﻿figure it out - a statistical consultancy from the Institute of Work Psychology, University of Sheffield

## Mplus code for mediation, moderation, and moderated mediation models

Model 36: 1 or more mediators, in parallel if multiple (example uses 1), 3 moderators, 1 moderating the IV-Mediator path, 2 moderating both the Mediator-DV path and the IV-DV path

Example Variables: 1 predictor X, 1 mediator M, 3 moderators W, V, Q, 1 outcome Y

Preliminary notes:

The code below assumes that

• The primary IV (variable X) is continuous or dichotomous.
• Any moderators (variables W, V, Q, Z) are continuous, though the only adaptation required to handle dichotomous moderators is in the MODEL CONSTRAINT: and loop plot code - an example of how to do this is given in model 1b. Handling categorical moderators with > 2 categories is demonstrated in model 1d.
• Any mediators (variable M, or M1, M2, etc.) are continuous and satisfy the assumptions of standard multiple regression. An example of how to handle a dichotomous mediator is given in model 4c.
• The DV (variable Y) is continuous and satisfies the assumptions of standard multiple regression - an example of how to handle a dichotomous DV is given in model 1e (i.e. a moderated logistic regression) and in model 4d (i.e. an indirect effect in a logistic regression).

Model Diagram:

Statistical Diagram:

Model Equation(s):

Y = b0 + b1M + b2MV + b3MQ + c1'X + c2'V + c3'Q + c4'XV + c5'XQ
M = a0 + a1X + a2W + a3XW

Algebra to calculate indirect and/or conditional effects by writing model as Y = a + bX:

Y = b0 + b1M + b2MV + b3MQ + c1'X + c2'V + c3'Q + c4'XV + c5'XQ
M = a0 + a1X + a2W + a3XW

Hence... substituting in equation for M

Y = b0 + b1(a0 + a1X + a2W + a3XW) + b2(a0 + a1X + a2W + a3XW)V + b3(a0 + a1X + a2W + a3XW)Q + c1'X + c2'V + c3'Q + c4'XV + c5'XQ

Hence... multiplying out brackets

Y = b0 + a0b1 + a1b1X + a2b1W + a3b1XW + a0b2V + a1b2XV + a2b2WV + a3b2XWV + a0b3 + a1b3XQ + a2b3WQ + a3b3XWQ + c1'X + c2'V + c3'Q + c4'XV + c5'XQ

Hence... grouping terms into form Y = a + bX

Y = (b0 + a0b1 + a2b1W + a0b2V + a2b2WV + a0b3 + a2b3WQ + c2'V + c3'Q) + (a1b1 + a3b1W + a1b2V + a3b2WV + a1b3Q + a3b3WQ + c1' + c4'V + c5'Q)X

Hence...

One indirect effect(s) of X on Y, conditional on W, V, Q:

a1b1 + a3b1W + a1b2V + a3b2WV + a1b3Q + a3b3WQ = (a1 + a3W) (b1 + b2V + b3Q)

One direct effect of X on Y, conditional on V, Q:

c1' + c4'V + c5'Q

Mplus code for the model:

! Predictor variable - X
! Mediator variable(s) – M
! Moderator variable(s) – W, V, Q
! Outcome variable - Y

USEVARIABLES = X M W V Q Y XW XV XQ MV MQ;

! Create interaction terms
! Note that they have to be placed at end of USEVARIABLES subcommand above

DEFINE:
MQ = M*Q;
MV = M*V;
XW = X*W;
XQ = X*Q;
XV = X*V;

ANALYSIS:
TYPE = GENERAL;
ESTIMATOR = ML;
BOOTSTRAP = 10000;

! In model statement name each path and intercept using parentheses

MODEL:
[Y] (b0);
Y ON M (b1);
Y ON MV (b2);
Y ON MQ (b3);

Y ON X (cdash1);
Y ON V (cdash2);
Y ON Q (cdash3);
Y ON XV (cdash4);
Y ON XQ (cdash5);

[M] (a0);
M ON X (a1);
M ON W (a2);
M ON XW (a3);

! Use model constraint subcommand to test conditional indirect effects
! You need to pick low, medium and high moderator values for W, V, Q
! for example, of 1 SD below mean, mean, 1 SD above mean

! 3 moderators, 3 values for each, gives 27 combinations
! arbitrary naming convention for conditional indirect and total effects used below:
! HWMVLQ = high value of W, medium value of V and low value of Q, etc.

MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W LOW_V MED_V HIGH_V LOW_Q MED_Q HIGH_Q
ILWLVLQ IMWLVLQ IHWLVLQ ILWMVLQ IMWMVLQ IHWMVLQ
ILWHVLQ IMWHVLQ IHWHVLQ
ILWLVMQ IMWLVMQ IHWLVMQ ILWMVMQ IMWMVMQ IHWMVMQ
ILWHVMQ IMWHVMQ IHWHVMQ
ILWLVHQ IMWLVHQ IHWLVHQ ILWMVHQ IMWMVHQ IHWMVHQ
ILWHVHQ IMWHVHQ IHWHVHQ
DLOV_LOQ DMEV_LOQ DHIV_LOQ DLOV_MEQ DMEV_MEQ DHIV_MEQ
DLOV_HIQ DMEV_HIQ DHIV_HIQ
TLWLVLQ TMWLVLQ THWLVLQ TLWMVLQ TMWMVLQ THWMVLQ
TLWHVLQ TMWHVLQ THWHVLQ
TLWLVMQ TMWLVMQ THWLVMQ TLWMVMQ TMWMVMQ THWMVMQ
TLWHVMQ TMWHVMQ THWHVMQ
TLWLVHQ TMWLVHQ THWLVHQ TLWMVHQ TMWMVHQ THWMVHQ
TLWHVHQ TMWHVHQ THWHVHQ);

LOW_W = #LOWW;   ! replace #LOWW in the code with your chosen low value of W
MED_W = #MEDW;   ! replace #MEDW in the code with your chosen medium value of W
HIGH_W = #HIGHW;   ! replace #HIGHW in the code with your chosen high value of W

LOW_V = #LOWV;   ! replace #LOWV in the code with your chosen low value of V
MED_V = #MEDV;   ! replace #MEDV in the code with your chosen medium value of V
HIGH_V = #HIGHV;   ! replace #HIGHV in the code with your chosen high value of V

LOW_Q = #LOWQ;   ! replace #LOWQ in the code with your chosen low value of Q
MED_Q = #MEDQ;   ! replace #MEDQ in the code with your chosen medium value of Q
HIGH_Q = #HIGHQ;   ! replace #HIGHQ in the code with your chosen high value of Q

! Calc conditional indirect effects for each combination of moderator values

ILWLVLQ = a1*b1 + a3*b1*LOW_W + a1*b2*LOW_V + a3*b2*LOW_W*LOW_V +
a1*b3*LOW_Q + a3*b3*LOW_W*LOW_Q;
IMWLVLQ = a1*b1 + a3*b1*MED_W + a1*b2*LOW_V + a3*b2*MED_W*LOW_V +
a1*b3*LOW_Q + a3*b3*MED_W*LOW_Q;
IHWLVLQ = a1*b1 + a3*b1*HIGH_W + a1*b2*LOW_V + a3*b2*HIGH_W*LOW_V +
a1*b3*LOW_Q + a3*b3*HIGH_W*LOW_Q;

ILWMVLQ = a1*b1 + a3*b1*LOW_W + a1*b2*MED_V + a3*b2*LOW_W*MED_V +
a1*b3*LOW_Q + a3*b3*LOW_W*LOW_Q;
IMWMVLQ = a1*b1 + a3*b1*MED_W + a1*b2*MED_V + a3*b2*MED_W*MED_V +
a1*b3*LOW_Q + a3*b3*MED_W*LOW_Q;
IHWMVLQ = a1*b1 + a3*b1*HIGH_W + a1*b2*MED_V + a3*b2*HIGH_W*MED_V +
a1*b3*LOW_Q + a3*b3*HIGH_W*LOW_Q;

ILWHVLQ = a1*b1 + a3*b1*LOW_W + a1*b2*HIGH_V + a3*b2*LOW_W*HIGH_V +
a1*b3*LOW_Q + a3*b3*LOW_W*LOW_Q;
IMWHVLQ = a1*b1 + a3*b1*MED_W + a1*b2*HIGH_V + a3*b2*MED_W*HIGH_V +
a1*b3*LOW_Q + a3*b3*MED_W*LOW_Q;
IHWHVLQ = a1*b1 + a3*b1*HIGH_W + a1*b2*HIGH_V + a3*b2*HIGH_W*HIGH_V +
a1*b3*LOW_Q + a3*b3*HIGH_W*LOW_Q;

ILWLVMQ = a1*b1 + a3*b1*LOW_W + a1*b2*LOW_V + a3*b2*LOW_W*LOW_V +
a1*b3*MED_Q + a3*b3*LOW_W*MED_Q;
IMWLVMQ = a1*b1 + a3*b1*MED_W + a1*b2*LOW_V + a3*b2*MED_W*LOW_V +
a1*b3*MED_Q + a3*b3*MED_W*MED_Q;
IHWLVMQ = a1*b1 + a3*b1*HIGH_W + a1*b2*LOW_V + a3*b2*HIGH_W*LOW_V +
a1*b3*MED_Q + a3*b3*HIGH_W*MED_Q;

ILWMVMQ = a1*b1 + a3*b1*LOW_W + a1*b2*MED_V + a3*b2*LOW_W*MED_V +
a1*b3*MED_Q + a3*b3*LOW_W*MED_Q;
IMWMVMQ = a1*b1 + a3*b1*MED_W + a1*b2*MED_V + a3*b2*MED_W*MED_V +
a1*b3*MED_Q + a3*b3*MED_W*MED_Q;
IHWMVMQ = a1*b1 + a3*b1*HIGH_W + a1*b2*MED_V + a3*b2*HIGH_W*MED_V +
a1*b3*MED_Q + a3*b3*HIGH_W*MED_Q;

ILWHVMQ = a1*b1 + a3*b1*LOW_W + a1*b2*HIGH_V + a3*b2*LOW_W*HIGH_V +
a1*b3*MED_Q + a3*b3*LOW_W*MED_Q;
IMWHVMQ = a1*b1 + a3*b1*MED_W + a1*b2*HIGH_V + a3*b2*MED_W*HIGH_V +
a1*b3*MED_Q + a3*b3*MED_W*MED_Q;
IHWHVMQ = a1*b1 + a3*b1*HIGH_W + a1*b2*HIGH_V + a3*b2*HIGH_W*HIGH_V +
a1*b3*MED_Q + a3*b3*HIGH_W*MED_Q;

ILWLVHQ = a1*b1 + a3*b1*LOW_W + a1*b2*LOW_V + a3*b2*LOW_W*LOW_V +
a1*b3*HIGH_Q + a3*b3*LOW_W*HIGH_Q;
IMWLVHQ = a1*b1 + a3*b1*MED_W + a1*b2*LOW_V + a3*b2*MED_W*LOW_V +
a1*b3*HIGH_Q + a3*b3*MED_W*HIGH_Q;
IHWLVHQ = a1*b1 + a3*b1*HIGH_W + a1*b2*LOW_V + a3*b2*HIGH_W*LOW_V +
a1*b3*HIGH_Q + a3*b3*HIGH_W*HIGH_Q;

ILWMVHQ = a1*b1 + a3*b1*LOW_W + a1*b2*MED_V + a3*b2*LOW_W*MED_V +
a1*b3*HIGH_Q + a3*b3*LOW_W*HIGH_Q;
IMWMVHQ = a1*b1 + a3*b1*MED_W + a1*b2*MED_V + a3*b2*MED_W*MED_V +
a1*b3*HIGH_Q + a3*b3*MED_W*HIGH_Q;
IHWMVHQ = a1*b1 + a3*b1*HIGH_W + a1*b2*MED_V + a3*b2*HIGH_W*MED_V +
a1*b3*HIGH_Q + a3*b3*HIGH_W*HIGH_Q;

ILWHVHQ = a1*b1 + a3*b1*LOW_W + a1*b2*HIGH_V + a3*b2*LOW_W*HIGH_V +
a1*b3*HIGH_Q + a3*b3*LOW_W*HIGH_Q;
IMWHVHQ = a1*b1 + a3*b1*MED_W + a1*b2*HIGH_V + a3*b2*MED_W*HIGH_V +
a1*b3*HIGH_Q + a3*b3*MED_W*HIGH_Q;
IHWHVHQ = a1*b1 + a3*b1*HIGH_W + a1*b2*HIGH_V + a3*b2*HIGH_W*HIGH_V +
a1*b3*HIGH_Q + a3*b3*HIGH_W*HIGH_Q;

! Calc conditional direct effects for each combination of moderator values

DLOV_LOQ = cdash1 + cdash4*LOW_V + cdash5*LOW_Q;
DMEV_LOQ = cdash1 + cdash4*MED_V + cdash5*LOW_Q;
DHIV_LOQ = cdash1 + cdash4*HIGH_V + cdash5*LOW_Q;

DLOV_MEQ = cdash1 + cdash4*LOW_V + cdash5*MED_Q;
DMEV_MEQ = cdash1 + cdash4*MED_V + cdash5*MED_Q;
DHIV_MEQ = cdash1 + cdash4*HIGH_V + cdash5*MED_Q;

DLOV_HIQ = cdash1 + cdash4*LOW_V + cdash5*HIGH_Q;
DMEV_HIQ = cdash1 + cdash4*MED_V + cdash5*HIGH_Q;
DHIV_HIQ = cdash1 + cdash4*HIGH_V + cdash5*HIGH_Q;

! Calc conditional total effects for each combination of moderator values

TLWLVLQ = ILWLVLQ + DLOV_LOQ;
TMWLVLQ = IMWLVLQ + DLOV_LOQ;
THWLVLQ = IHWLVLQ + DLOV_LOQ;

TLWMVLQ = ILWMVLQ + DMEV_LOQ;
TMWMVLQ = IMWMVLQ + DMEV_LOQ;
THWMVLQ = IHWMVLQ + DMEV_LOQ;

TLWHVLQ = ILWHVLQ + DHIV_LOQ;
TMWHVLQ = IMWHVLQ + DHIV_LOQ;
THWHVLQ = IHWHVLQ + DHIV_LOQ;

TLWLVMQ = ILWLVMQ + DLOV_MEQ;
TMWLVMQ = IMWLVMQ + DLOV_MEQ;
THWLVMQ = IHWLVMQ + DLOV_MEQ;

TLWMVMQ = ILWMVMQ + DMEV_MEQ;
TMWMVMQ = IMWMVMQ + DMEV_MEQ;
THWMVMQ = IHWMVMQ + DMEV_MEQ;

TLWHVMQ = ILWHVMQ + DHIV_MEQ;
TMWHVMQ = IMWHVMQ + DHIV_MEQ;
THWHVMQ = IHWHVMQ + DHIV_MEQ;

TLWLVHQ = ILWLVHQ + DLOV_HIQ;
TMWLVHQ = IMWLVHQ + DLOV_HIQ;
THWLVHQ = IHWLVHQ + DLOV_HIQ;

TLWMVHQ = ILWMVHQ + DMEV_HIQ;
TMWMVHQ = IMWMVHQ + DMEV_HIQ;
THWMVHQ = IHWMVHQ + DMEV_HIQ;

TLWHVHQ = ILWHVHQ + DHIV_HIQ;
TMWHVHQ = IMWHVHQ + DHIV_HIQ;
THWHVHQ = IHWHVHQ + DHIV_HIQ;

! Use loop plot to plot conditional indirect effect of X on Y for each combination of low, med, high moderator values
! Could be edited to show conditional direct or conditional total effects instead
! NOTE - values of 1,5 in LOOP() statement need to be replaced by
! logical min and max limits of predictor X used in analysis

PLOT(PLWLVLQ PMWLVLQ PHWLVLQ PLWMVLQ PMWMVLQ PHWMVLQ
PLWHVLQ PMWHVLQ PHWHVLQ
PLWLVMQ PMWLVMQ PHWLVMQ PLWMVMQ PMWMVMQ PHWMVMQ
PLWHVMQ PMWHVMQ PHWHVMQ
PLWLVHQ PMWLVHQ PHWLVHQ PLWMVHQ PMWMVHQ PHWMVHQ
PLWHVHQ PMWHVHQ PHWHVHQ);

LOOP(XVAL,1,5,0.1);

PLWLVLQ = ILWLVLQ*XVAL;
PMWLVLQ = IMWLVLQ*XVAL;
PHWLVLQ = IHWLVLQ*XVAL;

PLWMVLQ = ILWMVLQ*XVAL;
PMWMVLQ = IMWMVLQ*XVAL;
PHWMVLQ = IHWMVLQ*XVAL;

PLWHVLQ = ILWHVLQ*XVAL;
PMWHVLQ = IMWHVLQ*XVAL;
PHWHVLQ = IHWHVLQ*XVAL;

PLWLVMQ = ILWLVMQ*XVAL;
PMWLVMQ = IMWLVMQ*XVAL;
PHWLVMQ = IHWLVMQ*XVAL;

PLWMVMQ = ILWMVMQ*XVAL;
PMWMVMQ = IMWMVMQ*XVAL;
PHWMVMQ = IHWMVMQ*XVAL;

PLWHVMQ = ILWHVMQ*XVAL;
PMWHVMQ = IMWHVMQ*XVAL;
PHWHVMQ = IHWHVMQ*XVAL;

PLWLVHQ = ILWLVHQ*XVAL;
PMWLVHQ = IMWLVHQ*XVAL;
PHWLVHQ = IHWLVHQ*XVAL;

PLWMVHQ = ILWMVHQ*XVAL;
PMWMVHQ = IMWMVHQ*XVAL;
PHWMVHQ = IHWMVHQ*XVAL;

PLWHVHQ = ILWHVHQ*XVAL;
PMWHVHQ = IMWHVHQ*XVAL;
PHWHVHQ = IHWHVHQ*XVAL;

PLOT:
TYPE = plot2;

OUTPUT:
STAND CINT(bcbootstrap);