﻿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 38 (latent variable version): 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, with all 2-way and 3-way interactions

Example Variables: 1 latent predictor X measured by 4 observed variables X1-X4, 1 latent mediator M measured by 4 observed variables M1-M4, 3 latent moderators W, V, and Q, each measured by sets of 4 observed variables W1-W4, V1-V4, and Q1-Q4 respectively, 1 latent outcome Y measured by 4 observed variables Y1-Y4

Preliminary notes:

The code below assumes that

• The latent IV (factor X) is measured by continuous observed variables X1-X4.
• Any latent moderator(s) (factors W, V, Q, Z) are measured by continuous observed variables W1-W4, Z1-Z4, V1-V4, Q1-Q4 respectively.
• Any latent mediator(s) (factor M, or factors M1, M2, etc.) are measured by continuous observed variables M1-M4 or M1_1-M1-4, M2_1-M2_4 respectively.
• The latent outcome Y is measured by continuous observed variables Y1-Y4.

Model Diagram (factor indicator variables omitted for space/clarity reasons):

Statistical Diagram (factor indicator variables omitted for space/clarity reasons):

Model Equation(s):

Y = b0 + b1M + b2MV + b3MQ + b4MVQ + c1'X + c2'V + c3'Q + c4'XV + c5'XQ + c6'VQ + c7'XVQ
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 + b4MVQ + c1'X + c2'V + c3'Q + c4'XV + c5'XQ + c6'VQ + c7'XVQ
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 + b4(a0 + a1X + a2W + a3XW)VQ + c1'X + c2'V + c3'Q + c4'XV + c5'XQ + c6'VQ + c7'XVQ

Hence... multiplying out brackets

Y = b0 + a0b1 + a1b1X + a2b1W + a3b1XW + a0b2V + a1b2XV + a2b2WV + a3b2XWV + a0b3Q + a1b3XQ + a2b3WQ + a3b3XWQ + a0b4VQ + a1b4XVQ + a2b4WVQ + a3b4XWVQ + c1'X + c2'V + c3'Q + c4'XV + c5'XQ + c6'VQ + c7'XVQ

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

Y = (b0 + a0b1 + a2b1W + a0b2V + a2b2WV + a0b3Q + a2b3WQ + a0b4VQ + a2b4WVQ + c2'V + c3'Q + c6'VQ) + (a1b1 + a3b1W + a1b2V + a3b2WV + a1b3Q + a3b3WQ + a1b4VQ + a3b4WVQ + c1' + c4'V + c5'Q + c7'VQ)X

Hence...

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

a1b1 + a3b1W + a1b2V + a3b2WV + a1b3Q + a3b3WQ + a1b4VQ + a3b4WVQ = (a1 + a3W)(b1 + b2V + b3Q + b4VQ)

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

c1' + c4'V + c5'Q + c7'VQ

Mplus code for the model:

! Latent predictor variable X measured by X1-X4
! Latent mediator M measured by 4 observed variables M1-M4
! Latent moderators W, V, and Q, each measured by sets of 4 observed variables W1-W4, V1-V4, and Q1-Q4 respectively
! Latent outcome variable Y measured by Y1-Y4

USEVARIABLES = X1 X2 X3 X4 M1 M2 M3 M4
W1 W2 W3 W4 V1 V2 V3 V4 Q1 Q2 Q3 Q4
Y1 Y2 Y3 Y4;

ANALYSIS:
TYPE = GENERAL RANDOM;
ESTIMATOR = ML;
ALGORITHM = INTEGRATION;

! In model statement first state measurement model
! Then create any latent interactions required
! Then state structural model naming each path and intercept using parentheses

MODEL:

! Measurement model
! This makes these factors standardised
X BY X1 X2 X3 X4;
M BY M1 M2 M3 M4;
W BY W1* W2 W3 W4;
V BY V1* V2 V3 V4;
Q BY Q1* Q2 Q3 Q4;
Y BY Y1 Y2 Y3 Y4;

W@1;   V@1;   Q@1;

! Create latent interactions
MV | M XWITH V;
MQ | M XWITH Q;
XW | X XWITH W;
XV | X XWITH V;
XQ | X XWITH Q;
VQ | V XWITH Q;
MVQ | M XWITH VQ;
XVQ | X XWITH VQ;

! Fit structural model and name parameters
! Note that intercepts of M, Y are fixed = 0 since they are latent vars
! so no code to state and name them as parameters
Y ON M (b1);
Y ON MV (b2);
Y ON MQ (b3);
Y ON MVQ (b4);

Y ON X (cdash1);
Y ON V (cdash2);
Y ON Q (cdash3);
Y ON XV (cdash4);
Y ON XQ (cdash5);
Y ON VQ (cdash6);
Y ON XVQ (cdash7);

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 = -1;   ! -1 SD below mean value of W
MED_W = 0;   ! mean value of W
HIGH_W = 1;   ! +1 SD above mean value of W

LOW_V = -1;   ! -1 SD below mean value of V
MED_V = 0;   ! mean value of V
HIGH_V = 1;   ! +1 SD above mean value of V

LOW_Q = -1;   ! -1 SD below mean value of Q
MED_Q = 0;   ! mean value of Q
HIGH_Q = 1;   ! +1 SD above mean 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 + a1*b4*LOW_V*LOW_Q +
a3*b4*LOW_W*LOW_V*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 + a1*b4*LOW_V*LOW_Q +
a3*b4*MED_W*LOW_V*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 + a1*b4*LOW_V*LOW_Q +
a3*b4*HIGH_W*LOW_V*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 + a1*b4*MED_V*LOW_Q +
a3*b4*LOW_W*MED_V*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 + a1*b4*MED_V*LOW_Q +
a3*b4*MED_W*MED_V*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 + a1*b4*MED_V*LOW_Q +
a3*b4*HIGH_W*MED_V*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 + a1*b4*HIGH_V*LOW_Q +
a3*b4*LOW_W*HIGH_V*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 + a1*b4*HIGH_V*LOW_Q +
a3*b4*MED_W*HIGH_V*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 + a1*b4*HIGH_V*LOW_Q +
a3*b4*HIGH_W*HIGH_V*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 + a1*b4*LOW_V*MED_Q +
a3*b4*LOW_W*LOW_V*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 + a1*b4*LOW_V*MED_Q +
a3*b4*MED_W*LOW_V*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 + a1*b4*LOW_V*MED_Q +
a3*b4*HIGH_W*LOW_V*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 + a1*b4*MED_V*MED_Q +
a3*b4*LOW_W*MED_V*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 + a1*b4*MED_V*MED_Q +
a3*b4*MED_W*MED_V*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 + a1*b4*MED_V*MED_Q +
a3*b4*HIGH_W*MED_V*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 + a1*b4*HIGH_V*MED_Q +
a3*b4*LOW_W*HIGH_V*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 + a1*b4*HIGH_V*MED_Q +
a3*b4*MED_W*HIGH_V*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 + a1*b4*HIGH_V*MED_Q +
a3*b4*HIGH_W*HIGH_V*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 + a1*b4*LOW_V*HIGH_Q +
a3*b4*LOW_W*LOW_V*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 + a1*b4*LOW_V*HIGH_Q +
a3*b4*MED_W*LOW_V*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 + a1*b4*LOW_V*HIGH_Q +
a3*b4*HIGH_W*LOW_V*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 + a1*b4*MED_V*HIGH_Q +
a3*b4*LOW_W*MED_V*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 + a1*b4*MED_V*HIGH_Q +
a3*b4*MED_W*MED_V*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 + a1*b4*MED_V*HIGH_Q +
a3*b4*HIGH_W*MED_V*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 + a1*b4*HIGH_V*HIGH_Q +
a3*b4*LOW_W*HIGH_V*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 + a1*b4*HIGH_V*HIGH_Q +
a3*b4*MED_W*HIGH_V*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 + a1*b4*HIGH_V*HIGH_Q +
a3*b4*HIGH_W*HIGH_V*HIGH_Q;

! Calc conditional direct effects for each combination of moderator values

DLOV_LOQ = cdash1 + cdash4*LOW_V + cdash5*LOW_Q + cdash7*LOW_V*LOW_Q;
DMEV_LOQ = cdash1 + cdash4*MED_V + cdash5*LOW_Q + cdash7*MED_V*LOW_Q;
DHIV_LOQ = cdash1 + cdash4*HIGH_V + cdash5*LOW_Q + cdash7*HIGH_V*LOW_Q;

DLOV_MEQ = cdash1 + cdash4*LOW_V + cdash5*MED_Q + cdash7*LOW_V*MED_Q;
DMEV_MEQ = cdash1 + cdash4*MED_V + cdash5*MED_Q + cdash7*MED_V*MED_Q;
DHIV_MEQ = cdash1 + cdash4*HIGH_V + cdash5*MED_Q + cdash7*HIGH_V*MED_Q;

DLOV_HIQ = cdash1 + cdash4*LOW_V + cdash5*HIGH_Q + cdash7*LOW_V*HIGH_Q;
DMEV_HIQ = cdash1 + cdash4*MED_V + cdash5*HIGH_Q + cdash7*MED_V*HIGH_Q;
DHIV_HIQ = cdash1 + cdash4*HIGH_V + cdash5*HIGH_Q + cdash7*HIGH_V*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 from -3 to 3 in LOOP() statement since
! X is factor with mean set at default of 0

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,-3,3,0.1);

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;