Comparing Conditional Effects in Moderated Multiple Regression: Implementation using PROCESS for SPSS and SAS

Transcription

1 Comparing Conditional Effects in Moderated Multiple Regression: Implementation using PROCESS for SPSS and SAS Andrew F. Hayes The Ohio State University Department of Psychology This document describes a method for testing the difference between any two conditional effects of X on Y in a moderated multiple regression model. It is based in part on the approach outlined by Dawson and Richter (2006) in the Journal of Applied Psychology. It works for continuous or dichotomous moderators in any combination. Mean centering or standardization is not required, although substituting mean centered or standardized values in the formulas below does not harm or alter the derivation and discussion. I conclude with instructions for implementation in PROCESS for SPSS and SAS as of version Version 2.12 is scheduled for release toward the end of May Moderation of X s Effect by a Single Moderator M Consider the model In this model, the conditional effect of X on Y is Y = i Y + b 1 X + b 2 M + b 3 XM + e Y X Y = b 1 + b 3 M We wish to compare the conditional effect of X on Y when M = m 2 to the conditional effect of X on Y when M = m 1. The difference between these conditional effects is X Y = X Y m2 X Y m1 = b 3 (m 2 m 1 ) The sampling variance of X Y is V( X Y ) = (m 2 m 1 ) 2 V(b 3 ) The ratio of X Y to the square root of V( X Y ) is distributed as t(df residual ) under the null hypothesis that the two conditional effects are equal, where df residual is the residual degrees of freedom for the model. But this ratio contains (m 2 m 1 ) in both the numerator and the denominator. These cancel each other and so the ratio simplifies to b 3 / V(b 3 ), which is the t-ratio for b 3 from the regression analysis. Thus, an inference that M moderates the effect of X on Y with a test of significance for the regression coefficient for XM means that any two conditional effects of X defined by different values of M are significantly different from each other, with the same p-value as the p- value for b 3. Conversely, a failure of M to moderate X s effect from the test of the regression 1

2 coefficient for XM implies that no two conditional effects of X defined by different values of M differ from each other. As no test is needed to compare conditional effects in this model, this model is not discussed further in this document. Moderation of X s Effect Additively by Two Moderators M and W Next consider the model In this model, the conditional effect of X on Y is Y = i Y + b 1 X + b 2 M + b 3 W + b 4 XM + b 5 XW + e Y X Y = b 1 + b 4 M + b 5 W We wish to compare the conditional effect of X on Y when M and W = m 2 and w 2 to the conditional effect of X on Y when M and W = m 1 and w 1. The difference between these conditional effects is The sampling variance of X Y is X Y = X Y (m2,w2) X Y (m1,w1) = b 4 (m 2 m 1 ) + b 5 (w 2 w 1 ) V( X Y ) = (m 2 m 1 ) 2 V(b 4 ) + (w 2 w 1 ) 2 V(b 5 ) + 2(m 2 m 1 )(w 2 w 1 )COV(b 4 b 5 ) The ratio of X Y to the square root of V( X Y ) is distributed as t(df residual ) under the null hypothesis that the two conditional effects are equal, where df residual is the residual degrees of freedom for the model. Two special cases are worth highlighting. If W is held fixed any value, such that w 1 = w 2 = w, then X Y = X Y (m2,w) X Y (m1,w) = b 4 (m 2 m 1 ) and the sampling variance of X Y is V( X Y ) = (m 2 m 1 ) 2 V(b 4 ) By the argument in the prior section, the ratio of the difference in conditional effects of X when W is held fixed to the standard error of this difference is equivalent to the t statistic for b 4. Thus, the inference about b 4 results in an equivalent inference for the difference between any two conditional effects of X defined by different values of M, regardless of the common value of W. By the same argument, when M is held fixed at any value such that m 1 = m 2 = m 2

3 and the sampling variance of X Y is just X Y = X Y (m,w2) X Y (m,w1) = b 5 (w 2 w 1 ) V( X Y ) = (w 2 w 1 ) 2 V(b 5 ) The ratio of the difference in conditional effects of X when M is held fixed to the standard error of this difference is equivalent to the t statistic for b 5. Thus, the inference about b 5 results in an equivalent inference for the difference between any two conditional effects of X defined by different values of W, regardless of the common value of M. Moderation of X s Effect Multiplicatively by Two Moderators M and W ( Moderated Moderation ) Finally, consider the model Y = i Y + b 1 X + b 2 M + b 3 W + b 4 XM + b 5 XW + b 6 MW + b 7 XMW + e Y In this model, the conditional effect of X on Y is X Y = b 1 + b 4 M + b 5 W + b 7 MW We wish to compare the conditional effect of X on Y when M and W = m 2 and w 2 to the conditional effect of X on Y when M and W = m 1 and w 1. The difference between these conditional effects is X Y = X Y (m2,w2) X Y (m1,w1) = b 4 (m 2 m 1 ) + b 5 (w 2 w 1 ) + b 7 (m 2 w 2 m 1 w 1 ) The sampling variance of X Y is (see the derivation at the end) V( X Y ) = (m 2 m 1 ) 2 V(b 4 ) + (w 2 w 1 ) 2 V(b 5 ) + (m 2 w 2 m 1 w 1 ) 2 V(b 7 ) + 2(m 2 m 1 )(w 2 w 1 )COV(b 4 b 5 ) + 2(m 2 m 1 )(m 2 w 2 m 1 w 1 )COV(b 4 b 7 ) + 2(w 2 w 1 )(m 2 w 2 m 1 w 1 )COV(b 5 b 7 ) The ratio of X Y to the square root of V( X Y ) is distributed as t(df residual ) under the null hypothesis that the two conditional effects are equal, where df residual is the residual degrees of freedom for the model. Implementation in PROCESS PROCESS v2.12 or higher (scheduled for release in May of 2014) can be used to conduct a test of difference between any two conditional effects of X on Y in additive multiple moderation 3

4 (PROCESS model 2) or moderated moderation (PROCESS model 3). To do so, first center M and W around m 1 and w 1, respectively. Then execute PROCESS model 2 or 3, specifying contrast=1 and using the mmodval and wmodval commands with mmodval = m and wmodval = w where m = m 2 m 1 and w = w 2 w 1. If you want to fix m 2 or w 2 to m 1 or w 1, then use mmodval = 0 or wmodval = 0, respectively. Because this procedure relies on the mmodval and wmodval options, it is not available in the custom dialog version of PROCESS for SPSS. The generic form of the code that accomplishes the analysis in SPSS is compute mvarc=mvar-m1. compute wvarc=wvar-w1. process vars=yvar mvarc wvarc xvar cvarlist/y=yvar/m=mvarc/w=wvarc/model=3/mmodval=mdiff/ /wmodval=wdiff/contrast=1. where m1 and w1 are m 1 and w 1, mdiff is a numerical argument set to m 2 m 1 and wdiff if a numerical argument set to w 2 w 1, mvar, wvar, xvar, and yvar are the variable names in the data corresponding to M, W, X, and Y, and cvarlist is an optional list of covariates. Model=3 can be replaced with model=2 if desired. The equivalent code in SAS in generic form is data datafile;set datafile;mvarc=mvar-m1;wvarc=wvar-w1;run; %process (data=datafile,vars=yvar mvarc wvarc xvar cvarlist,y=yvar,x=xvar,m=mvarc,w=wvarc, model=3,mmodval=mdiff,wmodval=wdiff,contrast=1);.3308/contrast=1. Following this procedure will override the defaults in PROCESS such that it will no longer print the conditional effects of X on Y for various combinations of M and W. Instead, PROCESS will generate the difference between conditional effects ( X Y = X Y (m2,w2) X Y (m1,w1) ), the standard error of the difference ( V( X Y ), their ratio as a t statistic, a p-value for testing the null hypothesis of no difference, and a confidence interval for the difference. I illustrate using the example from section 9.4 (pp ) of Introduction to Mediation, Moderation, and Conditional Process Analysis (Hayes, 2013). In this example, Y is support for government actions to mitigate the effects of global climate change (govact), X is negative emotions about climate change (negemot), M is sex (0 = female, 1 = male), and W is age. The model also includes two covariates: political ideology (ideology) and positive emotions about climate change (posemot). Of interest is the relationship between support for government action and negative emotions about climate change with age and sex as moderators. The model estimated includes a three-way interaction (PROCESS model 3) between X, M, and Y that is statistically significant. See Figure 9.4 in Hayes (2013) for PROCESS output. Because age is continuous and sex is dichotomous, PROCESS automatically produces conditional effects of negative emotions for each group among people relatively younger (1SD below the mean = years), moderate in age (the mean = years) and relatively older (1SD above the mean = years). The relevant output can be found below. 4

5 Conditional effect of X on Y at values of the moderator(s): age sex Effect se t p LLCI ULCI In this example, we test whether the relationship between negative emotions and support for government action differs between relatively younger women (m 1 = 0, w 1 = ; the first row above) and men who are moderate in age (m 2 = 1, w 2 = ; the fourth row above). As can be seen from the table of conditional effects above, X Y (m2 = 1, w2 = ) = and X Y (m1 = 0, w1 = ) = Their difference is X Y (m2,w2) X Y (m1,w1) = = The SPSS code below conducts an inferential test that the difference between these conditional effects of X is equal to zero against the alternative that it is different from zero. compute sexc=sex-0. compute agec=age process vars=govact negemot posemot agec sexc ideology/y=govact/x=negemot/m=sexc/w=agec/model=3 /mmodval=1/wmodval= /contrast=1. The equivalent command in PROCESS for SAS is data glbwarm;set glbwarm;sexc=sex-0;agec=age ;run; %process (data=glbwarm,vars=govact negemot posemot agec sexc ideology,y=govact,x=negemot, m=sexc,w=agec,model=3,mmodval=1,wmodval= ,contrast=1); This code first centers sex and age around m 1 = 0 and w 1 = , respectively. Obviously, centering sex around zero doesn t do anything to sex in this case. I include this line of code to illustrate how the centering is conducted in general. Using these centered moderators in the code and specifying mmodval = m 2 m 1 = 1 0 = 1 and wmodval = w 2 w 1 = = , along with the contrast=1 option, produces an inferential test of the difference between X Y (m2 = 1, w2 = ) and X Y (m1 = 0, w1 = ). Most of the output PROCESS generates is not pertinent to the test of interest. What is pertinent is the section that reads Contrast of conditional effects of X on Y Contrast se t p LLCI ULCI As can be seen, PROCESS says this difference of 0.215, with a standard error of 0.060, is statistically significant, t(805) = 3.616, p <.001, with a 95% confidence interval of to It is possible using PROCESS to generate the conditional effect of X on Y for any two combinations of moderators M and W you choose and then conduct a test of the difference between these conditional effects. You don t have to use the values that PROCESS picks for you by default. For instance, the SPSS code below estimates the conditional effect of negative 5

6 emotions on support for government action among 30 year old males (m 1 = 1, w 1 = 30) as well as among 50-year old females (m 2 = 0, w 2 = 50). process vars=govact negemot posemot age sex ideology/y=govact/x=negemot/m=sex/w=age/model=3 /mmodval=1/wmodval=30. process vars=govact negemot posemot age sex ideology/y=govact/x=negemot/m=sex/w=age/model=3 /mmodval=0/wmodval=50. In SAS, use In %process (data=glbwarm,vars=govact negemot posemot age sex ideology,y=govact,x=negemot, m=sex,w=age,model=3,mmodval=1,wmodval=30); %process (data=glbwarm,vars=govact negemot posemot age sex ideology,y=govact,x=negemot, m=sex,w=age,model=3,mmodval=0,wmodval=50); The relevant sections of output are below. Conditional effect of X on Y at values of the moderator(s): age sex Effect se t p LLCI ULCI Conditional effect of X on Y at values of the moderator(s): age sex Effect se t p LLCI ULCI The effect of negative emotions on support for government action is among 30 year-old males (m 1 = 1, w 1 = 30) and among 50 year-old females (m 1 = 0, w 2 = 50). Both of these are statistically different from zero. The difference between them is X Y (m2,w2) X Y (m1,w1) = = A statistical test of the between these two conditional effects is conducted in PROCESS with the code below. In SPSS, use compute sexc=sex-1. compute agec=age-30. process vars=govact negemot posemot agec sexc ideology/y=govact/x=negemot/m=sexc/w=agec/model=3 /mmodval=-1/wmodval=20/contrast=1. The equivalent code in SAS is data glbwarm;set glbwarm;sexc=sex-1;agec=age-30;run; %process (data=glbwarm,vars=govact negemot posemot agec sexc ideology,y=govact,x=negemot, m=sexc,w=agec,model=3,mmodval=-1,wmodval=20,contrast=1); This code first centers sex and age around m 1 = 1 and w 1 = 30, respectively. Using these centered moderators in the code and specifying mmodval = m 2 m 1 = 0 1 = -1 and wmodval = w 2 w 1 = = 20 along with the contrast=1 option produces a test of the difference between X Y (m2,w2) and X Y (m1,w1). The resulting output is Contrast of conditional effects of X on Y Contrast se t p LLCI ULCI This difference of has an estimated standard error of and is not statistically different from zero, t(805) = , p = 0.471, with a 95% confidence interval from to

7 References Dawson, J. F., & Richter, A. W. (2006). Probing three-way interactions in moderated multiple regression: Development and application of a slope difference test. Journal of Applied Psychology, 91, Hayes, A. F. (2013). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach. New York: The Guilford Press. 7

8 Derivation of the Variance of the Difference between Conditional Effects of X for the Moderated Moderation Model We seek the sampling variance of b 4 (m 2 m 1 ) + b 5 (w 2 w 1 ) + b 7 (m 2 w 2 m 1 w 1 ). We know from covariance algebra that V ( ax by cz) = a b c V ( X ) COVXY COVXZ COV V ( Y ) COV XY YZ COVXZ COV YZ V ( Z) a b c = a b c av ( X ) bcovxy ccovxz acovxy bv ( Y ) ccovyz acov bcov cv ( Z) XZ YZ = 2 a V ( X ) abcov abcov accov XY XZ XY 2 b V ( Y ) bccov bccov YZ accov YZ XZ 2 c V ( Z) = a 2 V(X) + b 2 V(Y) + c 2 V(Z) + 2abCOV XY + 2acCOV XZ + 2bcCOV YZ Substitution of values below into the above yields the desired variance. X = b 4 Y = b 5 Z = b 7 a = (m 2 m 1 ) b = (w 2 w 1 ) c = (m 2 w 2 m 1 w 1 ) 8

9 Note: If one were to center M and W around m 1 and w 1 prior to model estimation and express the conditioning on the centered metric, such that one of the conditional effects is therefore X Y (0,0) the expressions simplify to X Y = b 4 m 2 + b 5 w 2 V( X Y ) = m 2 2 V(b 4 ) + w 2 2 V(b 5 ) + 2m 2 w 2 COV(b 4 b 5 ) ===== Note: If one were to center M and W around m 1 and w 1 prior to model estimation and express the conditioning on the centered metric, such that one of the conditional effects is therefore X Y (0,0) the expressions simplify to X Y = b 4 m 2 + b 5 w 2 + b 7 m 2 w 2 V( X Y ) = m 2 2 V(b 4 ) + w 2 2 V(b 5 ) + (m 2 w 2 ) 2 V(b 7 ) + 2m 2 w 2 COV(b 4 b 5 ) + 2m 2 2 w 2 COV(b 4 b 7 ) + 2m 2 w 2 2 COV(b 5 b 7 ) PROCESS v or higher can be used to conduct the test. First center M and W around m 1 and w 1, respectively. Then execute PROCESS model 3, specifying contrast=1 and using the mmodval and wmodval commands with mmodval = m and wmodval = w where m = m 2 m 1 and w = w 2 w 1. What PROCESS displays as the conditional effect of X on Y along with its test of significance will actually be the difference between the two conditional effects desired and a test that their difference equals zero along with a confidence interval for the difference. 9