﻿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 34: 1 or more mediators, in parallel if multiple (example uses 1), 3 moderators, 2 moderating the IV-Mediator path with all 2-way and 3-way interactions, one of which also moderates the direct IV-DV path, 1 moderating the Mediator-DV path and the direct IV-DV path

Example Variables: 1 predictor X, 1 mediator M, 3 moderators W, Z, V, 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 + c1'X + c2'W + c3'XW + c4'V + c5'XV
M = a0 + a1X + a2W + a3Z + a4XW + a5XZ + a6WZ + a7XWZ

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

Y = b0 + b1M + b2MV + c1'X + c2'W + c3'XW + c4'V + c5'XV
M = a0 + a1X + a2W + a3Z + a4XW + a5XZ + a6WZ + a7XWZ

Hence... substituting in equation for M

Y = b0 + b1(a0 + a1X + a2W + a3Z + a4XW + a5XZ + a6WZ + a7XWZ) + b2(a0 + a1X + a2W + a3Z + a4XW + a5XZ + a6WZ + a7XWZ)V + c1'X + c2'W + c3'XW + c4'V + c5'XV

Hence... multiplying out brackets

Y = b0 + a0b1 + a1b1X + a2b1W + a3b1Z + a4b1XW + a5b1XZ + a6b1WZ + a7b1XWZ + a0b2V + a1b2XV + a2b2WV + a3b2ZV + a4b2XWV + a5b2XZV + a6b2WZV + a7b2XWZV + c1'X + c2'W + c3'XW + c4'V + c5'XV

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

Y = (b0 + a0b1 + a2b1W + a3b1Z + a6b1WZ + a0b2V + a2b2WV + a3b2ZV + a6b2WZV + c2'W + c4'V) + (a1b1 + a4b1W + a5b1Z + a7b1WZ + a1b2V + a4b2WV + a5b2ZV + a7b2WZV + c1' + c3'W + c5'V)X

Hence...

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

a1b1 + a4b1W + a5b1Z + a7b1WZ + a1b2V + a4b2WV + a5b2ZV + a7b2WZV = (a1 + a4W + a5Z + a7WZ)(b1 + b2V)

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, Z, V
! Outcome variable - Y

USEVARIABLES = X M W Z V Y XW XZ WZ XV MV XWZ;

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

DEFINE:
MV = M*V;
XW = X*W;
XZ = X*Z;
XV = X*V;
WZ = W*Z;
XWZ = X*W*Z;

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 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 Z (a3);
M ON XW (a4);
M ON XZ (a5);
M ON WZ (a6);
M ON XWZ (a7);

! Use model constraint subcommand to test conditional indirect effects
! You need to pick low, medium and high moderator values for W, Z, V
! 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_Z MED_Z HIGH_Z LOW_V MED_V HIGH_V
ILWLZLV IMWLZLV IHWLZLV ILWMZLV IMWMZLV IHWMZLV
ILWHZLV IMWHZLV IHWHZLV
ILWLZMV IMWLZMV IHWLZMV ILWMZMV IMWMZMV IHWMZMV
ILWHZMV IMWHZMV IHWHZMV
ILWLZHV IMWLZHV IHWLZHV ILWMZHV IMWMZHV IHWMZHV
ILWHZHV IMWHZHV IHWHZHV
DLOW_LOV DMEW_LOV DHIW_LOV DLOW_MEV DMEW_MEV DHIW_MEV
DLOW_HIV DMEW_HIV DHIW_HIV
TLWLZLV TMWLZLV THWLZLV TLWMZLV TMWMZLV THWMZLV
TLWHZLV TMWHZLV THWHZLV
TLWLZMV TMWLZMV THWLZMV TLWMZMV TMWMZMV THWMZMV
TLWHZMV TMWHZMV THWHZMV
TLWLZHV TMWLZHV THWLZHV TLWMZHV TMWMZHV THWMZHV
TLWHZHV TMWHZHV THWHZHV);

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_Z = #LOWZ;   ! replace #LOWZ in the code with your chosen low value of Z
MED_Z = #MEDZ;   ! replace #MEDZ in the code with your chosen medium value of Z
HIGH_Z = #HIGHZ;   ! replace #HIGHZ in the code with your chosen high value of Z

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

! Calc conditional indirect effects for each combination of moderator values

ILWLZLV = a1*b1 + a4*b1*LOW_W + a5*b1*LOW_Z + a7*b1*LOW_W*LOW_Z +
a1*b2*LOW_V + a4*b2*LOW_W*LOW_V + a5*b2*LOW_Z*LOW_V +
a7*b2*LOW_W*LOW_Z*LOW_V;
IMWLZLV = a1*b1 + a4*b1*MED_W + a5*b1*LOW_Z + a7*b1*MED_W*LOW_Z +
a1*b2*LOW_V + a4*b2*MED_W*LOW_V + a5*b2*LOW_Z*LOW_V +
a7*b2*MED_W*LOW_Z*LOW_V;
IHWLZLV = a1*b1 + a4*b1*HIGH_W + a5*b1*LOW_Z + a7*b1*HIGH_W*LOW_Z +
a1*b2*LOW_V + a4*b2*HIGH_W*LOW_V + a5*b2*LOW_Z*LOW_V +
a7*b2*HIGH_W*LOW_Z*LOW_V;

ILWMZLV = a1*b1 + a4*b1*LOW_W + a5*b1*MED_Z + a7*b1*LOW_W*MED_Z +
a1*b2*LOW_V + a4*b2*LOW_W*LOW_V + a5*b2*MED_Z*LOW_V +
a7*b2*LOW_W*MED_Z*LOW_V;
IMWMZLV = a1*b1 + a4*b1*MED_W + a5*b1*MED_Z + a7*b1*MED_W*MED_Z +
a1*b2*LOW_V + a4*b2*MED_W*LOW_V + a5*b2*MED_Z*LOW_V +
a7*b2*MED_W*MED_Z*LOW_V;
IHWMZLV = a1*b1 + a4*b1*HIGH_W + a5*b1*MED_Z + a7*b1*HIGH_W*MED_Z +
a1*b2*LOW_V + a4*b2*HIGH_W*LOW_V + a5*b2*MED_Z*LOW_V +
a7*b2*HIGH_W*MED_Z*LOW_V;

ILWHZLV = a1*b1 + a4*b1*LOW_W + a5*b1*HIGH_Z + a7*b1*LOW_W*HIGH_Z +
a1*b2*LOW_V + a4*b2*LOW_W*LOW_V + a5*b2*HIGH_Z*LOW_V +
a7*b2*LOW_W*HIGH_Z*LOW_V;
IMWHZLV = a1*b1 + a4*b1*MED_W + a5*b1*HIGH_Z + a7*b1*MED_W*HIGH_Z +
a1*b2*LOW_V + a4*b2*MED_W*LOW_V + a5*b2*HIGH_Z*LOW_V +
a7*b2*MED_W*HIGH_Z*LOW_V;
IHWHZLV = a1*b1 + a4*b1*HIGH_W + a5*b1*HIGH_Z + a7*b1*HIGH_W*HIGH_Z +
a1*b2*LOW_V + a4*b2*HIGH_W*LOW_V + a5*b2*HIGH_Z*LOW_V +
a7*b2*HIGH_W*HIGH_Z*LOW_V;

ILWLZMV = a1*b1 + a4*b1*LOW_W + a5*b1*LOW_Z + a7*b1*LOW_W*LOW_Z +
a1*b2*MED_V + a4*b2*LOW_W*MED_V + a5*b2*LOW_Z*MED_V +
a7*b2*LOW_W*LOW_Z*MED_V;
IMWLZMV = a1*b1 + a4*b1*MED_W + a5*b1*LOW_Z + a7*b1*MED_W*LOW_Z +
a1*b2*MED_V + a4*b2*MED_W*MED_V + a5*b2*LOW_Z*MED_V +
a7*b2*MED_W*LOW_Z*MED_V;
IHWLZMV = a1*b1 + a4*b1*HIGH_W + a5*b1*LOW_Z + a7*b1*HIGH_W*LOW_Z +
a1*b2*MED_V + a4*b2*HIGH_W*MED_V + a5*b2*LOW_Z*MED_V +
a7*b2*HIGH_W*LOW_Z*MED_V;

ILWMZMV = a1*b1 + a4*b1*LOW_W + a5*b1*MED_Z + a7*b1*LOW_W*MED_Z +
a1*b2*MED_V + a4*b2*LOW_W*MED_V + a5*b2*MED_Z*MED_V +
a7*b2*LOW_W*MED_Z*MED_V;
IMWMZMV = a1*b1 + a4*b1*MED_W + a5*b1*MED_Z + a7*b1*MED_W*MED_Z +
a1*b2*MED_V + a4*b2*MED_W*MED_V + a5*b2*MED_Z*MED_V +
a7*b2*MED_W*MED_Z*MED_V;
IHWMZMV = a1*b1 + a4*b1*HIGH_W + a5*b1*MED_Z + a7*b1*HIGH_W*MED_Z +
a1*b2*MED_V + a4*b2*HIGH_W*MED_V + a5*b2*MED_Z*MED_V +
a7*b2*HIGH_W*MED_Z*MED_V;

ILWHZMV = a1*b1 + a4*b1*LOW_W + a5*b1*HIGH_Z + a7*b1*LOW_W*HIGH_Z +
a1*b2*MED_V + a4*b2*LOW_W*MED_V + a5*b2*HIGH_Z*MED_V +
a7*b2*LOW_W*HIGH_Z*MED_V;
IMWHZMV = a1*b1 + a4*b1*MED_W + a5*b1*HIGH_Z + a7*b1*MED_W*HIGH_Z +
a1*b2*MED_V + a4*b2*MED_W*MED_V + a5*b2*HIGH_Z*MED_V +
a7*b2*MED_W*HIGH_Z*MED_V;
IHWHZMV = a1*b1 + a4*b1*HIGH_W + a5*b1*HIGH_Z + a7*b1*HIGH_W*HIGH_Z +
a1*b2*MED_V + a4*b2*HIGH_W*MED_V + a5*b2*HIGH_Z*MED_V +
a7*b2*HIGH_W*HIGH_Z*MED_V;

ILWLZHV = a1*b1 + a4*b1*LOW_W + a5*b1*LOW_Z + a7*b1*LOW_W*LOW_Z +
a1*b2*HIGH_V + a4*b2*LOW_W*HIGH_V + a5*b2*LOW_Z*HIGH_V +
a7*b2*LOW_W*LOW_Z*HIGH_V;
IMWLZHV = a1*b1 + a4*b1*MED_W + a5*b1*LOW_Z + a7*b1*MED_W*LOW_Z +
a1*b2*HIGH_V + a4*b2*MED_W*HIGH_V + a5*b2*LOW_Z*HIGH_V +
a7*b2*MED_W*LOW_Z*HIGH_V;
IHWLZHV = a1*b1 + a4*b1*HIGH_W + a5*b1*LOW_Z + a7*b1*HIGH_W*LOW_Z +
a1*b2*HIGH_V + a4*b2*HIGH_W*HIGH_V + a5*b2*LOW_Z*HIGH_V +
a7*b2*HIGH_W*LOW_Z*HIGH_V;

ILWMZHV = a1*b1 + a4*b1*LOW_W + a5*b1*MED_Z + a7*b1*LOW_W*MED_Z +
a1*b2*HIGH_V + a4*b2*LOW_W*HIGH_V + a5*b2*MED_Z*HIGH_V +
a7*b2*LOW_W*MED_Z*HIGH_V;
IMWMZHV = a1*b1 + a4*b1*MED_W + a5*b1*MED_Z + a7*b1*MED_W*MED_Z +
a1*b2*HIGH_V + a4*b2*MED_W*HIGH_V + a5*b2*MED_Z*HIGH_V +
a7*b2*MED_W*MED_Z*HIGH_V;
IHWMZHV = a1*b1 + a4*b1*HIGH_W + a5*b1*MED_Z + a7*b1*HIGH_W*MED_Z +
a1*b2*HIGH_V + a4*b2*HIGH_W*HIGH_V + a5*b2*MED_Z*HIGH_V +
a7*b2*HIGH_W*MED_Z*HIGH_V;

ILWHZHV = a1*b1 + a4*b1*LOW_W + a5*b1*HIGH_Z + a7*b1*LOW_W*HIGH_Z +
a1*b2*HIGH_V + a4*b2*LOW_W*HIGH_V + a5*b2*HIGH_Z*HIGH_V +
a7*b2*LOW_W*HIGH_Z*HIGH_V;
IMWHZHV = a1*b1 + a4*b1*MED_W + a5*b1*HIGH_Z + a7*b1*MED_W*HIGH_Z +
a1*b2*HIGH_V + a4*b2*MED_W*HIGH_V + a5*b2*HIGH_Z*HIGH_V +
a7*b2*MED_W*HIGH_Z*HIGH_V;
IHWHZHV = a1*b1 + a4*b1*HIGH_W + a5*b1*HIGH_Z + a7*b1*HIGH_W*HIGH_Z +
a1*b2*HIGH_V + a4*b2*HIGH_W*HIGH_V + a5*b2*HIGH_Z*HIGH_V +
a7*b2*HIGH_W*HIGH_Z*HIGH_V;

! 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

TLWLZLV = ILWLZLV + DLOW_LOV;
TMWLZLV = IMWLZLV + DMEW_LOV;
THWLZLV = IHWLZLV + DHIW_LOV;

TLWMZLV = ILWMZLV + DLOW_LOV;
TMWMZLV = IMWMZLV + DMEW_LOV;
THWMZLV = IHWMZLV + DHIW_LOV;

TLWHZLV = ILWHZLV + DLOW_LOV;
TMWHZLV = IMWHZLV + DMEW_LOV;
THWHZLV = IHWHZLV + DHIW_LOV;

TLWLZMV = ILWLZMV + DLOW_MEV;
TMWLZMV = IMWLZMV + DMEW_MEV;
THWLZMV = IHWLZMV + DHIW_MEV;

TLWMZMV = ILWMZMV + DLOW_MEV;
TMWMZMV = IMWMZMV + DMEW_MEV;
THWMZMV = IHWMZMV + DHIW_MEV;

TLWHZMV = ILWHZMV + DLOW_MEV;
TMWHZMV = IMWHZMV + DMEW_MEV;
THWHZMV = IHWHZMV + DHIW_MEV;

TLWLZHV = ILWLZHV + DLOW_HIV;
TMWLZHV = IMWLZHV + DMEW_HIV;
THWLZHV = IHWLZHV + DHIW_HIV;

TLWMZHV = ILWMZHV + DLOW_HIV;
TMWMZHV = IMWMZHV + DMEW_HIV;
THWMZHV = IHWMZHV + DHIW_HIV;

TLWHZHV = ILWHZHV + DLOW_HIV;
TMWHZHV = IMWHZHV + DMEW_HIV;
THWHZHV = IHWHZHV + 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(PLWLZLV PMWLZLV PHWLZLV PLWMZLV PMWMZLV PHWMZLV
PLWHZLV PMWHZLV PHWHZLV
PLWLZMV PMWLZMV PHWLZMV PLWMZMV PMWMZMV PHWMZMV
PLWHZMV PMWHZMV PHWHZMV
PLWLZHV PMWLZHV PHWLZHV PLWMZHV PMWMZHV PHWMZHV
PLWHZHV PMWHZHV PHWHZHV);

LOOP(XVAL,1,5,0.1);

PLWLZLV = ILWLZLV*XVAL;
PMWLZLV = IMWLZLV*XVAL;
PHWLZLV = IHWLZLV*XVAL;

PLWMZLV = ILWMZLV*XVAL;
PMWMZLV = IMWMZLV*XVAL;
PHWMZLV = IHWMZLV*XVAL;

PLWHZLV = ILWHZLV*XVAL;
PMWHZLV = IMWHZLV*XVAL;
PHWHZLV = IHWHZLV*XVAL;

PLWLZMV = ILWLZMV*XVAL;
PMWLZMV = IMWLZMV*XVAL;
PHWLZMV = IHWLZMV*XVAL;

PLWMZMV = ILWMZMV*XVAL;
PMWMZMV = IMWMZMV*XVAL;
PHWMZMV = IHWMZMV*XVAL;

PLWHZMV = ILWHZMV*XVAL;
PMWHZMV = IMWHZMV*XVAL;
PHWHZMV = IHWHZMV*XVAL;

PLWLZHV = ILWLZHV*XVAL;
PMWLZHV = IMWLZHV*XVAL;
PHWLZHV = IHWLZHV*XVAL;

PLWMZHV = ILWMZHV*XVAL;
PMWMZHV = IMWMZHV*XVAL;
PHWMZHV = IHWMZHV*XVAL;

PLWHZHV = ILWHZHV*XVAL;
PMWHZHV = IMWHZHV*XVAL;
PHWHZHV = IHWHZHV*XVAL;

PLOT:
TYPE = plot2;

OUTPUT:
STAND CINT(bcbootstrap);