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I have obtained a significant interaction effect using the GLM procedure in SPSS and now I want to further decompose the interaction. In other words, I want to try to identify the source or root of the interaction via a more fine-grained analysis as a follow-up to the significant interaction effect. My design has a two-level between-subject factor that is an experimental condition where subjects are exposed to an anxiety-provoking situation prior to the experiment or they are in the control group. There is a repeated measures variable that measures subjects' reaction to a stressful situation on four separate measurement occasions. Thus, the design is a 2 (anxiety-provoking situation: yes versus no) by 4 (measurement occasion: trial 1 vs. trial 2 vs. trial 3 vs. trial 4). After obtaining a significant interaction between the experimental condition and the measurement occasions, I want to know within which measurement occasions there were differences between experimental groups.
A typical method used to address your question is called the method of simple main effects (Winer et al., 1991). This analysis cannot be performed with dialog boxes in SPSS, but simple main effects tests can be performed using syntax. The syntax shown below illustrates the use of the GLM procedure to obtain contrasts between levels of a variable within all other levels of the other variable in an interaction. The four levels of the within-subject factor are trial 1, trial2, trial3, and trial4. Anxiety is the name of the variable representing the control versus the experimental group. The subcommand that specifies the contrasts is the /EMMEANS subcommand. In the example below, the /EMMEANS subcommand is used twice. In the first example, we are requesting a comparison of the between-subjects factor, anxiety, within each level of the within-subjects factor, trial. In the second /EMMEANS subcommand, we are requesting a comparison of levels of the within-subject factor, trial, within each level of the between-subjects factor.
GLM
trial1 trial2 trial3 trial4 BY anxiety
/WSFACTOR = trial 4 SIMPLE
/METHOD = SSTYPE(3)
/CRITERIA = ALPHA(.05)
/EMMEANS = TABLES (anxiety*trial) COMPARE (anxiety)
/EMMEANS = TABLES (anxiety*trial) COMPARE (trial)
/WSDESIGN = trial
/DESIGN = anxiety .
The decomposed interaction table appears following the TABLES keyword. In the example syntax above, the interaction is designated as anxiety*trial following the TABLES keyword. Following the interaction specification, the COMPARE keyword appears followed by the name of the variable for which comparisons will be generated. In the above example, the variable anxiety occurs after the COMPARE keyword in the first example of the EMMEANS subcommand. Thus, the first instance of the EMMEANS will produce contrasts between levels of anxiety within each level of the within-subjects factor, trial. In this example, there will be four one-degree-of-freedom univariate F-tests that appear on the EMMEANS output: anxiety level 1 vs. anxiety level 2 at trial 1, anxiety level 1 vs. anxiety level 2 at trial 2, anxiety level 1 vs. anxiety level 2 at trial 3, and anxiety level 1 vs. anxiety level 2 at trial 4. For example, comparing anxiety level 1 vs. anxiety level 2 within trial 1 results in an F-value of 0.292 and a significance level of 0.601 as can be seen in the table below.
Univariate Tests
Measure: MEASURE_1;
| TRIAL | Sum of
Squares |
df | Mean
Square |
F | Sig. | |
|---|---|---|---|---|---|---|
| 1 | Contrast | 1.333 | 1 | 1.333 | .292 | .601 |
| Error | 45.667 | 10 | 4.567 | |||
| 2 | Contrast | 3.000 | 1 | 3.000 | .484 | .503 |
| Error | 62.000 | 10 | 6.200 | |||
| 3 | Contrast | 8.333E-02 | 1 | 8.333E-02 | .013 | .912 |
| Error | 64.167 | 10 | 6.417 | |||
| 4 | Contrast | 14.083 | 1 | 14.083 | 1.849 | .204 |
| Error | 76.167 | 10 | 7.617 | |||
| Each F tests the simple effects of Anxiety within each level combination of the other effects shown. These tests are based on the linearly independent pairwise comparisons among the estimated marginal means. | ||||||
The second instance of the EMMEANS subcommand is identical to the first with the exception being that the within-subjects factor, trial, is included where anxiety appeared previously which will produce output containing contrasts between levels of trial within each level of anxiety. In this example, there will be two three-degree-of-freedom multivariate F-tests that appear on the EMMEANS output: trial 1 vs. trial 2 vs. trial 3 vs. trial 4 at anxiety level 1 and trial 1 vs. trial 2 vs. trial 3 vs. trial 4 at anxiety level 2. Unlike the first group of univariate F-tests mentioned above, this group of tests are multivariate tests due to the repeated measures nature of these latter comparisons. For example, in the table below, the difference between levels of trials within the Control Group produces an F-value of 38.379 with a significance level of 0.000, indicating that there are differences between levels of the variable trial within the Control Group.
Multivariate Tests
| Anxiety | Value | F | Hypothesis df | Error df | Sig. | |
|---|---|---|---|---|---|---|
| Control Group | Pillai's trace | .935 | 38.379(a) | 3.000 | 8.000 | .000 |
| Wilks' lambda | .065 | 38.379(a) | 3.000 | 8.000 | .000 | |
| Hotelling's trace | 14.392 | 38.379(a) | 3.000 | 8.000 | .000 | |
| Roy's largest root | 14.392 | 38.379(a) | 3.000 | 8.000 | .000 | |
| Experimental Group | Pillai's trace | .916 | 28.927(a) | 3.000 | 8.000 | .000 |
| Wilks' lambda | .084 | 28.927(a) | 3.000 | 8.000 | .000 | |
| Hotelling's trace | 10.847 | 28.927(a) | 3.000 | 8.000 | .000 | |
| Roy's largest root | 10.847 | 28.927(a) | 3.000 | 8.000 | .000 | |
| Each F tests the multivariate simple effects of TRIAL within each level combination of the other effects shown. These tests are based on the linearly independent pairwise comparisons among the estimated marginal means. | ||||||
| a Exact statistic | ||||||
In addition to the appropriate F-tests of simple main effects, SPSS also produces tables of pairwise comparisons for each EMMEANS specification. The pairwise comparison output is useful when the simple main effect test contains more than one degree of freedom. For instance, the second EMMEANS specification used in the anxiety example results in two three-degree-of-freedom multivariate simple main effect tests. If you obtain a statistically significant multivariate simple main effect test result, you may wish to further explore pairwise comparisons among mean trial levels within each level of anxiety group to determine where measurement occasions differ. The pairwise comparison output allows you to do this. If you plan to examine more pairwise comparisons than the number of degrees of freedom in the original contrast (in this case 3 DF), you should employ a post-hoc correction factor to control type 1 error. Fortunately, SPSS offers several options for post-hoc type 1 error control, including Bonferroni and Sidak adjustments. You can select these when you specify the EMMEANS statement in the OPTIONS dialog box of the GLM procedure. The within-subjects table of pairwise comparisons is shown below, containing contrasts between all levels of trial within all levels of anxiety. For example, the table below contains a comparison of trial 1 with all other levels of trial within the Control Group of the anxiety variable producing output containing a mean difference of 5.167 between trial 1 and trial 2, a mean difference of 8.333 between trial 1 and trial 3 and a mean difference of 13.000 between trial 1 and trial 4 with a a significance value of 0.000 for all three of these comparisons.
Pairwise Comparisons
Measure: MEASURE_1
| Mean
Difference (I-J) |
Std. Error | Sig.(a) | 95% Confidence Interval for Difference(a) | ||||
|---|---|---|---|---|---|---|---|
| Anxiety | (I) TRIAL | (J) TRIAL | Bound |
Upper Bound |
|||
| Control Group | 1 | 2 | 5.167(*) | .980 | .000 | 2.982 | 7.351 |
| 3 | 8.333(*) | 1.170 | .000 | 5.726 | 10.941 | ||
| 4 | 13.000(*) | 1.301 | .000 | 10.102 | 15.898 | ||
| 2 | 1 | -5.167(*) | .980 | .000 | -7.351 | -2.982 | |
| 3 | 3.167(*) | .580 | .000 | 1.875 | 4.458 | ||
| 4 | 7.833(*) | .685 | .000 | 6.307 | 9.360 | ||
| 3 | 1 | -8.333(*) | 1.170 | .000 | -10.941 | -5.726 | |
| 2 | -3.167(*) | .580 | .000 | -4.458 | -1.875 | ||
| 4 | 4.667(*) | .558 | .000 | 3.424 | 5.909 | ||
| 4 | 1 | -13.000(*) | 1.301 | .000 | -15.898 | -10.102 | |
| 2 | -7.833(*) | .685 | .000 | -9.360 | -6.307 | ||
| 3 | -4.667(*) | .558 | .000 | -5.909 | -3.424 | ||
| Experimental Group | 1 | 2 | 4.833(*) | .980 | .001 | 2.649 | 7.018 |
| 3 | 9.167(*) | 1.170 | .000 | 6.559 | 11.774 | ||
| 4 | 11.500(*) | 1.301 | .000 | 8.602 | 14.398 | ||
| 2 | 1 | -4.833(*) | .980 | .001 | -7.018 | -2.649 | |
| 3 | 4.333(*) | .580 | .000 | 3.042 | 5.625 | ||
| 4 | 6.667(*) | .685 | .000 | 5.140 | 8.193 | ||
| 3 | 1 | -9.167(*) | 1.170 | .000 | -11.774 | -6.559 | |
| 2 | -4.333(*) | .580 | .000 | -5.625 | -3.042 | ||
| 4 | 2.333(*) | .558 | .002 | 1.091 | 3.576 | ||
| 4 | 1 | -11.500(*) | 1.301 | .000 | -14.398 | -8.602 | |
| 2 | -6.667(*) | .685 | .000 | -8.193 | -5.140 | ||
| 3 | -2.333(*) | .558 | .002 | -3.576 | -1.091 | ||
| Based on estimated marginal means | |||||||
| * The mean difference is significant at the .05 level. | |||||||
| a Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments). | |||||||
For more information about simple main effect tests, see:
Winer, B. J., Brown, D. R., & Michels, K. M. (1991). Statistical Principles in Experimental Design. New York, NY: McGraw-Hill.
If you have further questions, send E-mail to stats@ssc.utexas.edu.