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">n</span><span class="texatom" id="MathJax-Span-402" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-403" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mo" id="MathJax-Span-404" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: -0.592em; line-height: normal;"><span style="box-sizing: border-box; transition: none 0s ease 0s; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span></span></span></span><span class="mfrac" id="MathJax-Span-405" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px 0.122em; vertical-align: 0px; line-height: normal; width: 4.11em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px 0px 0px -0.83em; vertical-align: 0px; line-height: normal; clip: rect(3.158em, 1001.61em, 4.348em, -999.997em); top: -4.64em; left: 34.5234375px;"><span class="mrow" id="MathJax-Span-406" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-407" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">π<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span class="mo" id="MathJax-Span-408" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="mi" id="MathJax-Span-409" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-410" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px 0px 0px -1.961em; vertical-align: 0px; line-height: normal; clip: rect(3.158em, 1003.932em, 4.348em, -999.997em); top: -3.271em; left: 34.5234375px;"><span class="mrow" id="MathJax-Span-411" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mo" id="MathJax-Span-412" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="mn" id="MathJax-Span-413" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">1</span><span class="mo" id="MathJax-Span-414" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;">−</span><span class="mi" id="MathJax-Span-415" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">π<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span class="mo" id="MathJax-Span-416" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="mi" id="MathJax-Span-417" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-418" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="mo" id="MathJax-Span-419" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(0.836em, 1004.11em, 1.253em, -999.997em); top: -1.307em; left: 0em;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border-width: 1.3px 0px 0px; border-top-style: solid; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; overflow: hidden; width: 4.11em; height: 0px;"></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 1.074em;"></span></span></span></span><span class="texatom" id="MathJax-Span-420" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-421" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mo" id="MathJax-Span-422" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: -0.592em; line-height: normal;"><span style="box-sizing: border-box; transition: none 0s ease 0s; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span></span></span></span><span class="mo" id="MathJax-Span-423" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">=</span><span class="msubsup" id="MathJax-Span-424" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.955em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.158em, 1000.539em, 4.348em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-425" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">β<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.539em;"><span class="mn" id="MathJax-Span-426" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">0</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-427" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;">+</span><span class="msubsup" id="MathJax-Span-428" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.955em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.158em, 1000.539em, 4.348em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-429" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">β<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.539em;"><span class="mn" id="MathJax-Span-430" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">1</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mi" id="MathJax-Span-431" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 4.17em;"></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border-width: 0px; border-left-style: solid; padding: 0px; margin: 0px; vertical-align: -1.211em; line-height: normal; overflow: hidden; width: 0px; height: 3.075em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation" style="box-sizing: border-box; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; position: static; padding: 0px; border: 0px; width: 194.203125px; display: inline; transition: none; margin: 0px; vertical-align: 0px; line-height: normal; height: 1px !important; overflow: hidden !important;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi></mi><mi></mi><mrow class="MJX-TeXAtom-ORD"><mo maxsize="2.047em" minsize="2.047em"></mo></mrow><mfrac><mrow><mi></mi><mo stretchy="false"></mo><mi></mi><mo stretchy="false"></mo></mrow><mrow><mo stretchy="false"></mo><mn></mn><mo></mo><mi></mi><mo stretchy="false"></mo><mi></mi><mo stretchy="false"></mo><mo stretchy="false"></mo></mrow></mfrac><mrow class="MJX-TeXAtom-ORD"><mo maxsize="2.047em" minsize="2.047em"></mo></mrow><mo></mo><msub><mi></mi><mn></mn></msub><mo></mo><msub><mi></mi><mn></mn></msub><mi></mi></math></span></span></div>
<p><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">Due to the binary nature of STA, linear regression analysis is not recommended.</span></p>
<p><span style="box-sizing: border-box; font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">b. Form a scatterplot of STA versus AGE.</span></p>
<p><span style="box-sizing: border-box; font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">c. Write down an expression for the likelihood and log likelihood for the logistic regression model in part (a) using the ungrouped, n = 200 , data. Obtain expressions for the two likelihood equations.</span></p>
<pre class="r" style="box-sizing: border-box; overflow: auto; font-family: monospace; font-size: 13px; padding: 9.5px; margin-top: 0px; margin-bottom: 10px; line-height: 1.42857143; color: #333333; word-break: break-all; background-color: #f5f5f5; border: 1px solid #cccccc; border-top-left-radius: 4px; border-top-right-radius: 4px; border-bottom-right-radius: 4px; border-bottom-left-radius: 4px;"><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;"><code class="r" style="box-sizing: border-box; font-family: monospace; font-size: inherit; color: inherit; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px;"><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="paren" style="box-sizing: border-box; color: #687687;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="string" style="box-sizing: border-box; color: #dd1144;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="number" style="box-sizing: border-box; color: #009999;"></span><span class="paren" style="box-sizing: border-box; color: #687687;"></span></code></span></pre>
<p><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;"><img 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" title="Scatterplot of STA & AGE" alt="Scatterplot of STA & AGE" width="672" style="box-sizing: border-box; border: 0px; vertical-align: middle; height: auto;" /></span></p>
<p><span class="math" style="box-sizing: border-box; font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;"><span class="MathJax" id="MathJax-Element-12-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x2113;</mi><mo stretchy="false">(</mo><mi>&#x03B2;</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>&#x220F;</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></munderover><mo stretchy="false">[</mo><mi>&#x03C0;</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>y</mi><mi>i</mi></msub></mrow></msup><mo stretchy="false">[</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x03C0;</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mo>&#x2212;</mo><msub><mi>y</mi><mi>i</mi></msub></mrow></msup></math>" role="presentation" style="box-sizing: border-box; display: inline; line-height: normal; font-size: 14px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;"><nobr aria-hidden="true" style="box-sizing: border-box; transition: none; border: 0px; padding: 0px; margin: 0px; max-width: 5000em; max-height: 5000em; min-width: 0px; min-height: 0px; vertical-align: 0px; line-height: normal;"><span class="math" id="MathJax-Span-432" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 16.967em;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 14.11em; height: 0px; font-size: 16.799999237060547px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.551em, 1014.11em, 3.039em, -999.997em); top: -2.557em; left: 0em;"><span class="mrow" id="MathJax-Span-433" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-434" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">ℓ</span><span class="mo" id="MathJax-Span-435" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="mi" id="MathJax-Span-436" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">β<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-437" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="mo" id="MathJax-Span-438" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">=</span><span class="munderover" id="MathJax-Span-439" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.086em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.039em, 1000.955em, 4.408em, -999.997em); top: -3.985em; left: 0em;"><span class="mo" id="MathJax-Span-440" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0.003em; line-height: normal;">∏</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.515em, 1000.42em, 4.17em, -999.997em); top: -4.461em; left: 1.015em;"><span class="texatom" id="MathJax-Span-441" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-442" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-443" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">n</span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.336em, 1001.134em, 4.17em, -999.997em); top: -3.688em; left: 1.015em;"><span class="texatom" id="MathJax-Span-444" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-445" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-446" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">i</span><span class="mo" id="MathJax-Span-447" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">=</span><span class="mn" id="MathJax-Span-448" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">1</span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-449" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">[</span><span class="mi" id="MathJax-Span-450" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">π<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span class="mo" id="MathJax-Span-451" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="msubsup" id="MathJax-Span-452" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.717em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.479em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-453" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.42em;"><span class="mi" id="MathJax-Span-454" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">i</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-455" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="msubsup" id="MathJax-Span-456" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.896em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.158em, 1000.241em, 4.348em, -999.997em); top: -3.985em; left: 0em;"><span class="mo" id="MathJax-Span-457" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">]</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -4.342em; left: 0.36em;"><span class="texatom" id="MathJax-Span-458" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-459" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="msubsup" id="MathJax-Span-460" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.539em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.515em, 1000.301em, 4.289em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-461" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">y</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.301em;"><span class="mi" id="MathJax-Span-462" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 8.4px; font-style: italic;">i</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-463" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">[</span><span class="mn" id="MathJax-Span-464" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">1</span><span class="mo" id="MathJax-Span-465" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;">−</span><span class="mi" id="MathJax-Span-466" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">π<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span class="mo" id="MathJax-Span-467" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="msubsup" id="MathJax-Span-468" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.717em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.479em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-469" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.42em;"><span class="mi" id="MathJax-Span-470" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">i</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-471" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="msubsup" id="MathJax-Span-472" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 1.729em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.158em, 1000.241em, 4.348em, -999.997em); top: -3.985em; left: 0em;"><span class="mo" id="MathJax-Span-473" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">]</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -4.342em; left: 0.36em;"><span class="texatom" id="MathJax-Span-474" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-475" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mn" id="MathJax-Span-476" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">1</span><span class="mo" id="MathJax-Span-477" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">−</span><span class="msubsup" id="MathJax-Span-478" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.539em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.515em, 1000.301em, 4.289em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-479" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">y</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.301em;"><span class="mi" id="MathJax-Span-480" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 8.4px; font-style: italic;">i</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 2.562em;"></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border-width: 0px; border-left-style: solid; padding: 0px; margin: 0px; vertical-align: -0.425em; line-height: normal; overflow: hidden; width: 0px; height: 1.504em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation" style="box-sizing: border-box; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; position: static; padding: 0px; border: 0px; display: inline; transition: none; margin: 0px; vertical-align: 0px; line-height: normal; height: 1px !important; width: 1px !important; overflow: hidden !important;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi></mi><mo stretchy="false"></mo><mi></mi><mo stretchy="false"></mo><mo></mo><munderover><mo></mo><mrow class="MJX-TeXAtom-ORD"><mi></mi><mo></mo><mn></mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi></mi></mrow></munderover><mo stretchy="false"></mo><mi></mi><mo stretchy="false"></mo><msub><mi></mi><mi></mi></msub><mo stretchy="false"></mo><msup><mo stretchy="false"></mo><mrow class="MJX-TeXAtom-ORD"><msub><mi></mi><mi></mi></msub></mrow></msup><mo stretchy="false"></mo><mn></mn><mo></mo><mi></mi><mo stretchy="false"></mo><msub><mi></mi><mi></mi></msub><mo stretchy="false"></mo><msup><mo stretchy="false"></mo><mrow class="MJX-TeXAtom-ORD"><mn></mn><mo></mo><msub><mi></mi><mi></mi></msub></mrow></msup></math></span></span> <span class="math" style="box-sizing: border-box;"><span class="MathJax" id="MathJax-Element-13-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>e</mi><mspace width="thinmathspace" /><mi>x</mi><mo>=</mo><mi>A</mi><mi>G</mi><mi>E</mi><mspace width="thinmathspace" /><mi>a</mi><mi>n</mi><mi>d</mi><mspace width="thinmathspace" /><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><mn>0</mn><mspace width="thinmathspace" /><mi>i</mi><mi>f</mi><mspace width="thinmathspace" /><mi>t</mi><mi>h</mi><mi>e</mi><mspace width="thinmathspace" /><mi>p</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi><mspace width="thinmathspace" /><mi>l</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mspace width="thinmathspace" /><mi>o</mi><mi>r</mi><mspace width="thinmathspace" /><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><mn>1</mn><mspace width="thinmathspace" /><mi>i</mi><mi>f</mi><mspace width="thinmathspace" /><mi>t</mi><mi>h</mi><mi>e</mi><mspace width="thinmathspace" /><mi>p</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi><mspace width="thinmathspace" /><mi>d</mi><mi>i</mi><mi>e</mi><mi>d</mi></math>" role="presentation" style="box-sizing: border-box; display: inline; line-height: normal; font-size: 14px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;"><nobr aria-hidden="true" style="box-sizing: border-box; transition: none; border: 0px; padding: 0px; margin: 0px; max-width: 5000em; max-height: 5000em; min-width: 0px; min-height: 0px; vertical-align: 0px; line-height: normal;"><span class="math" id="MathJax-Span-481" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 35.122em;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 29.229em; height: 0px; font-size: 16.799999237060547px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.729em, 1029.229em, 2.92em, -999.997em); top: -2.557em; left: 0em;"><span class="mrow" id="MathJax-Span-482" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-483" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">w</span><span class="mi" id="MathJax-Span-484" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">h</span><span class="mi" id="MathJax-Span-485" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">e</span><span class="mi" id="MathJax-Span-486" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">r<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-487" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">e</span><span class="mspace" id="MathJax-Span-488" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="mi" id="MathJax-Span-489" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-490" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">=</span><span class="mi" id="MathJax-Span-491" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">A</span><span class="mi" id="MathJax-Span-492" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">G</span><span class="mi" id="MathJax-Span-493" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">E<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mspace" id="MathJax-Span-494" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="mi" id="MathJax-Span-495" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">a</span><span class="mi" id="MathJax-Span-496" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">n</span><span class="mi" id="MathJax-Span-497" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">d<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mspace" id="MathJax-Span-498" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="msubsup" id="MathJax-Span-499" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.717em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.42em, 4.348em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-500" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">y</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.42em;"><span class="mi" id="MathJax-Span-501" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">i</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-502" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">=</span><span class="mn" id="MathJax-Span-503" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">0</span><span class="mspace" id="MathJax-Span-504" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="mi" id="MathJax-Span-505" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">i</span><span class="mi" id="MathJax-Span-506" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">f<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.122em;"></span></span><span class="mspace" id="MathJax-Span-507" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="mi" id="MathJax-Span-508" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">t<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-509" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">h</span><span class="mi" id="MathJax-Span-510" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">e</span><span class="mspace" id="MathJax-Span-511" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="mi" id="MathJax-Span-512" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">p</span><span class="mi" id="MathJax-Span-513" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">a</span><span class="mi" id="MathJax-Span-514" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">t<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-515" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">i</span><span class="mi" id="MathJax-Span-516" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">e</span><span class="mi" id="MathJax-Span-517" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">n</span><span class="mi" id="MathJax-Span-518" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">t<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mspace" id="MathJax-Span-519" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="mi" id="MathJax-Span-520" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">l<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-521" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">i</span><span class="mi" id="MathJax-Span-522" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">v</span><span class="mi" id="MathJax-Span-523" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">e</span><span class="mi" id="MathJax-Span-524" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">d<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mspace" id="MathJax-Span-525" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="mi" id="MathJax-Span-526" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">o</span><span class="mi" id="MathJax-Span-527" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">r<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mspace" id="MathJax-Span-528" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="msubsup" id="MathJax-Span-529" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.717em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.42em, 4.348em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-530" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">y</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.42em;"><span class="mi" id="MathJax-Span-531" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">i</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-532" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">=</span><span class="mn" id="MathJax-Span-533" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">1</span><span class="mspace" id="MathJax-Span-534" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="mi" id="MathJax-Span-535" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">i</span><span class="mi" id="MathJax-Span-536" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">f<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.122em;"></span></span><span class="mspace" id="MathJax-Span-537" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="mi" id="MathJax-Span-538" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">t<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-539" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">h</span><span class="mi" id="MathJax-Span-540" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">e</span><span class="mspace" id="MathJax-Span-541" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="mi" id="MathJax-Span-542" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">p</span><span class="mi" id="MathJax-Span-543" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">a</span><span class="mi" id="MathJax-Span-544" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">t<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-545" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">i</span><span class="mi" id="MathJax-Span-546" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">e</span><span class="mi" id="MathJax-Span-547" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">n</span><span class="mi" id="MathJax-Span-548" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">t<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mspace" id="MathJax-Span-549" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; height: 0em; width: 0.182em; overflow: hidden;"></span><span class="mi" id="MathJax-Span-550" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">d<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-551" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">i</span><span class="mi" id="MathJax-Span-552" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">e</span><span class="mi" id="MathJax-Span-553" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">d<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 2.562em;"></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border-width: 0px; border-left-style: solid; padding: 0px; margin: 0px; vertical-align: -0.282em; line-height: normal; overflow: hidden; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation" style="box-sizing: border-box; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; position: static; padding: 0px; border: 0px; display: inline; transition: none; margin: 0px; vertical-align: 0px; line-height: normal; height: 1px !important; width: 1px !important; overflow: hidden !important;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi></mi><mi></mi><mi></mi><mi></mi><mi></mi><mspace width="thinmathspace"></mspace><mi></mi><mo></mo><mi></mi><mi></mi><mi></mi><mspace width="thinmathspace"></mspace><mi></mi><mi></mi><mi></mi><mspace width="thinmathspace"></mspace><msub><mi></mi><mi></mi></msub><mo></mo><mn></mn><mspace width="thinmathspace"></mspace><mi></mi><mi></mi><mspace width="thinmathspace"></mspace><mi></mi><mi></mi><mi></mi><mspace width="thinmathspace"></mspace><mi></mi><mi></mi><mi></mi><mi></mi><mi></mi><mi></mi><mi></mi><mspace width="thinmathspace"></mspace><mi></mi><mi></mi><mi></mi><mi></mi><mi></mi><mspace width="thinmathspace"></mspace><mi></mi><mi></mi><mspace width="thinmathspace"></mspace><msub><mi></mi><mi></mi></msub><mo></mo><mn></mn><mspace width="thinmathspace"></mspace><mi></mi><mi></mi><mspace width="thinmathspace"></mspace><mi></mi><mi></mi><mi></mi><mspace width="thinmathspace"></mspace><mi></mi><mi></mi><mi></mi><mi></mi><mi></mi><mi></mi><mi></mi><mspace width="thinmathspace"></mspace><mi></mi><mi></mi><mi></mi><mi></mi></math></span></span></span></span></p>
<p><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">applying natural logarithm to expression above</span></p>
<p></p>
<div class="MathJax_Display" style="box-sizing: border-box; text-align: center; margin: 1em 0em; position: relative; max-width: none; max-height: none; min-width: 0px; min-height: 0px; width: 345px;"><span class="MathJax" id="MathJax-Element-14-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>L</mi><mo stretchy="false">(</mo><mi>&#x03B2;</mi><mo stretchy="false">)</mo><mo>=</mo><mi>l</mi><mi>n</mi><mo stretchy="false">(</mo><mi>&#x2113;</mi><mo stretchy="false">(</mo><mi>&#x03B2;</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></munderover><msub><mi>y</mi><mi>i</mi></msub><mi>l</mi><mi>n</mi><mo stretchy="false">[</mo><mi>&#x03C0;</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>l</mi><mi>n</mi><mo stretchy="false">[</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x03C0;</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math>" role="presentation" style="box-sizing: border-box; display: inline; line-height: normal; font-size: 14px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;"><nobr aria-hidden="true" style="box-sizing: border-box; transition: none; border: 0px; padding: 0px; margin: 0px; max-width: 5000em; max-height: 5000em; min-width: 0px; min-height: 0px; vertical-align: 0px; line-height: normal;"><span class="math" id="MathJax-Span-554" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 27.146em;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 22.622em; height: 0px; font-size: 16.799999237060547px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(0.777em, 1022.562em, 3.991em, -999.997em); top: -2.557em; left: 0em;"><span class="mrow" id="MathJax-Span-555" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-556" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">L<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-557" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="mi" id="MathJax-Span-558" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">β<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-559" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="mo" id="MathJax-Span-560" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">=</span><span class="mi" id="MathJax-Span-561" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">l<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-562" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">n</span><span class="mo" id="MathJax-Span-563" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="mi" id="MathJax-Span-564" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">ℓ</span><span class="mo" id="MathJax-Span-565" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="mi" id="MathJax-Span-566" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">β<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-567" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="mo" id="MathJax-Span-568" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="mo" id="MathJax-Span-569" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">=</span><span class="munderover" id="MathJax-Span-570" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 1.313em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(2.86em, 1001.193em, 4.646em, -999.997em); top: -3.985em; left: 0em;"><span class="mo" id="MathJax-Span-571" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: -0.533em; line-height: normal;">∑</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.336em, 1000.955em, 4.289em, -999.997em); top: -2.854em; left: 0.122em;"><span class="texatom" id="MathJax-Span-572" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-573" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-574" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">i</span><span class="mo" id="MathJax-Span-575" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">=</span><span class="mn" id="MathJax-Span-576" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">1</span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.36em, 4.17em, -999.997em); top: -5.176em; left: 0.479em;"><span class="texatom" id="MathJax-Span-577" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-578" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-579" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">n</span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-580" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.182em; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.717em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.42em, 4.348em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-581" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">y</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.42em;"><span class="mi" id="MathJax-Span-582" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">i</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mi" id="MathJax-Span-583" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">l<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-584" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">n</span><span class="mo" id="MathJax-Span-585" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">[</span><span class="mi" id="MathJax-Span-586" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">π<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span class="mo" id="MathJax-Span-587" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="msubsup" id="MathJax-Span-588" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.717em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.479em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-589" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.42em;"><span class="mi" id="MathJax-Span-590" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">i</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-591" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="mo" id="MathJax-Span-592" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">]</span><span class="mo" id="MathJax-Span-593" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;">+</span><span class="mo" id="MathJax-Span-594" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="mn" id="MathJax-Span-595" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">1</span><span class="mo" id="MathJax-Span-596" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;">−</span><span class="msubsup" id="MathJax-Span-597" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.717em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.42em, 4.348em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-598" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">y</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.42em;"><span class="mi" id="MathJax-Span-599" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">i</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-600" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="mi" id="MathJax-Span-601" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">l<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-602" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">n</span><span class="mo" id="MathJax-Span-603" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">[</span><span class="mn" id="MathJax-Span-604" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">1</span><span class="mo" id="MathJax-Span-605" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;">−</span><span class="mi" id="MathJax-Span-606" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">π<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span class="mo" id="MathJax-Span-607" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="msubsup" id="MathJax-Span-608" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.717em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.479em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-609" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.42em;"><span class="mi" id="MathJax-Span-610" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">i</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-611" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="mo" id="MathJax-Span-612" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">]</span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 2.562em;"></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border-width: 0px; border-left-style: solid; padding: 0px; margin: 0px; vertical-align: -1.568em; line-height: normal; overflow: hidden; width: 0px; height: 3.575em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation" style="box-sizing: border-box; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; position: static; padding: 0px; border: 0px; width: 380.03125px; display: inline; transition: none; margin: 0px; vertical-align: 0px; line-height: normal; height: 1px !important; overflow: hidden !important;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi></mi><mo stretchy="false"></mo><mi></mi><mo stretchy="false"></mo><mo></mo><mi></mi><mi></mi><mo stretchy="false"></mo><mi></mi><mo stretchy="false"></mo><mi></mi><mo stretchy="false"></mo><mo stretchy="false"></mo><mo></mo><munderover><mo></mo><mrow class="MJX-TeXAtom-ORD"><mi></mi><mo></mo><mn></mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi></mi></mrow></munderover><msub><mi></mi><mi></mi></msub><mi></mi><mi></mi><mo stretchy="false"></mo><mi></mi><mo stretchy="false"></mo><msub><mi></mi><mi></mi></msub><mo stretchy="false"></mo><mo stretchy="false"></mo><mo></mo><mo stretchy="false"></mo><mn></mn><mo></mo><msub><mi></mi><mi></mi></msub><mo stretchy="false"></mo><mi></mi><mi></mi><mo stretchy="false"></mo><mn></mn><mo></mo><mi></mi><mo stretchy="false"></mo><msub><mi></mi><mi></mi></msub><mo stretchy="false"></mo><mo stretchy="false"></mo></math></span></span></div>
<p><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">The likelihood equations are:</span></p>
<p></p>
<div class="MathJax_Display" style="box-sizing: border-box; text-align: center; margin: 1em 0em; position: relative; max-width: none; max-height: none; min-width: 0px; min-height: 0px; width: 345px;"><span class="MathJax" id="MathJax-Element-15-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></munderover><mo stretchy="false">(</mo><msub><mi>y</mi><mi>i</mi></msub><mo>&#x2212;</mo><mi>&#x03C0;</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math>" role="presentation" style="box-sizing: border-box; display: inline; line-height: normal; font-size: 14px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;"><nobr aria-hidden="true" style="box-sizing: border-box; transition: none; border: 0px; padding: 0px; margin: 0px; max-width: 5000em; max-height: 5000em; min-width: 0px; min-height: 0px; vertical-align: 0px; line-height: normal;"><span class="math" id="MathJax-Span-613" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 9.229em;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 7.682em; height: 0px; font-size: 16.799999237060547px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(0.777em, 1007.682em, 3.991em, -999.997em); top: -2.557em; left: 0em;"><span class="mrow" id="MathJax-Span-614" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="munderover" id="MathJax-Span-615" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 1.313em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(2.86em, 1001.193em, 4.646em, -999.997em); top: -3.985em; left: 0em;"><span class="mo" id="MathJax-Span-616" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: -0.533em; line-height: normal;">∑</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.336em, 1000.955em, 4.289em, -999.997em); top: -2.854em; left: 0.122em;"><span class="texatom" id="MathJax-Span-617" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-618" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-619" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">i</span><span class="mo" id="MathJax-Span-620" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">=</span><span class="mn" id="MathJax-Span-621" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">1</span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.36em, 4.17em, -999.997em); top: -5.176em; left: 0.479em;"><span class="texatom" id="MathJax-Span-622" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-623" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-624" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">n</span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-625" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="msubsup" id="MathJax-Span-626" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.717em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.42em, 4.348em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-627" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">y</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.42em;"><span class="mi" id="MathJax-Span-628" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">i</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-629" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;">−</span><span class="mi" id="MathJax-Span-630" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">π<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span class="mo" id="MathJax-Span-631" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="msubsup" id="MathJax-Span-632" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.717em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.479em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-633" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.42em;"><span class="mi" id="MathJax-Span-634" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">i</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-635" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="mo" id="MathJax-Span-636" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="mo" id="MathJax-Span-637" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">=</span><span class="mn" id="MathJax-Span-638" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">0</span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 2.562em;"></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border-width: 0px; border-left-style: solid; padding: 0px; margin: 0px; vertical-align: -1.568em; line-height: normal; overflow: hidden; width: 0px; height: 3.575em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation" style="box-sizing: border-box; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; position: static; padding: 0px; border: 0px; width: 129.203125px; display: inline; transition: none; margin: 0px; vertical-align: 0px; line-height: normal; height: 1px !important; overflow: hidden !important;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><munderover><mo></mo><mrow class="MJX-TeXAtom-ORD"><mi></mi><mo></mo><mn></mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi></mi></mrow></munderover><mo stretchy="false"></mo><msub><mi></mi><mi></mi></msub><mo></mo><mi></mi><mo stretchy="false"></mo><msub><mi></mi><mi></mi></msub><mo stretchy="false"></mo><mo stretchy="false"></mo><mo></mo><mn></mn></math></span></span></div>
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clip: rect(0.777em, 1008.574em, 3.991em, -999.997em); top: -2.557em; left: 0em;"><span class="mrow" id="MathJax-Span-640" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="munderover" id="MathJax-Span-641" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 1.313em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(2.86em, 1001.193em, 4.646em, -999.997em); top: -3.985em; left: 0em;"><span class="mo" id="MathJax-Span-642" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: -0.533em; line-height: normal;">∑</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.336em, 1000.955em, 4.289em, -999.997em); top: -2.854em; left: 0.122em;"><span class="texatom" id="MathJax-Span-643" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-644" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; 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height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.36em, 4.17em, -999.997em); top: -5.176em; left: 0.479em;"><span class="texatom" id="MathJax-Span-648" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-649" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-650" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">n</span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-651" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.182em; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.717em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.479em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-652" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; 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<p><span style="box-sizing: border-box; font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">d. Obtain the maximum likelihood estimates of the parameters of the logistic regression model in part (a). These estimates should be based on the ungrouped, n = 200 , data. Using these estimates, write down the equation for the fitted values, that is, the estimated logistic probabilities. Plot the equation for the fitted values on the axes used in the scatterplots in part (b).</span></p>
<pre class="r" style="box-sizing: border-box; overflow: auto; font-family: monospace; font-size: 13px; padding: 9.5px; margin-top: 0px; margin-bottom: 10px; line-height: 1.42857143; color: #333333; word-break: break-all; background-color: #f5f5f5; border: 1px solid #cccccc; border-top-left-radius: 4px; border-top-right-radius: 4px; border-bottom-right-radius: 4px; border-bottom-left-radius: 4px;"></pre>
<p><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">Substituting with the estimated values we get:</span></p>
<p></p>
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height: 0px; font-size: 16.799999237060547px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(0.836em, 1013.991em, 3.693em, -999.997em); top: -2.557em; left: 0em;"><span class="mrow" id="MathJax-Span-669" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="msubsup" id="MathJax-Span-670" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.896em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.539em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-671" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">π<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.063em;"></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -3.807em; left: 0.479em;"><span class="texatom" id="MathJax-Span-672" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-673" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-674" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">e</span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-675" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="mi" id="MathJax-Span-676" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-677" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="mo" id="MathJax-Span-678" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">=</span><span class="mfrac" id="MathJax-Span-679" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px 0.122em; vertical-align: 0px; line-height: normal; width: 10.42em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px 0px 0px -3.985em; vertical-align: 0px; line-height: normal; clip: rect(2.979em, 1007.92em, 4.17em, -999.997em); top: -4.64em; left: 87.5234375px;"><span class="msubsup" id="MathJax-Span-680" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 7.92em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.42em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-681" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">e</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -4.342em; left: 0.42em;"><span class="texatom" id="MathJax-Span-682" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-683" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mo" id="MathJax-Span-684" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">(</span><span class="mo" id="MathJax-Span-685" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">−</span><span class="mn" id="MathJax-Span-686" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">3.05851</span><span class="mo" id="MathJax-Span-687" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">+</span><span class="mn" id="MathJax-Span-688" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">0.02754</span><span class="mi" id="MathJax-Span-689" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">A</span><span class="mi" id="MathJax-Span-690" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">G</span><span class="mi" id="MathJax-Span-691" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">E<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-692" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">)</span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px 0px 0px -5.116em; vertical-align: 0px; line-height: normal; clip: rect(3.039em, 1010.241em, 4.348em, -999.997em); top: -3.211em; left: 87.5234375px;"><span class="mrow" id="MathJax-Span-693" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mo" id="MathJax-Span-694" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="mn" id="MathJax-Span-695" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">1</span><span class="mo" id="MathJax-Span-696" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;">+</span><span class="msubsup" id="MathJax-Span-697" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 7.92em; height: 0px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.396em, 1000.42em, 4.17em, -999.997em); top: -3.985em; left: 0em;"><span class="mi" id="MathJax-Span-698" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">e</span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; top: -4.283em; left: 0.42em;"><span class="texatom" id="MathJax-Span-699" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mrow" id="MathJax-Span-700" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mo" id="MathJax-Span-701" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">(</span><span class="mo" id="MathJax-Span-702" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">−</span><span class="mn" id="MathJax-Span-703" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">3.05851</span><span class="mo" id="MathJax-Span-704" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">−</span><span class="mn" id="MathJax-Span-705" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">0.02754</span><span class="mi" id="MathJax-Span-706" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">A</span><span class="mi" id="MathJax-Span-707" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">G</span><span class="mi" id="MathJax-Span-708" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px; font-style: italic;">E<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-709" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.8776px;">)</span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-710" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 3.991em;"></span></span><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(0.836em, 1010.42em, 1.253em, -999.997em); top: -1.307em; left: 0em;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border-width: 1.3px 0px 0px; border-top-style: solid; padding: 0px; margin: 0px; vertical-align: 0em; line-height: normal; overflow: hidden; width: 10.42em; height: 0px;"></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 1.074em;"></span></span></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 2.562em;"></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border-width: 0px; border-left-style: solid; padding: 0px; margin: 0px; vertical-align: -1.211em; line-height: normal; overflow: hidden; width: 0px; height: 3.075em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation" style="box-sizing: border-box; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; position: static; padding: 0px; border: 0px; width: 235.03125px; display: inline; transition: none; margin: 0px; vertical-align: 0px; line-height: normal; height: 1px !important; overflow: hidden !important;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi></mi><mrow class="MJX-TeXAtom-ORD"><mi></mi></mrow></msub><mo stretchy="false"></mo><mi></mi><mo stretchy="false"></mo><mo></mo><mfrac><msup><mi></mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false"></mo><mo></mo><mn></mn><mo></mo><mn></mn><mi></mi><mi></mi><mi></mi><mo stretchy="false"></mo></mrow></msup><mrow><mo stretchy="false"></mo><mn></mn><mo></mo><msup><mi></mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false"></mo><mo></mo><mn></mn><mo></mo><mn></mn><mi></mi><mi></mi><mi></mi><mo stretchy="false"></mo></mrow></msup><mo stretchy="false"></mo></mrow></mfrac></math></span></span></div>
<p><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">And the logit transformation:</span></p>
<p></p>
<div class="MathJax_Display" style="box-sizing: border-box; text-align: center; margin: 1em 0em; position: relative; max-width: none; max-height: none; min-width: 0px; min-height: 0px; width: 345px;"><span class="MathJax" id="MathJax-Element-18-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>&#x2212;</mo><mn>3.05851</mn><mo>&#x2212;</mo><mn>0.02754</mn><mi>A</mi><mi>G</mi><mi>E</mi></math>" role="presentation" style="box-sizing: border-box; display: inline; line-height: normal; font-size: 14px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;"><nobr aria-hidden="true" style="box-sizing: border-box; transition: none; border: 0px; padding: 0px; margin: 0px; max-width: 5000em; max-height: 5000em; min-width: 0px; min-height: 0px; vertical-align: 0px; line-height: normal;"><span class="math" id="MathJax-Span-711" style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 16.074em;"><span style="box-sizing: border-box; transition: none; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 13.396em; height: 0px; font-size: 16.799999237060547px;"><span style="box-sizing: border-box; transition: none; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.67em, 1013.396em, 2.92em, -999.997em); top: -2.557em; left: 0em;"><span class="mrow" id="MathJax-Span-712" style="box-sizing: border-box; transition: none; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;"><span class="mi" id="MathJax-Span-713" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">g</span><span class="mo" id="MathJax-Span-714" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">(</span><span class="mi" id="MathJax-Span-715" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">x<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-716" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">)</span><span class="mo" id="MathJax-Span-717" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">=</span><span class="mo" id="MathJax-Span-718" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.301em; margin: 0px; vertical-align: 0px; line-height: normal;">−</span><span class="mn" id="MathJax-Span-719" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal;">3.05851</span><span class="mo" id="MathJax-Span-720" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;">−</span><span class="mn" id="MathJax-Span-721" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.241em; margin: 0px; vertical-align: 0px; line-height: normal;">0.02754</span><span class="mi" id="MathJax-Span-722" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">A</span><span class="mi" id="MathJax-Span-723" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">G</span><span class="mi" id="MathJax-Span-724" style="box-sizing: border-box; transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-style: italic;">E<span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0px; height: 2.562em;"></span></span></span><span style="box-sizing: border-box; transition: none; display: inline-block; position: static; border-width: 0px; border-left-style: solid; padding: 0px; margin: 0px; vertical-align: -0.282em; line-height: normal; overflow: hidden; width: 0px; height: 1.218em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation" style="box-sizing: border-box; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); -webkit-touch-callout: none; -webkit-user-select: none; position: static; padding: 0px; border: 0px; width: 225.03125px; display: inline; transition: none; margin: 0px; vertical-align: 0px; line-height: normal; height: 1px !important; overflow: hidden !important;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi></mi><mo stretchy="false"></mo><mi></mi><mo stretchy="false"></mo><mo></mo><mo></mo><mn></mn><mo></mo><mn></mn><mi></mi><mi></mi><mi></mi></math></span></span></div>
<pre class="r" style="box-sizing: border-box; overflow: auto; font-family: monospace; font-size: 13px; padding: 9.5px; margin-top: 0px; margin-bottom: 10px; line-height: 1.42857143; color: #333333; word-break: break-all; background-color: #f5f5f5; border: 1px solid #cccccc; border-top-left-radius: 4px; border-top-right-radius: 4px; border-bottom-right-radius: 4px; border-bottom-left-radius: 4px;"><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;"><code class="r" style="box-sizing: border-box; font-family: monospace; font-size: inherit; color: inherit; border-top-left-radius: 0px; border-top-right-radius: 0px; border-bottom-right-radius: 0px; border-bottom-left-radius: 0px;"><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="paren" style="box-sizing: border-box; color: #687687;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="string" style="box-sizing: border-box; color: #dd1144;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="number" style="box-sizing: border-box; color: #009999;"></span><span class="paren" style="box-sizing: border-box; color: #687687;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="paren" style="box-sizing: border-box; color: #687687;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="paren" style="box-sizing: border-box; color: #687687;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="paren" style="box-sizing: border-box; color: #687687;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="number" style="box-sizing: border-box; color: #009999;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="number" style="box-sizing: border-box; color: #009999;"></span><span class="identifier" style="box-sizing: border-box; color: #000000;"></span><span class="operator" style="box-sizing: border-box; color: #687687;"></span><span class="string" style="box-sizing: border-box; color: #dd1144;"></span><span class="paren" style="box-sizing: border-box; color: #687687;"></span></code></span></pre>
<p><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;"><img 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title="Scatterplot of STA & AGE and probabilities values from Logistic regression" alt="Scatterplot of STA & AGE and probabilities values from Logistic regression" width="672" style="box-sizing: border-box; border: 0px; vertical-align: middle; height: auto;" /></span></p>
<p><span style="box-sizing: border-box; font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">e. Summarize (describe in words) the results presented in the plot obtained from parts (b) and (d).</span></p>
<p></p>
<p><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">Given the dichotomous nature of the outcome variable (STA = patients live or die) logistic Regression is appropriate to understand the impact of age on the probability of surviving hospital discharge. From the fitted model we can conclude that as the age increases the probability of survival to hospital discharge reduces.</span></p>
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<h3>Introduction</h3>
<p>There is robust literature on the properties and application of regression methods, and these methods often perform quite well. Another powerful family of methods important to highlight is tree-based methods. </p>
<p>We will be using Introduction to Statistical Learning (ISLR, <a href="https://faculty.marshall.usc.edu/gareth-james/ISL/ISLR%20Seventh%20Printing.pdf" target="[object Object]">available freely online</a>) as the core reference text. For those interested in a deeper statistical understanding of these methods,<span style="font-size: 1em;"> Elements of Statistical Learning (ESLR) is also </span><a href="https://web.stanford.edu/~hastie/Papers/ESLII.pdf" target="[object Object]" style="font-size: 1em;">freely available</a>.<span style="font-size: 1em;"> </span></p>
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<h3>Classification and Regression Trees</h3>
<p>Decision trees at its simplest are sometimes also referred to as classification and regression trees, depending on whether the tree is being used for a classification (output is binary or categorical) or regression (output is numeric) task. Phrased another way, we can use trees for <strong>both </strong>classification and regression tasks. </p>
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<h2 class="hd hd-2 unit-title">Trees: Bias-Variance Tradeoff</h2>
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<h3 style="text-align: justify;">Bias vs. Variance</h3>
<p style="text-align: justify;"></p>
<p style="text-align: justify;">This is a canonical visual explaining what we mean when we refer to bias and variance. </p>
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<p style="text-align: justify;">The reason that we are not able to develop a completely accurate model is due to modeling error, which consists of three parts:</p>
<ol>
<li>Bias</li>
<li>Variance</li>
<li>Irreducible noise</li>
</ol>
<p style="text-align: justify;">Bias refers to the idea that we may be able to fit a model that works pretty well on the dataset that we have, but it is <em>biased</em> because it overfits our specific data too well and does not reflect the true relationship. Within the context of model flexibility, we think of this as related to flexible models because of their tendency to over-specify the relationship (and therefore not as easy to generalize). </p>
<p style="text-align: justify;">Variance refers to the idea that as we fit models on different datasets, we may get different models. Put another way, we may perform worse on any specific dataset, but we better approximate the structural nature of the relationship. </p>
<p style="text-align: justify;">The tension between these two components should be apparent - on the one hand, we can train a model that does well on our current dataset (and lose out on the ability to generalize the relationship across models) or we can train a model that performs consistently with lower variance but higher bias (so we could perform quite terribly on certain datasets and potentially lose out on the ability to describe particular trends within the data we train on). There will always be a tension between these two components, where sometimes it may be worth increasing one particular component to decrease the overall error, which is what we're describing with the term <em>bias-variance tradeoff</em>. </p>
<p style="text-align: justify;">Finally, irreducible noise refers to the remaining error that is always present in attempting to describe any empirical problem (as opposed to an analytically generated dataset and solution). </p>
<p style="text-align: justify;">These are important concepts that will be relevant as we continue, especially as we think about how we might evaluate the performance of our modeling approach. </p>
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<h2 class="hd hd-2 unit-title">Trees: Bootstrapping</h2>
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<h3>Exercise</h3>
<p style="font-size: 16px;">Here are two examples to work through to understand how to use tree-based methods and boosting:</p>
<ol>
<li>An example using the iris dataset as Alistair demonstrated in his video to best understand the concept. The ensemble methods tutorial provides a more applied exploration of the 506 census tracts of Boston from the 1970 census. <br />Please follow this tutorial walking through how you build trees on the <a href="https://ademos.people.uic.edu/Chapter24.html" target="[object Object]">iris dataset</a> and for an <a href="https://daviddalpiaz.github.io/r4sl/ensemble-methods.html" target="[object Object]">introduction to ensemble methods</a> (bagging & boosting).</li>
<li>An example from <a href="https://dataplatform.cloud.ibm.com/data/notebooks/converter/assets/de824290-6811-455b-b2a4-678fd6ae06ee?access_token=cb51caafa25bb0e596867d19fd0e96c77baeffca4b561091e38f0cd4476dc682&project=18a1ec4b-baaa-4cd8-b9c9-28001e5147ee" target="[object Object]">IBM Watson using XGBoost</a> to classify breast cancer as benign or malignant based on features from 569 diagnostic images from the Wisconsin dataset. <span style="font-size: 1em;">Go through steps 2 and 3.</span></li>
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<h3>Take-Home Message</h3>
<p>Overall we have given you a foundation of the most common data analysis techniques in health studies. It is important to clearly identify your objectives in a study as it identifies key aspects for planning and executing your analysis. For your desired outcome it is important to decide which analysis technique is most appropriate.</p>
<p>Here is a recap of the techniques we investigated:</p>
<ul>
<li>Linear Regression: An estimate of the expected change in the outcome per one unit increase in the covariate while keeping all other covariates constant.</li>
<li>Logistic Regression: An estimate of the fold change in the odds of the outcome per unit increase in the covariate, while keeping all other covariates constant.</li>
<li>Cox Proportional Hazard Model: An estimate of the fold change in the hazards of the outcome per unit increase in the covariate while keeping all other covariates constant.</li>
<li>Tree-based Models: A class of methods widely used for prediction that does not assume an underlying distribution of data.</li>
</ul>
<p>Within a given study, it is common to find yourself integrating multiple analysis techniques to accomplish different objectives within the study.</p>
<p></p>
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Question 1
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<legend id="06045c018efd4db0b07aa1d0f71a022f_2_1-legend" class="response-fieldset-legend field-group-hd">Which modeling/analysis method is always the best to use in a clinical setting?</legend>
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<h3>References</h3>
<p>As we mentioned at the beginning, these are among the most prevalent and useful methods in data analysis in a clinical setting, but are just a few of the many tools available in analyzing data.</p>
<p>Before concluding, we wanted to provide a brief overview to other references that may be helpful in exploring some of these topics further:</p>
<ul>
<li><a href="http://www-bcf.usc.edu/~gareth/ISL/">Introduction to Statistical Learning</a> provides a more gentle introduction to a variety of techniques and principles relevant to modern data analysis (book, slides and videos available)</li>
<li>For a more advanced treatment of topics:</li>
<ul>
<li><a href="https://web.stanford.edu/~hastie/ElemStatLearn/download.html">Elements of Statistical learning</a> contains more advanced to contents for some techniques studied in this module an other more complex techniques (e.x. neural nets, support vector machines (SVM)).</li>
<li><a href="https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-034-artificial-intelligence-spring-2005/syllabus/">Medical AI</a>is an open source course that introduces complex techniques of ML and artificial neural nets. It requires previous knowledge of calculus.</li>
<li>In case you want to analyze time variant information, try some <a href="https://www.stat.pitt.edu/stoffer/tsa4/tsa4.pdf">time series methods</a>.</li>
<li>We have <a href="http://introtodeeplearning.com/#schedule">another course</a> that provides an introduction to deep learning.</li>
<li>A more applied treatment of <a href="https://www.fast.ai/">various subjects</a> that may be of interest (machine learning, deep learning and computational linear algebra).</li>
</ul>
</ul>
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