﻿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 58: 1 or more mediators, in parallel if multiple (example uses 1), 1 moderators, which moderates both the IV- Mediator path and the Mediator-DV path

Example Variables: 1 predictor X, 1 mediator M, 1 moderator W, 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 + b2W + b3MW + c'X
M = a0 + a1X + a2W + a3XW

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

Y = b0 + b1M + b2W + b3MW + c'X
M = a0 + a1X + a2W + a3XW

Hence... substituting in equation for M

Y = b0 + b1(a0 + a1X + a2W + a3XW) + b2W + b3(a0 + a1X + a2W + a3XW)W + c'X

Hence... multiplying out brackets

Y = b0 + a0b1 + a1b1X + a2b1W + a3b1XW + b2W + a0b3W + a1b3XW + a2b3WW + a3b3XWW + c'X

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

Y = (b0 + a0b1 + a2b1W + b2W + a0b3W + a2b3WW) + (a1b1 + a3b1W + a1b3W + a3b3WW + c')X

Hence...

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

a1b1 + a3b1W + a1b3W + a3b3WW = (a1 + a3W)(b1 + b3W)

One direct effect of X on Y:

c'

Mplus code for the model:

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

USEVARIABLES = X M W Y XW MW;

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

DEFINE:
MW = M*W;
XW = X*W;

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 W (b2);
Y ON MW (b3);

Y ON X (cdash);

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

! 1 moderator, 3 values for it
! arbitrary naming convention for conditional indirect and total effects used below:
! MED_Q = medium value of Q, etc.

MODEL CONSTRAINT:
NEW(LOW_W MED_W HIGH_W
IND_LOWW IND_MEDW IND_HIW
IMM_LOW IMM_MEDW IMM_HIW
TOT_LOWW TOT_MEDW TOT_HIW);

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

! Calc conditional indirect effects for each combination of moderator values
! and index/indices of moderated mediation

IND_LOWW = a1*b1 + a3*b1*LOW_W + a1*b3*LOW_W + a3*b3*LOW_W*LOW_W;
IND_MEDW = a1*b1 + a3*b1*MED_W + a1*b3*MED_W + a3*b3*MED_W*MED_W;
IND_HIW = a1*b1 + a3*b1*HIGH_W + a1*b3*HIGH_W + a3*b3*HIGH_W*HIGH_W;

IMM_LOW = a3*b1 + a1*b3 + a3*b3*LOW_W;
IMM_MEDW = a3*b1 + a1*b3 + a3*b3*MED_W;
IMM_HIW = a3*b1 + a1*b3 + a3*b3*HIGH_W;

! Calc conditional total effects for each combination of moderator values

TOT_LOWW = IND_LOWW + cdash;
TOT_MEDW = IND_MEDW + cdash;
TOT_HIW = IND_HIW + cdash;

! 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(LOMOD MEDMOD HIMOD);

LOOP(XVAL,1,5,0.1);

LOMOD = IND_LOWW*XVAL;
MEDMOD = IND_MEDW*XVAL;
HIMOD = IND_HIW*XVAL;

PLOT:
TYPE = plot2;

OUTPUT:
STAND CINT(bcbootstrap);