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<p>The linear regression model assumes that there is a straight line (linear) relationship between the predictors and the response. It doesn’t ask for the straight line relationship to be perfect, but rather on average the cloud of points has a linear shape. If that is not true, then our prediction is going to be less accurate.</p>
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<p>We can check this relationship by seeing whether predictor is linear with the predicted response value, but this is cumbersome with multiple predictors. Rather, we typically calculate the <strong>residual</strong>, which is the difference between the response value and the predicted response value (similar to a type of model performance metrics we examined last week). Then, we can make a <strong>residual plot</strong> of the predicted response vs. residual. Ideally, this residual plot should have no pattern - some residuals above 0, some below 0, but no strong trend.</p>
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<p>If there’s a trend in the data, that means there are non-linear associations between some of the predictors and the response.</p>
<p>We see there’s a slight curve in our residual plot. We will look at ways to deal with this later in this lecture.</p>
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<p>In a model with more predictors, we can dig into more details by making a residual plot of a predictor vs. residual. This is often used to figure out which predictor is contributing to the shape of the predicted response vs. residual plot.</p>
<li><p>When there is a collinear relationship between three or more predictors, pairwise methods will fail. We may consider the Variance Inflation Factor to detect them, but doesn’t necessarily recommend which variables to remove.</p></li>
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</ul>
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<p>Suppose that we are consider the predictors of our training set:</p>
MeanBloodPressure= \beta_0 + \beta_1 \cdot Age + \beta_2 \cdot Age^2
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\]</span></p>
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<p>This is <em>still</em> a linear model – we have added a new predictor that gives us a quadratic shape. We use the <ahref="https://matthewwardrop.github.io/formulaic/latest/guides/splines/#poly"><code>poly()</code> function</a> to generate our polynomial predictor.</p>
@@ -516,7 +516,7 @@ <h2 data-number="3.4" class="anchored" data-anchor-id="interactions"><span class
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<p>Here is another way to extend the Linear Model:</p>
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<p>Suppose we think that <spanclass="math inline">\(BMI\)</span> and <spanclass="math inline">\(Gender\)</span> may be good predictors of <spanclass="math inline">\(MeanBloodPressure\)</span>:</p>
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<p>Let’s explore the relationship between <spanclass="math inline">\(MeanBloodPressure\)</span> and <spanclass="math inline">\(BMI\)</span> separately for values of <spanclass="math inline">\(Gender\)</span>.</p>
<p><spanclass="math inline">\(\beta_0\)</span> is a parameter describing the intercept of the line, and <spanclass="math inline">\(\beta_1\)</span> is a parameter describing the slope of the line.</p>
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<p>Suppose that from fitting the model on the Training Set, <spanclass="math inline">\(\beta_1=2\)</span>. That means increasing <spanclass="math inline">\(BMI\)</span> by 1 will lead to an increase of <spanclass="math inline">\(BloodPressure\)</span> by 2. This measures the strength of association between a variable and the outcome.</p>
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