Detailed Within-Subjects Analysis

Basic Concepts

On this webpage, we provide a more detailed analysis of the between-subjects factor for Example 1 of One Between-Subjects Factor and One Within-Subjects Factor.

The data for Example 1 is repeated on the left side of Figure 1.

Figure 1 – Data for Example 1

One-way Repeated Measures ANOVA

Although the GG epsilon and HF epsilon are very useful in addressing sphericity issues, it is still better not to use MSE (cell Z12 in Figure 4 of One Between-Subjects Factor and One Within-Subjects Factor) as the error term in detailed analyses.

In fact, we suggest analyzing simple effects on an individual Age group basis. We do this by running a one-way repeated measures ANOVA for each Age group (or by skipping ANOVA and going directly to contrasts).

Figure 2 shows the analyses for each of the age groups (as described in ANOVA with Repeated Measures with One Within Subjects Factor).

Figure 2 – Within-subjects simple effects ANOVA

Sphericity Correction

We must also compute a sphericity correction for each of these analyses based on the covariance matrices in Figure 4 of Within-Subjects Assumptions. E.g. the correction factors for the Young simple effect are shown in Figure 3.

Figure 3 – Calculation of GG and HF epsilon for Young effect

In Figure 4, we display the GG and HF epsilon values for each age group.

Figure 4 – Epsilon for each age effect

Applying the appropriate correction factor to the within-subjects ANOVA in Figure 2 (GG epsilon for Young and Middle and HF epsilon for Old), we get the revised output shown in Figure 5.

Figure 5 – Corrected within-subjects ANOVA

Note that to perform the simple effect tests, adjustments for sphericity are necessary.  Equality of the covariance matrices, however, is not necessary.

Links

↑ One between-subjects and one within-subjects factor

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

Reference

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