Let’s read in an example dataset, crf24, adapted from Kirk (1968,
First Edition).

get file 'c:https://stats.idre.ucla.edu/wp-content/uploads/2016/02/crf24.sav'.

These data are from a 2×4 factorial design.  The variable y is the dependent variable.  The variable a is an independent variable with two levels, while b is an independent variable with four levels.  Let’s look at a table of cell means and standard deviations.

means table = y by a by b.

Case Processing Summary

	Cases
	Included		Excluded		Total
	N	Percent	N	Percent	N	Percent
Y * A * B	32	100.0%	0	.0%	32	100.0%

Report
Y

A	B	Mean	N	Std. Deviation
1	1	3.75	4	1.500
	2	4.00	4	.816
	3	7.00	4	.816
	4	8.00	4	.816
	Total	5.69	16	2.120
2	1	1.75	4	.500
	2	3.00	4	.816
	3	5.50	4	.577
	4	10.00	4	.816
	Total	5.06	16	3.316
Total	1	2.75	8	1.488
	2	3.50	8	.926
	3	6.25	8	1.035
	4	9.00	8	1.309
	Total	5.38	32	2.756

Now let’s run the ANOVA.

glm y by a b.

Between-Subjects Factors

		N
A	1	16
	2	16
B	1	8
	2	8
	3	8
	4	8

Tests of Between-Subjects Effects
Dependent Variable: Y

Source	Type III Sum of Squares	df	Mean Square	F	Sig.
Corrected Model	217.000(a)	7	31.000	40.216	.000
Intercept	924.500	1	924.500	1199.351	.000
A	3.125	1	3.125	4.054	.055
B	194.500	3	64.833	84.108	.000
A * B	19.375	3	6.458	8.378	.001
Error	18.500	24	.771
Total	1160.000	32
Corrected Total	235.500	31
a R Squared = .921 (Adjusted R Squared = .899)

We see that in addition to a significant main effect for b there is a significant a*b interaction effect.  Before we do any of the tests of simple main effects, let’s graph the cell means to get an idea of what the interaction looks like. 
You can include a /plot subcommand with the glm command to get a
plot of the cell means.

glm y by a b
 /plot = profile(b*a).

<output omitted>

The interaction is clearly shown where the two lines cross over between levels b3 and b4.  We will now do a test of simple main effects looking at differences in a at each level of b.  This is done by including a /emmeans subcommand with the glm
command. 

glm y by a b
 /emmeans = tables(a*b)compare(a).

Between-Subjects Factors

		N
A	1	16
	2	16
B	1	8
	2	8
	3	8
	4	8

Tests of Between-Subjects Effects
Dependent Variable: Y

Source	Type III Sum of Squares	df	Mean Square	F	Sig.
Corrected Model	217.000(a)	7	31.000	40.216	.000
Intercept	924.500	1	924.500	1199.351	.000
A	3.125	1	3.125	4.054	.055
B	194.500	3	64.833	84.108	.000
A * B	19.375	3	6.458	8.378	.001
Error	18.500	24	.771
Total	1160.000	32
Corrected Total	235.500	31
a R Squared = .921 (Adjusted R Squared = .899)

Estimates
Dependent Variable: Y

		Mean	Std. Error	95% Confidence Interval
A	B			Lower Bound	Upper Bound
1	1	3.750	.439	2.844	4.656
	2	4.000	.439	3.094	4.906
	3	7.000	.439	6.094	7.906
	4	8.000	.439	7.094	8.906
2	1	1.750	.439	.844	2.656
	2	3.000	.439	2.094	3.906
	3	5.500	.439	4.594	6.406
	4	10.000	.439	9.094	10.906

Pairwise Comparisons
Dependent Variable: Y

			Mean Difference (I-J)	Std. Error	Sig.(a)	95% Confidence Interval for Difference(a)
B	(I) A	(J) A				Lower Bound	Upper Bound
1	1	2	2.000(*)	.621	.004	.719	3.281
	2	1	-2.000(*)	.621	.004	-3.281	-.719
2	1	2	1.000	.621	.120	-.281	2.281
	2	1	-1.000	.621	.120	-2.281	.281
3	1	2	1.500(*)	.621	.024	.219	2.781
	2	1	-1.500(*)	.621	.024	-2.781	-.219
4	1	2	-2.000(*)	.621	.004	-3.281	-.719
	2	1	2.000(*)	.621	.004	.719	3.281
Based on estimated marginal means
* The mean difference is significant at the .050 level.
a Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments).

Univariate Tests
Dependent Variable: Y

B		Sum of Squares	df	Mean Square	F	Sig.
1	Contrast	8.000	1	8.000	10.378	.004
	Error	18.500	24	.771
2	Contrast	2.000	1	2.000	2.595	.120
	Error	18.500	24	.771
3	Contrast	4.500	1	4.500	5.838	.024
	Error	18.500	24	.771
4	Contrast	8.000	1	8.000	10.378	.004
	Error	18.500	24	.771
Each F tests the simple effects of A within each level combination of the other effects shown. These tests are based on the linearly independent pairwise comparisons among the estimated marginal means.

Note:
Statisticians do not universally approve of the use of tests of simple
main effects.  In particular, there are concerns over the conceptual error
rate.  Tests of simple main effects are one tool that can be useful in
interpreting interactions.  Caution should be exercised in interpreting
the results of analyses of simple main effects.  In general, the results of tests of simple
main effects should be considered suggestive and not definitive.

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