Moderated Multiple Regression

Class 23

STATS TAKE HOME EXERCISE IS DUE THURSDAY DEC. 12

Deliver to Kent’s Mailbox or Place under his door (Rm. 352)

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

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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 Y =.32X + 4.28 4.83 5.22 5.61ZM Y =.18X + 4.64 4.95 5.17 5.38ZL Y =.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 caffeine lead to more arguments, but mainly for people with hostile personalities?

Criterion: Weekly arguments Continuous Var. 0-10 Predictor: Caffeinated coffee Categorical Var.

0 = decaff, 1 = caffeinated Moderator: Hostility Continuous var. 1 - 7

Regression Models to Test Moderating Effect of Tenure on Salary Increase

Without Interaction

Arguments = b0 (ave.arguments) + b1 (coffee.type) + b2 (hositility.score) With Interaction

Salary increase = b0 (ave. salary) + b1 (coffee) + b2 (hostility) + b3 (coffee*hostility)

Coffee is categorical, therefore a "dummy variable", values = 0 or 1 These values are markers, do not convey quantity Interaction term = Predictor * moderator, = coffee*hositility. That simple. Conduct regression, plotting, simple slopes analyses same as when predictor and moderator are both continuous variables.

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.00 7.00 5.00 .001.00 2.00 2.00 2.001.00 3.00 3.00 3.001.00 4.00 4.00 4.001.00 5.00 3.00 5.001.00 2.00 2.00 2.001.00 3.00 3.00 3.001.00 4.00 2.00 4.001.00 5.00 1.00 5.001.00 1.00 3.00 1.001.00 7.00 3.00 7.00

Coffee Hostility Args. Coff.hostile

DATASET ACTIVATE DataSet1.COMPUTE coffee.hostile=coffee * hostile.personality.EXECUTE.

REGRESSION /DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA CHANGE /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT arguments /METHOD=ENTER coffee hostile.personality /METHOD=ENTER coffee.hostile .

Plotting of Arguments due to Caffeine & HostilityY (arguments) = b0 (args when all predictors = zero)

+ b1X (effect of coffee) + b2Z (effect of hostility) + b3XZ (effect of coffee X hostility).

Y = 0.84 + 1.71X+ 0.74Z + -0.73XZ.

Y = (b1 + b3Z)X + (b2Z + b0) [conversion for simple slopes] Y = (1.17 + -.73Z)X + (.74Z + .84)

Compute ZH, ZM, ZL via “Frequencies" for esteem, 3.95 = mean, .76 = SD

ZH, = (3.60 + 1.72) = 5.32 ZM = (3.60+ 0) = 3.60

ZL = (3.60 - 1.72 ) = 1.88

Slope at ZH = (1.17 -.73 * 5.32)X + ([.74 * 5.32] + .84) = 2.34X+ 4.78

Slope at ZM = (1.17 -.73 * 3.60)X + ([.74 * 3.60] + .84) = 1.58X + 3.50

Slope at ZL = (1.17 -.73 * 1.88)X + ([.74 * 1.88] + .84) = 0.83X + 2.23

Plotting Dummy Variable Interaction1. Main predictor has only 2 values, 0 and 1

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

No Caffeine = 0 Caffeine = 1

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

Simple Slope

Formula No Caff.(X = 0)

Caffeinated(X = 1)

ZH Y= 2.34X +4.78 4.78 7.12ZM Y =1.58X+ 3.50 3.50 5.08ZL Y =.83X + 2.23 2.23 3.06

4. Plot values into graph

Graph Displaying Simple Slopes

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

No Caff Caffeinated

Coffee Type

Argu

men

ts

Low HostileMed. HostileHigh Hostile

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.

Requirements and Assumptions (Continued)

Independent Errors: Residuals for Sub. 1 ≠ residuals for Sub. 2. For example Sub. 2 sees Sub 1 screaming as Sub 1 leaves experiment. Sub 1 might influence Sub 2. If each new sub isaffected by preceding sub, then this influence will reduce

independence of errors, i.e., create autocorrelation. Autocorrelation is bias due to temporal adjacency.

Assess: Durbin-Watson test. Values range from 0 - 4, "2" is ideal. Closer to 0 means neg. correl, closer to 4 = pos. correl.

Sub 1 Funny movieSub 2 Funny movieSub 3 Sad movieSub 4 Sad movieSub 5 Funny movieSub 6 Funny movie

r (s1 s2) +r (s2 s3) +r (s3 s4) -r (s4 s5) -r (s5 s6) +

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 crytotl /METHOD=ENTER age upset /RESIDUALS DURBIN.

Durbin-Watson Test of Autocorrelation

MulticollinearityIn multiple regression, statistic assumes that each new predictor is in fact a unique measure.

If two predictors, A and B, are very highly correlated, then a model testing the added effect of Predictors A and B might, in effect, be testing Predictor A twice.

If so, the slopes of each variable are not orthogonal (go in different directions, but instead run parallel to each other (i.e., they are co-linear).

OrthogonalNon-orthogonal

Mac Collinearity: A Multicollinearity Saga

Suffering negative publicity regarding the health risks of fast food, the fast food industry hires the research firm of Fryes, Berger, and Shayque (FBS) to show that there is no intrinsic harm in fast food.

FBS surveys a random sample, and asks:

a.To what degree are you a meat eater? (carnivore)b.How often do you purchase fast food? (fast.food)c.What is your health status? (health) FBS conducts a multiple regression, entering fast.food in step one and carnivore in step 2.

FBS Fast Food and Carnivore Analysis

“See! See!” the FBS researchers rejoiced “Fast Food negatively predicts health in Model 1, BUT the effect of fast food on health goes away in Model 2, when being a carnivore is considered.”

Not So Fast, Fast Food Flacks

Colinearity Diagnostics 1.Correlation table

2.Collinearity Statistics

VIF (should be < 10) and/orTolerance should be more than .20