GLM: Simple Main Effects

A significant interaction effect can be analyzed as the simple main effects of one variable within each level of the other variable. The means for interaction between reward and drive level are shown in Figure 1 (General Linear Model (GLM): Two-way, Between-Subjects Designs notes). Our earlier discussion of this interaction noted that it looked as though there was no effect of reward in the high drive (24-hour deprivation) condition and that the effect of reward looked significant in the low drive (1-hour deprivation) condition.

Figure 1. Profile Plot of the Reward by Drive Level Interaction

You can use a simple main effects analysis to directly test that interpretation. Lets do an analysis on the three reward means within the high drive condition and another analysis on the reward means within the low drive condition. That analysis is called a "simple main effects" analysis.

We can edit the syntax for the Estimated Marginal Means subcommand, /EMMEANS, to easily create simple main effect tests. We begin with the basic set of syntax commands used to run a 2-way ANOVA using the GLM procedure. An easy way to do this is to use the GLM:Univariate dialog boxes to create the basic syntax for the 2-way ANOVA and then to add the commands to run the simple main effects. Here is a summary of the steps to follow:

(a) Select the dependent and independent variables.

(b) Go to the Estimated Marginal Means section of the Options.. dialog box and move all the effects from the full factorial to the Display Means for... window.

(c) Check the Compare Main Effects box in the Estimated Marginal means Window.

By default the confidence intervals for the main effect tests and the simple main effect tests will not be adjusted, that is, the LSD (none) test will be used when creating the confidence intervals.  You can choose to have the confidence intervals adjusted using either the Bonferonni procedure or the Sidak procedure.  The Bonferonni procedure adjusts the alpha level as follows:

Bonferonni alpha = significance level/C

where C is the number of paired comparisons and significance level is the alpha level specified at the bottom of the Options... dialog box.   The Sidak procedure adjusts the alpha level as follows:

Sidak alpha = significance level - (1 - significance level)^1/C

where C is the number of paired comparisons and significance level is the alpha level specified at the bottom of the Options... dialog box.

Lets select the Sidak procedure.

(d) Go back to the GLM - General Factorial main dialog box and Paste the commands to the syntax window.

Table 1 shows the basic syntax as created by the GLM dialog boxes.

To run the simple main effects analysis you simply Run the syntax commands from the Syntax dialog box.

The output from the /EMMEANS simple main effects subcommand includes three tables: Estimates, the estimated interaction means, See Table 3; Univariate Tests of the simple effects of reward within low drive and the simple effects of reward within high drive, see Table 4; and Pairwise Comparisons for reward within each level of drive, see Table 5.

Table 3. Estimates

Dependent Variable: Number correct on the 20 training trials

Table 4. Univariate Tests

Dependent Variable: Number correct on the 20 training trials

To interpret these results you could begin by describing the two overall tests of the simple main effects. A parsimonious description might go like this:

Because there are three levels of reward we don't know which levels of reward within the low drive conditions are significantly different. We need one additional analysis, the pairwise comparison between the reward levels within low drive. That analysis is shown in Table 5.

Table 5. Pairwise Comparisons

Dependent Variable: Number correct on the 20 training trials

Based on the pairwise comparisons, the interpretation of the interaction could be stated as follows:

A simple main effects analysis of an interaction can be viewed from either of the perspectives described above, from the simple main effects of variable A within variable B or vice versa. Oftentimes, as in this example, one of the perspectives seems to capture the essence of the data better than the other perspective. Sometimes which perspective you choose will depend upon your theoretical basis for the study.

Simple main effects analyses of interaction effects are gaining in popularity. They are appearing in more and more statistics textbooks. However, there is a basic problem with this approach. An interaction occurs because the effects of one variable are not the same across the levels of the other variable( see Becker & Coolidge, 1991). Visually, an interaction is described as a set of nonparallel lines, such as shown in Figures 1 above. Computationally, a simple 2 x 2 interaction is computed as the difference of differences. For example, the following formula:

A x B interaction effect = (A₁B₁ - A₂B₁) - (A₁B₂ - A₂B₂),

describes the interaction as the difference of the differences between A₁ and A₂ within each level of B. In the example used in this set of notes the interaction was of the form shown in Figure 2, one of the simple main effects was significant while the other simple main effect was not.

It is possible for the interaction to be significant but for neither of the simple main effects to be significant. This sometimes happens when the slopes for the simple main effects are of the opposite sign and interaction is marginally significant, see the example in Figure 3. This can occur because the interaction is not a test of individual simple effects but a comparison of simple effect differences. That is, the difference of the differences can be significant even though none of the simple main effects are significant. As the overall interaction effect becomes larger, then one or more of the simple main effects will also become significant.

Another logical possibility is that the simple main effects have the same sign and both are significant. For example, in Figure 4 both simple main effects of A within B have positive slopes. The interpretation of this interaction is that the effects of A are stronger within B₂ than within B₁.

The /EMMEANS syntax for running simple main effects makes the simple main effects comparison and displays pairwise comparisons for each simple main effect. You may wish to make a specific contrast for the simple main effect rather than running all pairwise comparisons. You can use the /LMATRIX syntax to create your own contrasts. We begin with the basic set of syntax commands used to run a 2-way ANOVA using the GLM procedure. An easy way to do this is to use the GLM-General Factorial dialog boxes to create the basic syntax for the 2-way ANOVA and then to add the commands to run the simple main effects. Table 6 shows the basic syntax as created by the GLM dialog boxes.

The /METHOD = SSTYPE(3), /INTERCEPT = INCLUDE,  and  /CRITERIA = ALPHA(.05) specify the default values, they will not be included in the syntax commands for this discussion..

Consider the following model

LB = l₁� +
         l₂drive₁ + l₃drive₂ +
         l₃reward₁ + l₄reward₂ + l₅reward₃ +
         l₆drive₁reward₁ + l₇drive₁reward₂ + l₈drive₁reward₃ + l₉drive₂reward₁ + l₁₀drive₂reward₂ + l₁₁drive₂reward₃

The 11 ls in the model are coefficients. The first row in the model is the grand mean; the second row is the drive main effect; the third row is the reward main effect; and the last row is the drive by reward interaction. The order of the elements in equation LB is determined by the order of the independent variables in the GLM syntax. In Table 1 drive was specified prior to reward so the LB model specified the drive main effect prior to the reward main effect. The interaction was written as drive*reward, with drive coming prior to reward, for the same reason. We will make use of this model when we specify the contrasts for the simple main effects.

Lets begin by specifying the simple main effect of reward within low drive. The syntax for that simple main effect is shown in Table 7.

The syntax makes use of a set of coefficients called LMATRIX coefficients. The syntax has three rows. The first row

/LMATRIX 'Reward differences at low drive level'

tells SPSS that the coefficients that follow are LMATRIX coefficients. The text enclosed in quotes will be used to label the output.

The second and third row define the coefficients for the 2 degrees of freedom for the simple main effect. Each row includes the coefficients for the reward main effect (coefficients l₃, l₄, and l₅) and for the corresponding elements in the drive* reward interaction (coefficients l₆ trough l₁₁). The semicolon at the end of line two is mandatory, it separates the two sets of coefficients.

REWARD -1 0  1 DRIVE*REWARD -1 0  1 0 0 0 ;
REWARD -1 2 -1 DRIVE*REWARD -1 2 -1 0 0 0

The main effect of reward has 2 df (df = #levels -1 = 3 -1 = 2). The simple main effect of reward within low drive has the same number of degrees of freedom, 2. We must specify a contrast for each degree of freedom.

First, lets define the two contrasts for reward. For example, we could use orthogonal polynomial contrasts. Orthogonal polynomial contrasts seem like a reasonable set of contrasts for this interaction because the reward means within the low drive condition look to be linear. The orthogonal polynomial coefficients for l₃, l₄ and l₅ are

L1 for reward = (-1)reward₁ + (0)reward₂ + (1)reward₃

for the linear orthogonal polynomial contrast, and

L2 for reward = (-1)reward₁ + (2)reward₂ + (-1)reward₃ .

for the quadratic orthogonal polynomial contrast.

Next, recall that the coefficients for the drive*reward interaction are

   l₆drive₁reward₁ + l₇drive₁reward₂ + l₈drive₁reward₃ + l₉drive₂reward₁ + l₁₀drive₂reward₂ + l₁₁drive₂reward₃ .

Coefficients  l₆, l₇, and l₈ are the effects of reward levels 1, 2 and 3 within low drive, drive₁. Coefficients l₉, l₁₀ ,and l₁₁are the effects of reward levels 1, 2, and 3 within high drive, drive₂. The rule is that you apply the reward main effect coefficients to the corresponding reward coefficients in the drive*reward interaction. If you are testing the reward main effect within drive₁then all the drive₂ coefficients would be zero, and vice versa. So

REWARD -1 0  1 DRIVE*REWARD -1 0  1 0 0 0 ;

is the linear effect within the first level of drive and

REWARD -1 2 -1 DRIVE*REWARD -1 2 -1 0 0 0

is the quadratic effect within the first level of drive.

Finally, lets add the LMATRIX syntax for the simple main effect of reward within high drive (drive level 2), see Table 8.

Look at the LMATRIX for the reward differences at high drive level. The drive* reward coefficients at drive level 1 are all set to zero while the drive*reward coefficients at drive level 2 are set to the corresponding coefficients for the reward main effect.

To run the simple main effects analysis you simply Run, All the syntax commands from the Syntax dialog box.

The output for the simple main effect reward within low drive are shown in Tables 9 and 10. Table 10 displays the results of for the two contrasts separately. L1 is the first contrast, the linear effect and L2 is the second contrast, the quadratic effect. The linear effect is significant because the confidence interval does not include zero. The quadratic effect is not significant, the confidence interval does include zero.

Table 5 shows the combined results for the 2 df test of the simple main effects of reward within the low drive condition. The effect was significant, F(2, 18) = 6.764, p = .006. The higher the reward the better the performance. The single degree of freedom tests indicated that the relationship between reward and performance was linear.

The output for the simple main effect reward within low drive are shown in Tables 11 and 12. Table 11 displays the results for the two contrasts separately. Neither contrast is significant, the confidence intervals for both contrasts include zero.

The 2 df test of the simple main effect of reward within the high drive condition is shown in Table 12. There was no effect of reward for animals in the high drive condition, F(2, 18) = 0.218, p = .806.

In the previous example we looked at the simple main effects of reward within each drive level. You could also look at the simple main effects of drive within each reward level. The plot that emphasizes the effects of drive within reward are shown in figure 5. It was created by selecting drive for the Horizontal Axis: and reward for the Separate Lines: option.

Figure 5. Profile Plot of the Reward by Drive Level Interaction Emphasizing
the Effects of Drive Within Each Level of Reward

There is one degree of freedom for each of the drive within reward contrasts, so there is a single contrast for drive differences at each level. The contrast for drive is the difference between the two drive levels, so the drive coefficients are -1 and 1.

The syntax commands are shown in Table 13.

The l coefficients for the drive*reward interaction are

         l₆drive₁reward₁ + l₇drive₁reward₂ + l₈drive₁reward₃ + l₉drive₂reward₁ + l₁₀drive₂reward₂ + l₁₁drive₂reward₃

The l coefficients at reward level 1 are l₆ and l₉, the coefficients at reward level 2 are l₇ and l₁₀, and the coefficients at reward level 3 are l₈ and l₁₁.

The effects within reward levels 1, 2 and 3 are shown in Tables 14, 15, and 16, respectively.

This view of the interaction suggests that drive level influenced behavior in the low reward condition (1 grape), F(1, 18) = 6.98, p = .017, but not in the moderate reward condition (3 grapes), F(1, 18) = 0.44, p = .517, or the high reward conditions (5 grapes), F(1, 18) = 1.74, p = .203. In the low reward condition performance was better under high drive (M = 3.0, SD = 3.2) then under low drive (M = 11.0, SD = 3.9).

Becker, L. A., & Coolidge, F. L. (1991). On the proper interpretation of residualized interaction means in an analysis of variance: A reply to Rosnow and Rosenthal. Psychological Reports, 68, 483-490.

Rosnow, R. L. & Rosenthal R. (1989). Definition and interpretation of interaction effects. Psychological Bulletin, 105, 143-146.