---
title: "Beta-Select Demonstration: Regression by `lm()`"
date: "2024-11-08"
output:
rmarkdown::html_vignette:
number_sections: true
vignette: >
%\VignetteIndexEntry{Beta-Select Demonstration: Regression by `lm()`}
%\VignetteEngine{knitr::rmarkdown}
%\VignetteEncoding{UTF-8}
bibliography: "references.bib"
csl: apa.csl
---
# Introduction
This article demonstrates how to use
`lm_betaselect()` from the package
[`betaselectr`](https://sfcheung.github.io/betaselectr/)
to standardize
selected variables in a model fitted
by `lm()` and forming confidence
intervals for the parameters.
# Data and Model
The sample dataset from the package
`betaselectr` will be used for this
demonstration:
``` r
library(betaselectr)
head(data_test_mod_cat2)
#> dv iv mod cov1 cat1
#> 1 15.53 13.95 50.75 25.33 gp2
#> 2 17.69 15.07 49.67 20.96 gp1
#> 3 28.56 14.43 53.42 19.22 gp3
#> 4 25.00 11.22 42.55 20.18 gp2
#> 5 19.33 14.93 52.12 22.82 gp2
#> 6 20.62 10.22 39.36 18.41 gp1
```
This is the regression model, fitted by
`lm()`:
``` r
lm_out <- lm(dv ~ iv * mod + cov1 + cat1,
data = data_test_mod_cat2)
```
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(lm_out)
#>
#> Call:
#> lm(formula = dv ~ iv * mod + cov1 + cat1, data = data_test_mod_cat2)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -17.0892 -4.6312 0.0057 5.0186 18.7053
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 90.87211 34.04462 2.669 0.00803 **
#> iv -6.06932 2.33701 -2.597 0.00988 **
#> mod -1.61636 0.68840 -2.348 0.01954 *
#> cov1 0.09885 0.19433 0.509 0.61136
#> cat1gp2 1.71248 1.15064 1.488 0.13775
#> cat1gp3 2.47838 1.10562 2.242 0.02574 *
#> iv:mod 0.13230 0.04656 2.841 0.00481 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 6.743 on 293 degrees of freedom
#> Multiple R-squared: 0.1149, Adjusted R-squared: 0.09676
#> F-statistic: 6.338 on 6 and 293 DF, p-value: 2.759e-06
```
# Problems With Standardization
One common type of standardized coefficients,
called "betas" in some programs, is
computed by standardizing *all* terms
in the model.
First, all variables in the model,
including the product term and dummy
variables, are computed:
``` r
data_test_mod_cat2_z <- data_test_mod_cat2
data_test_mod_cat2_z$iv_x_mod <- data_test_mod_cat2_z$iv *
data_test_mod_cat2_z$mod
data_test_mod_cat2_z$cat_gp2 <- as.numeric(data_test_mod_cat2_z$cat1 == "gp2")
data_test_mod_cat2_z$cat_gp3 <- as.numeric(data_test_mod_cat2_z$cat1 == "gp3")
head(data_test_mod_cat2_z)
#> dv iv mod cov1 cat1 iv_x_mod cat_gp2 cat_gp3
#> 1 15.53 13.95 50.75 25.33 gp2 707.9625 1 0
#> 2 17.69 15.07 49.67 20.96 gp1 748.5269 0 0
#> 3 28.56 14.43 53.42 19.22 gp3 770.8506 0 1
#> 4 25.00 11.22 42.55 20.18 gp2 477.4110 1 0
#> 5 19.33 14.93 52.12 22.82 gp2 778.1516 1 0
#> 6 20.62 10.22 39.36 18.41 gp1 402.2592 0 0
```
All the variables are then standardized:
``` r
data_test_mod_cat2_z <- data.frame(scale(data_test_mod_cat2_z[, -5]))
head(data_test_mod_cat2_z)
#> dv iv mod cov1 iv_x_mod cat_gp2
#> 1 -0.9458226 -0.44874323 0.23147783 2.553777460 -0.24816181 1.331109
#> 2 -0.6414005 0.13926755 -0.03378874 0.390649940 0.06187143 -0.748749
#> 3 0.8905756 -0.19673861 0.88727574 -0.470641109 0.23249121 -0.748749
#> 4 0.3888429 -1.88201951 -1.78258317 0.004553953 -2.01026427 1.331109
#> 5 -0.4102652 0.06576621 0.56797339 1.311340372 0.28829267 1.331109
#> 6 -0.2284575 -2.40702913 -2.56610202 -0.871586942 -2.58464861 -0.748749
#> cat_gp3
#> 1 -0.9401258
#> 2 -0.9401258
#> 3 1.0601418
#> 4 -0.9401258
#> 5 -0.9401258
#> 6 -0.9401258
```
The regression model is then fitted to the
standardized variables:
``` r
lm_std_common <- lm(dv ~ iv + mod + cov1 + cat_gp2 + cat_gp3 + iv_x_mod,
data = data_test_mod_cat2_z)
```
The "betas" commonly reported are the
coefficients in this model:
``` r
lm_std_common_summary <- summary(lm_std_common)
printCoefmat(lm_std_common_summary$coefficients,
digits = 5,
zap.ind = 1,
P.values = TRUE,
has.Pvalue = TRUE,
signif.stars = TRUE)
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.000000 0.054871 0.0000 1.000000
#> iv -1.629280 0.627360 -2.5970 0.009878 **
#> mod -0.927480 0.395006 -2.3480 0.019539 *
#> cov1 0.028140 0.055329 0.5087 0.611359
#> cat_gp2 0.116040 0.077970 1.4883 0.137753
#> cat_gp3 0.174620 0.077901 2.2416 0.025735 *
#> iv_x_mod 2.439510 0.858601 2.8413 0.004809 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
However, for this model, there are several
problems:
- The product term is also
standardized (`iv_x_mod`, computed
using the standard deviations of
`dv` and `iv:mod`). This is inappropriate [@hayes_introduction_2022].
One simple but underused solution is
standardizing the variables *before*
forming the product term [@friedrich_defense_1982].
- The confidence intervals are formed using
ordinary least squares (OLS), which does not
take into account the sampling variation
of the standardizers (the sample standard
deviations used in standardization) and
so the standard errors may be
biased [@yuan_biases_2011].
Although there
are situations in which the OLS
confidence and the nonparametric
percentile bootstrap confidences can be
similar (e.g., sample size is large
and the population values are not extreme),
it is recommended to use bootstrap
confidence intervals when computation
cost is low [@jones_computing_2013].
- 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
[@darlington_regression_2016].
# Beta-Select by `lm_betaselect()`
The function `lm_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,
with the product
term computed after `iv`, `mod`, and `dv`
are standardized.
``` r
lm_beta_select <- lm_betaselect(dv ~ iv*mod + cov1 + cat1,
data = data_test_mod_cat2,
do_boot = FALSE)
```
The function `lm_betaselect()` can be
used as `lm()`, with applicable arguments
such as the model formula and `data` passed
to `lm()`.
By default, *all* numeric variables will
be standardized before fitting the models.
Terms such as product terms are created
*after* standardization.
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 `lm_betaselect()`:
``` r
summary(lm_beta_select)
#> Call to lm_betaselect():
#> betaselectr::lm_betaselect(formula = dv ~ iv * mod + cov1 + cat1,
#> data = data_test_mod_cat2, do_boot = FALSE)
#>
#> Variable(s) standardized: dv, iv, mod, cov1
#>
#> Call:
#> stats::lm(formula = dv ~ iv * mod + cov1 + cat1, data = betaselectr::std_data(data = data_test_mod_cat2,
#> to_standardize = c("dv", "iv", "mod", "cov1")))
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2.4085 -0.6527 0.0008 0.7073 2.6363
#>
#> Coefficients:
#> Estimate CI.Lower CI.Upper Std. Error t value Pr(>|t|)
#> (Intercept) -0.308 -0.576 -0.040 0.136 -2.259 0.02461 *
#> iv 0.140 0.021 0.258 0.060 2.324 0.02082 *
#> mod 0.196 0.078 0.315 0.060 3.264 0.00123 **
#> cov1 0.028 -0.081 0.137 0.055 0.509 0.61136
#> cat1gp2 0.241 -0.078 0.561 0.162 1.488 0.13775
#> cat1gp3 0.349 0.043 0.656 0.156 2.242 0.02574 *
#> iv:mod 0.145 0.044 0.245 0.051 2.841 0.00481 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.9504 on 293 degrees of freedom
#>
#> R-squared : 0.115
#> Adjusted R-squared : 0.097
#> ANOVA test of R-squared : F(6, 293) = 6.338, p < 0.001
#>
#> Note:
#> - Results *after* standardization are reported.
#> - Standard errors are least-squares standard errors.
#> - T values are computed by 'Estimate / Std. Error'.
#> - P-values are usual t-test p-values.
#> - Least squares standard errors, t values, p-values, and confidence
#> intervals (if reported) should not be used for coefficients involved
#> in standardization.
#> - Least squares 95.0% confidence interval reported.
```
Compared to the solution with the product
term standardized, the coefficient of
`iv:mod` changed substantially from
2.440 to
0.145. 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`).
## Estimates and Bootstrap Confidence Interval
Suppose we want to address
both the first and the second problems,
with
- the product term computed after `iv`,
`mod`, and `dv` are standardized, and
- bootstrap confidence interval used.
We can call `lm_betaselect()` again, with
additional arguments
set:
``` r
lm_beta_select_boot <- lm_betaselect(dv ~ iv*mod + cov1 + cat1,
data = data_test_mod_cat2,
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 in bootstrapping. Set
this to an integer to make the results
reproducible.
This is the output of `summary()`
``` r
summary(lm_beta_select_boot)
#> Call to lm_betaselect():
#> betaselectr::lm_betaselect(formula = dv ~ iv * mod + cov1 + cat1,
#> data = data_test_mod_cat2, bootstrap = 5000, iseed = 4567)
#>
#> Variable(s) standardized: dv, iv, mod, cov1
#>
#> Call:
#> stats::lm(formula = dv ~ iv * mod + cov1 + cat1, data = betaselectr::std_data(data = data_test_mod_cat2,
#> to_standardize = c("dv", "iv", "mod", "cov1")))
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2.4085 -0.6527 0.0008 0.7073 2.6363
#>
#> Coefficients:
#> Estimate CI.Lower CI.Upper Std. Error z value Pr(Boot)
#> (Intercept) -0.308 -0.536 -0.080 0.117 -2.636 0.0056 **
#> iv 0.140 0.009 0.268 0.066 2.109 0.0384 *
#> mod 0.196 0.075 0.317 0.061 3.208 0.0016 **
#> cov1 0.028 -0.075 0.131 0.052 0.537 0.5700
#> cat1gp2 0.241 -0.067 0.540 0.155 1.560 0.1276
#> cat1gp3 0.349 0.064 0.631 0.146 2.394 0.0152 *
#> iv:mod 0.145 0.059 0.228 0.043 3.356 0.0020 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.9504 on 293 degrees of freedom
#>
#> R-squared : 0.115
#> Adjusted R-squared : 0.097
#> ANOVA test of R-squared : F(6, 293) = 6.338, p < 0.001
#>
#> 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 `dv` and
`iv`,
this is the call to do
this, setting
`to_standardize` to `c("iv", "dv")`:
``` r
lm_beta_select_boot_1 <- lm_betaselect(dv ~ iv*mod + cov1 + cat1,
data = data_test_mod_cat2,
to_standardize = c("dv", "iv"),
bootstrap = 5000,
iseed = 4567)
```
If we want to standardize all
variables except for `mod`
and `cov1`, we can use
this call, and set
`not_to_standardize` to `c("mod", "cov1")`:
``` r
lm_beta_select_boot_2 <- lm_betaselect(dv ~ iv*mod + cov1 + cat1,
data = data_test_mod_cat2,
not_to_standardize = c("mod", "cov1"),
bootstrap = 5000,
iseed = 4567)
```
The results of these calls are identical,
and only those of the first version are
printed:
``` r
summary(lm_beta_select_boot_1)
#> Call to lm_betaselect():
#> betaselectr::lm_betaselect(formula = dv ~ iv * mod + cov1 + cat1,
#> data = data_test_mod_cat2, to_standardize = c("dv", "iv"),
#> bootstrap = 5000, iseed = 4567)
#>
#> Variable(s) standardized: dv, iv
#>
#> Call:
#> stats::lm(formula = dv ~ iv * mod + cov1 + cat1, data = betaselectr::std_data(data = data_test_mod_cat2,
#> to_standardize = c("dv", "iv")))
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2.4085 -0.6527 0.0008 0.7073 2.6363
#>
#> Coefficients:
#> Estimate CI.Lower CI.Upper Std. Error z value Pr(Boot)
#> (Intercept) -2.991 -4.769 -1.196 0.899 -3.326 0.0024 **
#> iv -1.629 -2.667 -0.573 0.539 -3.021 0.0036 **
#> mod 0.048 0.019 0.078 0.015 3.199 0.0016 **
#> cov1 0.014 -0.037 0.066 0.026 0.533 0.5700
#> cat1gp2 0.241 -0.067 0.540 0.155 1.560 0.1276
#> cat1gp3 0.349 0.064 0.631 0.146 2.394 0.0152 *
#> iv:mod 0.036 0.015 0.056 0.011 3.366 0.0020 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.9504 on 293 degrees of freedom
#>
#> R-squared : 0.115
#> Adjusted R-squared : 0.097
#> ANOVA test of R-squared : F(6, 293) = 6.338, p < 0.001
#>
#> 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 `lm_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
0.116 and
0.175:
``` r
printCoefmat(lm_std_common_summary$coefficients[5:6, ],
digits = 5,
zap.ind = 1,
P.values = TRUE,
has.Pvalue = TRUE,
signif.stars = TRUE)
#> Estimate Std. Error t value Pr(>|t|)
#> cat_gp2 0.116041 0.077970 1.4883 0.13775
#> cat_gp3 0.174623 0.077901 2.2416 0.02574 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
These two values are not interpretable
because it does not make sense to talk
about a "one-SD change" in the dummy variables.
The *beta*s-*Select* of the dummy variables,
with only the outcome variable standardized,
are
0.241 and
0.349.
``` r
lm_beta_select_boot_summary <- summary(lm_beta_select_boot)
printCoefmat(lm_beta_select_boot_summary$coefficients[5:6, ],
digits = 5,
zap.ind = 1,
P.values = TRUE,
has.Pvalue = TRUE,
signif.stars = TRUE)
#> Estimate CI.Lower CI.Upper Std. Error z value Pr(Boot)
#> cat1gp2 0.241350 -0.067312 0.540477 0.154668 1.5604 0.1276
#> cat1gp3 0.349290 0.064136 0.630539 0.145927 2.3936 0.0152 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
They
represent the difference between `gp2`
and `gp3` from the reference group,
`gp1`, on the *standardized* outcome variable.
That is, their meanings are the same before
and after standardization. The only difference
is in the unit of the outcome variable.
# Conclusion
In regression analysis, 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
`lm_betaselect()`, we hope to make it
easier for researchers to do appropriate
standardization when reporting regression
results.
# References