Moderator and Mediator Analysis

Moderator and Mediator Analysis Seminar General Statistics Marijtje van Duijn October 8, Overview What is moderation and mediation? What is their relation to statistical concepts? Example(s) October 8, Moderator and mediator analysis

Mediation and Moderation X Y X October 8, Moderator and mediator analysis 3 Examples Y : test score X : sex, SES, etc. X : ability (IQ score) Y : test score X : brain volume, SES parents X : ability More? October 8, Moderator and mediator analysis 4

i Multiple regression Y + = β + βx i + βx i Goal: to explain variation in Y using Xs Assumptions Independent observations Normality (of residuals) and constant variance Linearity (of relationship Y and Xs) E i October 8, Moderator and mediator analysis 5 Regression X β Y E X β October 8, Moderator and mediator analysis 6 3

Explained variance in regression Y X X Circles represent variances X and X explain different parts of Y and are independent October 8, Moderator and mediator analysis 7 Multicollinearity competition between variables for explaining Y Y X X Degree depends on correlation between X s Variance inflation factor (VIF): worse (= less precise) parameter estimation October 8, Moderator and mediator analysis 8 4

Association between all (3) variables Several correlations to describe association Partial correlation: correlation between two variables with the third variable fixed (i.e. corrected for the third variable) (partial in SPSS) unique portion of variance Semi-partial correlation: correlation between two variables with one variable s association corrected for the third variable (part in SPSS output) extra variance explained Important in regression when variables are entered in a certain order Usually the partial correlation is smaller than the uncorrected ( zero-order ) correlation, and larger than the semi-partial correlation October 8, Moderator and mediator analysis 9 Multicollinearity/Mediation X Y X October 8, Moderator and mediator analysis 5

Mediation as a special case of multicollinearity and/or model selection Causal model defines direction of arrows X is the mediator (M) Also: intervening or process variable Or: indirect causal relationship Relations between all variables are assumed to be positive Question is whether direct effect between X and Y disappears when M is added to the regression equation October 8, Moderator and mediator analysis Mediation (Baron & Kenny, 986), http://davidakenny.net/cm/mediate.htm) X c c Y a M b October 8, Moderator and mediator analysis 6

Mediation as prescribed by Baron and Kenny (986) Estimate regression of Y on only X Estimated parameter c Estimate regression of M on X Estimated parameter a Estimate effect of M on Y, together with X Estimated parameters b and c Complete mediation: c = Partial mediation: c < c (can be tested) October 8, Moderator and mediator analysis 3 Testing of change in c Amount of mediation c-c Theoretically equal to ab (indirect path) Standard error of ab is approximately square root of b s a + a s b (Sobel test) see (do) http://quantpsy.org/sobel/sobel.htm Note: neither c nor c are needed! Some eye-balling also possible Or: nonparametric tests (based on bootstrapping) October 8, Moderator and mediator analysis 4 7

Example (Miles and Shevlin) Y: read, a measure of the number of books that people have read. X: enjoy, scale score to measure how much people enjoy reading books M: buy, a measure of how many books people have bought in the previous months Idea: how much people enjoy reading books -> the number of books bought -> the number of books read But: is the number of books a complete mediator.? (People could go to the library or borrow books from friends.) October 8, Moderator and mediator analysis 5 Descriptive Statistics Std. Mean Deviation N read 8,85 3,563 4 buy 5,73 8,65 4 enjoy 9,8 5,354 4 Pearson Correlation Correlations read buy enjoy read,,747,73 buy,747,,644 enjoy,73,644, Step a Unstandardized Standardized B Std. Error Beta t Sig. (Constant) 4,33,785 5,57, enjoy,487,74,73 6,65, a. Dependent Variable: read October 8, Moderator and mediator analysis 6 8

Step a Unstandardized Standardiz ed B Std. Error Beta t Sig. (Constant) 6,66, 3,74, enjoy,98,89,644 5,9, a. Dependent Variable: buy Step 3 Unstandardized a Standardized B Std. Error Beta t Sig. (Constant),973,765 3,887, Correlations Zeroorder Partial Part buy,5,54,47 3,786,,747,58,36 enjoy,86,83,49 3,45,,73,494,38 a. Dependent Variable: read October 8, Moderator and mediator analysis 7 Examples Baron/Kenny + Sobel a=.98 s a =.89 b=.487 s b =.74 ab=.478, s ab = (.487 *.89 +.98 *.74 )=. z-test 4.8; p<. Eye-balling c=.487 (.74), then roughly.34 < c <.64 c =.86 (.83): partial mediation (because. < c <.45) October 8, Moderator and mediator analysis 8 9

choice is important Many other patterns of association are possible Arrows between X and X may be reversed not always clear which variable mediates No causal relation, just association Explicit assumption of positive associations and ordering of (semi-) partial correlations. Not guaranteed October 8, Moderator and mediator analysis 9 Moderation X Y X October 8, Moderator and mediator analysis

Moderation is interaction The effect of X depends on the value of X (or vice versa) different slopes for different folks can be a way to tackle non-linearity in regression Interpretation depends on type of variable Interaction usually (but not necessarily) product of X and X X and X may be correlated Importance of model formulation October 8, Moderator and mediator analysis Regression model i = β + βx i + βx i + β3x i X i Y + E Centering is (usually) advised Facilitates interpretation Reduces the inevitable multicollinearity incurred with interaction terms c c c c i = β + βx i + βx i + β3x i X i Y + E i i October 8, Moderator and mediator analysis

X dichotomous and X continuous X = or X = (e.g. man/woman; control/experimental group) Y = β + β X + β X + β 3 X X X = : Y = β + β X X = : Y = (β + β ) + (β + β 3 )X so: intercept and regressioneffect of X change interaction effect represents the change in effect of X or, the difference in the effect of X between the two groups interpretation of β General formula: Y = (β + β X ) + (β + β 3 X )X October 8, Moderator and mediator analysis 3 X and X dichotomous X = or X = (e.g. man/woman) X = or X = (e.g. control/experimental grp) Y = β + β X + β X + β 3 X X X =, X = : Y = β X =, X = : Y = β + β X =, X = : Y = β + β X =, X = : Y = β + β + β + β 3 so: defines four groups with their own mean Interaction defines extra effect of X = and X = October 8, Moderator and mediator analysis 4

X nominal and X continuous X takes on more than (c) values (e.g. age groups, control/exp/exp) Make dummies, for each contrast, wrt reference group (e.g. controls) Leads to c- dichotomous variables And also c- interaction terms Also possible: dummies (indicators) for each group leaving out the intercept October 8, Moderator and mediator analysis 5 c=3; group reference group group D D 3 Y = β + β d D + β d D + β X +β 3 D X +β 4 D X group: Y = β + β X group: Y = (β + β d ) + (β + β 3 )X group3: Y = (β + β d ) + (β + β 4 )X October 8, Moderator and mediator analysis 6 3

X and X continuous Y = β + β X +β X +β 3 X X Y = (β + β X ) + (β +β 3 X )X Y = (β +β X ) + (β +β 3 X )X X X c c = X X = X c c * β = β β X β X + β 3 X X * β = β β 3 X * β = β β 3 X X * * Y = β + β X * + β X + β X 3 c X c X X = X = X : Y : Y = β * = β * + β * + β * X X c c October 8, Moderator and mediator analysis 7 Example (Miles and Shevlin) Y: grade in statistics course X: number of books read (-4) X: number of classes attended (-) Descriptive Statistics Mean Std. Deviation N grade 63,55 6,755 4 books,,43 4 attend 4, 4,78 4 October 8, Moderator and mediator analysis 8 4

October 8, Moderator and mediator analysis 9 Unstandardized Standardiz ed Coefficient s a B Std. Error Beta t Sig. (Constant) 63,4,3 8,534, Correlations Zeroorder Partial Part Collinearity Statistics Toleran ce VIF booksc 4,37,753,346,33,7,49,354,3,83,45 attendc,83,587,39,87,35,48,338,95,83,45 (Constant) 6,469,3 6,494, booksc 4,8,677,35,433,,49,376,34,83,45 attendc,333,56,34,37,3,48,368,36,8,47 bookscxatte ndc a. Dependent Variable: grade,735,349,7,4,4,4,33,7,997,3 October 8, Moderator and mediator analysis 3 5

Other example (missed) interaction: non-linearity? 3 groups with each a distinct linear relation October 8, Moderator and mediator analysis 3 October 8, Moderator and mediator analysis 3 6

October 8, Moderator and mediator analysis 33 Summary b Adjusted Std. Error of R R Square R Square the Estimate.45 a.64.34.5778 a. Predictors: (Constant), x b. Dependent Variable: y Predicted Value Residual Std. Predicted Value Std. Residual a. Dependent Variable: y Residuals Statistics a Minimum Maximum Mean Std. Deviation N 5.8864 9.797 7.8667.4 3-4.88636 3.646..5397 3 -.767.79.. 3 -.896.6..983 3 (Constant) x a. Dependent Variable: y Unstandardized Standardized a Correlations t Sig. Zero-order Partial Part Collinearity Statistics VIF B Std. Error Beta Tolerance 5.74.9 5.56..63.7.45.34.7.45.45.45.. Histogram Normal P-P Plot of Regression Standardized Residual Dependent Variable: y Dependent Variable: y, 6 Frequency 4 Expected Cum Prob,8,6,4, - Regression Standardized Residual Mean =-,78E-7 Std. Dev. =,983 N =3,,,,4,6,8 Observed Cum Prob, October 8, Moderator and mediator analysis 34 7

y Summary b Adjusted Std. Error of R R Square R Square the Estimate.98 a.96.957.57477 a. Predictors: (Constant), xkwadraat, x b. Dependent Variable: y Predicted Value Residual Std. Predicted Value Std. Residual a. Dependent Variable: y Residuals Statistics a Minimum Maximum Mean Std. Deviation N.5888.44 7.8667.736 3 -.8773.69..5546 3 -.68.949.. 3 -.44.34..965 3 (Constant) x xkwadraat a. Dependent Variable: y Unstandardized Standardized a Correlations t Sig. Zero-order Partial Part Collinearity Statistics VIF B Std. Error Beta Tolerance 8.5.59 3.839..4.6.349 9.4..45.867.348.996.4 -.54. -.894-3.56. -.96 -.976 -.89.996.4 Histogram Partial Regression Plot Dependent Variable: y Dependent Variable: y 6, Frequency 4, - Regression Standardized Residual Mean =-3,6E-6 Std. Dev. =,965 N =3 -, -5, -, -5,, x 5,, 5, October 8, Moderator and mediator analysis 35 Summary b Adjusted Std. Error of R R Square R Square the Estimate.4 a.68.7.6796 a. Predictors: (Constant), groep, x b. Dependent Variable: y Predicted Value Residual Std. Predicted Value Std. Residual a. Dependent Variable: y Residuals Statistics a Minimum Maximum Mean Std. Deviation N 5.73 9.9467 7.8667.3588 3-4.733 3.77..567 3 -.896.83.. 3 -.8...965 3 (Constant) x groep a. Dependent Variable: y Unstandardized Standardized a Correlations t Sig. Zero-order Partial Part Collinearity Statistics VIF B Std. Error Beta Tolerance 6..3 4.69...66.548.39.95.45.48.33.8 5.55 -.58.375 -.58 -.384.74.337 -.74 -.67.8 5.55 Partial Regression Plot Histogram Dependent Variable: y Dependent Variable: y 6,5 Frequency 4 y, -,5 - Mean =5,69E-6 Std. Dev. =,965 N =3-5, Regression Standardized Residual -4, -,, x, 4, October 8, Moderator and mediator analysis 36 8

y y Summary b Adjusted Std. Error of R R Square R Square the Estimate.75 a.56.5.935 a. Predictors: (Constant), groep3, groep, x b. Dependent Variable: y Predicted Value Residual Std. Predicted Value Std. Residual a. Dependent Variable: y Residuals Statistics a Minimum Maximum Mean Std. Deviation N 4.773.7 7.8667.779 3-3.777 3.777..83 3 -.5.568.. 3 -.96.96..947 3 (Constant) x groep groep3 a. Dependent Variable: y Unstandardized Standardized a Correlations t Sig. Zero-order Partial Part Collinearity Statistics VIF B Std. Error Beta Tolerance 4.556.9 4.994..7.3.47.396.74.45.64.8.8 5.55 3.476.3.6.653.3.645.46.344.37 3.57 -.36.38 -.56 -.6.874 -.3 -.3 -..35 7.394 Partial Regression Plot Histogram Dependent Variable: y Dependent Variable: y 4, 8, Frequency 6 4, -, -4, - Regression Standardized Residual Mean =3,47E-6 Std. Dev. =,947 N =3-4, -,, x, 4, October 8, Moderator and mediator analysis 37 Summary b Adjusted Std. Error of R R Square R Square the Estimate.997 a.993.99.55 a. Predictors: (Constant), intxgr3, intxgr, x, groep, groep3 b. Dependent Variable: y Predicted Value Residual Std. Predicted Value Std. Residual a. Dependent Variable: y Residuals Statistics a Minimum Maximum Mean Std. Deviation N..48 7.8667.763 3 -.488.65758..789 3 -.488.94.. 3 -.669.65..9 3 (Constant) x groep groep3 intxgr intxgr3 a. Dependent Variable: y Unstandardized Standardized a Correlations t Sig. Zero-order Partial Part Collinearity Statistics VIF B Std. Error Beta Tolerance 4.97E-4.7....8.485 36.59..45.99.69.6 6.657.45.47.756 4.39..645.98.48.54 8.55 8..596 3.6 3.. -.3.987.57.6 37.737 -.985.39 -.378-5.5..66 -.98 -.44.3 3.455 -.5.39-5.4-38.458. -.8 -.99 -.646.4 69.99 Histogram Partial Regression Plot Dependent Variable: y Dependent Variable: y 4,, Frequency 8 6 4, -, - Mean =,8E-5 Std. Dev. =,9 N =3-4, Regression Standardized Residual -4, -,,, 4, x October 8, Moderator and mediator analysis 38 9

choice important Missing interaction may result in Violations of linearity assumption non-constant variance (heterogeneity) Incorrect model choice and interpretation October 8, Moderator and mediator analysis 39 Conclusion choice and selection crucial in detecting mediation and moderation Substantive / theoretical considerations should guide the model selection process! October 8, Moderator and mediator analysis 4

Other methods/models Mediation sometimes too simple More refined path analysis Regression analysis sometimes too simple More elaborate models for causal modeling Structural Equation s October 8, Moderator and mediator analysis 4