Beta-Select Demonstration: Logistic Regression by glm()
Date: 2026-04-07

# Beta-Select Demonstration: Logistic Regression by `glm()`

#### 2026-04-07

# 1 Introduction

This article demonstrates how to use `glm_betaselect()` from the package `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.

# 2 Data and Model

The sample dataset from the package `betaselectr` will be used for in this demonstration:

    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()`:

    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:

    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

# 3 Problems With Standardization

In logistic regression, there are several ways to do standardization (Menard, 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:

    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:

    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:

    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:

    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, 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 & Chan, 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.

# 4 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 betas-select (\(\beta{s}_{Select}\), \(\beta_{Select}\) in singular form), to differentiate them from coefficients computed by standardizing all variables, including product terms.

## 4.1 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.

    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()`:

    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 et al., 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`).

## 4.2 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:

    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()`

    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   -1.864   -0.617      0.320  -3.622   <0.001 ***
    #> iv             0.140   -0.137    0.427      0.146   0.958   0.3404
    #> mod            0.194   -0.080    0.492      0.146   1.327   0.1684
    #> cov1          -0.046   -0.299    0.194      0.123  -0.374   0.6912
    #> cat1gp2        0.890    0.198    1.692      0.384   2.315   0.0096 **
    #> cat1gp3        1.283    0.648    2.072      0.360   3.567   <0.001 ***
    #> iv:mod         0.335    0.108    0.591      0.123   2.732   0.0036 **
    #> ---
    #> 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    0.155    0.539
    #> iv           1.150    0.872    1.532
    #> mod          1.214    0.923    1.636
    #> cov1         0.955    0.741    1.214
    #> cat1gp2      2.435    1.219    5.433
    #> cat1gp3      3.607    1.912    7.939
    #> iv:mod       1.398    1.114    1.807
    #>
    #> 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 & Muthén, 2021).

## 4.3 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")`:

    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"`:

    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:

    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   -7.293   -0.116      1.814  -1.907   0.0412 *
    #> iv            -3.832   -6.782   -1.163      1.417  -2.705   0.0068 **
    #> mod            0.046   -0.019    0.118      0.035   1.328   0.1684
    #> cov1          -0.046   -0.299    0.194      0.123  -0.374   0.6912
    #> cat1gp2        0.890    0.198    1.692      0.384   2.315   0.0096 **
    #> cat1gp3        1.283    0.648    2.072      0.360   3.567   <0.001 ***
    #> iv:mod         0.080    0.026    0.139      0.029   2.797   0.0036 **
    #> ---
    #> 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    0.890
    #> iv           0.022    0.001    0.312
    #> mod          1.047    0.981    1.125
    #> cov1         0.955    0.741    1.214
    #> cat1gp2      2.435    1.219    5.433
    #> cat1gp3      3.607    1.912    7.939
    #> iv:mod       1.083    1.026    1.149
    #>
    #> 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 betas-select, researchers need to state which variables are standardized and which are not. This can be done in table notes.

## 4.4 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:

    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.

# 5 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 betas-select using `glm_betaselect()`, we hope to make it easier for researchers to do appropriate standardization when reporting generalized linear modeling results.

# References

Asparouhov, T., & Muthén, B. O. (2021). Bootstrap p-value computation. https://www.statmodel.com/download/FAQ-Bootstrap%20-%20Pvalue.pdf
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
Friedrich, R. J. (1982). In defense of multiplicative terms in multiple regression equations. American Journal of Political Science, 26(4), 797–833. https://doi.org/10.2307/2110973
Menard, S. (2004). Six approaches to calculating standardized logistic regression coefficients. The American Statistician, 58(3), 218–223. https://doi.org/10.1198/000313004X946
Yuan, K.-H., & Chan, W. (2011). Biases and standard errors of standardized regression coefficients. Psychometrika, 76(4), 670–690. https://doi.org/10.1007/s11336-011-9224-6