Moderated Multiple Regression

1. Moderated Multiple Regression Class 18

2. Functions of Regression • 1. Establishing relations between variables • Do frustration and aggression co-occur? • 2. Establishing causality between variables • Does frustration (at Time 1) predict aggression (at Time 2)? • 3. Testing how multiple predictor variables relate to, or predict, an outcome variable. • Do frustration, and social class, and family income predict aggression? [additive effects] • 4. Test for moderating effects between predictors on outcomes. • Does frustration predict aggression, but mainly for people with low income? [interactive effect] • 5. Forecasting/trend analyses • If incomes continue to decline in the future, aggression will increase by X amount.

3. ANOVA VS. REGRESSION ANOVA: Is the mean of Group A different from the mean of Group B, after accounting for random error? Regression: Is the slope of predictor X on outcome Y significant, after accounting for random error?

4. Positive and Negative Regression Slopes

5. Scatter Plot With Regression Line Note: Line represents the "best fitting slope". Many points fall away from this line, above or below it Disparate points represent "error"

6. Error = Average Difference Between ??? and ???

7. Error = Average Difference Between Predicted Point (X88 - Ŷ88) and Actual Point (X88 - Y88)

8. Regression Assumes Errors are normally, independently, and identically Distributed at Every Level of the Predictor (X) X1 X2 X3

9. Regression Not Always Linear Different shapes 1. Curvalinear 2. J-shaped 3. Catastrophic or Exponential Regression can test for each of these shapes, but must be "informed" beforehand. Research must look at scatter plot to determine what pattern occurs

10. Regression Models Basic Linear Model Features: Intercept, one predictor Y = b0 + b1 + Error (residual) Do bullies aggress more after being reprimanded?  Multiple Linear Model Features: Intercept, two or more predictors Y = ??????   Do bullies aggress after reprimand and after nice kid is praised? Moderated Multiple Linear Model Features: Intercept, two or more predictors, interaction term(s) Y = ?????????? Aggress after reprimand, nice kid praised, and (reprimnd * praise)

11. Regression Models Basic Linear Model Features: Intercept, one predictor Y = b0 + b1 + Error (residual) Do bullies aggress more after being reprimanded?  Multiple Linear Model Features: Intercept, two or more predictors Y = b0 + b1 + b2 + Error (residual)   Do bullies aggress after reprimand and after nice kid is praised? Moderated Multiple Linear Model Features: Intercept, two or more predictors, interaction term(s) Y = b0 + b1 + b2 + b1b2 + Error (residual) Aggress after reprimand, nice kid praised, and (reprimnd * praise)

12. Does Self Esteem Moderate the Use of Emotion as Information? Harber, 2004, Personality and Social Psychology Bulletin, 31, 276-288 People use their emotions as information, especially when objective info. is lacking. Emotions are therefore persuasive messages from the self to the self. Are all people equally persuaded by their own emotions? Perhaps feeling good about oneself will affect whether to "believe" one's one emotions. Therefore, self-esteem should determine how much emotions affect judgment. In other worlds, when self-esteem is high, emotions should influence judgment more, and when self-esteem is low, emotions should influence judgments less.

13. Method: Studies 1 & 2 • 1. Collect self-esteem scores several weeks before experiment. • 2. Subjects listen to series of 12 disturbing baby cries. • Subjects rate how much the baby is conveying distress • through his cries, for each cry. • 4. After rating all 12 cries, subjects indicate how upsetting it was for them to listen to the cries.

14. Predictions Overall positive relation between personal upset and cry ratings (more upset subjects feel, more extremely they'll rate cries). This association will be moderated by self-esteem * For people w’ high esteem, association will be strongest * For people w’ low esteem, association will be weakest.

15. Developing Predictor and Outcome Variables PREDICTORS Upset = single item "How upset did baby cries make you feel?" COMPUTE esteem = (esteem1R + esteem2R + esteem3 + esteem4R + esteem5 + esteem6R + esteem7R + esteem8 + esteem9 + esteem10) / 10 . EXECUTE . COMPUTE upsteem = upset*esteem . EXECUTE . OUTCOME COMPUTE crytotl = (cry1 + cry2 + cry3 + cry4 + cry5 + cry6 + cry7 + cry8 + cry9 + cry10 + cry11 + cry12) / 12 . EXECUTE .

16. SPSS Syntax for Multiple Regression REGRESSION /DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE /STATISTICS COEFF OUTS BCOV R ANOVA CHANGE /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT crytotl /METHOD=ENTER upset esteem /METHOD=ENTER upset esteem upsteem .

17. Interpreting SPSS Regression Output (a) Regression page A1

18. page A2

19. page B1 R = ? R2= ? Adj. R2 = ? R sq. change = ? Sig. F Change = does new model explain ??? variance Note: “Residual” = ?

20. page B1 R = Power of regression R2= Amount var. explained Adj. R2 = Corrects for multiple predictors R sq. change = Impact of each added model Sig. F Change = does new model explain signif. amount added variance Note: ANOVA F must be significant, EXCEPT IF INTERACTION OUTCOME PREDICTED A-PRIORI “Residual” = random error, NOT interaction

21. page B2 Notes: 1. t = ???/ Std. Error 2. B and t change for upset, esteem when interaction term (upsteem) included. WHY? 3. Does Model 2 shows that interaction term is significant?

22. page B2 Notes: 1. t = B / Std. Error 2. B and t change for upset, esteem when interaction term (upsteem) included. 3. Model 2 shows that interaction effect is significant.

23. Regression Model for Esteem and Affect as Information Model: Y = b0 + b1X+ b2Z+ b3XZ Where Y = cry rating X = upset Z = esteem XZ = esteem*upset And b0 = X.XX = MEANING? b1 = = X.XX = MEANING? b2 = = X.XX = MEANING? b3 = =X.XX = MEANING?

24. Regression Model for Esteem and Affect as Information Model: Y = b0 + b1X+ b2Z+ b3XZ Where Y = cry rating X = upset Z = esteem XZ = esteem*upset And b0 = 6.53 = intercept (average score when upset, esteem, upsetXexteem = 0) b1 = -0.57 = slope (influence) of upset b2 = -0.48 = slope (influence) of esteem b3 = 0.18 = slope (influence) of upset X esteem interaction

25. ??? ??? ??? Plotting Outcome: Baby Cry Ratings as a Function of Listener's Upset and Listener's Self Esteem

26. cry rating Plotting Outcome: Baby Cry Ratings as a Function of Listener's Upset and Listener's Self Esteem Self Esteem Upset

27. Plotting Interactions with Two Continuous Variables Y = b0 + b1X+ b2Z+ b3XZ equals Y = (b1 + b3Z)X+ (b2Z+ b0) Y = (b1 + b3Z)X is simple slope of Y on X at Z. Means "the effect X has on Y, conditioned by the interactive contribution of Z." Thus, when Z is one value, the X slope takes one shape, when Z is another value, the X slope takes other shape.

28. Plotting Simple Slopes • Compute regression to obtain values of • Y = b0 + b1X+ b2Z+ b3XZ • 2. Transform Y = b0 + b1X+ b2Z+ b3XZ into • Y = (b1 + b3Z)X+ (b2Z+ b0) • Y = (? + ?Z)X + (?Z + ?) • 3. Select 3 values of Z that demonstrate the simple slopes of • X when Z is low, when Z is average, and when Z is high. • Standard practice: • Z at one SD above the mean = ZH • Z at the mean = ZM • Z at one SD below the mean = ZL

29. Plotting Simple Slopes • Compute regression to obtain values of • Y = b0 + b1X+ b2Z+ b3XZ • 2. Transform Y = b0 + b1X+ b2Z+ b3XZ into • Y = (b1 + b3Z)X+ (b2Z+ b0) • Y = (-.53 + .18Z)X + (-.48Z + 6.53) • 3. Select 3 values of Z that demonstrate the simple slopes of • X when Z is low, when Z is average, and when Z is high. • Standard practice: • Z at one SD above the mean = ZH • Z at the mean = ZM • Z at one SD below the mean = ZL

30. Plotting Simple Slopes (continued) • Insert values for all the regression coefficients (i.e., b1, b2, b3) and the intercept (i.e., b0), from computation (i.e., SPSS print-out). • Insert ZH into(b1 + b3Z)X + (b2Z+ b0) to get slope when Z is high • Insert ZM into(b1 + b3Z)X + (b2Z+ b0) to get slope when Z is • moderate • Insert ZL into(b1 + b3Z)X + (b2Z+ b0) to get slope when Z is low

31. Example of Plotting, Baby Cry Study, Part I Y (cry rating) = b0 (rating when all predictors = zero) + b1X (effect of upset) + b2Z (effect of esteem) + b3XZ (effect of upset X esteem interaction). Y = 6.53 + -.53X -.48Z + .18XZ. Y= (b1 + b3Z)X+ (b2Z+ b0) [conversion for simple slopes] Y = (-.53 + .18Z)X + (-.48Z + 6.53) Compute ZH, ZM, ZL via “Frequencies" for esteem, 3.95 = mean, .76 = SD ZH, = (3.95 + .76) = 4.71 ZM = (3.95 + 0) = 3.95 ZL = (3.95 - .76) = 3.19 Slope at ZH = (-.53 + .18 * 4.71)X + ([-.48 * 4.71] + 6.53) = .32X + 4.27 Slope at ZM = (-.53 + .18 * 3.95)X + ([-.48 * 3.95] + 6.53) = .18X + 4.64 Slope at ZL = (-.53 + .18 * 3.19)X + ([-.48 * 3.19] + 6.53) = .04X + 4.99

32. Example of Plotting, Baby Cry Study, Part II Simple Slope Formula Low Upset (X = 1.73) Medium Upset (X = 2.94) High Upset (X = 4.15) ZH .32X + 4.28 4.83 5.22 5.61 ZM .18X + 4.64 4.95 5.17 5.38 ZL .04X + 4.99 5.06 5.11 5.16 • 1. Compute mean and SD of main predictor ("X") i.e., Upset • Upset mean = 2.94, SD = 1.21 • Select values on the X axis displaying main predictor, e.g. upset at: • Low upset = 1 SD below mean` = 2.94 – 1.21 = 1.73 • Medium upset = mean = 2.94 – 0.00 = 2.94 • High upset = 1SD above mean = 2.94 + 1.21 = 4.15 • Plug these values into ZH, ZM, ZL simple slope equations 4. Plot values into graph

33. Graph Displaying Simple Slopes

34. Are the Simple Slopes Significant? Question: Do the slopes of each of the simple effects lines (ZH, ZM, ZL) significantly differ from zero? Procedure to test, using as an example ZH (the slope when esteem is high): 1. Transform Z to Zcvh (CV = conditional value) by subtracting ZH from Z. Zcvh = Z - ZH = Z – 4.71 Conduct this transformation in SPSS as: COMPUTE esthigh = esteem - 4.71. 2. Create new interaction term specific to Zcvh, i.e.,(X* Zcvh) COMPUTE upesthi = upset*esthigh . 3. Run regression, using same X as before, but substituting Zcvh for Z, and X* Zcvh forXZ

35. Are the Simple Slopes Significant?--Programming COMMENT SIMPLE SLOPES FOR CLASS DEMO COMPUTE esthigh = esteem - 4.71 . COMPUTE estmed = esteem - 3.95. COMPUTE estlow = esteem - 3.19 . COMPUTE upesthi = esthigh*upset . COMPUTE upestmed = estmed*upset . COMPUTE upestlow = estlow*upset . REGRESSION [for the simple effect of high esteem (esthigh)] /MISSING LISTWISE /STATISTICS COEFF OUTS BCOV R ANOVA CHANGE /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT crytotl /METHOD=ENTER upset esthigh /METHOD=ENTER upset esthigh upesthi .

36. Simple Slopes Significant?—Results Regression NOTE: Key outcome is B of "upset", Model 2. If significant, then the simple effect of upset for the high esteem slope is signif.

37. Moderated Multiple Regression with Continuous Predictor and Categorical Moderator (Aguinis, 2004) Problem: Does performance affect faculty salary for tenured versus untenured professors? Criterion: Salary increase Continuous Var. $13.00 -- $2148 Predictor: Performance Continuous Var. 1 -- 5 Moderator: Tenure Categorical Var. 0 (yes) 1 (no)

38. Regression Models to Test Moderating Effect of Tenure on Salary Increase Without Interaction Salary increase = b0 (ave. salary) + b1 (perf.) + b2 (tenure) With Interaction Salary increase = b0 (ave. salary) + b1 (perf.) + b2 (tenure) + b3 (perf. * tenure) Tenure is categorical, therefore a "dummy variable", values = 0 or 1 These values are markers, do not convey quantity Interaction term = Predictor * moderator, = perf. * tenure. That simple. Conduct regression, plotting, simple slopes analyses same as when predictor and moderator are both continuous variables.

39. Centering Data Centering data is done to standardize it. Aiken and West recommend doing it in all cases. * Makes zero score meaningful * Has other benefits Aguinas recommends doing it in some cases. * Sometimes uncentered scores are meaningful Procedure upset M = 2.94, SD = 1.19; esteem M = 3.94, SD = 0.75 COMPUTE upcntr = upset – 2.94. COMPUTE estcntr = esteem = 3.94 upcntr M = 0, SD = 1.19; esteem M = 0, SD = 0.75 Centering may affect the slopes of predictor and moderator, BUT it does not affect the interaction term.

40. THE KENT AND HERMAN DIALOGUE A Moderated Multiple Regression Drama With A Satisfactory Conclusion Appropriate for All Audiences

41. Interaction term IS significant Overall model IS NOT significant

42. Act 1, Scene 1: Kent contacts Herman regarding this vexing conundrum. Dear Dr. Aguinis, I am using your text in my graduate methods course. It is very clear and straightforward, which both my students and I appreciate. A question came up that I thought you might be able to answer. If an MMR model produces a significant interaction, but the ANOVA F is not itself significant, is the significant interaction still a valid result? My impression is that the F of the overall model (as indicated by the ANOVA F and/or by the R-sqr. change) must be significant. Thank you for your response, Kent Harber

43. Act 1, scene 2: Herman replies!

44. Act 1, scene 3: Are simple effects doomed???

45. Act 1, scene 4: Herman sustains the dramatic tension. Kent, Before I give you an answer and to make sure I understand the question. What do you mean precisely by "the ANOVA F test"? Regards, --Herman.

46. Act 1, scene 4: Herman drops the Big Clue Kent, Thanks for the clarification. Now, I understand your question perfectly. An article by Bedeian and Mossholder (1994), J. of Management, addresses this question directly. The full citation is on page 177 of my book. All the best, --Herman.

47. Finale: Simple effects are redeemed!!! [enter marching band, stage right]

48. Data Management Issues Setting up data file Checking accuracy of data Disposition of data Why obsess on these details? Murphy's Law If something can go wrong, it will go wrong, and at the worst possible time.

49. Creating a Coding Master 1. Get survey copy 2. Assign variable names 3. Assign variable values 4. Assign missing values 5. Proof master for accuracy 6. Make spare copy, keep in file drawer

50. Coding Master variable values variable names Note: Var. values not needed for scales