---
title: "Beta-Select Demonstration: SEM by 'lavaan'"
date: "2025-05-01"
output:
rmarkdown::html_vignette:
number_sections: true
vignette: >
%\VignetteIndexEntry{Beta-Select Demonstration: SEM by 'lavaan'}
%\VignetteEngine{knitr::rmarkdown}
%\VignetteEncoding{UTF-8}
bibliography: "references.bib"
csl: apa.csl
---
# Introduction
This article demonstrates how to use
`lav_betaselect()` from the package
[`betaselectr`](https://sfcheung.github.io/betaselectr/)
to standardize
selected variables in a model fitted
by `lavaan` and forming confidence
intervals for the parameters.
# Data and Model
The sample dataset from the package
`betaselectr` will be used in this
demonstration:
``` r
library(betaselectr)
head(data_test_medmod)
#> dv iv mod med cov1 cov2
#> 1 7.487873 11.42573 16.65805 42.28988 54.14051 15.56069
#> 2 8.474931 16.64790 22.66332 42.08692 39.21125 17.61286
#> 3 11.206539 14.81278 22.80955 32.76869 31.97963 20.77333
#> 4 10.148827 15.79632 22.94451 43.96807 42.72187 15.66971
#> 5 7.421606 14.29621 24.51562 37.10942 42.74174 21.97132
#> 6 6.846435 12.00819 25.22163 35.46051 30.85914 22.35444
```
Although mean-centering is not
necessary for testing an interaction,
the computation of the standardized
coefficients of variables involved
in an interaction requires that they
are mean-centered, for now.
We mean-center `iv` and `mod` first:
``` r
data_test_medmod$iv <- data_test_medmod$iv - mean(data_test_medmod$iv)
data_test_medmod$mod <- data_test_medmod$mod - mean(data_test_medmod$mod)
```
This is the path model, fitted by
`lavaan::sem()`:
``` r
library(lavaan)
#> This is lavaan 0.6-19
#> lavaan is FREE software! Please report any bugs.
mod <-
"
med ~ iv + mod + iv:mod + cov1 + cov2
dv ~ med + iv + cov1 + cov2
"
fit <- sem(mod,
data_test_medmod)
```
The model has a moderator, `mod`, posited
to moderate the effect from `iv` to
`med`. The product term is `iv:mod`.
These are the results:
``` r
summary(fit)
#> lavaan 0.6-19 ended normally after 1 iteration
#>
#> Estimator ML
#> Optimization method NLMINB
#> Number of model parameters 11
#>
#> Number of observations 200
#>
#> Model Test User Model:
#>
#> Test statistic 2.303
#> Degrees of freedom 2
#> P-value (Chi-square) 0.316
#>
#> Parameter Estimates:
#>
#> Standard errors Standard
#> Information Expected
#> Information saturated (h1) model Structured
#>
#> Regressions:
#> Estimate Std.Err z-value P(>|z|)
#> med ~
#> iv 0.626 0.215 2.909 0.004
#> mod 0.329 0.129 2.560 0.010
#> iv:mod 0.286 0.039 7.340 0.000
#> cov1 -0.093 0.070 -1.327 0.185
#> cov2 0.242 0.133 1.823 0.068
#> dv ~
#> med 0.092 0.011 8.098 0.000
#> iv 0.227 0.038 5.896 0.000
#> cov1 -0.006 0.013 -0.454 0.650
#> cov2 0.030 0.025 1.230 0.219
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|)
#> .med 60.292 6.029 10.000 0.000
#> .dv 2.087 0.209 10.000 0.000
```
# Problems With Standardization
We can request the standardized solution
using `lavaan::standardizedSolution()`:
``` r
standardizedSolution(fit,
output = "text")
#>
#> Regressions:
#> est.std Std.Err z-value P(>|z|) ci.lower ci.upper
#> med ~
#> iv 0.182 0.062 2.964 0.003 0.062 0.303
#> mod 0.165 0.064 2.597 0.009 0.040 0.290
#> iv:mod 0.432 0.051 8.390 0.000 0.331 0.533
#> cov1 -0.077 0.058 -1.332 0.183 -0.189 0.036
#> cov2 0.105 0.057 1.836 0.066 -0.007 0.218
#> dv ~
#> med 0.459 0.052 8.845 0.000 0.357 0.560
#> iv 0.331 0.053 6.279 0.000 0.228 0.434
#> cov1 -0.024 0.054 -0.454 0.650 -0.130 0.081
#> cov2 0.066 0.054 1.233 0.218 -0.039 0.171
#>
#> Variances:
#> est.std Std.Err z-value P(>|z|) ci.lower ci.upper
#> .med 0.656 0.050 13.243 0.000 0.559 0.753
#> .dv 0.569 0.050 11.353 0.000 0.471 0.667
```
However, for this model, there are several
problems:
- The product term, `iv:mod`, is also
standardized. This is inappropriate.
One simple but underused solution is
to standardize the variables *before*
forming the product term [@friedrich_defense_1982].
- The confidence intervals are formed using
the delta-method, which has been found
to be inferior to methods such as
nonparametric percentile bootstrap
confidence interval for the standardized
solution [@falk_are_2018]. Although there
are situations in which the delta-method
confidence and the nonparametric
percentile bootstrap confidences can be
similar (e.g., sample size is large
and the sample estimates are not extreme),
it is still safe to at least try both
methods and compare the results.
- There are cases in which some variables
are measured by meaningful units and
do not need to be standardized. for
example, if `cov1` is age measured by
year, then age is more
meaningful than "standardized age".
- In path analysis, categorical variables
are usually represented by dummy variables,
each of them having only two possible
values (0 or 1). It is not meaningful
to standardize the dummy variables.
# Beta-Select `lav_betaselect()`
The function `lav_betaselect()` can be used
to solve this problem by:
- standardizing variables before product
terms are formed,
- standardizing only variables for which
standardization can facilitate
interpretation, and
- forming confidence intervals that take
into account selected standardization.
We call the coefficients computed by
this kind of standardization *beta*s-select
($\beta{s}_{Select}$, $\beta_{Select}$
in singular form),
to differentiate them from coefficients
computed by standardizing all variables,
including product terms.
## Estimates Only
Suppose we only need to
solve the first problem, with the product
term computed after `iv` and `mod`
are standardized:
``` r
fit_beta <- lav_betaselect(fit)
```
``` r
fit_beta
```
This is the output
if printed using the
default options:
```
#>
#> Selected Standardization:
#>
#> Standard Error: Nil
#>
#> Parameter Estimates Settings:
#>
#> Standard errors: Standard
#> Information: Expected
#> Information saturated (h1) model: Structured
#>
#> Regressions:
#> BetaSelect
#> med ~
#> iv 0.182
#> mod 0.165
#> iv:mod 0.400
#> cov1 -0.077
#> cov2 0.105
#> dv ~
#> med 0.459
#> iv 0.331
#> cov1 -0.024
#> cov2 0.066
#>
#> Footnote:
#> - Variable(s) standardized: cov1, cov2, dv, iv, med, mod
#> - Call 'print()' and set 'standardized_only' to 'FALSE' to print both
#> original estimates and betas-select.
#> - Product terms (iv:mod) have variables standardized before computing
#> them. The product term(s) is/are not standardized.
```
Compared to the solution with the product
term standardized, the coefficient of
`iv:mod` changed substantially from
0.432 to
0.286. As shown by
@cheung_improving_2022, the coefficient
of *standardized* product term (`iv:mod`)
can be substantially different from the
properly standardized product term
(the product of standardized `iv` and
standardized `mod`).
The footnote will also indicate
variables that are standardized,
and remarked that product terms
are formed *after* standardization.
## Estimates and Bootstrap Confidence Interval
Suppose we want to address
both the first and the second problems,
with
- the product term computed after `iv` and `mod`
standardized, and
- bootstrap confidence intervals used, that
take into account the sampling variation
of the standardizers (the standard deviations).
We can call `lav_betaselect()`
again, with additional arguments
set:
``` r
fit_beta <- lav_betaselect(fit,
std_se = "bootstrap",
bootstrap = 5000,
iseed = 2345,
parallel = "snow",
ncpus = 20)
#> Finding product terms in the model ...
#> Finished finding product terms.
#>
#> Compute bootstrapping standardized solution:
```
These are the additional arguments:
- `std_se`: The method to compute the
standard errors as well as confidence
intervals. Set to `"bootstrap"` for
nonparametric bootstrapping.
- `iseed`: The seed for the random number
generator used for bootstrapping. Set
this to an integer to make the results
reproducible.
- `parallel`: The method to be used for
parallel processing. It will be passed
to `lavaan::bootstrapLavaan()`. Supported
values are `"none"`, `"snow"`, and
`"multicore"`.
- `ncpus`: The number of CPU cores to
use if `parallel` processing is not
`"none"`. Default is
`parallel::detectCores(logical = FALSE) - 1`,
or the number of *physical* cores
minus one.
This is the output if
printed with default
options:
``` r
fit_beta
```
```
#>
#> Selected Standardization:
#>
#> Standard Error: Nonparametric bootstrap
#> Bootstrap samples: 5000
#> Confidence Interval: Percentile
#> Level of Confidence: 95.0%
#>
#> Parameter Estimates Settings:
#>
#> Standard errors: Standard
#> Information: Expected
#> Information saturated (h1) model: Structured
#>
#> Regressions:
#> BetaSelect SE Z p-value Sig CI.Lo CI.Hi CI.Sig
#> med ~
#> iv 0.182 0.061 2.992 0.005 ** 0.063 0.300 Sig.
#> mod 0.165 0.069 2.389 0.015 * 0.028 0.303 Sig.
#> iv:mod 0.400 0.047 8.565 0.000 *** 0.298 0.481 Sig.
#> cov1 -0.077 0.057 -1.353 0.185 -0.186 0.038 n.s.
#> cov2 0.105 0.061 1.725 0.094 . -0.019 0.219 n.s.
#> dv ~
#> med 0.459 0.052 8.828 0.000 *** 0.348 0.553 Sig.
#> iv 0.331 0.051 6.480 0.000 *** 0.229 0.431 Sig.
#> cov1 -0.024 0.058 -0.418 0.686 -0.137 0.090 n.s.
#> cov2 0.066 0.058 1.139 0.259 -0.050 0.178 n.s.
#>
#> Footnote:
#> - Variable(s) standardized: cov1, cov2, dv, iv, med, mod
#> - Sig codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> - Standard errors, p-values, and confidence intervals are not computed
#> for betas-select which are fixed in the standardized solution.
#> - P-values for betas-select are asymmetric bootstrap p-value computed
#> by the method of Asparouhov and Muthén (2021).
#> - Call 'print()' and set 'standardized_only' to 'FALSE' to print both
#> original estimates and betas-select.
#> - Product terms (iv:mod) have variables standardized before computing
#> them. The product term(s) is/are not standardized.
```
In this dataset, with 200 cases,
the delta-method confidence
intervals are close to the
bootstrap confidence intervals,
except obviously for the
product term because the coefficient
of the product term has substantially
different values in the two
solutions.
## Estimates and Bootstrap Confidence Intervals, With Only Selected Variables Standardized
Suppose we want to address also the
the third issue, and standardize only
some of the variables. This can be
done using either `to_standardize`
or `not_to_standardize`.
- Use `to_standardize` when
the number of variables to standardize
is much fewer than the number of variables
not to standardize.
- Use `not_to_standardize`
when the number variables to standardize
is much more than the
the number of variables not to standardize.
For example, suppose we only
need to standardize `dv` and
`iv`, `cov1`, and `cov2`,
this is the call to do
this, setting
`to_standardize` to `c("iv", "dv", "cov1", "cov2")`:
``` r
fit_beta_select_1 <- lav_betaselect(fit,
std_se = "bootstrap",
to_standardize = c("iv", "dv", "cov1", "cov2"),
bootstrap = 5000,
iseed = 2345,
parallel = "snow",
ncpus = 20)
```
If we want to standardize all
variables except for `dv`
and `mod`, we can use
this call, and set
`not_to_standardize` to `c("mod", "dv")`:
``` r
fit_beta_select_2 <- lav_betaselect(fit,
std_se = "bootstrap",
not_to_standardize = c("mod", "dv"),
bootstrap = 5000,
iseed = 2345,
parallel = "snow",
ncpus = 20)
```
The results of these calls are identical,
and only those of the second version are
printed:
``` r
fit_beta_select_2
```
```
#> Selected Standardization:
#>
#> Standard Error: Nonparametric bootstrap
#> Bootstrap samples: 5000
#> Confidence Interval: Percentile
#> Level of Confidence: 95.0%
#>
#> Parameter Estimates Settings:
#>
#> Standard errors: Standard
#> Information: Expected
#> Information saturated (h1) model: Structured
#>
#> Regressions:
#> BetaSelect SE Z p-value Sig CI.Lo CI.Hi CI.Sig
#> med ~
#> iv 0.182 0.061 2.992 0.005 ** 0.063 0.300 Sig.
#> mod 0.034 0.014 2.389 0.015 * 0.006 0.063 Sig.
#> iv:mod 0.083 0.010 8.565 0.000 *** 0.062 0.100 Sig.
#> cov1 -0.077 0.057 -1.353 0.185 -0.186 0.038 n.s.
#> cov2 0.105 0.061 1.725 0.094 . -0.019 0.219 n.s.
#> dv ~
#> med 0.878 0.116 7.567 0.000 *** 0.635 1.092 Sig.
#> iv 0.634 0.100 6.337 0.000 *** 0.430 0.826 Sig.
#> cov1 -0.047 0.112 -0.418 0.686 -0.265 0.168 n.s.
#> cov2 0.126 0.111 1.137 0.259 -0.093 0.345 n.s.
#>
#> Footnote:
#> - Variable(s) standardized: cov1, cov2, iv, med
#> - Sig codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> - Standard errors, p-values, and confidence intervals are not computed
#> for betas-select which are fixed in the standardized solution.
#> - P-values for betas-select are asymmetric bootstrap p-value computed
#> by the method of Asparouhov and Muthén (2021).
#> - Call 'print()' and set 'standardized_only' to 'FALSE' to print both
#> original estimates and betas-select.
#> - Product terms (iv:mod) have variables standardized before computing
#> them. The product term(s) is/are not standardized.
```
The footnotes show that, by
specifying that `dv` and `mod` are not
standardized, all the other four variables
are standardized: `iv`, `med`, `cov1`, and `cov2`.
Therefore, in this case, it is more
convenient to use `not_to_standardize`.
When reporting *beta*s-*select*, researchers need
to state which variables
are standardized and which are not.
This can be done in table notes,
or in a column of the parameter estimate
tables. The output can of `lav_betaselect()`
can be printed with `show_Bs.by` set
to `TRUE` to demonstrate the second
approach:
``` r
print(fit_beta_select_2,
show_Bs.by = TRUE)
```
```
#> Regressions:
#> BetaSelect SE Z p-value Sig CI.Lo CI.Hi CI.Sig Selected
#> med ~
#> iv 0.182 0.061 2.992 0.005 ** 0.063 0.300 Sig. iv,med
#> mod 0.034 0.014 2.389 0.015 * 0.006 0.063 Sig. med
#> iv:mod 0.083 0.010 8.565 0.000 *** 0.062 0.100 Sig. iv,med
#> cov1 -0.077 0.057 -1.353 0.185 -0.186 0.038 n.s. cov1,med
#> cov2 0.105 0.061 1.725 0.094 . -0.019 0.219 n.s. cov2,med
#> dv ~
#> med 0.878 0.116 7.567 0.000 *** 0.635 1.092 Sig. med
#> iv 0.634 0.100 6.337 0.000 *** 0.430 0.826 Sig. iv
#> cov1 -0.047 0.112 -0.418 0.686 -0.265 0.168 n.s. cov1
#> cov2 0.126 0.111 1.137 0.259 -0.093 0.345 n.s. cov2
#>
#> Footnote:
#> - Variable(s) standardized: cov1, cov2, iv, med
#> - Sig codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> - Standard errors, p-values, and confidence intervals are not computed
#> for betas-select which are fixed in the standardized solution.
#> - P-values for betas-select are asymmetric bootstrap p-value computed
#> by the method of Asparouhov and Muthén (2021).
#> - Call 'print()' and set 'standardized_only' to 'FALSE' to print both
#> original estimates and betas-select.
#> - The column 'Selected' lists variable(s) standardized when computing
#> the standardized coefficient of a parameter. ('NA' for user-defined
#> parameters because they are computed from other standardized
#> parameters.)
#> - Product terms (iv:mod) have variables standardized before computing
#> them. The product term(s) is/are not standardized.
```
## Categorical Variables
When calling `lav_betaselect()`,
variables with only two values in
the dataset are assumed to be categorical
and will not be standardized by default.
This can be overriden by setting
`skip_categorical_x` to `FALSE`, though
not recommended.
# Conclusion
In structural equation modeling, there
are situations in which standardizing
all variables is not appropriate, or
when standardization needs to be done
before forming product terms. We are
not aware of tools that can do appropriate
standardization *and* form confidence
intervals that takes into account the
selective standardization. By promoting
the use of *beta*s-*select* using
`lav_betaselect()`, we hope to make it
easier for researchers to do appropriate
Standardization in when reporting SEM
results.
# References