GLM Repeated Measures:
Two Within, One Between

Reading: SPSS Advanced Models 9.0: 2. Repeated Measures
Download:  glm_w2b1.sav       (Download Tips)

1. Overview

In the earlier notes on repeated measures designs we looked at a design with a single within subject factor and at a design with one within- and one between-subjects factors. In this set of notes we look a design with two within-subjects factors and 1 between-subjects factor.

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2. The Data

The data for this set of notes comes from a study of the therapeutic effects of verbally disclosing thoughts and feelings about the loss of a spouse (Segal, Bogaards, Becker, and Chatman, in press).  The participants were 30 (7 males and 23 females) older adults (M age = 67.0, range = 51 to 85).  The research design was a delayed treatment design. Participants randomly assigned to the treatment condition  participated in four 20-minute sessions within a 2-week period.  In each session the participant was asked to talk about the loss of their spouse and to express their deepest thoughts and feelings for the entire 20 minutes. The experimental sessions were conducted in the participants home.  The participants completed an array of psychological measures prior to treatment (T1), immediately post treatment (T2) and at 1-month following treatment (T3).  Participants randomly assigned to the delayed treatment completed the same array of measures, however treatment occurred for the delayed treatment group between T2 and T3 and they were again tested (T4) at 1-month following T3.

The analysis we will be looking at are a set of "process" measures that looked at the mood of the participant at the beginning and end of each of the four disclosure sessions.  The process measures included the Positive and Negative Affect Schedule (PANAS, Watson, Clark, & Tellegen, 1988), and a self-report scale that asked, "How painful is it for you to think about the death of your spouse right now?"  We will be looking at the Negative Affect scale of the PANAS.   The design for the process measures is shown in Table 1.  The design of the study is a 2 x 4 x 2 ANOVA with gender (male vs. female) as a between subjects factor and session number (1 vs. 2 vs. 3 vs. 4) and prepost (beginning of the session vs. end of the session) as within subject factors.

The data are stored in the file glm_w2b1.sav. Both the negative and positive affect PANAS scores are included in the data file.  The eight negative affect scores of the PANAS are shown in Table 2 along with the age and gender variables. Note that the order of the variables in the datafile do not match the order of the variables in Table 2. 

Note the structure used to name the dependent variables.  The first three letters, PAN, indicate the PANAS scale.  The next three letters, PRE or POS, indicate whether this is a presession or a postsession score. The next to the last letter, N or P, indicates whether the score is the negative or positive subscore of the PANAS. The last digit, 1, 2, 3, or 4, indicates the session number.

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3. Select the GLM Repeated Measures Procedure

The GLM Repeated Measures dialog box is opened by clicking

Analysis
      General Linear Model
            Repeated Measures...

GLM - Repeated Measures Define Factor(s)

In this design there are two within-subjects factors, sessions with four levels and prepost with 2 levels. The 2 within-subject factors make up a 4 x 2 within-subjects design. There are eight within-subjects cells. The eight scores are the measures for each of the eight-within subjects cells. You need to tell SPSS how the eight measures are related to the eight cells in the 4 x 2 within-subjects design.

The within-subject factors are defined in the opening window. Enter the first factor, session, in the Within-Subjects Factor Name: window, enter 4 in the Number of Levels: window and then Add the factor to the window that lists the within-subject factors.. Do the same for the prepost factor : enter the factor name, prepost,  and then enter 2 for the number of levels.

Move the gender factor to the Between-Subjects Factors(s): window.

Lets look at the effect sizes: Options... Display, Estimates of effect size.

And the descriptive statistics: Options... Display, Descriptive Statistics.

It is unlikely that the orthogonal polynomial contrasts will be useful in interpreting any significant effects so lets selects the "Repeated" contrasts for the session within subject effect:  Contrasts... then highlight session[polynomial], then change contrast to Repeated, press Change and then press Continue to exit the contrast dialog box.

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4. The Basic GLM Output

The tests of sphericity for each of the within subject effects are shown in Table 5.

There is no sphericity problem for the session main effect, Mauchly's W (5) = 5.378, p = .372.  We will look at the "sphericity assumed" row the of the output for any effects tested by the "session" error term. 

The prepost effect has only 1 degree of freedom so the sphericity test is not relevant.   SPSS recognizes that is not relevant by providing a Mauchly's test value of 1.00 with 0 df, and all epsilon values equal to 1.00. We will look at the "sphericity assumed" row of the output for any effects tested by the "prepost" error term.

There is a small sphericity problem for the session by prepost interaction, Mauchly's W (5) = 12.724, p = .026.  We will use the Huynh-Feldt epsilon correction when looking at the values for any effect tested by the "session by prepost" error term.  

The univariate ANOVA table for the within-subjects effects are shown in Table 6.

Measure: PANAS_NG

There are several significant within-subjects effects: the session by gender interaction, F(3, 81) = 7.60, p < .0005, h ²  = .22; the prepost main effect, F(1, 27) = 16.70, p < .0005, h ² = .38; the prepost by gender interaction, F(1, 27) = 5.69, p = .024, h ² = .17; and the session by prepost by gender interaction, F(2.670, 72.081) = 3.64, p = .020, h ² = .22.  Note that the Huynh-Feldt corrected degrees of freedom were used when describing the 3-way interaction because the 3-way interaction includes the session by prepost effect, which was deemed to have a sphericity problem.

The ANOVA table for the between-subjects effects are shown in Table 7. 

Transformed Variable: Average

There are no significant between-subject effects of interest.

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5. Interpreting Significant Effects: Displaying the Means

The general approach to interpreting ANOVA effects are to begin with the highest order interaction. In our case it is the 3-way interaction. The descriptive statistics are shown in Table 8.  The estimated marginal means for the 3-way interaction are shown in Table 9. And the profile plots are shown in Figure 1.

Descriptives

Table 8. Descriptive Statistics

Estimated Marginal Means

Table 9. Estimated Marginal Means for Gender of participant * SESSION * PREPOST

Measure: PANAS_NG

I don't know about you, but I find it difficult to see what is happening in the 3-way interaction by just looking at the table of means.  Back in the old days, prior to running SPSS for windows, my first instinct was to get out a piece of graph paper and plot the means. Fortunately, the profile plots procedure will that for us. 

Profile Plots

If you have an apriori hypothesis that involves an interaction, then the interpretation of the interaction should be informed by that hypothesis. If the interaction was not predicted, as in this instance, then you should look about your data from multiple perspectives. You may find that one perspective is better than the others at capturing the meaning of the interaction.

The essence of a three-way interaction is that the two-way interaction is not the same under each level of the third factor.  There are several possibilities here, you could look at the A x B interaction at each level of C, A x C interaction at each level of B,  or the B x C interaction at each level of A. In addition, for each of the two-way interactions, you could place one factor on either the horizontal or vertical axis.  Thus, for a three-way interaction there are 6 different ways to construct the profile plots.

In addition to looking at the various plots, you should think about the simple, simple main effects, and how they might be used to explain the interaction. For this three-way interaction there are three simple, simple main effects, the simple simple main effects of session, of prepost and of gender.

Session by Prepost Interaction within each Level of Gender

The profile plots in Figure 1 were created by selecting session as the horizontal axis variable, prepost as the separate lines variable, and gender as the separate plots variable (i.e., A x B within C). 

Plotting gender as separate plots focuses the readers attention on the session by prepost interaction within each gender. The relevant simple, simple main effects are session and prepost, the 2-way interaction factors.

For males, negative affect at the beginning of the session tended to decrease across the first three sessions and then increased at the fourth session.  Negative affect at the end of the session tended to increase across all four sessions. This interpretation could be statistically tested by looking at the simple main effects of session and the relevant pairwise comparisons..  The change in affect from the beginning to the end of the session tended to increase across the four sessions. This interpretation could be statistically tested by looking at the simple main effects of prepost. 

For females, negative affect decreased across sessions. This decrease occurred both at the beginning of the session and at the end of the session. This observation could be statistically tested by looking at the simple main effects of session and the relevant pairwise comparisons. At the first session, there may have been a small increase in negative affect from the beginning to the end of the session. By the fourth session there was very little increase in negative affect from the beginning to the end of the session.  This interpretation could be statistically tested by looking at the simple, simple main effects of prepost.

Figure 1.  Profile plots of the session by prepost by gender interaction means.

Session by Gender Interaction within each Level of Pre- and Postsession

The profile plots in Figure 2 use session as the horizontal axis, gender as the separate lines variable, prepost as the separate plots variable (i.e., A x C within B).  

Plotting the pre- and postsession scores as separate plots focuses the readers attention on the gender by session interaction. The relevant simple, simple main effects are session and gender, the 2-way interaction factors.

Negative affect scores at the beginning of each session tended to decrease for both males and females across the first three sessions.  At the fourth session the negative affect scores for females continued to decrease, but for males there was an increase in negative affect at the beginning of the fourth session.   These observations could be statistically tested by looking at the simple main effects of session and the relevant pairwise comparisons. There were no gender differences at the beginning of each of the first three sessions. At the fourth session males reported more negative affect at the beginning of the session than did the females.  These observations could be tested by the simple, simple main effects of gender.

Negative affect scores at the end of each session tended to increase for males across the four sessions and to decrease for females across the four sessions.  These observations could be statistically tested by looking at the simple main effects of session and the relevant pairwise comparisons. There were little or no gender differences at the end of session 1.  Negative affect scores at the end of the session tended to be higher for males than females at sessions 2, 3 and 4, with the largest difference occurring at session 4.  These observations could be tested by the simple, simple main effects of gender. 

Figure 2.  Profile plots of the session by gender by prepost interaction means.

Prepost by Gender Interaction within each level of Session

The profile plots in Figure 3 use prepost as the horizontal axis variable, gender as the separate lines variable and session as the separate plots variable (i.e., B x C within A). 

Plotting the session as separate plots focuses the readers attention on the prepost by gender interaction.  The relevant simple, simple main effects are prepost and gender, the 2-way interaction factors.

The interpretation of these four prepost by gender interactions within sessions is possible.  But, I don't even want to write up the possibilities for these notes.  Either of the previous two ways seem like better options to me. What do you think? 

Figure 3.  Profile plots of the prepost by gender by session interaction means.

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6. Interpreting Significant Effects: Simple, Simple Main Effects Analysis

The syntax commands that were to create the three sets of plots (see the /PLOT syntax) and the simple, simple main effects for session, prepost, and gender (see the /EMMEANS syntax) are shown below --

Statistical Tests: Session by Prepost Interaction within each Level of Gender

The multivariate tests of the simple, simple main effects of session within each combination of prepost and gender are shown in Table 10.  The table has been modified to present only the F test (the redundant multivariate tests have been omitted).

The pairwise comparisons for the simple, simple main effects of session are shown in Table 11.