﻿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 44: 1 or more mediators, in parallel if multiple (example uses 1), 3 moderators, 1 moderating both the IV- Mediator path and the direct IV-DV path, 2 moderating the Mediator-DV path with all 2-way and 3-way interactions, 1 of which also moderates the direct 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 + b2Q + b3MV + b4MQ + b5VQ + b6MVQ + c1'X + c2'W + c3'XW + c4'V + c5'XV
M = a0 + a1X + a2W + a3XW

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

Y = b0 + b1M + b2Q + b3MV + b4MQ + b5VQ + b6MVQ + c1'X + c2'W + c3'XW + c4'V + c5'XV
M = a0 + a1X + a2W + a3XW

Hence... substituting in equation for M

Y = b0 + b1(a0 + a1X + a2W + a3XW) + b2Q + b3(a0 + a1X + a2W + a3XW)V + b4(a0 + a1X + a2W + a3XW)Q + b5VQ + b6(a0 + a1X + a2W + a3XW)VQ + c1'X + c2'W + c3'XW + c4'V + c5'XV

Hence... multiplying out brackets

Y = b0 + a0b1 + a1b1X + a2b1W + a3b1XW + b2Q + a0b3V + a1b3XV + a2b3WV + a3b3XWV + a0b4Q + a1b4XQ + a2b4WQ + a3b4XWQ + b5VQ + a0b6VQ + a1b6XVQ + a2b6WVQ + a3b6XWVQ + c1'X + c2'W + c3'XW + c4'V + c5'XV

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

Y = (b0 + a0b1 + a2b1W + b2Q + a0b3V + a2b3WV + a0b4Q + a2b4WQ + b5VQ + a0b6VQ + a2b6WVQ + c2'W + c4'V) + (a1b1 + a3b1W + a1b3V + a3b3WV + a1b4Q + a3b4WQ + a1b6VQ + a3b6WVQ + c1' + c3'W + c5'V)X

Hence...

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

a1b1 + a3b1W + a1b3V + a3b3WV + a1b4Q + a3b4WQ + a1b6VQ + a3b6WVQ = (a1 + a3W)(b1 + b3V + b4Q + b6VQ)

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

c1' + c3'W + c5'V

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 VQ MV MQ MVQ;

! 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;
XV = X*V;
VQ = V*Q;
MVQ = M*V*Q;

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 Q (b2);
Y ON MV (b3);
Y ON MQ (b4);
Y ON VQ (b5);
Y ON MVQ (b6);

Y ON X (cdash1);
Y ON W (cdash2);
Y ON XW (cdash3);
Y ON V (cdash4);
Y ON XV (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
DLOW_LOV DMEW_LOV DHIW_LOV DLOW_MEV DMEW_MEV DHIW_MEV
DLOW_HIV DMEW_HIV DHIW_HIV
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*b3*LOW_V + a3*b3*LOW_W*LOW_V +
a1*b4*LOW_Q + a3*b4*LOW_W*LOW_Q + a1*b6*LOW_V*LOW_Q +
a3*b6*LOW_W*LOW_V*LOW_Q;
IMWLVLQ = a1*b1 + a3*b1*MED_W + a1*b3*LOW_V + a3*b3*MED_W*LOW_V +
a1*b4*LOW_Q + a3*b4*MED_W*LOW_Q + a1*b6*LOW_V*LOW_Q +
a3*b6*MED_W*LOW_V*LOW_Q;
IHWLVLQ = a1*b1 + a3*b1*HIGH_W + a1*b3*LOW_V + a3*b3*HIGH_W*LOW_V +
a1*b4*LOW_Q + a3*b4*HIGH_W*LOW_Q + a1*b6*LOW_V*LOW_Q +
a3*b6*HIGH_W*LOW_V*LOW_Q;

ILWMVLQ = a1*b1 + a3*b1*LOW_W + a1*b3*MED_V + a3*b3*LOW_W*MED_V +
a1*b4*LOW_Q + a3*b4*LOW_W*LOW_Q + a1*b6*MED_V*LOW_Q +
a3*b6*LOW_W*MED_V*LOW_Q;
IMWMVLQ = a1*b1 + a3*b1*MED_W + a1*b3*MED_V + a3*b3*MED_W*MED_V +
a1*b4*LOW_Q + a3*b4*MED_W*LOW_Q + a1*b6*MED_V*LOW_Q +
a3*b6*MED_W*MED_V*LOW_Q;
IHWMVLQ = a1*b1 + a3*b1*HIGH_W + a1*b3*MED_V + a3*b3*HIGH_W*MED_V +
a1*b4*LOW_Q + a3*b4*HIGH_W*LOW_Q + a1*b6*MED_V*LOW_Q +
a3*b6*HIGH_W*MED_V*LOW_Q;

ILWHVLQ = a1*b1 + a3*b1*LOW_W + a1*b3*HIGH_V + a3*b3*LOW_W*HIGH_V +
a1*b4*LOW_Q + a3*b4*LOW_W*LOW_Q + a1*b6*HIGH_V*LOW_Q +
a3*b6*LOW_W*HIGH_V*LOW_Q;
IMWHVLQ = a1*b1 + a3*b1*MED_W + a1*b3*HIGH_V + a3*b3*MED_W*HIGH_V +
a1*b4*LOW_Q + a3*b4*MED_W*LOW_Q + a1*b6*HIGH_V*LOW_Q +
a3*b6*MED_W*HIGH_V*LOW_Q;
IHWHVLQ = a1*b1 + a3*b1*HIGH_W + a1*b3*HIGH_V + a3*b3*HIGH_W*HIGH_V +
a1*b4*LOW_Q + a3*b4*HIGH_W*LOW_Q + a1*b6*HIGH_V*LOW_Q +
a3*b6*HIGH_W*HIGH_V*LOW_Q;

ILWLVMQ = a1*b1 + a3*b1*LOW_W + a1*b3*LOW_V + a3*b3*LOW_W*LOW_V +
a1*b4*MED_Q + a3*b4*LOW_W*MED_Q + a1*b6*LOW_V*MED_Q +
a3*b6*LOW_W*LOW_V*MED_Q;
IMWLVMQ = a1*b1 + a3*b1*MED_W + a1*b3*LOW_V + a3*b3*MED_W*LOW_V +
a1*b4*MED_Q + a3*b4*MED_W*MED_Q + a1*b6*LOW_V*MED_Q +
a3*b6*MED_W*LOW_V*MED_Q;
IHWLVMQ = a1*b1 + a3*b1*HIGH_W + a1*b3*LOW_V + a3*b3*HIGH_W*LOW_V +
a1*b4*MED_Q + a3*b4*HIGH_W*MED_Q + a1*b6*LOW_V*MED_Q +
a3*b6*HIGH_W*LOW_V*MED_Q;

ILWMVMQ = a1*b1 + a3*b1*LOW_W + a1*b3*MED_V + a3*b3*LOW_W*MED_V +
a1*b4*MED_Q + a3*b4*LOW_W*MED_Q + a1*b6*MED_V*MED_Q +
a3*b6*LOW_W*MED_V*MED_Q;
IMWMVMQ = a1*b1 + a3*b1*MED_W + a1*b3*MED_V + a3*b3*MED_W*MED_V +
a1*b4*MED_Q + a3*b4*MED_W*MED_Q + a1*b6*MED_V*MED_Q +
a3*b6*MED_W*MED_V*MED_Q;
IHWMVMQ = a1*b1 + a3*b1*HIGH_W + a1*b3*MED_V + a3*b3*HIGH_W*MED_V +
a1*b4*MED_Q + a3*b4*HIGH_W*MED_Q + a1*b6*MED_V*MED_Q +
a3*b6*HIGH_W*MED_V*MED_Q;

ILWHVMQ = a1*b1 + a3*b1*LOW_W + a1*b3*HIGH_V + a3*b3*LOW_W*HIGH_V +
a1*b4*MED_Q + a3*b4*LOW_W*MED_Q + a1*b6*HIGH_V*MED_Q +
a3*b6*LOW_W*HIGH_V*MED_Q;
IMWHVMQ = a1*b1 + a3*b1*MED_W + a1*b3*HIGH_V + a3*b3*MED_W*HIGH_V +
a1*b4*MED_Q + a3*b4*MED_W*MED_Q + a1*b6*HIGH_V*MED_Q +
a3*b6*MED_W*HIGH_V*MED_Q;
IHWHVMQ = a1*b1 + a3*b1*HIGH_W + a1*b3*HIGH_V + a3*b3*HIGH_W*HIGH_V +
a1*b4*MED_Q + a3*b4*HIGH_W*MED_Q + a1*b6*HIGH_V*MED_Q +
a3*b6*HIGH_W*HIGH_V*MED_Q;

ILWLVHQ = a1*b1 + a3*b1*LOW_W + a1*b3*LOW_V + a3*b3*LOW_W*LOW_V +
a1*b4*HIGH_Q + a3*b4*LOW_W*HIGH_Q + a1*b6*LOW_V*HIGH_Q +
a3*b6*LOW_W*LOW_V*HIGH_Q;
IMWLVHQ = a1*b1 + a3*b1*MED_W + a1*b3*LOW_V + a3*b3*MED_W*LOW_V +
a1*b4*HIGH_Q + a3*b4*MED_W*HIGH_Q + a1*b6*LOW_V*HIGH_Q +
a3*b6*MED_W*LOW_V*HIGH_Q;
IHWLVHQ = a1*b1 + a3*b1*HIGH_W + a1*b3*LOW_V + a3*b3*HIGH_W*LOW_V +
a1*b4*HIGH_Q + a3*b4*HIGH_W*HIGH_Q + a1*b6*LOW_V*HIGH_Q +
a3*b6*HIGH_W*LOW_V*HIGH_Q;

ILWMVHQ = a1*b1 + a3*b1*LOW_W + a1*b3*MED_V + a3*b3*LOW_W*MED_V +
a1*b4*HIGH_Q + a3*b4*LOW_W*HIGH_Q + a1*b6*MED_V*HIGH_Q +
a3*b6*LOW_W*MED_V*HIGH_Q;
IMWMVHQ = a1*b1 + a3*b1*MED_W + a1*b3*MED_V + a3*b3*MED_W*MED_V +
a1*b4*HIGH_Q + a3*b4*MED_W*HIGH_Q + a1*b6*MED_V*HIGH_Q +
a3*b6*MED_W*MED_V*HIGH_Q;
IHWMVHQ = a1*b1 + a3*b1*HIGH_W + a1*b3*MED_V + a3*b3*HIGH_W*MED_V +
a1*b4*HIGH_Q + a3*b4*HIGH_W*HIGH_Q + a1*b6*MED_V*HIGH_Q +
a3*b6*HIGH_W*MED_V*HIGH_Q;

ILWHVHQ = a1*b1 + a3*b1*LOW_W + a1*b3*HIGH_V + a3*b3*LOW_W*HIGH_V +
a1*b4*HIGH_Q + a3*b4*LOW_W*HIGH_Q + a1*b6*HIGH_V*HIGH_Q +
a3*b6*LOW_W*HIGH_V*HIGH_Q;
IMWHVHQ = a1*b1 + a3*b1*MED_W + a1*b3*HIGH_V + a3*b3*MED_W*HIGH_V +
a1*b4*HIGH_Q + a3*b4*MED_W*HIGH_Q + a1*b6*HIGH_V*HIGH_Q +
a3*b6*MED_W*HIGH_V*HIGH_Q;
IHWHVHQ = a1*b1 + a3*b1*HIGH_W + a1*b3*HIGH_V + a3*b3*HIGH_W*HIGH_V +
a1*b4*HIGH_Q + a3*b4*HIGH_W*HIGH_Q + a1*b6*HIGH_V*HIGH_Q +
a3*b6*HIGH_W*HIGH_V*HIGH_Q;

! Calc conditional direct effects for each combination of moderator values

DLOW_LOV = cdash1 + cdash3*LOW_W + cdash5*LOW_V;
DMEW_LOV = cdash1 + cdash3*MED_W + cdash5*LOW_V;
DHIW_LOV = cdash1 + cdash3*HIGH_W + cdash5*LOW_V;

DLOW_MEV = cdash1 + cdash3*LOW_W + cdash5*MED_V;
DMEW_MEV = cdash1 + cdash3*MED_W + cdash5*MED_V;
DHIW_MEV = cdash1 + cdash3*HIGH_W + cdash5*MED_V;

DLOW_HIV = cdash1 + cdash3*LOW_W + cdash5*HIGH_V;
DMEW_HIV = cdash1 + cdash3*MED_W + cdash5*HIGH_V;
DHIW_HIV = cdash1 + cdash3*HIGH_W + cdash5*HIGH_V;

! Calc conditional total effects for each combination of moderator values

TLWLVLQ = ILWLVLQ + DLOW_LOV;
TMWLVLQ = IMWLVLQ + DMEW_LOV;
THWLVLQ = IHWLVLQ + DHIW_LOV;

TLWMVLQ = ILWMVLQ + DLOW_MEV;
TMWMVLQ = IMWMVLQ + DMEW_MEV;
THWMVLQ = IHWMVLQ + DHIW_MEV;

TLWHVLQ = ILWHVLQ + DLOW_HIV;
TMWHVLQ = IMWHVLQ + DMEW_HIV;
THWHVLQ = IHWHVLQ + DHIW_HIV;

TLWLVMQ = ILWLVMQ + DLOW_LOV;
TMWLVMQ = IMWLVMQ + DMEW_LOV;
THWLVMQ = IHWLVMQ + DHIW_LOV;

TLWMVMQ = ILWMVMQ + DLOW_MEV;
TMWMVMQ = IMWMVMQ + DMEW_MEV;
THWMVMQ = IHWMVMQ + DHIW_MEV;

TLWHVMQ = ILWHVMQ + DLOW_HIV;
TMWHVMQ = IMWHVMQ + DMEW_HIV;
THWHVMQ = IHWHVMQ + DHIW_HIV;

TLWLVHQ = ILWLVHQ + DLOW_LOV;
TMWLVHQ = IMWLVHQ + DMEW_LOV;
THWLVHQ = IHWLVHQ + DHIW_LOV;

TLWMVHQ = ILWMVHQ + DLOW_MEV;
TMWMVHQ = IMWMVHQ + DMEW_MEV;
THWMVHQ = IHWMVHQ + DHIW_MEV;

TLWHVHQ = ILWHVHQ + DLOW_HIV;
TMWHVHQ = IMWHVHQ + DMEW_HIV;
THWHVHQ = IHWHVHQ + DHIW_HIV;

! 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);