moderated multiple regression class 18. functions of regression 1. establishing relations between...

Moderated Multiple Regression

Class 18

Functions of Regression1. 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.

ANOVA VS. REGRESSION

ANOVA: Is the mean of Group A different from the mean of Group B, after accounting for random error?

0

5

10

15

20

25

Low Frustration High Frustration

Aggr

essio

n

Regression: Is the slope of predictor X on outcome Y significant, after accounting for random error?

0

2

4

6

8

10

12

low medium high veryhigh

extreme

Aggr

essio

n

Positive and Negative Regression Slopes

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"

Error = Average Difference Between ??? and ???

Error = Average Difference Between

Predicted Point (X88 - Ŷ88) and Actual Point (X88 - Y88)

Regression Assumes Errors are normally, independently, and identically Distributed at Every Level of the Predictor (X)

X1 X2 X3

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

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)

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)

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.

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 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.

1

2

3

4

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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 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.



Dependent Variable: crytotlc. Note: “Residual” = ?

R = ?

R2 = ?

Adj. R2 = ?

R sq. change = ?

Sig. F Change = does new model explain ??? variance

Model Summary

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

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

Model1

2




Change Statistics



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.



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 = ???/ 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?

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


Beta


t Sig.


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)

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

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)

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

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




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


Beta


t Sig.


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 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]

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.

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

Coding Master

variable names

variable values

Note: Var. values not needed for scales

Cleaning Data Set

1. Exercise in delay of gratification 2. Purpose: Reduce random error 3. Improve power of inferential stats.

Complete Data Set

Note: Are any cases missing data?

Are any “Minimums” too low? Are any “Maximums” too high?

Do Ns indicate missing data?

Do SDs indicate extreme outliers?

Do variables correlate in the expected manner?

Using Cross Tabs to Check for Missing or Erroneous Data Entry

Case A: Expect equal cell sizesGender

Oldest Youngest Only Child

Males 10 10 20

Females 5 15 20

TOTAL 15 25 40

Case B: Impossible outcomeNumber of Siblings

Oldest Youngest Only Child

None 4 3 6

One 3 4 0

More than one 3 4 2

TOTAL 10 10 8

Check a Sub Sample 1. Determine acceptable error rate 2. Number of cases to randomly sample, by rate of acceptable error rate:

Acceptable Error Rate

Number of Cases to Randomly Review

If No Errors Detected, chances

are good that:.50 5 50% or fewer errors

.40 10 40% or fewer errors



Storing Data

Raw Data

1. Hold raw data in secure place

2. File raw data by ID #

3. Hold raw date for at least 5 years post publication, per APA Automated Data

1. One pristine source, one working file, one syntax file

2. Back up, Back up, Back up

` 3. Use external hard drive as back-up for PC

File Raw Data Records By ID Number

01-20 21-40 41-60 61-80 81-100 101-120

COMMENT SYNTAX FILE GUN CONTROL STUDY SPRING 2007

COMMENT DATA MANAGEMENT

IF (gender = 1 & party = 1) genparty = 1 .EXECUTE .IF (gender = 1 & party = 2) genparty = 2 .EXECUTE .IF (gender = 2 & party = 1) genparty = 3 .EXECUTE .IF (gender = 2 & party = 2) genparty = 4 .EXECUTE .

COMMENT ANALYSES

UNIANOVA gunctrl BY gender party /METHOD = SSTYPE(3) /INTERCEPT = INCLUDE /PRINT = DESCRIPTIVE /CRITERIA = ALPHA(.05) /DESIGN = gender party gender*party .

ONEWAY gunctrl BY genparty /CONTRAST= -1 -1 -1 3 /STATISTICS DESCRIPTIVES /MISSING ANALYSIS /POSTHOC = TUKEY ALPHA(.05).

Save Syntax File

moderated multiple regression class 18. functions of regression 1. establishing relations between...

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