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Standardized Moderation Effect in a Path Model by stdmod_lavaan()

Shu Fai Cheung and David Weng Ngai Vong

2024-09-22

Purpose

This document demonstrates how to use stdmod_lavaan() from the package stdmod to compute the standardized moderation effect in a path model fitted by lavaan::sem().

More about this package can be found in vignette("stdmod", package = "stdmod") or at https://sfcheung.github.io/stdmod/.

Setup the Environment

library(stdmod) # For computing the standardized moderation effect conveniently
library(lavaan) # For doing path analysis in lavaan.
#> This is lavaan 0.6-18
#> lavaan is FREE software! Please report any bugs.

Load the Dataset

data(test_mod1)
round(head(test_mod1, 3), 3)
#>       dv     iv    mod    med   cov1   cov2
#> 1 23.879 -0.133 -0.544 10.310 -0.511 -0.574
#> 2 23.096  1.456  1.539 11.384  0.094 -0.264
#> 3 23.201  0.319  1.774  9.615 -0.172  0.488

This test data set has 300 cases, six variables, all continuous.

Fit the Model by `lavaan::sem()`

The product term can be formed manually or by the colon operator, :. stdmod_lavaan() will work in both cases.

This is the model to be tested:

mod <-
"
med ~ iv + mod + iv:mod + cov1
dv ~ med + cov2
"
fit <- sem(mod, test_mod1, fixed.x = FALSE)
summary(fit)
#> lavaan 0.6-18 ended normally after 1 iteration
#> 
#>   Estimator                                         ML
#>   Optimization method                           NLMINB
#>   Number of model parameters                        23
#> 
#>   Number of observations                           300
#> 
#> Model Test User Model:
#>                                                       
#>   Test statistic                                 1.058
#>   Degrees of freedom                                 5
#>   P-value (Chi-square)                           0.958
#> 
#> Parameter Estimates:
#> 
#>   Standard errors                             Standard
#>   Information                                 Expected
#>   Information saturated (h1) model          Structured
#> 
#> Regressions:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>   med ~                                               
#>     iv                0.221    0.030    7.264    0.000
#>     mod               0.104    0.030    3.489    0.000
#>     iv:mod            0.257    0.025   10.169    0.000
#>     cov1              0.104    0.025    4.099    0.000
#>   dv ~                                                
#>     med               0.246    0.041    5.962    0.000
#>     cov2              0.191    0.023    8.324    0.000
#> 
#> Covariances:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>   iv ~~                                               
#>     mod               0.481    0.063    7.606    0.000
#>     iv:mod           -0.149    0.059   -2.501    0.012
#>     cov1             -0.033    0.058   -0.575    0.565
#>     cov2             -0.071    0.059   -1.216    0.224
#>   mod ~~                                              
#>     iv:mod           -0.180    0.062   -2.923    0.003
#>     cov1             -0.060    0.059   -1.010    0.313
#>     cov2             -0.107    0.061   -1.763    0.078
#>   iv:mod ~~                                           
#>     cov1             -0.051    0.061   -0.837    0.403
#>     cov2              0.063    0.063    1.001    0.317
#>   cov1 ~~                                             
#>     cov2              0.071    0.061    1.158    0.247
#> 
#> Variances:
#>                    Estimate  Std.Err  z-value  P(>|z|)
#>    .med               0.201    0.016   12.247    0.000
#>    .dv                0.169    0.014   12.247    0.000
#>     iv                0.954    0.078   12.247    0.000
#>     mod               1.017    0.083   12.247    0.000
#>     iv:mod            1.088    0.089   12.247    0.000
#>     cov1              1.039    0.085   12.247    0.000
#>     cov2              1.076    0.088   12.247    0.000

The results show that mod significantly moderates the effect of iv on med.

Compute the Standardized Moderation Effect

As in the case of regression, the coefficient of iv:mod in the standardized solution is not the desired standardized coefficient because it standardizes the product term.

standardizedSolution(fit)[3, ]
#>   lhs op    rhs est.std    se      z pvalue ci.lower ci.upper
#> 3 med  ~ iv:mod   0.466 0.043 10.842      0    0.382     0.55

After fitting the path model by lavaan::lavaan(), we can use stdmod_lavaan() to compute the standardized moderation effect using the standard deviations of the focal variable, the moderator, and the outcome variable (Cheung, Cheung, Lau, Hui, & Vong, 2022).

The minimal arguments are:

fit: The output from lavaan::lavaan() and its wrappers, such as lavaan::sem().
x: The focal variable, the variable with its effect on the outcome variable being moderated.
y: The outcome variable.
w: The moderator.
x_w: The product term.

fit_iv_mod_std <- stdmod_lavaan(fit = fit,
                                x = "iv",
                                y = "med",
                                w = "mod",
                                x_w = "iv:mod")
fit_iv_mod_std
#> 
#> Call:
#> stdmod_lavaan(fit = fit, x = "iv", y = "med", w = "mod", x_w = "iv:mod")
#> 
#>                  Variable
#> Focal Variable         iv
#> Moderator             mod
#> Outcome Variable      med
#> Product Term       iv:mod
#> 
#>              lhs op    rhs   est    se      z pvalue ci.lower ci.upper
#> Original     med  ~ iv:mod 0.257 0.025 10.169      0    0.208    0.307
#> Standardized med  ~ iv:mod 0.440    NA     NA     NA       NA       NA

The standardized moderation effect of mod on the iv-med path is 0.440.

Form Bootstrap Confidence Interval

stdmod_lavaan() can also be used to form nonparametric bootstrap confidence interval for the standardized moderation effect.

There are two approaches to do this. First, if bootstrap confidence intervals was requested when fitting the model, the stored bootstrap estimates will be used. This is efficient because there is no need to do bootstrapping again.

We fit the model again, with bootstrapping:

fit <- sem(mod, test_mod1, fixed.x = FALSE,
           se = "boot",
           bootstrap = 2000,
           iseed = 987543)

If bootstrapping has been done when fitting the model, just adding boot_ci = TRUE is enough to request nonparametric percentile bootstrap confidence interval:

fit_iv_mod_std_ci <- stdmod_lavaan(fit = fit,
                                   x = "iv",
                                   y = "med",
                                   w = "mod",
                                   x_w = "iv:mod",
                                   boot_ci = TRUE)
fit_iv_mod_std_ci
#> 
#> Call:
#> stdmod_lavaan(fit = fit, x = "iv", y = "med", w = "mod", x_w = "iv:mod", 
#>     boot_ci = TRUE)
#> 
#>                  Variable
#> Focal Variable         iv
#> Moderator             mod
#> Outcome Variable      med
#> Product Term       iv:mod
#> 
#>              lhs op    rhs   est    se     z pvalue ci.lower ci.upper
#> Original     med  ~ iv:mod 0.257 0.035 7.298      0    0.184    0.322
#> Standardized med  ~ iv:mod 0.440    NA    NA     NA    0.322    0.539
#> 
#> Confidence interval of standardized moderation effect:
#> - Level of confidence: 95%
#> - Bootstrapping Method: Nonparametric
#> - Type: Percentile
#> - Number of bootstrap samples requests: 
#> - Number of bootstrap samples with valid results: 2000
#> 
#> NOTE: Bootstrapping conducted by the method in 0.2.7.5 or later. To use
#> the method in the older versions for reproducing previous results, set
#> 'use_old_version' to 'TRUE'.

The 95% confidence interval of the standardized moderation effect is 0.322 to 0.539.

The second approach, not covered here, uses do_boot() from the manymome package. to generate bootstrap estimates. To use the stored bootstrap estimates, set boot_out to the output of do_boot(). The stored bootstrap estimates will then be used. This method can be used when non-bootstrapping confidence intervals are needed when fitting the model.

Remarks

The function stdmod_lavaan() can be used for more complicated path models. The computation of the standardized moderation effect in a path model depends only on the standard deviations of the three variables involved (x, w, and y).

Reference(s)

The computation of the standardized moderation effect is based on the simple formula presented in the following manuscript, using the standard deviations of the outcome variable, focal variable, and the moderator:

Cheung, S. F., Cheung, S.-H., Lau, E. Y. Y., Hui, C. H., & Vong, W. N. (2022) Improving an old way to measure moderation effect in standardized units. Health Psychology, 41(7), 502-505. https://doi.org/10.1037/hea0001188.

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