Latent Variable Moderated Mediation by SAM
Shu Fai Cheung & Sing-Hang Cheung
2026-05-30
Source:vignettes/articles/mome_sam.Rmd
mome_sam.RmdIntroduction
This article is a brief illustration of how to use
cond_indirect_effects() from the package manymome (Cheung & Cheung, 2024) to estimate the
conditional indirect effects among latent variables when the model
parameters are estimated by the SAM (structural-after-measurement)
method by Rosseel et al. (2025).
Data Set and Model
This is the sample data set used for illustration:
library(manymome)
dat <- data_sem_mome
print(head(dat), digits = 3)
#> x1 x2 x3 x4 w1 w2 w3 w4 m1 m2
#> 1 0.594 0.3010 0.836 0.1931 2.354 0.185 0.851 1.145 0.636 0.528
#> 2 -0.656 -0.0448 0.687 0.0455 -1.838 -0.724 -0.157 -1.078 -1.198 -0.675
#> 3 0.302 -0.0953 0.347 -0.2332 0.744 -0.734 1.855 0.264 0.257 -0.159
#> 4 -2.022 -0.1876 -1.483 -0.6256 -0.519 0.202 0.727 0.382 -0.517 -1.221
#> 5 0.922 0.1873 0.381 -0.0365 -1.779 -0.992 -2.368 -1.089 -0.163 -0.982
#> 6 0.873 1.9889 0.956 1.2211 0.680 0.437 1.750 0.940 1.329 0.573
#> m3 m4 y1 y2 y3 y4
#> 1 0.280 0.3634 -0.239 0.924 0.168 0.8163
#> 2 -1.124 0.0211 0.408 -1.062 -0.281 -0.0422
#> 3 -0.294 -0.6764 -0.138 0.914 0.177 0.7359
#> 4 -0.340 0.4333 -0.552 0.334 -0.172 0.1154
#> 5 -0.978 -0.0132 -1.268 -1.430 -0.696 -1.5051
#> 6 1.591 0.4417 -0.284 -0.124 -0.351 0.7251This dataset has indicators of the following four latent variables:
one predictor (fx), one mediators (fm), one
outcome variable (fy), and one moderator
(fw).
Suppose this is the model being fitted:
The path from fx to fm is moderated by
fw, the moderator.
If this model is fitted to the scale scores, then a product term
fx:fw is used to model the moderation.
If the model is for the latent variables, the new approach, SAM
(structural-after-measurement), presented in Rosseel et al. (2025) can be used, using the
function sam() from lavaan. This is the model
syntax:
mod <- "
# Measurement model:
fx =~ x1 + x2 + x3 + x4
fw =~ w1 + w2 + w3 + w4
fm =~ m1 + m2 + m3 + m4
fy =~ y1 + y2 + y3 + y4
# Structural model:
fm ~ fx + fw + fx:fw
fy ~ fm + fx
"The moderation effect is modelled by fx:fw. To fit this
model by SAM, use sam() from lavaan.
As recommended by Rosseel et al. (2025), nonparametric bootstrapping will be used to compute the standard errors and form the confidence intervals.
fit <- sam(
model = mod,
data = data_sem_mome,
se = "bootstrap",
bootstrap.args = list(
R = 2000
),
iseed = 2345,
parallel = "snow",
ncpus = 20
)For details on the SAM approach, see Rosseel et al. (2025) and Rosseel & Loh (2024).
Generating Bootstrap Estimates (Optional)
Although bootstrapping has already been conducted when calling
sam(), it is recommended to call do_boot() to
compute additional statistics, such as implied variances and
covariances, to be used in other functions in manymome.
Otherwise, this step needs to be repeated every time those functions are
called.
boot_out <- do_boot(fit)Because bootstrap estimates have already been stored, only the output
of sam() is sufficient.
Conditional Indirect Effects
We can now use cond_indirect_effects() to estimate the
indirect effects for different levels of the moderator (fw)
and form their bootstrap confidence intervals. Information stored by
do_boot() can be reused. There is no need to repeat the
resampling.
Suppose we want to estimate the indirect effect from fx
to fy through fm, conditional on
fw:
(Refer to vignette("manymome") and the help page of
cond_indirect_effects() on the arguments.)
out_xmy_on_w <- cond_indirect_effects(
wlevels = "fw",
x = "fx",
y = "fy",
m = "fm",
fit = fit,
boot_ci = TRUE,
boot_out = boot_out
)
out_xmy_on_w
#>
#> == Conditional indirect effects ==
#>
#> Path: fx -> fm -> fy
#> Conditional on moderator(s): fw
#> Moderator(s) represented by: fw
#>
#> [fw] (fw) ind CI.lo CI.hi Sig fm~fx fy~fm
#> 1 M+1.0SD 0.834 0.361 0.141 0.568 Sig 0.810 0.446
#> 2 Mean -0.000 0.179 0.069 0.288 Sig 0.401 0.446
#> 3 M-1.0SD -0.834 -0.003 -0.042 0.034 -0.007 0.446
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 2000 samples.
#> - The 'ind' column shows the conditional indirect effects.
#> - 'fm~fx','fy~fm' is/are the path coefficient(s) along the path
#> conditional on the moderator(s).When fw is one standard deviation below mean, the
indirect effect is -0.003, with 95% confidence interval [-0.042,
0.034].
When fw is one standard deviation above mean, the
indirect effect is 0.361, with 95% confidence interval [0.141,
0.568].
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.
Standardized Conditional Indirect effects
The standardized conditional indirect effect from fx to
fy through fm conditional on fw
can be estimated by setting standardized_x and
standardized_y to TRUE:
std_xmy_on_w <- cond_indirect_effects(
wlevels = "fw",
x = "fx",
y = "fy",
m = "fm",
fit = fit,
boot_ci = TRUE,
boot_out = boot_out,
standardized_x = TRUE,
standardized_y = TRUE
)
std_xmy_on_w
#>
#> == Conditional indirect effects ==
#>
#> Path: fx -> fm -> fy
#> Conditional on moderator(s): fw
#> Moderator(s) represented by: fw
#>
#> [fw] (fw) std CI.lo CI.hi Sig fm~fx fy~fm ind
#> 1 M+1.0SD 0.834 0.435 0.170 0.680 Sig 0.810 0.446 0.361
#> 2 Mean -0.000 0.215 0.083 0.342 Sig 0.401 0.446 0.179
#> 3 M-1.0SD -0.834 -0.004 -0.050 0.041 -0.007 0.446 -0.003
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 2000 samples.
#> - std: The standardized conditional indirect effects.
#> - ind: The unstandardized conditional indirect effects.
#> - 'fm~fx','fy~fm' is/are the path coefficient(s) along the path
#> conditional on the moderator(s).When fw is one standard deviation below mean, the
standardized indirect effect is -0.004, with 95% confidence interval
[-0.050, 0.041].
When fw is one standard deviation above mean, the
indirect effect is 0.435, with 95% confidence interval [0.170,
0.680].