1 G Lect 7M Statistical power for regression Statistical interaction G Multiple Regression Week 7 (Monday)

2 G Lect 7M Statistical Power Issues in Regression Suppose you are interested in the relation of Conscientiousness (a Big Five measure of personality) on truthfulness in internet surveys. Suppose a previous study of 100 people showed that for every unit change of C (on a 0-10 scale) T (on a scale) increased by 5 points. The regression equation: »T= *C + e Suppose the C effect was just significant at the.05 level. What power would you have if you replicated the study EXACTLY with the same population, same measures, and same N?

3 G Lect 7M Power considerations The power is a function of »Effect size »Standard error The standard error is a function of »Sample size »Amount of residual variation »Distribution of predictor

4 G Lect 7M Power Tables and Programs For regression, we tend to assume that predictors have normal distributions Often use R 2 as a summary of effect size Increase in R 2 already takes into account »Distribution of predictor »Amount of error variation »Covariation among predictors

5 G Lect 7M Three Designs: Same N, Different Power Let’s study the process with structural model: »Y = X + e Suppose further than three studies were carried out. »The first samples 25 persons each from X=1 and X=10. »The second samples 5 persons each in each interval between X=1 and X=10. »The third takes a random sample of 50 persons.

6 G Lect 7M The Results All Three studies give comparable regression estimates, but standard errors and R 2 values are very different.

7 G Lect 7M Calculating Power for Regression Programs and tables stress R 2 and R 2 change. »These values depend on distribution of X »Conventions of small, medium and large effect size might not map on substantive beliefs about what is large or subtle. Example: »Power and Precision Software Recommendation: Study R 2 for different designs using simulation.

8 G Lect 7M Statistical Interaction: Moderation of Effects Example »In many studies of perceived social support, stress and distress, a three- way picture is found: For persons who are not experiencing an important stress, the relation between social support and distress is small or zero. For persons who have a modest amount of stress, social support seems to reduce distress. For persons with high stress, social support has large inverse association with distress »This is called the stress-buffering effect of support.

9 G Lect 7M Two Pictures of Interaction Stress Distress High Support Some Support No Support Stress Distress Support e The second emphasizes moderation (Baron & Kenny)

10 G Lect 7M Representing Interaction in the Regression Equation One way to model how the effect of X 1 is systematically affected by X 2 is to include a multiplicative term in the regression equation »Y=b 0 +b 1 X 1 +b 2 X 2 +b 3 (X 1 *X 2 )+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.

11 G Lect 7M Interpreting the Multiplicative Model Y=b 0 +b 1 X 1 +b 2 X 2 +b 3 (X 1 *X 2 )+e The effect (slope) of X 1 varies with different values of X 2 »For X 2 =0, the effect is b 1 »For X 2 =1, the effect is b 1 +b 3 »For X 2 =2, the effect is b 1 +2b 3 Because the coefficients b 1 and b 2 can be easily interpreted when X 1 and X 2 are zero, it is advisable to CENTER variables involved in interactions to make values of zero easy to understand.

12 G Lect 7M 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.

13 G Lect 7M 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.