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‎book/_build/.doctrees/discussions/linear-models.doctree

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‎book/_build/.doctrees/exercises/linear-models.doctree

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‎book/_build/html/_sources/discussions/linear-models.md

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# Discussion questions
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1. Linear regression can answer research questions typically reserved for specialty statistics like the t-test and ANOVA. For example, Pearson correlation coefficient has the form of $r = \frac{COV[X,Y]}{STD[X]STD[Y]}$, how does this compare to the ordinary least squares solution for $\hat{\beta}$? Provide a quantitative comparison of the two methods.
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1. Linear regression can answer research questions typically reserved for specialty statistics like the t-test and ANOVA. For example, Pearson correlation coefficient has the form of $r = \frac{COV[X,Y]}{STD[X]STD[Y]}$, how does this compare to the ordinary least squares solution for $\hat{\beta}$? Provide a quantitative comparison of the two methods. Note: The unstandardized beta (B) represents the change in Y for a one-unit change in X. The standardized beta (β) represents the change in Y in standard deviation units for a one standard deviation change in X.
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2. Provide an explanation for why polynomial models still meet the assumptions of normal linear regression. Justify against each of the 4 assumptions for ordinary least squares regression.
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‎book/_build/html/_sources/exercises/linear-models.ipynb

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‎book/_build/html/_sources/exercises/the-ordinary-least-squares-solution.ipynb

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‎book/_build/html/discussions/linear-models.html

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<h1>Discussion questions<a class="headerlink" href="#discussion-questions" title="Link to this heading">#</a></h1>
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<ol class="arabic simple">
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<li><p>Linear regression can answer research questions typically reserved for specialty statistics like the t-test and ANOVA. For example, Pearson correlation coefficient has the form of <span class="math notranslate nohighlight">\(r = \frac{COV[X,Y]}{STD[X]STD[Y]}\)</span>, how does this compare to the ordinary least squares solution for <span class="math notranslate nohighlight">\(\hat{\beta}\)</span>? Provide a quantitative comparison of the two methods.</p></li>
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<li><p>Linear regression can answer research questions typically reserved for specialty statistics like the t-test and ANOVA. For example, Pearson correlation coefficient has the form of <span class="math notranslate nohighlight">\(r = \frac{COV[X,Y]}{STD[X]STD[Y]}\)</span>, how does this compare to the ordinary least squares solution for <span class="math notranslate nohighlight">\(\hat{\beta}\)</span>? Provide a quantitative comparison of the two methods. Note: The unstandardized beta (B) represents the change in Y for a one-unit change in X. The standardized beta (β) represents the change in Y in standard deviation units for a one standard deviation change in X.</p></li>
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<li><p>Provide an explanation for why polynomial models still meet the assumptions of normal linear regression. Justify against each of the 4 assumptions for ordinary least squares regression.</p></li>
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‎book/_build/html/exercises/linear-models.html

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<p>Use the <code class="docutils literal notranslate"><span class="pre">head</span></code> function to look at the first few rows of each data frame.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># If you are running this on your local computer, wet your workign directory to </span>
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<span class="c1"># and change the location to where it is on your computer. </span>
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<p>Use the <code class="docutils literal notranslate"><span class="pre">pairs</span></code> function to look at all the pairwise scatterplots of the variables in <code class="docutils literal notranslate"><span class="pre">d1</span></code>. Describe which variables seem positively correlated, negatively correlated, or not correlated at all.</p>
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<p>Show the results using the <code class="docutils literal notranslate"><span class="pre">summary</span></code> function, and report the mean coefficient values for <span class="math notranslate nohighlight">\(beta_0\)</span> &amp; <span class="math notranslate nohighlight">\(\beta_1\)</span> (<code class="docutils literal notranslate"><span class="pre">coef</span></code> function) and their 95% confidence intervals (<code class="docutils literal notranslate"><span class="pre">confint</span></code> function). Is grey matter volume significantly associated with Flanker Task performance?</p>
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<p><strong>DUE:</strong> 5pm EST, February 26, 2024</p>
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<p><strong>DUE:</strong> 5pm EST, February 24, 2025</p>
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<p><strong>IMPORTANT</strong> Did you collaborate with anyone on this assignment? If so, list their names here.</p>
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‎book/_build/html/exercises/the-ordinary-least-squares-solution.html

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<p>b) Now we’re going to see if the solutions we proved above actually line up with the model fit that R gives us (it should…). Calculate what the <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span> coefficients should be for a simple linear regression model using <code class="docutils literal notranslate"><span class="pre">Flanker_Unadj</span></code> as <span class="math notranslate nohighlight">\(Y\)</span> and <code class="docutils literal notranslate"><span class="pre">FS_Total_GM_Vol</span></code> as <span class="math notranslate nohighlight">\(X\)</span>. Use the formulas we derived above (<span class="math notranslate nohighlight">\(\beta_1 = Cov[XY]/Var[X]\)</span> , <span class="math notranslate nohighlight">\(\beta_0 = E[Y] - \beta_1E[X]\)</span>). Then use <code class="docutils literal notranslate"><span class="pre">lm()</span></code> to compare the coefficients you calculated with the ones R gives you.</p>
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<p><strong>DUE:</strong> 5pm EST, Feb 28, 2024</p>
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<p><strong>DUE:</strong> 5pm EST, Feb 26, 2025</p>
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<p><strong>IMPORTANT</strong> Did you collaborate with anyone on this assignment? If so, list their names here.</p>
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‎book/_build/html/searchindex.js

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‎book/_build/jupyter_execute/exercises/linear-models.ipynb

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