G89.2229 Multiple Regression Week 7 Wednesday

1. G89.2229 Lect 7W Statistical interaction Extended Example Considering alternative models G89.2229 Multiple Regression Week 7 (Wednesday)

2. G89.2229 Lect 7W Representing Interaction in the Regression Equation If we believe that the effect of X1 varies as a function of level of a second variable, X2, we can build a simple multiplicative interactive effect. Y=b0+b1X1+b2X2+b3(X1*X2)+e This multiplicative term creates a curved surface in the predicted Y If the multiplicative term is needed, but left out, the residuals may display heteroscedasticity This multiplicative model is related to the polynomial models studied last week.

3. G89.2229 Lect 7W Interpreting the Multiplicative Model Y=b0+b1X1+b2X2+b3(X1*X2)+e The effect (slope) of X1 varies with different values of X2 For X2=0, the effect of X1 is b1 For X2=1, the effect of X1 is b1+b3 For X2=2, the effect of X1 is b1+2b3 Because the coefficients b1 and b2 can be easily interpreted when X1 and X2 are zero, it is advisable to CENTER variables involved in interactions to make values of zero easy to understand.

4. G89.2229 Lect 7W Neuroticism (Emotional Stability) and Stress In a study by Kennedy (2000), 200 persons were asked to report about their own personalities, and to fill out a daily diary regarding troublesome events, and their current mood. For our analysis, we average the counts of troublesome events over days, and also average daily depressed mood. What do you expect the relation of troublesome events to depressed mood to be? Will the relation vary according to how emotionally stable people seem to be?

5. G89.2229 Lect 7W Measures and Sample Measures (Variables) POMS depressed mood (M) Sad, blue Emotional Stability (E) Saucier's short Goldberg form "Moody" vs. "Serene" Troublesome things (T) A lot of work, negative feedback, headache, bureaucracy Sample Graduate students in intimate relationships, plus snowball contacts.

6. G89.2229 Lect 7W Analysis Plan Specify Model: M=b0+b1E +b2T +b3(E*T)+e Describe distributions Estimate and evaluate model Examine residuals Plot interaction Consider alternative models Polynomial Rescaled outcome Estimate and evaluate alternative models Form conclusion Report results

7. G89.2229 Lect 7W Moderation issues Scaling of the outcome variable can affect whether an interaction term is needed. If we have a simple multiplicative model in Y, it will be additive in Ln(Y). E(Y|XW) = bXW E(ln(Y)|XW) = ln(b)+ln(X)+ln(W) Scaling is especially important if the trajectories of interest do not cross in the region where data is available.

8. G89.2229 Lect 7W Detecting and testing for scaling effects When the variance seems to be related to the level of Y, the hypothesis of interactions being simple scaling functions needs to be considered. Showing that the theoretically interesting interaction remains when Y is transformed to ln(Y) is good evidence Showing that ln(Y) increases heteroscedasticity also helps (if it is true) Often our theory predicts interaction, and scientists are motivated to demonstrate it.

9. G89.2229 Lect 7W Quadratic trends and interaction: Ganzach (1997) Ganzach (Psych Methods, 1997, Vol 2, page 235) argues that an alternative to the interactive model that should be considered is one with quadratic �main effects�. He suggests always centering IVs fitting the model If the quadratic terms are not needed, then they can be eliminated.

10. G89.2229 Lect 7W Two Interaction Plots Model of Depressed Mood Model of SQRT(d. mood)