One between subjects factor and two within subjects factors

Introduction

On this webpage, we will focus on repeated measures ANOVA where there are two within-subjects factors and one between-subjects factor. We will assume that the sphericity assumption has been met and so won’t deal with correction factors. We also won’t explore follow-up testing since these are described extensively elsewhere (click here or here for more details).

Example

Example 1: Suppose there are three groups (G1, G2, G3), and each subject in these three groups was given 4 types of dexterity tests (T1, T2, T3, T4), each repeated once using the left hand and the other using the right hand. All the subjects in the experiment were right-handed. The data for this example are as shown in Figure 1.

Figure 1 – Data for Repeated Measures ANOVA 2W+1B

The three groups make up the three levels of the between-subjects factor (Group), which we will label A. The tests make up the 4 levels of a within-subjects factor (Test), which we will label B. The two hands make up the levels of the other within-subjects factor (Hand), which we will label C.

We use the scheme described in Figure 2. Here a = # of Factor A levels, b = # of Factor B levels, c = # of Factor C and m = # of subjects.

Figure 2 – Sources of Variation

Data Analysis Tool

Real Statistics Data Analysis Tool: The repeated measures ANOVA analysis required for one between-subjects factor and two within-subjects factors can be performed using Real Statistics’ Mixed Three-way Repeated Measures Anova data analysis tool.

For Example 1, press Ctrl-m and click on the Anova tab (or choose the ANOVA option if using the original user interface). Next, choose the Mixed Three-way Repeated Measures Anova option. Fill in the dialog box that appears as shown in Figure 3.

Figure 3 – Mixed Three-way Repeated Measures Anova dialog box

After pressing the OK button, the output shown in Figures 4, 5 and 6 are displayed.

Figure 4 – Repeated Measures ANOVA output (part 1)

Figure 5 – Repeated Measures ANOVA output (part 2)

Figure 6 – Repeated Measures ANOVA output (part 3)

Representative Formulas

Representative formulas from Figures 4 and 5 are shown in Figure 7.

Figure 7 – Formulas from Figures 4 and 5

Representative formulas from Figure 6 about degrees of freedom are shown in Figure 8, while those about SS are shown in Figure 9.

Figure 8 – Df formulas from Figure 6

Figure 9 – SS formulas from Figure 6

References

Howell, D. C. (2010) Statistical methods for psychology (7^th ed.). Wadsworth, Cengage Learning.
https://labs.la.utexas.edu/gilden/files/2016/05/Statistics-Text.pdf

Winer, B. J. (1962) Statistical principles in experimental design. McGraw-Hill
https://psycnet.apa.org/record/2008-16855-000