Latent Variable Moderation 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 effects of a latent variable when its effect is moderated by another latent variable, given 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).

Only the latent variables fx, fx, and fy will be used in this illustration.

Suppose this is the model being fitted:

The path from fx to fy 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
fy =~ y1 + y2 + y3 + y4

# Structural model:
fy ~ fx + fw + fx:fw
"

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, consult @rosseel_structural_2025 and @rosseel_structural_2024.

This is a summary of the results:

summary(fit,
        ci = TRUE)
#> This is lavaan 0.6-21 -- using the SAM approach to SEM
#> 
#>   SAM method                                     LOCAL
#>   Mapping matrix M method                           ML
#>   Number of measurement blocks                       3
#>   Estimator measurement part                        ML
#>   Estimator  structural part                        ML
#> 
#>   Number of observations                           500
#> 
#> Summary Information Measurement + Structural:
#> 
#>   Block Latent Nind Chisq Df
#>       1     fw    4 3.416  2
#>       2     fx    4 1.782  2
#>       3     fy    4 2.332  2
#> 
#>   Model-based reliability latent variables:
#> 
#>      fx    fw    fy
#>   0.887 0.881 0.831
#> 
#>   Summary Information Structural part:
#> 
#>   chisq df cfi rmsea srmr
#>       0  0   1     0    0
#> 
#> Parameter Estimates:
#> 
#>   Standard errors                            Bootstrap
#>   Number of requested bootstrap draws             2000
#>   Number of successful bootstrap draws            2000
#> 
#> Regressions:
#>                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
#>   fy ~                                                                  
#>     fx                0.208    0.045    4.622    0.000    0.120    0.296
#>     fw               -0.008    0.042   -0.183    0.855   -0.090    0.074
#>     fx:fw             0.288    0.058    5.000    0.000    0.184    0.413
#> 
#> Covariances:
#>                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
#>   fx ~~                                                                 
#>     fw               -0.016    0.033   -0.486    0.627   -0.083    0.047
#>     fx:fw            -0.005    0.049   -0.109    0.913   -0.100    0.091
#>   fw ~~                                                                 
#>     fx:fw             0.036    0.045    0.814    0.415   -0.055    0.121
#> 
#> Intercepts:
#>                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
#>     fx:fw            -0.016    0.033   -0.486    0.627   -0.083    0.047
#> 
#> Variances:
#>                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
#>     fx                0.643    0.064    9.989    0.000    0.520    0.772
#>     fw                0.695    0.065   10.667    0.000    0.564    0.826
#>    .fy                0.381    0.039    9.741    0.000    0.302    0.458
#>     fx:fw             0.436    0.089    4.913    0.000    0.288    0.624

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 Effects

We can now use cond_effects() to estimate the effect of fx on fy 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 conditional effects from fx to fy, conditional on fw:

(Refer to vignette("manymome") and the help page of cond_effects() on the arguments.)

out_xy_on_w <- cond_effects(
  wlevels = "fw",
  x = "fx",
  y = "fy",
  fit = fit,
  boot_ci = TRUE,
  boot_out = boot_out
)
out_xy_on_w
#> 
#> == Conditional effects ==
#> 
#>  Path: fx -> fy
#>  Conditional on moderator(s): fw
#>  Moderator(s) represented by: fw
#> 
#>      [fw]   (fw)    ind  CI.lo CI.hi Sig
#> 1 M+1.0SD  0.834  0.448  0.322 0.593 Sig
#> 2 Mean     0.000  0.208  0.120 0.296 Sig
#> 3 M-1.0SD -0.834 -0.032 -0.154 0.092    
#> 
#>  - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#>    nonparametric bootstrapping with 2000 samples.
#>  - The 'ind' column shows the conditional effects.
#> 

When fw is one standard deviation below mean, the indirect effect is -0.032, with 95% confidence interval [-0.154, 0.092].

When fw is one standard deviation above mean, the indirect effect is 0.448, with 95% confidence interval [0.322, 0.593].

Standardized Conditional Indirect effects

The standardized conditional indirect effect from fx to fy conditional on fw can be estimated by setting standardized_x and standardized_y to TRUE:

std_xy_on_w <- cond_effects(
  wlevels = "fw",
  x = "fx",
  y = "fy",
  fit = fit,
  boot_ci = TRUE,
  boot_out = boot_out,
  standardized_x = TRUE,
  standardized_y = TRUE
)
std_xy_on_w
#> 
#> == Conditional effects ==
#> 
#>  Path: fx -> fy
#>  Conditional on moderator(s): fw
#>  Moderator(s) represented by: fw
#> 
#>      [fw]   (fw)    std  CI.lo CI.hi Sig    ind
#> 1 M+1.0SD  0.834  0.539  0.405 0.696 Sig  0.448
#> 2 Mean     0.000  0.250  0.147 0.345 Sig  0.208
#> 3 M-1.0SD -0.834 -0.038 -0.187 0.112     -0.032
#> 
#>  - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#>    nonparametric bootstrapping with 2000 samples.
#>  - std: The standardized conditional effects. 
#>  - ind: The unstandardized conditional effects.
#> 

When fw is one standard deviation below mean, the standardized indirect effect is -0.038, with 95% confidence interval [-0.187, 0.112].

When fw is one standard deviation above mean, the indirect effect is 0.539, with 95% confidence interval [0.405, 0.696].

Plotting the Conditional Effects

The plot() method can be used on the output of cond_effects().

This is the plot of the conditional effects:

plot(out_xy_on_w)

This is the plot of the standardized conditional effects:

plot(std_xy_on_w)

The two plots are expected to be identical except for the values on the axes. It is because they differ merely only the units of the variables, one standardized and one unstandardized.

Please the help page of the plot() method for the output of cond_effects() on how to customize the plot.

Reference(s)

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