Example for Factorial Design

Suppose an investigator has a 2X3 factorial design where the first factor is gender (male versus female) and the second factor is ethnicity (Latino versus African American versus European American).  The outcome measure is a person's score on a test measuring depression that ranges from 0 to 100, with higher scores indicating greater levels of depression.  There are 50 individuals in each cell of the design (yielding a total of 300 research participants).  The investigator is interested in gender differences in depression and whether gender differences are the same for Latinos, African Americans and European Americans.  The mean scores from this hypothetical study are as follows:

                                  African         European

                 Latinos      Americans      Americans

Females        50              52                  60

Males           40              41                  45

Simple Main Effects Analysis

Simple main effects analysis asks if the there is a significant effect for one  factor at each level of the other factor.  In this case, the investigator wants to know (1) is there a gender difference for Latinos, (2) is there a gender difference for African Americans and (3) is there a gender difference for European Americans?

The question of gender differences for Latinos compares the mean for female Latinos with the mean for male Latinos, 50 - 40 = 10.  The statistical significance of this comparison represents a single degree of freedom simple main effect and is tested using information from the analysis of variance summary table.

The question of gender differences for African Americans compares the mean for female African Americans with the mean for male African Americans, 52 - 41 = 11.  The statistical significance of this comparison is another single degree of freedom simple main effect and also is tested using information from the analysis of variance summary table.

The question of gender differences for European Americans compares the mean for female European Americans with the mean for male European Americans, 60 - 45 = 15.  The statistical significance of this comparison also is a single degree of freedom simple main effect and also is tested using information from the analysis of variance summary table.

Single Degree of Freedom Interaction Contrasts

Although simple main effects are informative, they do not formally compare the gender differences for one of the ethnic groups with the gender differences for another ethnic group.  For example, the gender difference for Latinos is 50 - 40 = 10 and for African Americans it is 52 - 41 = 11.  The gender difference appears to be larger in African Americans than Latinos (i.e., 11 is larger than 10), but this difference could just reflect sampling error.  We need to test the statistical significance of the difference between the two differences (i.e. 10 - 11).  This is an interaction contrast.

In this example, there are three such interaction contrasts of interest: (1) is the gender difference for Latinos the same as the gender difference for African Americans, (2) is the gender difference for Latinos the same as the gender difference for European Americans, and (3) is the gender difference for African Americans the same as the gender difference for European Americans?

Each of these comparisons can be captured in a single number:

1.  Latino gender difference - African American gender difference: (50-40) - (52-41)  = 10 - 11 = -1

2.  Latino gender difference - European American gender difference: (50-40) - (60-45)  = 10 - 15 = -5

3.  African American gender difference - European gender difference: (52-41) - (60-45)  = 11 - 15 = -4

ZumaStat provides formal significance tests and confidence intervals for the above three comparisons.

The Difference Between Simple Main Effect Contrasts and Interaction Contrasts

Both simple main effect contrasts and interaction contrasts are informative but they address different questions.  In the above analysis, the simple main effects analysis answered if there was a gender difference for a single ethnic group.  By comparison, the interaction contrasts compares the gender difference for one ethnic group with that of another ethnic group.  Simple main effects do not focus on such comparisons.