1. Moderated Multiple Regression
Class 25

2. ANNOUNCEMENTS Voluntary Review Session Thursday, 10:00-11:20
Also Available Monday, Dec. 16, by Appointment: 11:00 – 5:00
Final Tuesday, Dec :00 Room 302
Only covers material in scheduled classes
Covers Mid-term on.
Multiple Choice and Short Answer
Take-home Stats due Tuesday, Dec. 17

3. Dummy Coding for Multiple (more than 2) Conditions
More complete description of dummy coding for multiple conditions at review meeting. But mainly FYI.
What you need to know for final:
1. Number of dummy variables = number of groups – 1.
2. Select one group as comparison: It is always coded as “0”
3. Comparison group selection is based on:
a. Most neutral group OR
b. Theoretically appropriate comparison OR
c. Largest group

4. Does Self Esteem Moderate the Use of Emotion as Information?
Harber, 2004, Personality and Social Psychology Bulletin, 31,
Emotions as Information Theory: People use their emotions as information,
especially when objective info. is lacking (Clore, et al., 2001).
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
own emotions.
If so, then self-esteem should determine how much emotions affect judgment.
Thus, when self-esteem is high, emotions should influence judgment more,
when self-esteem is low, emotions should influence judgments less.

5. Self Esteem and Emotions as Info: Method
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.
Main Predictor: Personal Upset; Moderator: Self Esteem
Outcome: Ratings of baby’s distress

6. Upset That Subjects Felt
Predictions
Overall positive relation between personal upset and cry ratings
(more upset subjects feel, more extremely they'll rate baby cries), BUT:
The relation between own upset and baby cries 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.
Upset That Subjects Felt

7. Regression Models Basic Linear Model
Basic Linear Model
Features: Intercept, one predictor
Y = b0 + b1 + Error (residual)
Does personal upset affect ratings of baby cries? 
Multiple Linear Model
Features: Intercept, two or more predictors
Y = b0 + b1 + b2 + Error (residual)
  Do personal upset and self esteem affect ratings of baby cries?
Moderated Multiple Linear Model
Features: Intercept, two or more predictors, and interaction term(s)
Y = b0 + b1 + b2 + b1b2 + Error (residual)
Do upset and esteem and the interaction of upset * esteem affect ratings of baby cries?

8. 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 . [Esteem Measure: Rosenberg, 1965]
COMPUTE upsteem = upset*esteem .
EXECUTE .
OUTCOME
COMPUTE crytotl = (cry1 + cry2 + cry3 + cry4 + cry5 +
cry6 + cry7 + cry8 + cry9 + cry10 + cry11 + cry12) / 12 .
EXECUTE . 
Interaction Term

9. SPSS Syntax for MMR _____ Stepwise _____ Hierarchical
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 .
_____ Stepwise
_____ Hierarchical
_____ Forced entry
What regression method used here? X
Why upset and esteem entered in Model 1, and upsteem in Model 2?
To test the unique contribution of interaction term (upsteem).

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

11. page A2
Correls Show:
Upset is positively related to cry ratings, as expected.
Upset and Esteem are is only moderately related to each other; co-linearity not a likely problem

12. page B1 “Residual” = random error R = Power of regression R2 =
Adj. R2 =
R sq. change =
Power of regression
Amount var. explained
Sig. F Change =
Corrects for multiple predictors
Does new model explain signif. amount added variance
Impact of each added model
“Residual” = random error

13. SPSS Regression Output: Predictor Effects
Constant refers to what?
Intercept; Value of DV when IVs = 0
Slope; influence of IV on DV
B refers to what?
Variance around the slope
Std. Error refers to what?
Beta refers to what?
Standardization of B
t refers to what?
B / Std. Error
Sig. refers to what?
Significance of effect of IV on DV, sig. of slope

14. Understanding the Interaction in MMR
Pos. B for upsteem means pos. relation between (upset * esteem) and cry ratings;
NOTES: b3 = 0.183; Graphs below do not include constant (intercept)
Lisa: Lowest upset (1) and lowest esteem (1) Upsteem = 1*1 = 1 * b3 = 0.18
Joe: Highest upset (5) and lowest esteem (1) Upsteem = 5*1 = 5 * b3 =
Tim: Lowest upset (1) and highest esteem (4) Upsteem = 1*4 = 4 * b3 = 0.73
Jane: Highest upset (5) and highest esteem (5) Upsteem = 5*5 = 25 * b3 = 4.58
High Esteem People
Low Esteem People
Upset
Cry Ratings
Joe
.92
Lisa
.18
Jane
4.58
Cry Ratings
Tim
.73
Upset

15. Esteem and Affect as Information
Regression Model for
Esteem and Affect as Information
Model: Y = b0 + b1X + b2Z + b3XZ
Where Y = cry rating
X = upset score
Z = esteem score
XZ = esteem*upset score
And b0 = =
b1 = =
b2 = =
b3 = =
Intercept (average cry-rating score when upset, esteem, upset X esteem all = 0
Slope (influence) of upset
Slope (influence) of esteem
Slope (influence) of upset X esteem interaction
Y = (upset score) (esteem score) (esteem score X upset score)

16. DV? Moderator? Predictor?
Plotting Outcome: How Does Personal Upset Affect Baby Cry Ratings as a Function of One’s Self Esteem
DV?
Self Esteem
Moderator?
cry rating
Upset
Predictor?

17. Plotting Interactions with Two Continuous Variables
What is slope of upset when esteem is: Low, Average, High?
Need to compute slope lines (“simple slopes”) for these levels of esteem.
Regression formula provides way to do so.
Y = b0 + b1X + b2Z + b3XZ
equals
Y = (b1 + b3Z)X + (b2Z + b0)
Y = (b1 + b3Z)X is simple slope of Y due to X at Z (low, ave., high).
Means "the effect predictor X has on outcome Y, due to value of moderator Z." i.e., the effect upset has on cry ratings, due to
level of esteem (low, ave., high).
Thus, when Z is one value, the slope of X takes one shape,
when Z is another value, the slope of X takes other shape.

18. 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) 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, three levels of moderator (Z):
Z at one SD above the mean = ZH
Z at the mean = ZM
Z at one SD below the mean = ZL
Y = (upset) (esteem) (esteem X upset)
Y = ( Z)X + (-.48Z )

19. 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 of X (upset) when
Z (esteem) is high
Insert ZM into (b1 + b3Z)X + (b2Z + b0) to get slope of X (upset) when
Z (esteem) is moderate
Insert ZL into (b1 + b3Z)X + (b2Z + b0) to get slope of X (upset) when
Z (esteem) is low

20. Example of Plotting Baby Cry Study, Part I
Y (cry rating) = b (cry rating when all predictors = zero)
+ b1X (effect of upset) + b2Z (effect of esteem)
+ b3XZ (effect of upset X esteem interaction).
Y = X -.48Z + .18XZ.
Y = ( Z)X + (-.48Z )
Go to “Descriptive Statistics”, Page 1 of SPSS Regression output
Compute ZH, ZM, ZL via “Descriptives” for esteem: = mean, .76 = SD
ZH, = Mean esteem + 1 SD = ( ) = 4.71
ZM = Mean Esteem + 0 SD = ( ) = 3.95
ZL = Mean esteem - 1 SD = ( ) = 3.19
Slope at ZH = ( * 4.71)X + ([-.48 * 4.71] ) = .32X
Slope at ZM = ( * 3.95)X + ([-.48 * 3.95] ) = .18X
Slope at ZL = ( * 3.19)X + ([-.48 * 3.19] ) = .04X

21. Example of Plotting, Baby Cry Study, Part II
1. Get mean and SD of main predictor ("X") i.e., Upset from “Descriptives”
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 = = 4.15
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
4.83
5.22
5.61 ZM
.18X
4.95
5.17
5.38 ZL
.04X
5.06
5.11
5.16
4. Plot values into graph

22. Graph Displaying Simple Slopes

23. 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 (cvh = conditional value / high) by subtracting ZH from Z.
Zcvh = Z - ZH = Z – 4.71
Conduct this transformation in SPSS as:
COMPUTE esthigh = esteem
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

24. Are the Simple Slopes Significant?--Programming
COMMENT SIMPLE SLOPES FOR CLASS DEMO
COMPUTE esthigh = esteem
COMPUTE estmed = esteem
COMPUTE estlow = esteem
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 .

25. 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.

26. Moderated Multiple Regression with
Continuous Predictor and Categorical Moderator
(Aguinis, 2004)
Problem: How does performance affect faculty salary for
tenured versus untenured professors?   
Criterion: Salary increase Continuous Var: $ $180
Predictor: Performance Continuous Var: 1 (low) to 5 (high)
Moderator: Tenure Categorical Var: 1 (Tenured)
0 (Non Tenured)
NOTE: THESE ARE NOT VALUES FROM AGUINIS;
NOT SAME AS TAKE-HOME TASK.

27. Regression Models to Test Moderating Effect of Tenure on Salary Increase
Without Interaction
Salary increase = b0 + b1 (perf.) + b2 (tenure)
With Interaction
Salary increase = b0 + b1 (perf.) + b2 (tenure) + b3 (perf. * tenure)
Tenure is categorical, therefore a "dummy variable", values = 0 or 1
Interaction term = Predictor * moderator, = perf. * tenure. That simple.
Conduct regression, plotting, simple slopes analyses same as when
predictor and moderator are both continuous variables.

28. Interaction Term and Syntax for Regression with Dummy Variable
COMPUTE perf.tenure=Performance * Tenure.
EXECUTE.
Regression Syntax
DATASET ACTIVATE DataSet1.
REGRESSION
/DESCRIPTIVES MEAN STDDEV CORR SIG N
/MISSING LISTWISE
/STATISTICS COEFF OUTS R ANOVA CHANGE
/CRITERIA=PIN(.05) POUT(.10)
/NOORIGIN
/DEPENDENT Salary
/METHOD=ENTER Performance Tenure
/METHOD=ENTER perf.tenure.

29. Regression Output for Interaction with Dummy Variable

31. Graphing Dummy Variable Interactions
Regression transformation same as before:
Y = b0 + b1X + b2Z + b3XZ Y = X + 11Z + 17 XZ
equals equals
Y = (b1 + b3Z)X + (b2Z + b0) Y = (5 + (17*Z))X + (11Z + 55)
However, easier computations because Z (tenure) has only two values, 0 and 1!
Tenured slope =
Salary = (5 + (17 * 1))X + ((11*1) + 55) = 22X + 66
Non-Tenured slope =
Salary = (5 + (17 * 0))X + (11 * 0) + 55) = 5X + 55

32. Dummy Variable Graphing
Performance (X): Mean = 3, SD = 1.45
Low
Performance
=1.55
Ave Performance
3 + 0 = 3
High
= 4.45
Tenured
Y = 22X + 66
100
132
166
Non-Tenured
Y = 5X + 55 63 70 77