Regression Analyses II Mediation & Moderation

Review of Regression Multiple IVs but single DV Y’ = a+b1X1 + b2X2 + b3X3...bkXk Where k is the number of predictors Find solution where Sum(Y-Y’) 2 minimized Interactions & Regression Y’ = a + b1X + b2Z + b3XZ Curvilinearity & Regression Y’ = a + b1X + b2X 2 + b3X 3

Testing Significance of R 2 With df = k 1 -k 2 & N - k k 2 is subset of k 1 Significance of R 2 Significance of Increment in R 2 With df = k eff and N - k tot - 1

Researchers often confuse Two completely different processes and analytical approaches Mediation - effect of an IV on DV occurs through another variable Moderation - effect of IV on DV depends on the level of another variable Mediation and Moderation

Indirect causal – Z is a mediator of X and Y X Y Z X Y Moderated causal – Z is moderator of X and Y Z

Implies processes or mechanisms by which an IV influences a DV Often interpreted as “causal” mechanisms Direct effects: Effect of IV on DV outside the mediator (c) Indirect effects: Effect of IV on the DV through the mediator (a * b) Total effects: Effect of IV on the DV through both indirect and direct effects (c + (a*b)) Mediation

IV related to mediator (X and M: path a) Mediator related to DV (M and Y: path b) IV related to DV (X and Y: path c) Relationship between the IV and DV is weakened or n.s. when mediator is controlled (X not related to Y when controlling for M: no path c when controlling for paths a and b) b c a M X Y

(1) Regress mediator on IV (test for path a) IV must be related to mediator beta = direct effect of X on M (2) Regress DV on IV (test for path c) IV must relate to DV beta = total effect of X on Y (3) Regress DV on both IV and mediator Mediator must affect DV after controlling for IV Full mediation if effect of IV disappears (beta n.s.) Partial mediation if effect of X remains but beta is reduced but significant Steps to Test for Mediation b c a M X Y

Example of Mediation b c a Positive Affect Job Sat Work SE Regression #1. PA predicts WSE Beta PA =.279* Regression #2. PA predicts JS Beta PA =.463* Regression #3. PA and WSE predict JS Beta PA =.345* (compare to beta from regression #2) Beta WSE =.372* Is there evidence of mediation?

Mediation: Order of Causality –With three variable systems, difficult to determine proper causal order –Use issues of timing, logic, and theory to help determine causal order –If data collected at single point in time, not a test of causality a b X M Y a b M X Y

Moderation A test of moderation is a test of interaction Multiplicative effects of IVs on a DV Low High Y X Low Z High Z Medium Z Low

Moderated & Curvilinear Effects Enter main effects first –Significance test for increment in R 2 –Interpretation of  must occur in this step –Can enter main effects all at once or one at a time Enter curvilinear or interaction terms second –Significance test for increment in R 2 –Interpretation of  must occur at point when interactions entered –Can enter in any order, but lower order must precede higher order interactions –Two-way interactions must be entered before testing three- way interactions –Curvilinear effects and interaction effects may be confounded when IVs are intercorrelated

Testing Moderation (1) Create cross-product of two IVs; Compute XM = X * M (2) Partial main effects first; Interpreted at Step 1 When interaction term not included, b weights for main effects indicate “general effects” When interaction term included, b weights for main effects indicate effect of one variable on Y when the other is zero

Slopes (1) Y’ = a + b 1 X + b 2 M + b 3 XM Rewritten: (2) Y’ = a + b 1 X + b 3 XM + b 2 M Rewritten: (3) Y’ = a + (b 1 + b 3 M)X + b 2 M *You can clearly see that the value for b 1 is the value when M = 0 (no moderation) so that b 3 M cancels out.

Scale Invariance Low High Z = 20 Y X High Low Z = 10 Z = 0 b weight of X with interaction term in model

Scale Invariance Low High Z = 10 Y X High Low Z = 0 Z = -10 b weight of X with interaction term in model Now subtract 10 points from all Z scores

Lack of Scale Invariance Why main effects are not scale invariant (1) Y’ = a + b 1 X + b 2 M + b 3 XM Now, let’s subtract a constant c from X and a constant f from M and rewrite equation 1: (2a) Y’ = a + b 1 (X - c) + b 2 (M - f) + b 3 (X - c)(M - f) Solving: (2b) Y’ = (a - b 1 c - b 2 f + b 3 cf) + (b 1 - b 3 f)X + (b 2 - b 3 c)M + b 3 XM

Simple Slopes Y’ = a + b 1 X + b 2 M + b 3 XM Rewritten: Y’ = a + b 1 X + b 3 XM + b 2 M Rewritten: Y’ = a + (b 1 + b 3 M)X + b 2 M You can now compute a slope for X at a given value of M. This is known as a simple slope. If you choose meaningful points for M, then you can interpret the simple slopes. That’s what the graph does for you visually.

Moderation - Interpretation Interpretation of interactions –Simple slopes –Plotting Plotting interactions –Continuous (pick -1 SD, mean, +1 SD) –Categorical (codes representing group)

Moderation - Interpretation Low High Y X Low Z High Z Medium Z Low Group 1 Low High X Low Group 2 Y

Moderation - Issues Predictors and interaction terms will be highly correlated unless centered –The high correlation does not create problems with collinearity or interpretation (unless extremely high) b/c partial main effects first and findings are scale invariant –But if you did not partial main effects first, it would screw up the regression weights & they would not be interpretable

Measurement error influences detection and interpretation of moderating effects –Low reliability has complex influences on tests of moderation Testing for moderation often has low power (unreliability, error heterogeneity, etc.) This is particularly true in field research –Small effect size (1% to 3% of variance) –Requires “X” pattern of the moderating IVs –Testing for moderation and curvilinearity together requires “filled” pattern of the moderating IVs Moderation - Issues

Do positive and negative affect interact in predicting work-family conflict? REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA CHA /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT wfc /METHOD=ENTER positaff negaff /METHOD=ENTER paxna.

Following up on the interaction Step 1. Regression Equation from Final Step Y’ = (-.197*PA) + (-.256*NA) + (.014*NA*PA) Step 2. Moderator take mean, +1SD, -1SD PA Mean = PA +1SD = PA -1SD = Step 3. Insert points from last step to create 3 regression lines Mean (-.197*33.06)+(-.256*NA)+(.014* NA *33.06) +1SD (-.197*39.82)+ (-.256*NA)+(.014* NA *39.82) -1SD (-.197*26.30)+(-.256*NA)+(.014* NA * 26.30) Reduces: Y’ = (.207*NA) Y’ = (.557*NA) Y’ = (.368*NA)

Plot the following lines Y’ = (.207*NA) [Mean] Y’ = (.557*NA) [+1] Y’ = (.368*NA) [-1] Useful to choose several points on line Interpret Following up on the interaction

Presence of interactions qualifies the interpretation of main effects Presence of higher order interactions qualifies the interpretation of lower order interactions df are used up quickly as more potential interactions are added Interpreting interactions with more than 3 variables is very difficult Interaction Effects

Moderation/Mediation Models Testing mediation & moderation models together If Z is a categorical variable, can test model using multiple groups analysis in SEM If Z is continuous, can test model using special SEM models, but very difficult X M Y Z

Moderation/Mediation Models (1) Enter X, enter Z, enter XZ, enter M, enter MZ –If increment R 2 for neither XZ nor MZ significant, no evidence for moderation –If MZ significant, suggests moderated mediation (MM) –If MZ not significant after controlling for XZ, but XZ is signif, then suggests XZ has direct moderating effect (not mediated through MZ) (2) Enter M, enter Z, enter MZ, enter X, enter XZ –If MZ significant, and XZ was signif in step 1 but no longer signif here, suggests MM –If MZ not signif, no evidence for MM X M Y Z

Curvilinearity Curvilinearity can be considered moderation of a variable by itself Tested the same way as moderation Most of the same issues regarding moderation apply to curvilinearity Low High Y X Low X High X Medium X Low

Nonlinear relationships. The quadratic effect of Publications: Publications 2 This new variable would be tested after original variable, Publications, had been entered. Publications 2 is just the product of Publications with itself. It looks like any other product used to test an interaction. How can this variable be interpreted as an interaction? Non-linear regression

What would a significant quadratic effect of publications mean in addition to a significant linear effect?

A quadratic effect indicates that the linear relation between a variable and the outcome changes slope across levels of the variable.

REGRESSION /DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE /STATISTICS COEFF R CHA ANOVA COLLIN TOL /NOORIGIN /DEPENDENT y /METHOD=ENTER x /RESIDUALS /CASEWISE ALL ZRESID SRESID LEVER COOK /SCATTERPLOT=(*RESID,y) (*RESID, x) (*ZRESID,*ZPRED ) Syntax to Examine Residuals

Output from Syntax

Syntax Polynomial Regression REGRESSION /DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE /STATISTICS COEFF R CHA ANOVA COLLIN TOL /NOORIGIN /DEPENDENT y /ENTER x /ENTER x2 /RESIDUALS /CASEWISE ALL ZRESID SRESID LEVER COOK /SCATTERPLOT=(*RESID,y) (*RESID, x) (*ZRESID,*ZPRED ) Block Number 2. Method: Enter X2

Variable(s) Entered on Step Number 1.. X Multiple R R Square R Square Change Adjusted R Square F Change Standard Error Signif F Change.4291 Block Number 2. Method: Enter X2 Variable(s) Entered on Step Number 2.. X2 Multiple R R Square R Square Change Adjusted R Square F Change Standard Error Signif F Change.0000

Residuals with X2 in the equation