14. Correlation and Regression

So far, most of our inferential tests have focused on group differences or differences between observed and expected frequencies. In this chapter, we shift to statistical relationships between variables.

Correlation and regression are both used to examine relationships among variables, but they answer different kinds of questions. Correlation describes the strength and direction of the relationship between two variables. Regression uses one or more predictor variables to predict or explain variation in a continuous outcome variable.

A key difference is that correlation treats the two variables symmetrically. We ask whether two variables are related, but we do not label one variable as the predictor and the other as the outcome. Regression is directional in its setup: one variable is the outcome, and one or more variables are used as predictors.

Analysis	Variables and Design	Research Question
Pearson correlation	Two continuous variables	Are the two variables linearly related?
Spearman correlation	Two ordinal or non-normally distributed variables, or a monotonic relationship	Are the two variables related in rank order?
Simple linear regression	One predictor variable and one continuous outcome variable	Does one variable predict the outcome?
Multiple regression	Two or more predictor variables and one continuous outcome variable	How well do the predictors work together to predict the outcome?
Regression with categorical predictors	Continuous outcome with categorical and/or continuous predictors	Do groups differ on the outcome after being represented in a regression model?
Hierarchical regression	Predictors entered in blocks or steps	Does adding a new set of predictors improve the model?
General linear model	A broad framework that includes many analyses we have learned	How are correlation, t-tests, ANOVA, and regression connected?

This chapter begins with correlation in ?sec-correlation, where we focus on describing the strength and direction of relationships between variables. We then move to regression in ?sec-regression, where we use predictors to estimate or explain a continuous outcome. Finally, ?sec-general-linear-model shows how many of the statistical tests we have learned are connected through the general linear model.

As always, we will continue using the same four-step hypothesis-testing process: look at the data, check assumptions, perform the test, and interpret the results. The main difference is that we are now asking about relationships among variables rather than comparing groups or frequencies.

One important caution applies throughout this chapter: relationships do not automatically imply causation. A correlation or regression coefficient can show that variables are related, but causal conclusions depend on the research design, measurement quality, and alternative explanations.