# 13: Linear Regression

• 13.1: Line of Best Fit
In correlations, we referred to a linear trend in the data. That is, we assumed that there was a straight line we could draw through the middle of our scatterplot that would represent the relation between our two variables, X and Y . Regression involves solving for the equation of that line, which is called the Line of Best Fit.
• 13.2: Prediction
In regression, we most frequently talk about prediction, specifically predicting our outcome variable Y from our explanatory variable X, and we use the line of best fit to make our predictions.
• 13.3: ANOVA Table
Our ANOVA table in regression follows the exact same format as it did for ANOVA (hence the name). Our top row is our observed effect, our middle row is our error, and our bottom row is our total. The columns take on the same interpretations as well: from left to right, we have our sums of squares, our degrees of freedom, our mean squares, and our F statistic.
• 13.4: Hypothesis Testing in Regression
Regression, like all other analyses, will test a null hypothesis in our data. In regression, we are interested in predicting Y scores and explaining variance using a line, the slope of which is what allows us to get closer to our observed scores than the mean of Y can. Thus, our hypotheses concern the slope of the line, which is estimated in the prediction equation by b .
• 13.5: Happiness and Well-Being
Researchers are interested in explaining differences in how happy people are based on how healthy people are. They gather data on each of these variables from 18 people and fit a linear regression model to explain the variance. We will follow the four-step hypothesis testing procedure to see if there is a relation between these variables that is statistically significant.
• 13.6: Multiple Regression and Other Extensions
The next step in regression is to study multiple regression, which uses multiple X variables as predictors for a single Y variable at the same time. The math of multiple regression is very complex but the logic is the same: we are trying to use variables that are statistically significantly related to our outcome to explain the variance we observe in that outcome. Other forms of regression include curvilinear models that can explain curves in the data rather than straight lines.
• 13.E: Linear Regression (Exercises)