# 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)

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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 = b 0 + b 1 X + b 2 Z + b 3 XZ Where Y = cry rating X = upset Z = esteem XZ = esteem*upset And b 0 = X.XX = MEANING? b 1 = = X.XX = MEANING? b 2 = = X.XX = MEANING? b 3 = =X.XX = MEANING?

Regression Model for Esteem and Affect as Information Model: Y = b 0 + b 1 X + b 2 Z + b 3 XZ Where Y = cry rating X = upset Z = esteem XZ = esteem*upset And b 0 = 6.53 = intercept (average score when upset, esteem, upsetXexteem = 0) b 1 = -0.57 = slope (influence) of upset b 2 = -0.48 = slope (influence) of esteem b 3 = 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 = b 0 + b 1 X + b 2 Z + b 3 XZ equals Y = (b 1 + b 3 Z)X + (b 2 Z + b 0 ) Y = (b 1 + b 3 Z)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 = b 0 + b 1 X + b 2 Z + b 3 XZ 2. Transform Y = b 0 + b 1 X + b 2 Z + b 3 XZ into Y = (b 1 + b 3 Z)X + (b 2 Z + b 0 ) 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 = Z H Z at the mean= Z M Z at one SD below the mean = Z L

Interpreting SPSS Regression Output (a) Regression page A1

4.Insert values for all the regression coefficients (i.e., b 1, b 2, b 3 ) and the intercept (i.e., b 0 ), from computation (i.e., SPSS print-out). 5.Insert Z H into (b 1 + b 3 Z)X + (b 2 Z + b 0 ) to get slope when Z is high Insert Z M into (b 1 + b 3 Z)X + (b 2 Z + b 0 ) to get slope when Z is moderate Insert Z L into (b 1 + b 3 Z)X + (b 2 Z + b 0 ) to get slope when Z is low Plotting Simple Slopes (continued)

Example of Plotting Baby Cry Study, Part I Y (cry rating) = b 0 (rating when all predictors = zero) + b 1 X (effect of upset) + b 2 Z (effect of esteem) + b 3 XZ (effect of upset X esteem interaction). Y= 6.53 + -.53X + -.48Z +.18XZ. Y = (b 1 + b 3 Z)X + (b 2 Z + b 0 ) [conversion for simple slopes] Y= (-.53 +.18 Z )X + (-.48 Z + 6.53) Compute Z H, Z M, Z L via “Frequencies" for esteem, 3.95 = mean,.76 = SD Z H, = (3.95 +.76) = 4.71 Z M = (3.95 + 0) = 3.95 Z L = (3.95 -.76) = 3.19 Slope at Z H = (-.53 +.18 * 4.71 )X + ([-.48 * 4.71 ] + 6.53) =.32X + 4.27 Slope at Z M = (-.53 +.18 * 3.95 )X + ([-.48 * 3.95 ] + 6.53) =. 18X + 4.64 Slope at Z L = (-.53 +.18 * 3.19 )X + ([-.48 * 3.19 ] + 6.53) =. 04X + 4.99

Example of Plotting, Baby Cry Study, Part II 1. 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 = 1.73 Medium upset = mean = 2.94 – 0.00 = 2.94 High upset = 1SD above mean = 2.94 + 1.21 = 4.15 3.Plug these values into Z H, Z M, Z L simple slope equations Simple Slope FormulaLow Upset (X = 1.73) Medium Upset (X = 2.94) High Upset (X = 4.15) ZHZH Y =.32X + 4.284.835.225.61 ZMZM Y =.18X + 4.644.955.175.38 ZLZL Y =.04X + 4.995.065.115.16 4. Plot values into graph

Graph Displaying Simple Slopes

Are the Simple Slopes Significant? Question: Do the slopes of each of the simple effects lines (Z H, Z M, Z L ) significantly differ from zero? Procedure to test, using as an example Z H (the slope when esteem is high): 1. Transform Z to Z cvh (CV = conditional value) by subtracting Z H from Z. Z cvh = Z - Z H = Z – 4.71 Conduct this transformation in SPSS as: COMPUTE esthigh = esteem - 4.71. 2. Create new interaction term specific to Z cvh, i.e., (X* Z cvh ) COMPUTE upesthi = upset*esthigh. 3. Run regression, using same X as before, but substituting Z cvh for Z, and X* Z cvh 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 NOTE: Key outcome is B of "upset", Model 2. If significant, then the simple effect of upset for the high esteem slope is signif.

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 = b 0 (ave.arguments) + b 1 (coffee.type) + b 2 (hositility.score) With Interaction Salary increase = b 0 (ave. salary) + b 1 (coffee) + b 2 (hostility) + b 3 (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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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 & Hostility Y (arguments) = b 0 (args when all predictors = zero) + b 1 X (effect of coffee) + b 2 Z (effect of hostility) + b 3 XZ (effect of coffee X hostility). Y= 0.84 + 1.71X+ 0.74Z + -0.73XZ. Y = (b 1 + b 3 Z)X + (b 2 Z + b 0 ) [conversion for simple slopes] Y= (1.17 + -.73 Z )X + (.74 Z +.84) Compute Z H, Z M, Z L via “Frequencies" for esteem, 3.95 = mean,.76 = SD Z H, = (3.60 + 1.72) = 5.32 Z M = (3.60+ 0) = 3.60 Z L = (3.60 - 1.72 ) = 1.88 Slope at Z H = (1.17 -.73 * 5.32 )X + ([.74 * 5.32 ] +.84) = 2.34X+ 4.78 Slope at Z M = (1.17 -.73 * 3.60 )X + ([.74 * 3.60 ] +.84) = 1.58X + 3.50 Slope at Z L = (1.17 -.73 * 1.88 )X + ([.74 * 1.88 ] +.84) = 0.83X + 2.23

Plotting Dummy Variable Interaction 1. 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 Z H, Z M, Z L simple slope equations Simple Slope FormulaNo Caff. (X = 0) Caffeinated (X = 1) ZHZH Y= 2.34X +4.78 4.787.12 ZMZM Y =1.58X + 3.50 3.505.08 ZLZL Y =.83X + 2.23 2.233.06 4. Plot values into graph

Graph Displaying Simple Slopes

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.

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 is affected 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 movie Sub 2 Funny movie Sub 3 Sad movie Sub 4 Sad movie Sub 5 Funny movie Sub 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

Multicollinearity In 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 ). Orthogonal Non-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/or Tolerance should be more than. 20

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