Chapter 8
Introduction to Linear Regression
Learning Outcomes
 Define the explanatory variable as the independent variable (predictor), and the response variable as the dependent variable (predicted).
 Plot the explanatory variable ($x$) on the xaxis and the response variable ($y$) on the yaxis, and fit a linear regression model
$y = \beta_0 + \beta_1 x$
where $\beta_0$ is the intercept, and $\beta_1$ is the slope. Note that the point estimates (estimated from observed data) for $\beta_0$ and $\beta_1$ are $b_0$ and $b_1$, respectively.
 When describing the association between two numerical variables, evaluate
 direction: positive ($x \uparrow, y \uparrow$), negative ($x \downarrow, y \uparrow$)
 form: linear or not
 strength: determined by the scatter around the underlying relationship
 Define correlation as the \emph{linear} association between two numerical variables.
 Note that a relationship that is nonlinear is simply called an association.
 Note that correlation coefficient ($r$, also called Pearson's $r$) the following properties:
 the magnitude (absolute value) of the correlation coefficient measures the strength of the linear association between two numerical variables
 the sign of the correlation coefficient indicates the direction of association
 the correlation coefficient is always between 1 and 1, inclusive, with 1 indicating perfect negative linear association, +1 indicating perfect positive linear association, and 0 indicating no \emph{linear} relationship
 the correlation coefficient is unitless
 since the correlation coefficient is unitless, it is not affected by changes in the center or scale of either variable (such as unit conversions)
 the correlation of X with Y is the same as of Y with X
 the correlation coefficient is sensitive to outliers
 Recall that correlation does not imply causation.
 Define residual ($e$) as the difference between the observed ($y$) and predicted ($\hat{y}$) values of the response variable.
$e_i = y_i  \hat{y}_i$
 Define the least squares line as the line that minimizes the sum of the squared residuals, and list conditions necessary for fitting such line:
 linearity
 nearly normal residuals
 constant variability
 Define an indicator variable as a binary explanatory variable (with two levels).
 Calculate the estimate for the slope ($b_1$) as
$b_1 = R\frac{s_y}{s_x}$
, where $r$ is the correlation coefficient, $s_y$ is the standard deviation of the response variable, and $s_x$ is the standard deviation of the explanatory variable.  Interpret the slope as
 “For each unit increase in $x$, we would expect $y$ to increase/decrease on average by $b_1$ units” when $x$ is numerical.
 “The average increase/decrease in the response variable when between the baseline level and the other level of the explanatory variable is $b_1$.” when $x$ is categorical.
 Note that whether the response variable increases or decreases is determined by the sign of $b_1$.
 Note that the least squares line always passes through the average of the response and explanatory variables ($\bar{x},\bar{y}$).
 Use the above property to calculate the estimate for the slope ($b_0$) as
$b_0 = \bar{y}  b_1 \bar{x}$
, where $b_1$ is the slope, $\bar{y}$ is the average of the response variable, and $\bar{x}$ is the average of explanatory variable.  Interpret the intercept as
 “When $x = 0$, we would expect $y$ to equal, on average, $b_0$.” when $x$ is numerical.
 “The expected average value of the response variable for the reference level of the explanatory variable is $b_0$.” when $x$ is categorical.
 Predict the value of the response variable for a given value of the explanatory variable, $x^\star$, by plugging in $x^\star$ in the in the linear model:
$\hat{y} = b_0 + b_1 x^\star$
 Only predict for values of $x^\star$ that are in the range of the observed data.
 Do not extrapolate beyond the range of the data, unless you are confident that the linear pattern continues.
 Define $R^2$ as the percentage of the variability in the response variable explained by the the explanatory variable.
 For a good model, we would like this number to be as close to 100% as possible.
 This value is calculated as the square of the correlation coefficient, and is between 0 and 1, inclusive.
 Define a leverage point as a point that lies away from the center of the data in the horizontal direction.
 Define an influential point as a point that influences (changes) the slope of the regression line.
 This is usually a leverage point that is away from the trajectory of the rest of the data.
 Do not remove outliers from an analysis without good reason.
 Be cautious about using a categorical explanatory variable when one of the levels has very few observations, as these may act as influential points.
 Determine whether an explanatory variable is a significant predictor for the response variable using the $t$test and the associated pvalue in the regression output.
 Set the null hypothesis testing for the significance of the predictor as
$H_0: \beta_1 = 0$
, and recognize that the standard software output yields the pvalue for the twosided alternative hypothesis. Note that $\beta_1 = 0$ means the regression line is horizontal, hence suggesting that there is no relationship between the explanatory and the response variables.
 Calculate the T score for the hypothesis test as
$T_{df}=\frac { b_{ 1 }{ null\quad value } }{ SE_{ b_{ 1 } } }$
with $df = n  2$. Note that the T score has $n  2$ degrees of freedom since we lose one degree of freedom for each parameter we estimate, and in this case we estimate the intercept and the slope.
 Note that a hypothesis test for the intercept is often irrelevant since it's usually out of the range of the data, and hence it is usually an extrapolation.
 Calculate a confidence interval for the slope as
$b_1 \pm t^\star_{df} SE_{b_1}$
where $df = n  2$ and$t^\star_{df}$
is the critical score associated with the given confidence level at the desired degrees of freedom. Note that the standard error of the slope estimate
$SE_{b_1}$
can be found on the regression output.
 Note that the standard error of the slope estimate
Supplemental Readings

Linear regression with SAT scores  This document outlines the implementation of linear regression stepbystep emphasizing visualizations.