Moderated Mediation Analysis by Multiple
Regression
Shu Fai Cheung & Sing-Hang Cheung
2024-10-04
Introduction
This article is a brief illustration of how to use
cond_indirect_effects() from the package manymome (Cheung & Cheung,
2023) to estimate the conditional indirect effects when the model
parameters are estimate by ordinary least squares (OLS) multiple
regression using lm().
Data Set and Model
This is the sample data set used for illustration:
library(manymome)
dat <- data_med_mod_a
print(head(dat), digits = 3)
#> x w m y c1 c2
#> 1 8.58 1.57 28.9 36.9 6.03 4.82
#> 2 10.36 1.10 24.8 24.5 5.19 5.34
#> 3 10.38 2.88 37.3 38.1 4.63 5.02
#> 4 9.53 3.16 32.6 37.9 2.94 6.01
#> 5 11.34 3.84 49.2 59.0 6.12 5.05
#> 6 9.66 2.22 26.4 35.4 4.02 5.03
This dataset has 6 variables: one predictor (x), one
mediators (m), one outcome variable (y), one
moderator (w) and two control variables (c1
and c2).
Suppose this is the model being fitted:
plot of chunk mome_lm_draw_model
The path parameters can be estimated by two multiple regression
models:
lm_m <- lm(m ~ x*w + c1 + c2, dat)
lm_y <- lm(y ~ m + x + c1 + c2, dat)
These are the estimates of the regression coefficient of the
paths:
# ###### Predict m ######
#
summary(lm_m)
#>
#> Call:
#> lm(formula = m ~ x * w + c1 + c2, data = dat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -8.5621 -2.0065 -0.2142 1.7618 10.4270
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 16.4910 12.1039 1.362 0.1763
#> x 0.0959 1.1958 0.080 0.9362
#> w -3.4871 4.7907 -0.728 0.4685
#> c1 0.5372 0.4162 1.291 0.2000
#> c2 -0.1533 0.4211 -0.364 0.7165
#> x:w 0.9785 0.4794 2.041 0.0441 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 3.998 on 94 degrees of freedom
#> Multiple R-squared: 0.7479, Adjusted R-squared: 0.7345
#> F-statistic: 55.79 on 5 and 94 DF, p-value: < 2.2e-16
#
# ###### Predict y ######
#
summary(lm_y)
#>
#> Call:
#> lm(formula = y ~ m + x + c1 + c2, data = dat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -9.4396 -2.8156 -0.3145 2.3231 11.2849
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 4.50635 5.40625 0.834 0.407
#> m 0.95867 0.05806 16.512 <2e-16 ***
#> x -0.01980 0.50578 -0.039 0.969
#> c1 0.68241 0.44110 1.547 0.125
#> c2 -0.49573 0.44565 -1.112 0.269
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 4.229 on 95 degrees of freedom
#> Multiple R-squared: 0.7669, Adjusted R-squared: 0.7571
#> F-statistic: 78.15 on 4 and 95 DF, p-value: < 2.2e-16
Although not mandatory, it is recommended to combine the models into
one object (a system of regression models) using
lm2list():
fit_lm <- lm2list(lm_m, lm_y)
fit_lm
#>
#> The model(s):
#> m ~ x * w + c1 + c2
#> y ~ m + x + c1 + c2
Simply use the lm() outputs as arguments. Order does not
matter. To ensure that the regression outputs can be validly combined,
lm2list() will also check:
whether the same sample is used in all regression analysis (not
just same sample size, but the same set of cases), and
whether the models are “connected”, to ensure that the regression
outputs can be validly combined.
Generating Bootstrap
Estimates
To form nonparametric bootstrap confidence interval for effects to be
computed, do_boot() can be used to generate bootstrap
estimates for all regression coefficients first. These estimates can be
reused for any effects to be estimated.
boot_out_lm <- do_boot(fit_lm,
R = 100,
seed = 54532,
ncores = 1)
Please see vignette("do_boot") or the help page of
do_boot() on how to use this function. In real research,
R, the number of bootstrap samples, should be set to 2000
or even 5000. The argument ncores can usually be omitted
unless users want to manually control the number of CPU cores used in
parallel processing.
Conditional Indirect
Effects
We can now use cond_indirect_effects() to estimate the
indirect effects for different levels of the moderator (w)
and form their bootstrap confidence interval. By reusing the generated
bootstrap estimates, there is no need to repeat the resampling.
Suppose we want to estimate the indirect effect from x
to y through m, conditional on
w:
(Refer to vignette("manymome") and the help page of
cond_indirect_effects() on the arguments.)
out_xmy_on_w <- cond_indirect_effects(wlevels = "w",
x = "x",
y = "y",
m = "m",
fit = fit_lm,
boot_ci = TRUE,
boot_out = boot_out_lm)
out_xmy_on_w
#>
#> == Conditional indirect effects ==
#>
#> Path: x -> m -> y
#> Conditional on moderator(s): w
#> Moderator(s) represented by: w
#>
#> [w] (w) ind CI.lo CI.hi Sig m~x y~m
#> 1 M+1.0SD 3.164 3.060 2.168 4.039 Sig 3.192 0.959
#> 2 Mean 2.179 2.136 1.407 2.925 Sig 2.228 0.959
#> 3 M-1.0SD 1.194 1.212 -0.288 2.564 1.265 0.959
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 100 samples.
#> - The 'ind' column shows the conditional indirect effects.
#> - 'm~x','y~m' is/are the path coefficient(s) along the path conditional
#> on the moderator(s).
When w is one standard deviation below mean, the
indirect effect is 1.212, with 95% confidence interval [-0.288,
2.564].
When w is one standard deviation above mean, the
indirect effect is 3.060, with 95% confidence interval [2.168,
4.039].
Note that any conditional indirect path in the model can be estimated
this way. There is no limit on the path to be estimated, as long as all
required path coefficients are in the model.
cond_indirect_effects() will also check whether a path is
valid. However, for complicated models, structural equation modelling
may be a more flexible approach than multiple regression.
Not covered here, but the index of moderated moderated mediation can
also be estimated in models with two moderators on the same path,
estimated by regression. See vignette("manymome") for an
example.
Standardized
Conditional Indirect effects
The standardized conditional indirect effect from x to
y through m conditional on w can
be estimated by setting standardized_x and
standardized_y to TRUE:
std_xmy_on_w <- cond_indirect_effects(wlevels = "w",
x = "x",
y = "y",
m = "m",
fit = fit_lm,
boot_ci = TRUE,
boot_out = boot_out_lm,
standardized_x = TRUE,
standardized_y = TRUE)
std_xmy_on_w
#>
#> == Conditional indirect effects ==
#>
#> Path: x -> m -> y
#> Conditional on moderator(s): w
#> Moderator(s) represented by: w
#>
#> [w] (w) std CI.lo CI.hi Sig m~x y~m ind
#> 1 M+1.0SD 3.164 0.318 0.220 0.437 Sig 3.192 0.959 3.060
#> 2 Mean 2.179 0.222 0.134 0.309 Sig 2.228 0.959 2.136
#> 3 M-1.0SD 1.194 0.126 -0.031 0.260 1.265 0.959 1.212
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 100 samples.
#> - std: The standardized conditional indirect effects.
#> - ind: The unstandardized conditional indirect effects.
#> - 'm~x','y~m' is/are the path coefficient(s) along the path conditional
#> on the moderator(s).
The standardized indirect effect is 0.126, with 95% confidence
interval [-0.031, 0.260].
More Complicated
Models
After the regression coefficients are estimated,
cond_indirect_effects(), indirect_effect(),
and related functions are used in the same way as for models fitted by
lavaan::sem(). The levels for the moderators are controlled
by mod_levels() and related functions in the same way
whether a model is fitted by lavaan::sem() or
lm(). Pplease refer to other articles (e.g.,
vignette("manymome") and
vignette("mod_levels")) on how to estimate effects in other
model analyzed by multiple regression.
Reference
Cheung, S. F., & Cheung, S.-H. (2023). manymome: An R
package for computing the indirect effects, conditional effects, and
conditional indirect effects, standardized or unstandardized, and their
bootstrap confidence intervals, in many (though not all) models.
Behavior Research Methods. https://doi.org/10.3758/s13428-023-02224-z