﻿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 71: 1 or more mediators, in parallel if multiple (example uses 1), 2 moderators, both moderating both of the Mediator-DV path and the direct IV-DV path, with all 2-way and 3-way interactions, 1 of which also moderates the IV-Mediator path

Example Variables: 1 predictor X, 1 mediator M, 2 moderators W, 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 + b2MW + b3MV + b4MWV + c1'X + c2'W + c3'V + c4'XW + c5'XV + c6'WV + c7'XWV
M = a0 + a1X + a2W + a3XW

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

Y = b0 + b1M + b2MW + b3MV + b4MWV + c1'X + c2'W + c3'V + c4'XW + c5'XV + c6'WV + c7'XWV
M = a0 + a1X + a2W + a3XW

Hence... substituting in equation for M

Y = b0 + b1(a0 + a1X + a2W + a3XW) + b2(a0 + a1X + a2W + a3XW)W + b3(a0 + a1X + a2W + a3XW)V + b4(a0 + a1X + a2W + a3XW)WV + c1'X + c2'W + c3'V + c4'XW + c5'XV + c6'WV + c7'XWV

Hence... multiplying out brackets

Y = b0 + a0b1 + a1b1X + a2b1W + a3b1XW + a0b2W + a1b2XW + a2b2WW + a3b2XWW + a0b3V + a1b3XV + a2b3WV + a3b3XWV + a0b4WV + a1b4XWV + a2b4WWV + a3b4XWWV + c1'X + c2'W + c3'V + c4'XW + c5'XV + c6'WV + c7'XWV

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

Y = (b0 + a0b1 + a2b1W + a0b2W + a2b2WW + a0b3V + a2b3WV + a0b4WV + a2b4WWV + c2'W + c3'V + c6'WV) + (a1b1 + a3b1W + a1b2W + a3b2WW + a1b3V + a3b3WV + a1b4WV + a3b4WWV + c1' + c4'W + c5'V + c7'WV)X

Hence...

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

a1b1 + a3b1W + a1b2W + a3b2WW + a1b3V + a3b3WV + a1b4WV + a3b4WWV = (a1 + a3W)(b1 + b2W + b3V + b4WV)

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

c1' + c4'W + c5'V + c7'WV

Mplus code for the model:

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

USEVARIABLES = X M W V Y XW XV WV MW MV XWV MWV;

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

DEFINE:
MW = M*W;
MV = M*V;
XW = X*W;
XV = X*V;
WV = W*V;
MWV = M*W*V;
XWV = X*W*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 MW (b2);
Y ON MV (b3);
Y ON MWV (b4);

Y ON X(cdash1);
Y ON W (cdash2);
Y ON V (cdash3);
Y ON XW (cdash4);
Y ON XV (cdash5);
Y ON WV (cdash6);
Y ON XWV (cdash7);

[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
! for example, of 1 SD below mean, mean, 1 SD above mean

! 2 moderators, 3 values for each, gives 9 combinations
! arbitrary naming convention for conditional indirect and total effects used below:
! MEV_LOQ = 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
ILOW_LOV IMEW_LOV IHIW_LOV ILOW_MEV IMEW_MEV IHIW_MEV
ILOW_HIV IMEW_HIV IHIW_HIV
DLOW_LOV DMEW_LOV DHIW_LOV DLOW_MEV DMEW_MEV DHIW_MEV
DLOW_HIV DMEW_HIV DHIW_HIV
TLOW_LOV TMEW_LOV THIW_LOV TLOW_MEV TMEW_MEV THIW_MEV
TLOW_HIV TMEW_HIV THIW_HIV);

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

! Calc conditional indirect effects for each combination of moderator values

ILOW_LOV = a1*b1 + a3*b1*LOW_W + a1*b2*LOW_W + a3*b2*LOW_W*LOW_W +
a1*b3*LOW_V + a3*b3*LOW_W*LOW_V + a1*b4*LOW_W*LOW_V +
a3*b4*LOW_W*LOW_W*LOW_V;
IMEW_LOV = a1*b1 + a3*b1*MED_W + a1*b2*MED_W + a3*b2*MED_W*MED_W +
a1*b3*LOW_V + a3*b3*MED_W*LOW_V + a1*b4*MED_W*LOW_V +
a3*b4*MED_W*MED_W*LOW_V;
IHIW_LOV = a1*b1 + a3*b1*HIGH_W + a1*b2*HIGH_W + a3*b2*HIGH_W*HIGH_W +
a1*b3*LOW_V + a3*b3*HIGH_W*LOW_V + a1*b4*HIGH_W*LOW_V +
a3*b4*HIGH_W*HIGH_W*LOW_V;

ILOW_MEV = a1*b1 + a3*b1*LOW_W + a1*b2*LOW_W + a3*b2*LOW_W*LOW_W +
a1*b3*MED_V + a3*b3*LOW_W*MED_V + a1*b4*LOW_W*MED_V +
a3*b4*LOW_W*LOW_W*MED_V;
IMEW_MEV = a1*b1 + a3*b1*MED_W + a1*b2*MED_W + a3*b2*MED_W*MED_W +
a1*b3*MED_V + a3*b3*MED_W*MED_V + a1*b4*MED_W*MED_V +
a3*b4*MED_W*MED_W*MED_V;
IHIW_MEV = a1*b1 + a3*b1*HIGH_W + a1*b2*HIGH_W + a3*b2*HIGH_W*HIGH_W +
a1*b3*MED_V + a3*b3*HIGH_W*MED_V + a1*b4*HIGH_W*MED_V +
a3*b4*HIGH_W*HIGH_W*MED_V;

ILOW_HIV = a1*b1 + a3*b1*LOW_W + a1*b2*LOW_W + a3*b2*LOW_W*LOW_W +
a1*b3*HIGH_V + a3*b3*LOW_W*HIGH_V + a1*b4*LOW_W*HIGH_V +
a3*b4*LOW_W*LOW_W*HIGH_V;
IMEW_HIV = a1*b1 + a3*b1*MED_W + a1*b2*MED_W + a3*b2*MED_W*MED_W +
a1*b3*HIGH_V + a3*b3*MED_W*HIGH_V + a1*b4*MED_W*HIGH_V +
a3*b4*MED_W*MED_W*HIGH_V;
IHIW_HIV = a1*b1 + a3*b1*HIGH_W + a1*b2*HIGH_W + a3*b2*HIGH_W*HIGH_W +
a1*b3*HIGH_V + a3*b3*HIGH_W*HIGH_V + a1*b4*HIGH_W*HIGH_V +
a3*b4*HIGH_W*HIGH_W*HIGH_V;

! Calc conditional direct effects for each combination of moderator values

DLOW_LOV = cdash1 + cdash4*LOW_W + cdash5*LOW_V + cdash7*LOW_W*LOW_V;
DMEW_LOV = cdash1 + cdash4*MED_W + cdash5*LOW_V + cdash7*MED_W*LOW_V;
DHIW_LOV = cdash1 + cdash4*HIGH_W + cdash5*LOW_V + cdash7*HIGH_W*LOW_V;

DLOW_MEV = cdash1 + cdash4*LOW_W + cdash5*MED_V + cdash7*LOW_W*MED_V;
DMEW_MEV = cdash1 + cdash4*MED_W + cdash5*MED_V + cdash7*MED_W*MED_V;
DHIW_MEV = cdash1 + cdash4*HIGH_W + cdash5*MED_V + cdash7*HIGH_W*MED_V;

DLOW_HIV = cdash1 + cdash4*LOW_W + cdash5*HIGH_V + cdash7*LOW_W*HIGH_V;
DMEW_HIV = cdash1 + cdash4*MED_W + cdash5*HIGH_V + cdash7*MED_W*HIGH_V;
DHIW_HIV = cdash1 + cdash4*HIGH_W + cdash5*HIGH_V + cdash7*HIGH_W*HIGH_V;

! Calc conditional total effects for each combination of moderator values

TLOW_LOV = ILOW_LOV + DLOW_LOV;
TMEW_LOV = IMEW_LOV + DMEW_LOV;
THIW_LOV = IHIW_LOV + DHIW_LOV;

TLOW_MEV = ILOW_MEV + DLOW_MEV;
TMEW_MEV = IMEW_MEV + DMEW_MEV;
THIW_MEV = IHIW_MEV + DHIW_MEV;

TLOW_HIV = ILOW_HIV + DLOW_HIV;
TMEW_HIV = IMEW_HIV + DMEW_HIV;
THIW_HIV = IHIW_HIV + 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(PLOW_LOV PMEW_LOV PHIW_LOV PLOW_MEV PMEW_MEV PHIW_MEV
PLOW_HIV PMEW_HIV PHIW_HIV);

LOOP(XVAL,1,5,0.1);

PLOW_LOV = ILOW_LOV*XVAL;
PMEW_LOV = IMEW_LOV*XVAL;
PHIW_LOV = IHIW_LOV*XVAL;

PLOW_MEV = ILOW_MEV*XVAL;
PMEW_MEV = IMEW_MEV*XVAL;
PHIW_MEV = IHIW_MEV*XVAL;

PLOW_HIV = ILOW_HIV*XVAL;
PMEW_HIV = IMEW_HIV*XVAL;
PHIW_HIV = IHIW_HIV*XVAL;

PLOT:
TYPE = plot2;

OUTPUT:
STAND CINT(bcbootstrap);