6.1 Moderated Mediation
6.2.1 Effect sizes for prototypical moderated mediation cases
For clarity of presentation, models for the five prototypical cases of moderated mediation are considered to be for manifest rather than latent variables. For Case 1 (the predictor moderates the effect of the mediator on the outcome), the matrix of conditional effects BstMOD is expressed as
0 0 0 0 0 0
0 0 0 0 0 0
0 1 0 0 0
0 0 0
0 0 .
0
st st st st
MOD
st
x ymx
st
mx m
st
yx ym y
mx
st
yx ym ymx x
= +
= +
=
+
x
x x
x x x
B B jη Ω
(6.9)
The matrix of indirect effects MstMOD from BstMOD is
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( ) 1
0 0 0
0 0 0 .
( ) 0 0
st st st
MOD MOD MOD
st
mx ym ymx x
= − − − −
=
+
x x
M I B I B
(6.10)
MOD is then calculated from MstMOD
2 2
2
2 2
0 0 0 0 0 0
0 0 0 0 0 0 0
( ) 0 0 0 0 ( ) 0 0
0 0 0
0 0 0 .
0 0 ( ( ))
st st st
MOD MOD MOD
st x
st m
st st st
mx ym ymx x y mx ym ymx x
st st
mx ym ymx x x
=
=
+ +
=
+
x x x x
x x
M Ψ M
(6.11)
As described in Chapter 5, the effect size ( mx( ymx +ymxxxst))2xst2 consists of products of xst , which are the covariance between xst and xst2 (x x,2), and the variance of xst2 ( 2
2
x ).
Substituting these findings into Equation 6.11 yields the for Case 1 as
2 2
2 2 2 2 2 2 2
2 , ,
st st
MOD mx ym mx ymx x x mx ym ymx x x x
= x+ x + x x (6.12)
where x is the conditional value of the predictor at which MOD is evaluated. This means that, because x is standardized, MOD at the mean of x is equivalent in form to for a simple three- variable mediation model. However, when mx2 0, MOD = only if ym2 x =ym x2 , or when there is no effect of the interaction.
has several desirable properties as an effect size measure for conditional indirect effects in moderated mediation models illustrated in Case 1. As demonstrated in Lachowicz et al.
96
(2018), is interpretable as the variance explained indirectly in an outcome by a predictor (or set of predictors) through a mediator (or set of mediators). In a moderated mediation model, the variance explained indirectly is conditional on the values of the moderator. is standardized, so it is invariant to changes in the scales of the predictor, mediator, and outcome. is also
monotonically related to its respective conditional indirect effect. In addition, 95% CIs can be obtained for using a bootstrapping procedure. Finally, the matrix-based approach allows for to be obtained for moderated mediation models more complex than the five cases previously illustrated (e.g., moderated mediation with covariates, multiple mediators, multiple moderators).
The derivation of for Case 2 (covariate w moderates the effect of x on m) is not as straightforward as for Case 1. Specifically, following the same steps as in the derivation for for Case 1 results in an effect size for a quantity that is not the squared standardized conditional indirect effect. Repeating the procedure followed in Case 1, the reduced form matrix of
standardized conditional regression coefficients BstMOD (x is designated the focal predictor) is
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0
0 0 0 0
0 0
0 0
st x st
w mxw
st
MOD st
mx mw m
st
yx ym y
st
mx mxw w mw
yx ym
x w m y x w m y
= +
= +
x
x x
x x
x x x
x x
B
.
(6.13)
The matrix of standardized conditional indirect effects MstMOD is calculated from BstMOD as
97
0 0 0 0
0 0 0 0
0 0 0 0 ,
( ) 0 0
st MOD
st
mx mxw w ym mw ym
=
+
x x x x x
M (6.14)
and MOD is calculated from MstMOD is
2 2 2 2
0 0 0 0
0 0 0 0
0 0 0 0 .
(( ) ) ( )
0 0 0
2( )
MOD
st st st
mx mxw w ym x mw ym w
st st
mx mxw w ym mw ym xw
=
+ + +
+
x x x x x
x x x x x
(6.15)
in this case consists of variance explained y indirectly by x through m conditional on w (
2 2 2
(mxx+mxwxwst) ymxxst ), variance explained in y indirectly by w through m ((mwxymx)2), and the covariance of the indirect effects of x and w on y through m (
2(mxx+mxwx wst) ymxmwxymxxwst ). This shows that straightforward application of the matrix method results in that quantifies the total variance explained indirectly from all of the
conditional indirect effects on a specific outcome. However, in many cases it is of interest to report an effect size for a specific conditional indirect effect, and obtained here consists of variance explained indirectly from several sources.
is derived for a specific conditional indirect effect using a modification of the matrix method described in Chapter 4. To obtain the specific for the conditional indirect effect
2 2 2
(mxx+mxwxwst) ymxxst , BstMOD is modified by pre- and post- multiplication by an elementary matrix O that replaces the regression coefficient mxx with zero, resulting in a modified matrix of coefficients BstMOD*
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*
0 0 0 0
1 0 0 0 1 0 0 0
0 0 0 0
0 0 0 0 0 0 0 0
0 0
0 0 1 0 0 0 1 0
0 0
0 0 0 1 0 0 0 1
0 0 0 0
0 0 0 0
0 0 0 .
0 0
st st
MOD MOD
st
mx mxw w mw
yx ym
st
mx mxw w
yx ym
=
= +
= +
x x x
x x
x x
x x
B OB O
(6.16)
The matrix of indirect effects MstMOD* is calculated from BstMOD* as
*
0 0 0 0
0 0 0 0
0 0 0 0 ,
( ) 0 0 0
st MOD
st
mx mxw w ym
=
+
x x x
M (6.17)
and *MOD is calculated from MstMOD* as
*
2
0 0 0 0
0 0 0 0
0 0 0 0 .
0 0 0 (( ) )
MOD
st
mx mxw w ym
=
+
x x x
(6.18)
As in Case 1, MOD incorporates products of variances and covariances of product terms, expressed as
2 2 2 2 2 2 2
2 , ,
st st
MOD mx ym mx ymx w xw mx ym ymx w xw x
= x+ x + x x (6.19)
99
where xw2 is the variance of the product term xw, and xw x, is the covariance of x with the product term. In addition, as in Case 1, the conditional variance in y explained indirectly by x through w at the mean of w is equivalent to from the simple three-variable mediation model.
for Cases 1 – 5 can be found in Table 13. For models where the specific and total differ as in Case 2, the specific and total are provided assuming that the indirect effect of x is of primary interest. Derivations for in the remaining Cases 3 – 5 follow the same procedures as in Cases 1 and 2.