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<!DOCTYPE html>
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<title>Regression diagnostics and prediction</title>
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<div id="section-intro" class="section level2">
<h2>Intro</h2>
<p>We use models to explain the relationship between variables and to make predictions. For now we will focus on <strong>linear</strong> models (but remember there are other types of models too!)</p>
<p>Today, we will focus on fitting and interpreting models with a continuous outcome variable and <strong>multiple</strong> predictors. This type of model is often called a <strong>multivariable</strong> (<em>not multivariate</em>) model. We will again be using functions from the <code>broom</code> package. In particular, we will be using the <code>augment</code> function to get model predictions and check assumptions for our linear regression model.</p>
</div>
<div id="section-todays-data" class="section level2">
<h2>Today’s data</h2>
<p>Today’s data come from a series of atmospheric measurements taken at the Agriculture Exhibition Hall in Beijing from 2013 - 2017. The following variables are in the dataset:</p>
<ul>
<li><code>year</code>, <code>month</code>, <code>day</code>, and <code>hour</code>: when the observation was taken</li>
<li><code>PM2.5</code> and <code>PM10</code>: levels of fine particulate matter in the atmosphere, in <span class="math inline">\(\mu\)</span>g/m<span class="math inline">\(^3\)</span></li>
<li><code>SO2</code>, <code>NO2</code>, <code>CO</code>, <code>O3</code>: sulfur dioxide, nitrogen dioxide, carbon monoxide, and ozone concentrations, in <span class="math inline">\(\mu\)</span>g/m<span class="math inline">\(^3\)</span>, respectivel</li>
<li><code>TEMP</code>: temperature, in degrees C</li>
<li><code>PRES</code>: barometric pressure, in hPa</li>
<li><code>DEWP</code>: dew point temperature, in degrees C</li>
<li><code>RAIN</code>: precipitation levels, in mm</li>
<li><code>wd</code>: wind direction (eight compass directions)</li>
<li><code>SWPM</code>: wind speed, in m/s</li>
</ul>
<p>These data are adapted from a dataset released by Song Xi Chen as referenced in Zhang S., et al (2017), Proc. Royal Soc. A. Let’s explore some of these data.</p>
<p>Last time, we created a linear model using ordinary least squares that predicts the dew point given the current barometric pressure, and interpreted the slope and intercept parameter estimates as follows:</p>
<pre class="r"><code>mod1 <- lm(DEWP ~ PRES, data = beijing)
tidy(mod1) %>%
mutate_if(is.numeric, round, 3)</code></pre>
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<ul>
<li><p><strong>Intercept</strong>: if the barometric pressure is 0, we would expect the dew point to be 1071 degrees Celsius, on average.</p></li>
<li><p><strong>Slope</strong>: for each additional hectopascal of barometric pressure, we would expect the dew point to decrease by 1.08 degrees Celsius.</p></li>
</ul>
</div>
<div id="section-adding-more-predictors" class="section level2">
<h2>Adding more predictors</h2>
<p>We have a pretty good idea of what the dew point would be, <strong>conditional</strong> on the value of the barometric pressure. But if we knew the temperature and<br />
precipitation, we might do even better! Let’s add some more predictors to our model:</p>
<pre class="r"><code>mod2 <- lm(DEWP ~ PRES + TEMP + RAIN, data = beijing)</code></pre>
<p>In general, the underlying linear model with multiple predictors is given by</p>
<p><span class="math display">\[y_i = \beta_0 + \beta_1 x_{1i} + \beta_2x_{2i} + \cdots + \beta_px_{pi} +
\epsilon_i\]</span></p>
<ul>
<li><span class="math inline">\(p\)</span> is the total number of predictor or explanatory variables</li>
<li><span class="math inline">\(y_i\)</span> is the outcome variable for individual <span class="math inline">\(i\)</span></li>
<li><span class="math inline">\(\beta_0\)</span> is the intercept parameter</li>
<li><span class="math inline">\(\beta_1, \beta_2, \cdots, \beta_p\)</span> are the slope parameters</li>
<li><span class="math inline">\(x_{1i}, \cdots, x_{pi}\)</span> are the predictor variables</li>
<li><span class="math inline">\(\epsilon_i\)</span> is the unobserved error term</li>
</ul>
<p>Interpretations for slope estimates are made <strong>conditionally</strong> on other predictors in the model. In this way, we can adjust for potential confounders in our model:</p>
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<p><span class="math display">\[\widehat{DEWP} = 316 - 0.364 PRES + 0.723 TEMP + 1.15 RAIN\]</span></p>
<ul>
<li><p>If the barometric pressure, temperature, and precipitation are all 0, then we would expect a dew point of 361 degrees Celsius (hmm…).</p></li>
<li><p>For each additional hPa increase in barometric pressure, we would expect a 0.364 C <em>decrease</em> in dew point (on average), <strong>holding temperature and precipiation constant</strong>.</p></li>
<li><p>For each additional degree C increase in temperature, we would expect a 0.723 C increase in dew point (on average), <strong>holding barometric pressure and precipiation constant</strong>.</p></li>
<li><p>For each additional mm increase in precipitation, we would expect a 1.15 C increase in dew point (on average), <strong>holding barometric pressure and temperature constant</strong>.</p></li>
</ul>
</div>
<div id="section-inference-with-multiple-predictors" class="section level2">
<h2>Inference with multiple predictors</h2>
<p>We may again test hypotheses that <span class="math inline">\(\beta\)</span> coefficients are equal to 0 (vs. the alternative that they are not equal to 0). Be sure that any interpretation for slopes are made <strong>conditionally</strong> on the other predictors in the model (that is, holding them constant/adjusting for them, etc.). Under the null hypothesis, each slope coefficient should follow a <span class="math inline">\(t\)</span> distribution, with <span class="math inline">\(n - p - 1\)</span> degrees of freedom (where <span class="math inline">\(n\)</span> is the total sample size and <span class="math inline">\(p\)</span> is the number of estimated parameters in your model).</p>
<p>Let’s check the <code>tidy</code> model output from before:</p>
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<p><strong>Is there sufficient evidence that barometric pressure has a linear association</strong> <strong>with dew point, while adjusting for temperature and precipitation?</strong></p>
<p><span class="spoiler">Yes. At the <span class="math inline">\(\alpha\)</span> = 0.05 level, we reject the null hypothesis. There is sufficient evidence of such an association (i.e., that the <span class="math inline">\(\beta\)</span> coefficient is not equal to 0).</span></p>
</div>
<div id="section-r2-and-r2_adj" class="section level2">
<h2><span class="math inline">\(R^2\)</span> and <span class="math inline">\(R^2_{adj}\)</span></h2>
<p>We can once again <code>glance</code> at the model output to obtain model-specific summaries. Let’s take a look at the <span class="math inline">\(R^2\)</span> for our model:</p>
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<p>We see that approximately 70% of the variability in dew point explained by its linear relationship with barometric pressure, temperature, and dew point, which is higher than the 60% explained by barometric pressure alone.</p>
<p>However, <span class="math inline">\(R^2\)</span> can never decrease when variables are added to a model, <em>even if they are useless for prediction</em>. We adjust for the number of predictors in our model to obtain <strong>adjusted</strong> <span class="math inline">\(R^2\)</span>, where the adjustment is made to account for the number of predictors. <span class="math inline">\(R^2_{adj}\)</span> incorporates a penalty for each additional predictor, so it will go down if a new variable does not meaningfully improve prediction. However, note that <span class="math inline">\(R^2_{adj}\)</span> can <strong>not</strong> be interpreted as explained variability.</p>
</div>
<div id="section-predicting-new-values" class="section level2">
<h2>Predicting new values</h2>
<p>Our fitted model is given by</p>
<p><span class="math display">\[\widehat{DEWP} = 360.7 - 0.36PRES + 0.72 TEMP + 1.15RAIN,\]</span></p>
<p>and can be used to predict dew points given the barometric pressure, temperature, and precipitation (simply plug into the model).</p>
<div id="section-exercise-1" class="section level3">
<h3>Exercise 1</h3>
<p>Suppose the barometric pressure, temperature, and precipitation are equal to the mean values in our dataset. What would the predicted dew point be? You may use the following space to code your answer.</p>
<div class="tutorial-exercise" data-label="ex-1" data-completion="1" data-diagnostics="1" data-startover="1" data-lines="0">
<pre class="text"><code>mod2 <- lm(DEWP ~ PRES + TEMP + RAIN, data = beijing)
...</code></pre>
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</div>
<p><strong>Is this prediction close to the mean dew point in our dataset?</strong></p>
<p><span class="spoiler">Yes, our estimate of 2.37 degrees C is quite close to the mean dew point in the dataset, which is equal to 2.38.</span></p>
</div>
</div>
<div id="section-augmenting-model-results" class="section level2">
<h2>Augmenting model results</h2>
<p>We can actually obtain the predicted values using code, without having to deal with messy algebra. We will use the <code>augment</code> function. Let’s try it out on our model object:</p>
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<p>Notice that the <code>augment</code> function provides fitted values and residuals (and more) for every observation in our dataset.</p>
<p>The mean pressure, temperature, and precipitation in our dataset are 1012.6 hPa, 13.55 C, and 0.074 mm, respectively (did you obtain these results using the <code>summarize</code> function in a pipeline?). Let’s create a new <code>tibble</code> and “feed” it into our <code>mod2</code> model object, which contains our multiple linear regression model:</p>
<pre class="r"><code>mean_vals <- beijing %>%
summarize(PRES = mean(PRES, na.rm = T),
TEMP = mean(TEMP, na.rm = T),
RAIN = mean(RAIN, na.rm = T))
augment(mod2, newdata = mean_vals)</code></pre>
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<p>Note that the new dataset has the same variable names as found in our model. With our model we predict a dew point of 2.37 C and note that the standard error for this prediction is 0.195 C.</p>
<p><strong>How might we use these results to create interval estimates?</strong></p>
<p><span class="spoiler">We can construct a confidence interval by taking our point estimate <code>.fitted</code>, and adding/subtracting the margin of error given by the confidence multiplier times the standard error of our estimate, <code>.se.fit</code>.</span></p>
</div>
<div id="section-model-assumptions" class="section level2">
<h2>Model assumptions</h2>
<p>So far, we have blindly fit linear models without caring about any assumptions. However, like all statistical techniques, there are a few main assumptions to the linear model:</p>
<ul>
<li><p>The outcomes <span class="math inline">\(y_i\)</span> are <strong>independent</strong>. This means that observations do not “influence” each other in any way (e.g., repeated outcome measures are made on the same individual)</p></li>
<li><p>A <strong>linear relationship</strong> holds (though we can relax this to some extent)</p></li>
<li><p>The <strong>error</strong> terms are <font class = "vocab">normally distributed</font>, with mean zero and <font class = "vocab">constant variance</font> across all predicted/fitted values</p></li>
</ul>
<p>Many of the assumptions deal with the error term and fitted values, which we might estimate using the residuals and predictions from our model. Luckily, we can use the <code>augment</code> function from the <code>broom</code> package to calculate these. Let’s consider the multiple regression model from earlier, focusing on the <code>.fitted</code> and <code>.resid</code> variables in this data frame:</p>
<pre class="r"><code>augment(mod2) %>%
select(.fitted, .resid)</code></pre>
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</script>
</div>
<ul>
<li><p><code>.fitted</code> provides the predicted value of the response variable (for the given observation)</p></li>
<li><p><code>.resid</code> provides the residual for the observation</p></li>
</ul>
<p>Let’s evaluate these four assumptions.</p>
</div>
<div id="section-independence" class="section level2">
<h2>Independence</h2>
<p>Unfortunately, in many cases we must simply think about how the data are collected. In this case, we have atmospheric measurements through time at a single station. Independence might <strong>not</strong> be satisfied – we might imagine that if the dew point is high today, it’ll probably still be high tomorrow. Note that the data are already sorted in chronological order. Let’s plot the dew point through time, examining the first three hundred data points:</p>
<pre class="r"><code>beijing %>%
slice(1:300) %>%
ggplot(data = ., mapping = aes(x = 1:300, y = DEWP)) +
geom_line() +
labs(x = "Measurement number", y = "Dew point (C)")</code></pre>
<p><img src="tutorial-diag-pred-augment_files/figure-html/unnamed-chunk-10-1.png" width="624" /></p>
<p>We see that independence is violated. Although we used a graph to help us determine this visually, this was only because our data happened to be time- series data. In general, there is no plot that we will always be able to use to diagnose independence.</p>
</div>
<div id="section-normality-of-errors" class="section level2">
<h2>Normality of errors</h2>
<p>We can simply plot a histogram of the residuals and assess visually. There are more formal methods, but for our purposes this will suffice. Note our use of the <code>.resid</code> variable from the augmented model results:</p>
<pre class="r"><code>augment(mod2) %>%
ggplot(data = ., mapping = aes(x = .resid)) +
geom_histogram() +
labs(x = "Residual", y = "Count",
title = "Residuals are left-skewed")</code></pre>
<p><img src="tutorial-diag-pred-augment_files/figure-html/unnamed-chunk-11-1.png" width="624" /></p>
<p><strong>Do the residuals appear to be normally distributed?</strong></p>
<p><span class="spoiler">No - they are left skewed!</span></p>
</div>
<div id="section-the-residual-plot" class="section level2">
<h2>The residual plot</h2>
<p>One of the most useful diagnostic plots is the <strong>residual plot</strong>, which plots the fitted values against the residuals. Let’s take a look:</p>
<pre class="r"><code>augment(mod2) %>%
ggplot(data = ., mapping = aes(x = .fitted, y = .resid)) +
geom_point() +
geom_hline(yintercept = 0, color = "red") +
labs(x = "Fitted value", y = "Residual")</code></pre>
<p><img src="tutorial-diag-pred-augment_files/figure-html/unnamed-chunk-12-1.png" width="624" /></p>
<p>Ideally, the residual plot should be a “shapeless blob” with no real pattern. A pattern would be indicative of potential violation of our model assumptions.</p>
<ul>
<li><p>If the residuals are not symmetric about the x-axis, then we might have violation of linearity. This appears to be the case in our residual plot</p></li>
<li><p>If the residuals do not have similar vertical spread for all fitted values, then we might have violation of constant error variance (homoscedasticity of errors). This might result in a “fanning” pattern. Our residual plot doesn’t suggest that this assumption violated.</p></li>
</ul>
<p><strong>Do linearity and homogeneity of error variance appear satisfied in our plot?</strong> <strong>Why or why not?</strong></p>
<p><span class="spoiler">Linearity does not appear to be satisfied; we see evidence of assymetry around the x-axis (examine the ranges of the residuals above and below this line, for instance). However, constant variance does look to be an appropriate assumption - at every vertical “slice” of the data, the spread of the residuals appears to be relatively similar)</span></p>
</div>
<div id="section-interaction-effects" class="section level2">
<h2>Interaction effects</h2>
<p>Sometimes, the effect of one variable depends on the value of another. For example, the effect of exercise % on obesity may be different for smokers vs. non-smokers. To model such a relationship, we create an <strong>interaction term</strong>.</p>
<p>We fit interaction terms in the model by multiplying predictors together. Note the syntax used in the code below for the interaction between <code>TEMP</code> and <code>RAIN</code>:</p>
<pre class="r"><code>mod3 <- lm(DEWP ~ PRES + TEMP + RAIN + TEMP*RAIN, data = beijing)</code></pre>
<p>For a given PRES, TEMP, and RAIN, our predicted DEWP is given by <span class="math display">\[\begin{align*}
\widehat{DEWP} &= \hat{\beta}_0 + \hat{\beta}_1PRES + \hat{\beta}_2TEMP +\\
&\mathrel{\phantom{=}} \hat{\beta}_3RAIN + \hat{\beta}_4TEMP \times RAIN
\end{align*}\]</span></p>
<p>For a day with the same PRES AND TEMP but one higher unit of RAIN, the predicted DEWP is given by <span class="math display">\[\begin{align*}
\widehat{DEWP}^\star &= \hat{\beta}_0 + \hat{\beta}_1PRES + \hat{\beta}_2TEMP +\\
&\mathrel{\phantom{=}} \hat{\beta}_3(RAIN + 1) + \hat{\beta}_4TEMP \times (RAIN + 1)
\end{align*}\]</span></p>
<p>Subtracting the two equations, we have <span class="math display">\[\widehat{DEWP} - \widehat{DEWP}^\star = \hat{\beta}_3 + \hat{\beta}_4 TEMP\]</span> The expected change in dew point for a one mm increase in precipitation, if pressure is held constant, is <span class="math inline">\(\hat{\beta}_3 + \hat{\beta}_4 \times TEMP\)</span></p>
<p><strong>The effect of precipitation on the dew point, adjusting for pressure, depends on temperature!</strong></p>
<p>Let’s take a look at the tidy output:</p>
<pre class="r"><code>tidy(mod3)</code></pre>
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<p><strong>What IS the estimated expected change in dew point given a one mm</strong> <strong>increase in precipitation, if we hold barometric pressure constant?</strong> <strong>(Hint: as stated before, it depends on the temperature!)</strong></p>
<p><span class="spoiler">The answer is the main effect of precipitation plus the interaction effect with temperature, multiplied by the observed temperature. That is, 9.01 - 0.35 <span class="math inline">\(\times\)</span> TEMP</span></p>
</div>
<div id="section-on-your-own" class="section level2">
<h2>On your own!</h2>
<div id="section-exercise-2" class="section level3">
<h3>Exercise 2</h3>
<p>Recently, atmospheric pollution in terms of fine particulate matter (PM2.5) has been of concern. Fit a model that predicts PM2.5 levels based on current weather patterns, and assess any assumptions needed for your linear model. Is there evidence that any weather-related variables are linearly associated with PM2.5 levels, conditional on levels of the other predictors?</p>
<p>You may use the following space to code your answer:</p>
<div class="tutorial-exercise" data-label="ex-2" data-completion="1" data-diagnostics="1" data-startover="1" data-lines="0">
<pre class="text"><code>___ <- lm(___ ~ ___ + ___ + ..., data = ___)
tidy(___)
# Histogram of residuals
augment(___) %>%
ggplot(data = ., mapping = aes(x = ___)) +
geom_histogram() +
labs(x = "Residual", y = "Count",
title = "___")
# Residual plot
augment(___) %>%
ggplot(data = ., mapping = aes(x = ___, y = ___)) +
geom_point() +
geom_hline(yintercept = 0, color = "red") +
labs(x = "Fitted value", y = "Residual",
title = "___")</code></pre>
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<div id="section-return-to-learn-r-tutorials-page" class="section level2">
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<h2 class="title toc-ignore" style="display:none;">Regression diagnostics and prediction</h2>
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