Latent Variable Moderated Mediation by SAM"

In manymome: Mediation, Moderation and Moderated-Mediation After Model Fitting

(This document is for Version 0.3.4.25 or later, currently on GitHub only. The new version is scheduled to be available on CRAN in June 2026.)

Introduction

This article is a brief illustration of how to use cond_indirect_effects() from the package manymome [@cheung_manymome_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_structural_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  0.7251

This 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_structural_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_structural_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_structural_2025 and @rosseel_structural_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].

Reference(s)

manymome documentation built on June 8, 2026, 9:06 a.m.