13. ANOVA

ANOVA stands for ANalysis Of VAriance. ANOVA is used when we want to compare means across groups, conditions, or time points. Like the t-tests in ?sec-t-tests, ANOVA is used with a continuous outcome variable. The major difference is that ANOVA can handle research questions with three or more groups, repeated measurements, or more than one independent variable.

Although ANOVA is used to compare means, it does so by analyzing variability. ANOVA compares how much scores vary between groups or conditions to how much scores vary within groups or conditions. If the between-group variability is large relative to the within-group variability, the ANOVA is more likely to be statistically significant.

Most ANOVA tests are omnibus tests. This means they test whether there is a difference somewhere among the means, but they do not always tell us exactly which means differ. When an omnibus ANOVA is statistically significant, we often need follow-up analyses, such as post hoc tests or planned contrasts, to identify the specific group differences.

There are several types of ANOVA. The appropriate test depends on the number of independent variables and whether the groups or measurements are independent or related.

Test	Variables and Design	Research Question
One-way ANOVA	One continuous outcome and one categorical independent variable with three or more independent groups	Do three or more independent groups differ on the outcome?
Repeated-measures ANOVA	One continuous outcome measured across three or more related conditions or time points	Do scores differ across three or more related measurements?
Factorial ANOVA	One continuous outcome and two or more categorical independent variables	Do two or more independent variables predict the outcome, and do their effects depend on each other?
Mixed factorial ANOVA	One continuous outcome with at least one between-subjects independent variable and at least one within-subjects independent variable	Do group differences, repeated-measurement differences, or their interaction predict the outcome?
ANCOVA	One continuous outcome, at least one categorical independent variable, and at least one continuous covariate	Do groups differ on the outcome after statistically adjusting for a continuous covariate?

This chapter begins with the one-way ANOVA because it is the most direct extension of the independent t-test. We then examine how to follow up a significant ANOVA, how to analyze related measurements with repeated-measures ANOVA, and how to test more complex designs with factorial ANOVA. Finally, we briefly introduce ANCOVA as an extension of ANOVA that includes a continuous covariate.

In each case, we will continue using the same four-step hypothesis-testing process: look at the data, check assumptions, perform the test, and interpret the results.