1. Unit 16: Inferences about Two Dichotomous Predictors and their Interaction

2. Learning Objectives
For models with dichotomous intendant variables, you will learn:
Basic terminology from ANOVA framework
How to identify main effects, simple effects and interactions in table of means and figures
Two coding systems for dichotomous variables (centered vs. dummy)
How to link coefficients from interactive models with each coding system to table of means and figures (both directions)
How to calculate simple effects
How to write up and display results

3. An Example Catholic Jewish All Male 3 3 3.0 Female 5 7 6.0
Attitudes toward abortion as a function Sex and Religion
Sex: Male vs. female
Religion: Catholic vs. Jewish
Attitude: 1–10 with higher scores indicating more permissive
attitudes
Equal n=20 in each cell. N=80
Catholic Jewish All
Male
Female
All

4. Terms and Brief definitions
Catholic Jewish All
Male
Female
All
Common terminology: One-way ANOVA, Two-way ANOVA, Three-way ANOVA; 2 X 2 ANOVA; 2 X 3 X 2 ANOVA; Factorial ANOVA;
Cell Mean: The mean of a group of participants at specific levels on each factor (e.g., Catholic men, Jewish women, etc)
Marginal Mean: The mean of cell means across a row or column
Grand Mean: The mean of all cell means
Unweighted vs. weighted means

5. Terms and Brief definitions
Catholic Jewish All
Male
Female
All
Main effect: The “average” effect of an IV on the DV across the levels of another IV in the model [The effect of an IV at the average level of another IV]. Evaluated with marginal means for IV
Simple effect: The effect of an IV on the DV at a specific level of the other IV. Evaluated with cell means for focal IV at specific level of other IV (moderator)
Interaction: The simple effect of a (focal) IV on the DV differs across the levels of another (moderator) IV in the model

6. Describing Effects Catholic Jewish All Male 3 3 3.0 Female 5 7 6.0
Describe the magnitude of the main effect of Sex on Attitudes
Women have 3 points more permissive attitudes about abortion than men
Describe the magnitude of the main effect of Religion on Attitudes
Jews have 1 point more permissive attitudes about abortion than Catholics

7. Describing Effects Catholic Jewish All Male 3 3 3.0 Female 5 7 6.0
Describe the magnitude of the simple effects of Sex on Attitudes
Among Catholics, women have 2 points more permissive attitudes about abortion than men
Among Jews, women have 4 points more permissive attitudes about abortion than men

8. Describing Effects Catholic Jewish All Male 3 3 3.0 Female 5 7 6.0
Describe the magnitude of the simple effects of Religion on Attitudes
Among men, there is no different in attitudes between Jews and Catholics
Among women, Jews have 2 points more permissive attitudes about abortion than Catholics

9. Describing Effects Catholic Jewish All Male 3 3 3.0 Female 5 7 6.0
Does there appear to be an interaction between Sex and Religion? Why or why not?
Yes, there does appear to be an interaction. The simple effects of Sex appear to be different across Jews (4 points) than Catholics (2 points). Alternatively, the simple effects of Religion appear to be different across women (2 points) and men (0 points)
Of course, you cant tell if the interaction is significant by looking at the data descriptively, you have to test the interaction.

10. Describing Effects Catholic Jewish All Male 3 3 3.0 Female 5 7 6.0
How might you quantify the magnitude of the interaction?
The simple effect of Sex increases by 2 points from Catholics (2) to Jews (4)
The simple effect of Religion increases by 2 points from men (0) to women (2)
This will be the parameter estimate for the interaction!

11. There are two common options for coding the regressors for categorical IVs : Dummy codes and Centered codes.
Centered
cSex cRel
Male Catholic -0.5
Female Jewish 0.5
Dummy
Sex Rel
Male 0 Catholic 0
Female 1 Jewish 1
The two systems yield essentially the same result (except for b0) for additive models

12. Link parameter estimate b0, b1, & b2 to the figure
Centered codes:
cSex cRel
Male Catholic -0.5
Female Jewish 0.5
Attitudes = *cSexFvM + 1*cRelJvC
Link parameter estimate b0, b1, & b2 to the figure

13. Centered codes for Additive Model
Catholic Jewish All
Male
Female
All
Centered codes:
cSex cRel
Male Catholic -0.5
Female Jewish 0.5
Attitudes = *cSexFvM + 1*cRelJvC
Link parameter estimate b0, b1, & b2 to the table of means
b0 is the predicted value for attitudes for 0 on both regressors. This is the grand mean
b1 is the effect of Sex. It will be forced to be constant across religions (6 – 3 = 3)
b2 is the effect of Religion constant across sexes (5 – 4 = 1)

14. Additive Model Constraints
Catholic Jewish All
Male
Female
All
Centered: Attitudes = *cSexFvM + 1*cRelJvC
What constraints were imposed in the additive model?
The effect of Sex was constrained to be the same in both Catholics and Jews. 3 is the “best” parameter value b/c it is the average of the two simple effects of sex (2 vs. 4).
The effect of Religion was constrained to be the same in both men and women. 1 is the “best” parameter value b/c it is the average of the two simple effects of Religion(0 vs. 2).

15. Interactive Models Catholic Jewish All Male 3 3 3.0 Female 5 7 6.0
How do you relax this constraint and test the interaction between Sex and Religion?
Regress Attitudes on Sex, Religion and their interaction (calculated as the product of Sex * Religion)
Attitudes = b0 + b1*Sex + b2*Religion + b3*SexXReligion
The test of the b3 coefficient against zero is the test of the interaction (Alternatively: the comparison of this model to the compact model: Attitudes = b0 + b1*Sex + b2*Religion

16. In interactive models, centered codes and dummy codes yield very different results b/c they are testing different questions
You will use centered codes when you want to test main effects and interactions. The parameter estimate for the regressors coding for the IVs test the main effect of each IV. This is the approach you will almost always use for your primary analysis. [SPSS does this automatically for you for factors in their GLM function but not in Regression]
You will use dummy codes when you want to test simple effects. The parameter estimate for the regressors coding for the IVs will test a specific simple effect of each IV. You will often need to recode your IVs more than once to test all relevant simple effects.
The parameter estimate for the interaction and its interpretation is the same across these systems.

17. Centered codes for Interactive Model
Catholic Jewish All
Male
Female
All
Centered codes:
cSex cRel
Male Catholic -0.5
Female Jewish 0.5
Attitudes = *cSexFvM + 1*cRelJvC + 2*cSexFvMXcRelJvC
Link parameter estimate b0, b1, & b2 to the table of means
b0 is the predicted value for attitudes for 0 on all regressors. This is the grand mean
b1 is the main effect of Sex (6 – 3 = 3)
b2 is the main effect of Religion(5 – 4 = 1)

18. Centered codes for Interactive Model
Catholic Jewish All
Male
Female
All
Centered codes:
cSex cRel
Male Catholic -0.5
Female Jewish 0.5
Attitudes = *cSexFvM + 1*cRelJvC + 2*cSexFvMXcRelJvC
Link b3 to the table of means considering Sex as focal variable
b3 is the change in the magnitude of the (simple) Sex effect across religions
Jewish Catholic
(7 – 3) – (5 – 3) = 2

19. Centered codes for Interactive Model
Catholic Jewish All
Male
Female
All
Centered codes:
cSex cRel
Male Catholic -0.5
Female Jewish 0.5
Attitudes = *cSexFvM + 1*cRelJvC + 2*cSexFvMXcRelJvC
Link b3 to the table of means considering Religion as focal variable
b3 is the change in the magnitude of the (simple) Religion effect across Sexes
Female Male
(7 – 5) – (3 – 3) = 2

20. Centered codes for Interactive Model
cSex cReligion
Male Catholic -0.5
Female Jewish 0.5
Attitudes = *cSexFvM + 1*cRelJvC + 2*cSexXFvMcRelJvC
Link all coefficients to figure

21. Centered codes for Interactive Model
Catholic Jewish All
Male
Female
All
Centered codes:
cSex cRel
Male Catholic -0.5
Female Jewish 0.5
Attitudes = *cSexFvM + 1*cRelJvC + 2*cSexFvMXcRelJvC
You can use the regression equation to reproduce cell means
Catholic men: *(-.5) + 1*(-.5) + 2*(-.5)*(-.5) = 3
Jewish women: *(.5) + 1*(.5) + 2*(.5)*(-.5) = 7
Etc…..

22. Dummy codes for Interactive Model
Catholic Jewish All
Male
Female
All
Dummy codes:
Sex Rel
Male Catholic 0
Female 1 Jewish 1
Attitudes = *SexFvM + 0*RelJvC + 2*SexFvMXRelJvC
Link parameter estimate b0, b1, & b2 to the table of means
b0 is the predicted value for attitudes for 0 on all regressors. This is predicted value for male Catholics
b1 is the simple effect of Sex for Catholics (coded 0) (5 – 3 = 2)
b2 is the simple effect of Religion for men (3 – 3 = 0)

23. Dummy codes for Interactive Model
Catholic Jewish All
Male
Female
All
Dummy codes:
Sex Rel
Male Catholic 0
Female 1 Jewish 1
Attitudes = *SexFvM + 0*RelJvC + 2*SexFvMXRelJvC
Link b3 to the table of means considering Sex as focal variable
b3 is the change in the magnitude of the simple Sex effect across religions
Jewish Catholic
(7 – 3) – (5 – 3) = 2

24. Dummy codes for Interactive Model
Catholic Jewish All
Male
Female
All
Dummy codes:
Sex Rel
Male Catholic 0
Female 1 Jewish 1
Attitudes = *SexFvM + 0*RelJvC + 2*SexFvMXRelJvC
Link b3 to the table of means considering Religion as focal variable
b3 is the change in the magnitude of the Religion effect across Sexes
Female Male
(7 – 5) – (3 – 3) = 2

25. Dummy codes for Interactive Model
Sex Rel
Male Catholic 0
Female 1 Jewish 1
Attitudes = *SexFvM + 0*RelJvC + 2*SexFvMXRelJvC
Link all coefficients to figure

26. Dummy codes for Interactive Model
Catholic Jewish All
Male
Female
All
Dummy codes:
Sex Rel
Male Catholic 0
Female 1 Jewish 1
Attitudes = *SexFvM + 0*RelJvC + 2*SexFvMXRelJvC
You can use the regression equation to reproduce cell means
Catholic men: *(0) + 0*(0) + 2*(0)*(0) = 3
Jewish women: *(1) + 0*(1) + 2*(1)*(1) = 7
Etc…..

27. Dummy codes for Interactive Model
Catholic Jewish All
Male
Female
All
Dummy codes:
Sex Rel
Male Catholic 0
Female 1 Jewish 1
Attitudes = *SexFvM + 0*RelJvC + 2*SexFvMXRelJvC
The above dummy coding system gave us the above regression equation where b1 was the simple effect of Sex in Catholics. How do we get the simple effect of Sex in Jews?
Recode Religion with Dummy codes but set Jewish = 0 and Catholic = 1. Now the sex effect will be the simple effect in Jews because Jews = 0

28. Dummy codes for Interactive Model
Catholic Jewish All
Male
Female
All
Dummy codes:
Sex Rel
Male Catholic 1
Female 1 Jewish 0
What will the regression equation be now?
Attitudes = *SexFvM + 0*RelCvJ *SexFvMXRelCvJ
Recoding the moderator changes the simple effect for the focal variable. With dummy codes, the parameter estimate for the focal variable is always the simple effect when the moderator = 0

29. Text Results
We analyzed attitudes about abortion in a general linear model with centered, unit-weighted regressors for Sex (Female vs. Male), Religion (Jewish vs. Catholic) and their interaction. We report both raw GLM coefficients (b) and partial eta-squared (p2) to document effect sizes.
The main effect of Sex was significant, b=3, p2= 0.##, t(76) = #.##, p = 0.###, indicating that women’s attitudes about abortion were 3 points higher than men on average. However, the Sex X Religion interaction was also significant, b=2, p2= 0.##, t(76) = #.##, p = 0.### (see Figure 1). This indicates that the magnitude of the Sex effect was significantly greater in Jews (b = 4, p2= 0.##, p= 0.### ) than in Catholics (b=2, p2= 0.##, p=0.###).

30. Figure
What else should be included in this publication quality figure?
Confidence intervals (+1 SE)
Raw data points (4 columns, jittered)
Notation to indicate simple effects for focal variable?

31. On Your Own: Certain Uncertain All Sober 100 120 110 Drunk 90 90 90
Centered codes:
cGroup cThreat
Sober Certain -0.5
Drunk Uncertain 0.5
FPS = b0 + b1*cGroup + b2*cThreat + b3*cGroup*cThreat
FPS = *cGroup + 10*cThreat + -20*cGroup*cThreat

32. On Your Own: Certain Uncertain All Sober 100 120 110
Intoxicated
All
Centered codes:
Beverage Group Threat
Sober 0 Certain 0
Drunk 1 Uncertain 1
FPS = b0 + b1*Group + b2*Threat + b3*Group*Threat
FPS = *Group + 20*Threat + -20*Group*Threat

33. Learning Outcomes
For models with dichotomous intendant variables, you learned:
Basic terminology from ANOVA framework
How to identify main effects, simple effects and interactions in table of means and figures
Two coding systems for dichotomous variables (centered vs. dummy)
How to link coefficients from interactive models with each coding system to table of means and figures (both directions)
How to calculate simple effects
How to write up and display results