---
title: "Beta-Select Demonstration: Logistic Regression by `glm()`"
date: "2024-11-08"
output:
rmarkdown::html_vignette:
number_sections: true
vignette: >
%\VignetteIndexEntry{Beta-Select Demonstration: Logistic Regression by `glm()`}
%\VignetteEngine{knitr::rmarkdown}
%\VignetteEncoding{UTF-8}
bibliography: "references.bib"
csl: apa.csl
---
# Introduction
This article demonstrates how to use
`glm_betaselect()` from the package
[`betaselectr`](https://sfcheung.github.io/betaselectr/)
to standardize
selected variables in a model fitted
by `glm()` and forming confidence
intervals for the parameters.
Logistic regression is used in this
illustration.
# Data and Model
The sample dataset from the package
`betaselectr` will be used for in this
demonstration:
``` r
library(betaselectr)
head(data_test_mod_cat_binary)
#> dv iv mod cov1 cat1
#> 1 1 16.67 51.76 18.38 gp2
#> 2 1 17.36 56.85 21.52 gp3
#> 3 1 14.50 46.49 21.52 gp2
#> 4 0 16.16 48.25 16.28 gp3
#> 5 0 9.61 42.95 15.89 gp1
#> 6 0 13.14 48.65 21.03 gp3
```
This is the logistic regression model, fitted by
`glm()`:
``` r
glm_out <- glm(dv ~ iv * mod + cov1 + cat1,
data = data_test_mod_cat_binary,
family = binomial())
```
The model has a moderator, `mod`, posited
to moderate the effect from `iv` to
`med`. The product term is `iv:mod`.
The variable `cat1` is a categorical variable
with three groups: `gp1`, `gp2`, `gp3`.
These are the results:
``` r
summary(glm_out)
#>
#> Call:
#> glm(formula = dv ~ iv * mod + cov1 + cat1, family = binomial(),
#> data = data_test_mod_cat_binary)
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 24.36566 9.83244 2.478 0.013209 *
#> iv -1.83370 0.67576 -2.714 0.006657 **
#> mod -0.52322 0.19848 -2.636 0.008385 **
#> cov1 -0.02286 0.06073 -0.376 0.706562
#> cat1gp2 0.89002 0.36257 2.455 0.014100 *
#> cat1gp3 1.28291 0.34448 3.724 0.000196 ***
#> iv:mod 0.03815 0.01364 2.797 0.005163 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for binomial family taken to be 1)
#>
#> Null deviance: 415.03 on 299 degrees of freedom
#> Residual deviance: 390.91 on 293 degrees of freedom
#> AIC: 404.91
#>
#> Number of Fisher Scoring iterations: 4
```
# Problems With Standardization
In logistic regression, there are several
ways to do standardization [@menard_six_2004].
We use the
same approach in linear regression and
standardize all variables, except for the
binary response variable.
First, all variables in the model,
including the product term and dummy
variables, are computed:
``` r
data_test_mod_cat_binary_z <- data_test_mod_cat_binary
data_test_mod_cat_binary_z$iv_x_mod <- data_test_mod_cat_binary_z$iv *
data_test_mod_cat_binary_z$mod
data_test_mod_cat_binary_z$cat_gp2 <- as.numeric(data_test_mod_cat_binary_z$cat1 == "gp2")
data_test_mod_cat_binary_z$cat_gp3 <- as.numeric(data_test_mod_cat_binary_z$cat1 == "gp3")
head(data_test_mod_cat_binary_z)
#> dv iv mod cov1 cat1 iv_x_mod cat_gp2 cat_gp3
#> 1 1 16.67 51.76 18.38 gp2 862.8392 1 0
#> 2 1 17.36 56.85 21.52 gp3 986.9160 0 1
#> 3 1 14.50 46.49 21.52 gp2 674.1050 1 0
#> 4 0 16.16 48.25 16.28 gp3 779.7200 0 1
#> 5 0 9.61 42.95 15.89 gp1 412.7495 0 0
#> 6 0 13.14 48.65 21.03 gp3 639.2610 0 1
```
All the variables are then standardized:
``` r
data_test_mod_cat_binary_z <- data.frame(scale(data_test_mod_cat_binary_z[, -5]))
data_test_mod_cat_binary_z$dv <- data_test_mod_cat_binary$dv
head(data_test_mod_cat_binary_z)
#> dv iv mod cov1 iv_x_mod cat_gp2 cat_gp3
#> 1 1 0.8347403 0.4632131 -0.7895117 0.8142500 1.4553064 -0.9591663
#> 2 1 1.1648852 1.6757589 0.7727941 1.6887948 -0.6848501 1.0390968
#> 3 1 -0.2035415 -0.7922125 0.7727941 -0.5160269 1.4553064 -0.9591663
#> 4 0 0.5907202 -0.3729432 -1.8343660 0.2283915 -0.6848501 1.0390968
#> 5 0 -2.5432642 -1.6355154 -2.0284103 -2.3581688 -0.6848501 -0.9591663
#> 6 0 -0.8542619 -0.2776547 0.5289948 -0.7616218 -0.6848501 1.0390968
```
The logistic regression model is then fitted to the
standardized variables:
``` r
glm_std_common <- glm(dv ~ iv + mod + cov1 + cat_gp2 + cat_gp3 + iv_x_mod,
data = data_test_mod_cat_binary_z,
family = binomial())
```
The "betas" commonly reported are the
coefficients in this model:
``` r
glm_std_common_summary <- summary(glm_std_common)
printCoefmat(glm_std_common_summary$coefficients,
digits = 5,
zap.ind = 1,
P.values = TRUE,
has.Pvalue = TRUE,
signif.stars = TRUE)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) -0.11220 0.12083 -0.9284 0.353184
#> iv -3.83240 1.41234 -2.7135 0.006657 **
#> mod -2.19640 0.83316 -2.6362 0.008385 **
#> cov1 -0.04600 0.12206 -0.3765 0.706562
#> cat_gp2 0.41590 0.16942 2.4547 0.014100 *
#> cat_gp3 0.64200 0.17239 3.7242 0.000196 ***
#> iv_x_mod 5.41270 1.93542 2.7967 0.005163 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
However, for this model, there are several
problems:
- The product term, `iv:mod`, is also
standardized (`iv_x_mod`, computed
using the standard deviations of
`dv` and `iv:mod`).
This is inappropriate.
One simple but underused solution is
standardizing the variables *before*
forming the product term [see @friedrich_defense_1982
on the case of linear regression].
- The default confidence intervals are formed using
profiling in `glm()`. It does allow for
asymmetry. However, it does not
take into account the sampling variation
of the standardizers (the sample standard
deviations used in standardization).
It is unclear whether it will be
biased, as in the case of OLS standard
error [@yuan_biases_2011].
- 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 regression models, 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 by `glm_betaselect()`
The function `glm_betaselect()` can be used
to solve these problems by:
- standardizing variables before product
terms are formed,
- standardizing only variables for which
standardization can facilitate
interpretation, and
- forming bootstrap 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, standardizing all
numeric variables except for the
response variable (which is binary),
with the product
term computed after `iv` and `mod`
are standardized.
``` r
glm_beta_select <- glm_betaselect(dv ~ iv*mod + cov1 + cat1,
data = data_test_mod_cat_binary,
skip_response = TRUE,
family = binomial(),
do_boot = FALSE)
```
The function `glm_beta_iv_mod()` can be
used as `glm()`, with applicable arguments
such as the model formula and `data` passed
to `glm()`.
By default, *all* numeric variables will
be standardized before fitting the models.
Terms such as product terms are created
*after* standardization.
For `glm()`, standardizing the outcome
variable (`dv` in this example) may not
be meaningful or may even be not allowed.
In the case of logistic regression, the
outcome variable need to be 0 or 1 only.
Therefore, `skip_response` is set to
`TRUE`, to request that the response
(outcome) variable is *not* standardized.
Moreover, categorical variables (factors and
string variables) will not be standardized.
Bootstrapping is done by default. In this
illustration, `do_boot = FALSE` is added
to disable it because we only want to
address the first problem. We will do bootstrapping when
addressing the issue with confidence intervals.
The `summary()` method can be used
ont the output of `glm_betaselect()`:
``` r
summary(glm_beta_select)
#> Waiting for profiling to be done...
#> Call to glm_betaselect():
#> betaselectr::lm_betaselect(formula = dv ~ iv * mod + cov1 + cat1,
#> family = binomial(), data = data_test_mod_cat_binary, skip_response = TRUE,
#> do_boot = FALSE, model_call = "glm")
#>
#> Variable(s) standardized: iv, mod, cov1
#>
#> Call:
#> stats::glm(formula = dv ~ iv * mod + cov1 + cat1, family = binomial(),
#> data = betaselectr::std_data(data = data_test_mod_cat_binary,
#> to_standardize = c("iv", "mod", "cov1")))
#>
#> Coefficients:
#> Estimate CI.Lower CI.Upper Std. Error z value Pr(>|z|)
#> (Intercept) -1.158 -1.783 -0.584 0.304 -3.807 < 0.001 ***
#> iv 0.140 -0.125 0.409 0.136 1.027 0.30449
#> mod 0.194 -0.080 0.474 0.141 1.376 0.16878
#> cov1 -0.046 -0.287 0.193 0.122 -0.376 0.70656
#> cat1gp2 0.890 0.193 1.620 0.363 2.455 0.01410 *
#> cat1gp3 1.283 0.625 1.981 0.344 3.724 < 0.001 ***
#> iv:mod 0.335 0.108 0.578 0.120 2.797 0.00516 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for binomial family taken to be 1)
#>
#> Null deviance: 415.03 on 299 degrees of freedom
#> Residual deviance: 390.91 on 293 degrees of freedom
#> AIC: 404.9
#>
#> Number of Fisher Scoring iterations: 4
#>
#> Transformed Parameter Estimates:
#> Exp(B) CI.Lower CI.Upper
#> (Intercept) 0.314 0.168 0.558
#> iv 1.150 0.882 1.506
#> mod 1.214 0.923 1.607
#> cov1 0.955 0.751 1.213
#> cat1gp2 2.435 1.213 5.052
#> cat1gp3 3.607 1.868 7.251
#> iv:mod 1.398 1.114 1.782
#>
#> Note:
#> - Results *after* standardization are reported.
#> - Standard errors are least-squares standard errors.
#> - Z values are computed by 'Estimate / Std. Error'.
#> - P-values are usual z-test p-values.
#> - Default standard errors, z values, p-values, and confidence intervals
#> (if reported) should not be used for coefficients involved in
#> standardization.
#> - Default 95.0% confidence interval reported.
```
Compared to the solution with the product
term standardized, the coefficient of
`iv:mod` changed substantially from
5.413 to
0.335. Similar
to the case of linear regression
[@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`).
## 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` are
standardized, and
- bootstrap confidence interval used.
We can call `glm_betaselect()`
again, with additional arguments
set:
``` r
glm_beta_select_boot <- glm_betaselect(dv ~ iv*mod + cov1 + cat1,
data = data_test_mod_cat_binary,
family = binomial(),
skip_response = TRUE,
bootstrap = 5000,
iseed = 4567)
```
These are the additional arguments:
- `bootstrap`: The number of bootstrap
samples to draw. Default is 100. It should
be set to 5000 or even 10000.
- `iseed`: The seed for the random number
generator used for bootstrapping. Set
this to an integer to make the results
reproducible.
This is the output of `summary()`
``` r
summary(glm_beta_select_boot)
#> Call to glm_betaselect():
#> betaselectr::lm_betaselect(formula = dv ~ iv * mod + cov1 + cat1,
#> family = binomial(), data = data_test_mod_cat_binary, skip_response = TRUE,
#> bootstrap = 5000, iseed = 4567, model_call = "glm")
#>
#> Variable(s) standardized: iv, mod, cov1
#>
#> Call:
#> stats::glm(formula = dv ~ iv * mod + cov1 + cat1, family = binomial(),
#> data = betaselectr::std_data(data = data_test_mod_cat_binary,
#> to_standardize = c("iv", "mod", "cov1")))
#>
#> Coefficients:
#> Estimate CI.Lower CI.Upper Std. Error z value Pr(Boot)
#> (Intercept) -1.158 -1.869 -0.598 0.322 -3.598 <0.001 ***
#> iv 0.140 -0.134 0.420 0.142 0.982 0.336
#> mod 0.194 -0.083 0.486 0.145 1.337 0.169
#> cov1 -0.046 -0.287 0.193 0.122 -0.376 0.699
#> cat1gp2 0.890 0.193 1.722 0.386 2.306 0.012 *
#> cat1gp3 1.283 0.644 2.063 0.362 3.542 <0.001 ***
#> iv:mod 0.335 0.109 0.597 0.124 2.700 0.004 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for binomial family taken to be 1)
#>
#> Null deviance: 415.03 on 299 degrees of freedom
#> Residual deviance: 390.91 on 293 degrees of freedom
#> AIC: 404.9
#>
#> Number of Fisher Scoring iterations: 4
#>
#> Transformed Parameter Estimates:
#> Exp(B) CI.Lower CI.Upper
#> (Intercept) 0.314 0.154 0.550
#> iv 1.150 0.875 1.521
#> mod 1.214 0.920 1.625
#> cov1 0.955 0.750 1.213
#> cat1gp2 2.435 1.213 5.596
#> cat1gp3 3.607 1.904 7.867
#> iv:mod 1.398 1.115 1.816
#>
#> Note:
#> - Results *after* standardization are reported.
#> - Nonparametric bootstrapping conducted.
#> - The number of bootstrap samples is 5000.
#> - Standard errors are bootstrap standard errors.
#> - Z values are computed by 'Estimate / Std. Error'.
#> - The bootstrap p-values are asymmetric p-values by Asparouhov and
#> Muthén (2021).
#> - Percentile bootstrap 95.0% confidence interval reported.
```
By default, 95% percentile bootstrap
confidence intervals are printed
(`CI.Lower` and `CI.Upper`). The *p*-values
(`Pr(Boot)`) are asymmetric bootstrap
*p*-values [@asparouhov_bootstrap_2021].
## 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 number of the variables
not to standardize
- Use `not_to_standardize`
when the number of variables to standardize
is much more than the number of
variables not to standardize.
For example, suppose we only
need to standardize
`iv` and `cov1`,
this is the call to do
this, setting
`to_standardize` to `c("iv", "cov1")`:
``` r
glm_beta_select_boot_1 <- glm_betaselect(dv ~ iv*mod + cov1 + cat1,
data = data_test_mod_cat_binary,
to_standardize = c("iv", "cov1"),
skip_response = TRUE,
family = binomial(),
bootstrap = 5000,
iseed = 4567)
```
If we want to standardize all
variables except for `mod` (`dv`
is skipped by `skip_response`) we can use
this call, and set
`not_to_standardize` to `"mod"`:
``` r
glm_beta_select_boot_2 <- glm_betaselect(dv ~ iv*mod + cov1 + cat1,
data = data_test_mod_cat_binary,
not_to_standardize = c("mod"),
skip_response = TRUE,
family = binomial(),
bootstrap = 5000,
iseed = 4567)
```
The results of these calls are identical,
and only those of the first version are
printed:
``` r
summary(glm_beta_select_boot_1)
#> Call to glm_betaselect():
#> betaselectr::lm_betaselect(formula = dv ~ iv * mod + cov1 + cat1,
#> family = binomial(), data = data_test_mod_cat_binary, to_standardize = c("iv",
#> "cov1"), skip_response = TRUE, bootstrap = 5000, iseed = 4567,
#> model_call = "glm")
#>
#> Variable(s) standardized: iv, cov1
#>
#> Call:
#> stats::glm(formula = dv ~ iv * mod + cov1 + cat1, family = binomial(),
#> data = betaselectr::std_data(data = data_test_mod_cat_binary,
#> to_standardize = c("iv", "cov1")))
#>
#> Coefficients:
#> Estimate CI.Lower CI.Upper Std. Error z value Pr(Boot)
#> (Intercept) -3.460 -7.063 -0.061 1.798 -1.924 0.0460 *
#> iv -3.832 -6.807 -1.171 1.431 -2.678 0.0044 **
#> mod 0.046 -0.020 0.115 0.035 1.339 0.1692
#> cov1 -0.046 -0.287 0.193 0.122 -0.376 0.6988
#> cat1gp2 0.890 0.193 1.722 0.386 2.306 0.0120 *
#> cat1gp3 1.283 0.644 2.063 0.362 3.542 <0.001 ***
#> iv:mod 0.080 0.027 0.140 0.029 2.767 0.0040 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for binomial family taken to be 1)
#>
#> Null deviance: 415.03 on 299 degrees of freedom
#> Residual deviance: 390.91 on 293 degrees of freedom
#> AIC: 404.9
#>
#> Number of Fisher Scoring iterations: 4
#>
#> Transformed Parameter Estimates:
#> Exp(B) CI.Lower CI.Upper
#> (Intercept) 0.031 0.001 0.941
#> iv 0.022 0.001 0.310
#> mod 1.047 0.980 1.122
#> cov1 0.955 0.750 1.213
#> cat1gp2 2.435 1.213 5.596
#> cat1gp3 3.607 1.904 7.867
#> iv:mod 1.083 1.027 1.150
#>
#> Note:
#> - Results *after* standardization are reported.
#> - Nonparametric bootstrapping conducted.
#> - The number of bootstrap samples is 5000.
#> - Standard errors are bootstrap standard errors.
#> - Z values are computed by 'Estimate / Std. Error'.
#> - The bootstrap p-values are asymmetric p-values by Asparouhov and
#> Muthén (2021).
#> - Percentile bootstrap 95.0% confidence interval reported.
```
For *beta*s-*select*, researchers need
to state which variables
are standardized and which are not.
This can be done in table notes.
## Categorical Variables
When calling `glm_betaselect()`,
categorical variables (factors and
string variables) will never be standardized.
In the example above, the coefficients
of the two dummy variables when both
the dummy variables and the outcome
variables are standardized are
0.416 and
0.642:
``` r
printCoefmat(glm_std_common_summary$coefficients[5:6, ],
digits = 5,
zap.ind = 1,
P.values = TRUE,
has.Pvalue = TRUE,
signif.stars = TRUE)
#> Estimate Std. Error z value Pr(>|z|)
#> cat_gp2 0.41587 0.16941 2.4547 0.014100 *
#> cat_gp3 0.64201 0.17239 3.7242 0.000196 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
These two values are not interpretable
because it does not make sense to talk
about a "one-SD change" in the dummy variables.
# Conclusion
In generalized linear modeling, there
are many 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
`glm_betaselect()`, we hope to make it
easier for researchers to do appropriate
standardization when reporting generalized
linear modeling results.
# References