Moderated Multiple Regression

Class 22

STATS TAKE HOME EXERCISE IS DUE THURSDAY DEC. 12

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)

THE KENT AND HERMAN DIALOGUE

A Moderated Multiple Regression Drama

With A Satisfactory Conclusion

Appropriate for All Audiences

Overall model IS NOT

significant

Interaction term IS marginally

significant

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

Act 1, Scene 1: Kent contacts Herman regarding this vexing conundrum.

Kent, I believe you are referring to a test of a targeted interaction effect without looking at the overall (omnibus) effect. Please see pp. 134-135 of the book. Let me know if this does not answer your question and I will be delighted to follow up with you. Thanks for your kind words about my book! All the best, --Herman.

Act 1, scene 2: Herman replies!

Herman, thanks for getting back to me on this. Based on those pages of your text, it appears that the answer to my question is as follows:

If the omnibus F is itself not significant, then a significant interaction term within this non-significant model will itself not be interpretable.

Sadly (for some rather appealing interaction effects) this makes sense.

Again, very good of you to get back to me on this question. Best regards, Kent

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

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.

Act 1, scene 4: Herman sustains the dramatic tension.

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.

Act 1, scene 4: Herman drops the Big Clue

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

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. Thus, when self-esteem is high, emotions should influence judgment more, and when self-esteem is low, emotions should influence judgments less.

Method: Studies 1 & 2

1. Collect self-esteem scores several weeks before experiment.

2. Subjects listen to series of 12 disturbing baby cries.

3. 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.

Predictions Overall positive relation between personal upset and cry

ratings (more upset subjects feel, more extremely they'll rate cries).

This relation will be moderated by self-esteem

* For people w’ high esteem, the relation will be strongest

* For people w’ low esteem, the relation will be weakest.

1

2

3

4

5

6

7

low upset mod upset High upset

Cry

Ra

ting

s

Low EsteemMod. EsteemHigh Esteem

Developing Predictor and Outcome VariablesPREDICTORS

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 . 

SPSS Syntax for Moderated 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 .

Interpreting SPSS Regression Output (a) 

Regression 

Descriptive Statistics

5.1715 .53171 77

2.9351 1.20675 77

3.9519 .76168 77

11.3481 4.87638 77

crytotl

upset

esteem

upsteem

Mean Std. Deviation N

page A1

Correlations

1.000 .434 .031 .498

.434 1.000 -.277 .857

.031 -.277 1.000 .229

.498 .857 .229 1.000

. .000 .395 .000

.000 . .007 .000

.395 .007 . .023

.000 .000 .023 .

77 77 77 77

77 77 77 77

77 77 77 77

77 77 77 77

crytotl

upset

esteem

upsteem

crytotl

upset

esteem

upsteem

crytotl

upset

esteem

upsteem

Pearson Correlation

Sig. (1-tailed)

N

crytotl upset esteem upsteem

Variables Entered/Removedb

esteem,upset

a . Enter

upsteema . Enter

Model1

2

VariablesEntered

VariablesRemoved Method

All requested variables entered.a.

Dependent Variable: crytotlb.

page A2

Model Summary

.461a .213 .191 .47810 .213 9.999 2 74 .000

.545b .297 .269 .45473 .085 8.803 1 73 .004

Model1

2

R R SquareAdjustedR Square

Std. Error ofthe Estimate

R SquareChange F Change df1 df2 Sig. F Change

Change Statistics

Predictors: (Constant), esteem, upseta.

Predictors: (Constant), esteem, upset, upsteemb.

page B1

ANOVAc

4.571 2 2.286 9.999 .000a

16.915 74 .229

21.486 76

6.391 3 2.130 10.303 .000b

15.095 73 .207

21.486 76

Regression

Residual

Total

Regression

Residual

Total

Model1

2

Sum ofSquares df Mean Square F Sig.

Predictors: (Constant), esteem, upseta.

Predictors: (Constant), esteem, upset, upsteemb.

Dependent Variable: crytotlc.

Note: ANOVA F must be significant, EXCEPT IF INTERACTION OUTCOME PREDICTED A-PRIORI

“Residual” = random error, NOT interaction

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

Coefficientsa

4.101 .364 11.260 .000

.211 .047 .479 4.462 .000

.114 .075 .163 1.522 .132

6.529 .888 7.349 .000

-.527 .253 -1.196 -2.085 .041

-.478 .212 -.685 -2.256 .027

.183 .062 1.680 2.967 .004

(Constant)

upset

esteem

(Constant)

upset

esteem

upsteem

Model1

2

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: crytotla.

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.

Regression Model for Esteem and Affect as Information

Model Y = b0 + b1X + b2Z + b3XZ Where Y = cry rating

X = upsetZ = esteemXZ = esteem*upset

And b0 = X.XX = MEANING?

b1 = = X.XX = MEANING?b2 = = X.XX = MEANING?b3 = =X.XX = MEANING?

Regression Model for Esteem and Affect as Information

Model: Y = b0 + b1X + b2Z + b3XZ Where Y = cry rating

X = upsetZ = esteemXZ = esteem*upset

And b0 = 6.53 = intercept (average score when

upset, esteem, upsetXexteem = 0)b1 = -0.57 = slope (influence) of upsetb2 = -0.48 = slope (influence) of esteemb3 = 0.18 = slope (influence) of upset X

esteem interaction

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

???

???

???

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

cry rating

Upset

Self Esteem

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.

Plotting Simple Slopes

1.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) and insert values

Y = (? + ?Z)X + (?Z + ?)

3. Select 3 values of Z that display 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

Interpreting SPSS Regression Output (a) 

Regression 

Descriptive Statistics

5.1715 .53171 77

2.9351 1.20675 77

3.9519 .76168 77

11.3481 4.87638 77

crytotl

upset

esteem

upsteem

Mean Std. Deviation N

page A1

4.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).

5.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 ismoderate

Insert ZL into (b1 + b3Z)X + (b2Z + b0) to get slope when Z is low

Plotting Simple Slopes(continued)

Example of Plotting Baby Cry Study, Part IY (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

Example of Plotting, Baby Cry Study, Part II1. Compute mean and SD of main predictor ("X") i.e., Upset

Upset mean = 2.94, SD = 1.21

2. Select values on the X axis displaying main predictor, e.g. upset at:

Low upset = 1 SD below mean` = 2.94 – 1.21 = 1.73Medium upset = mean = 2.94 – 0.00 = 2.94High upset = 1SD above mean = 2.94 + 1.21 = 4.15

3. Plug these values into ZH, ZM, ZL simple slope equations

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

4. Plot values into graph

Graph Displaying Simple Slopes

4.6

5

5.4

5.8

Mild Upset Mod. Upset Extreme Upset

Participants' Level of Upset

Baby

Cry

Rat

ings

High EsteemMed. EsteemLow Esteem

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 for XZ

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 .

Simple Slopes Significant?—Results

Regression Model Summary

.461a .213 .191 .47810 .213 9.999 2 74 .000

.545b .297 .269 .45473 .085 8.803 1 73 .004

Model1

2

R R SquareAdjustedR Square

Std. Error ofthe Estimate

R SquareChange F Change df1 df2 Sig. F Change

Change Statistics

Predictors: (Constant), esthigh, upseta.

Predictors: (Constant), esthigh, upset, upesthib.

NOTE: Key outcome is B of "upset", Model 2. If significant, then the simple effect of upset for the high esteem slope is signif.Coefficientsa

4.639 .145 31.935 .000

.211 .047 .479 4.462 .000

.114 .075 .163 1.522 .132

4.277 .184 23.212 .000

.336 .062 .762 5.453 .000

-.478 .212 -.685 -2.256 .027

.183 .062 1.009 2.967 .004

(Constant)

upset

esthigh

(Constant)

upset

esthigh

upesthi

Model1

2

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: crytotla.

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)

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.

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, BUTit does not affect the interaction term.

THE KENT AND HERMAN DIALOGUE

A Moderated Multiple Regression Drama

With A Satisfactory Conclusion

Appropriate for All Audiences

Overall model IS NOT

significant

Interaction term IS marginally

significant

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

Act 1, Scene 1: Kent contacts Herman regarding this vexing conundrum.

Kent, I believe you are referring to a test of a targeted interaction effect without looking at the overall (omnibus) effect. Please see pp. 134-135 of the book. Let me know if this does not answer your question and I will be delighted to follow up with you. Thanks for your kind words about my book! All the best, --Herman.

Act 1, scene 2: Herman replies!

Herman, thanks for getting back to me on this. Based on those pages of your text, it appears that the answer to my question is as follows:

If the omnibus F is itself not significant, then a significant interaction term within this non-significant model will itself not be interpretable.

Sadly (for some rather appealing interaction effects) this makes sense.

Again, very good of you to get back to me on this question. Best regards, Kent

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

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.

Act 1, scene 4: Herman sustains the dramatic tension.

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.

Act 1, scene 4: Herman drops the Big Clue

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