Mixed Model Analysis of Variance

Simple ANOVAs

Between-units (e.g., between-subjects) ANOVAs are characterized by units that are “nested under” one or more categorical independent variables. Between-subjects ANOVAs examine the differences between two or more independent groups. For example, a simple one-way between-subjects ANOVA may test whether girls or boys have better grades in school. Here, there is one dichotomous independent variable that varies between-subjects (gender). The goal of the ANOVA is to examine whether the mean scores for each group (boys vs. girls) are reliably different from each other. The statistical model can be described as

$$Y=b_0+b_{1}X+e,$$

where Y is the dependent variable (scores), X is the dichotomous independent variable (gender), and e refers to the residuals (the errors). If the coefficient b1 is statistically significant, one would conclude that the data provide evidence for the idea that one of the two genders has better grades than the other. Between-subjects ANOVAs are more flexible than independent samples t tests because they allow for multiple independent variables with two or more levels each.

Within-units (e.g., within-subjects) ANOVAs are characterized by units that are “crossed with” one or more categorical independent variables. They frequently examine differences between one measurement of a particular variable and another measurement of the same variable for a given subject. In such cases, the observations are not independent of each other in that two data points from the same subject are likely to be more similar to each other than two data points from two different subjects. Within-subjects ANOVAs examine the differences between two or more dependent groups. Their goal is often to examine changes in an outcome variable over time. For example, a one-way, within-subjects ANOVA may test whether students have better grades in English or math. Here, there is one dichotomous independent variable that varies within-subjects (discipline: English vs. math). The statistical model can be described as

$$(Y_1-Y_2)=b_0+e,$$

where Y1 is subjects’ English grade and Y2 is subjects’ math grade. Like before, the e refers to the residuals (the errors). If the coefficient b0 is statistically significant, one would conclude that the data provide evidence for the idea that students’ English and math grades differ from each other. Compared to paired samples t tests, within-subjects ANOVAs are more flexible because they allow for multiple independent variables with two or more levels each.

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