Moderated Multiple Regression II Class 25

Regression Models Basic Linear Model Features: Intercept, one predictor Y = b 0 + b 1 + Error (residual) Do bullies aggress more after being reprimanded? Multiple Linear Model Features: Intercept, two or more predictors Y = b 0 + b 1 + b 2 + Error (residual) Do bullies aggress after reprimand and after family stress? Moderated Multiple Linear Model Features: Intercept, two or more predictors, and interaction term(s ) Y = b 0 + b 1 + b 2 + b 1 b 2 + Error (residual) Aggress after reprimand, family stress, and (reprimand * stress)

Does Self Esteem Moderate the Use of Emotion as Information? Harber, 2004, Personality and Social Psychology Bulletin, 31, 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 own emotions. Therefore, 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.

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.

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

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. COMPUTE upsteem = upset*esteem. EXECUTE. OUTCOME COMPUTE crytotl = (cry1 + cry2 + cry3 + cry4 + cry5 + cry6 + cry7 + cry8 + cry9 + cry10 + cry11 + cry12) / 12. EXECUTE. Interaction Term

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

Interpreting SPSS Regression Output (a) Regression page A1

page A2

page B1 “ Residual ” = random error, NOT interaction R = R 2 = Adj. R 2 = R sq. change = Sig. F Change = Power of regression Does new model explain signif. amount added variance Corrects for multiple predictors Amount var. explained Impact of each added model

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

Pos. B for upsteem means pos. relation between (upset * esteem) and cry ratings; NOTES: b3 = 0.183; Graphs below do not include constant (intercept) Tina: Low upset (1) and low esteem (1) Upsteem = 1*1 = 1 * b3 = 0.18 Joe: High upset (5) and low esteem (1) Upsteem = 5*1 = 5 * b3 = 0.92 Tim: Low upset (1) and high esteem (4) Upsteem = 1*4 = 4 * b3 = 0.73 Jane: High upset (5) and high esteem (5) Upsteem = 5*5 = 25 * b3 = 4.58 Understanding the Interaction in MMR Low Esteem People Upset Cry Ratings Joe.92 Tina.18 High Esteem People Upset Cry Ratings Jane 4.58 Tim.73

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 score Z = esteem score XZ = esteem*upset score 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 score Z = esteem score XZ = esteem*upset score And b 0 = 6.53 = b 1 = = b 2 = = b 3 = 0.18 = Intercept (average 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) (esteem) +.18 (esteem X upset)

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

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 = 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 (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.

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 Y = ( Z)X + (-.48Z ) Y = (upset) (esteem) +.18 (esteem X upset)

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 (cry 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= X -.48Z +.18XZ. Y = (b 1 + b 3 Z)X + (b 2 Z + b 0 ) [conversion for simple slopes] Y= ( Z )X + (-.48 Z ) Compute Z H, Z M, Z L via “Frequencies" for esteem, 3.95 = mean,.76 = SD Z H, = ( ) = 4.71 ( ) Z M = ( ) = 3.95 ( ) Z L = ( ) = 3.19 ( ) Slope at Z H = ( * 4.71 )X + ([-.48 * 4.71 ] ) =.32X Slope at Z M = ( * 3.95 )X + ([-.48 * 3.95 ] ) =. 18X Slope at Z L = ( * 3.19 )X + ([-.48 * 3.19 ] ) =. 04X

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 = 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 = = 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.32X ZMZM.18X ZLZL.04X 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 (cvh = conditional value / high) by subtracting Z H from Z. Z cvh = Z - Z H = Z – 4.71 Conduct this transformation in SPSS as: COMPUTE esthigh = esteem 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 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.

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 performance affect faculty salary for tenured versus untenured professors? Criterion: Salary increase Continuous Var. $ $2148 Predictor: Performance Continuous Var Moderator: Tenure Categorical Var. 0 (yes) 1 (no)

Regression Models to Test Moderating Effect of Tenure on Salary Increase Without Interaction Salary increase = b 0 (ave. salary) + b 1 (perf.) + b 2 (tenure) With Interaction Salary increase = b 0 (ave. salary) + b 1 (perf.) + b 2 (tenure) + b 3 (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.