Beta-Select Demonstration: Regression by `lm()`

Introduction

This article demonstrates how to use lm_betaselect() from the package 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:

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():

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:

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:

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:

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:

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:

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, 2022). One simple but underused solution is standardizing the variables before forming the product term (Friedrich, 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 & Chan, 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 & Waller, 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 & Hayes, 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 betas-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.

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():

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 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).

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:

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()

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 & Muthén, 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"):

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"):

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:

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 betas-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:

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 betas-Select of the dummy variables, with only the outcome variable standardized, are 0.241 and 0.349.

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 betas-select using lm_betaselect(), we hope to make it easier for researchers to do appropriate standardization when reporting regression 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

Darlington, R. B., & Hayes, A. F. (2016). Regression analysis and linear models: Concepts, applications, and implementation (Illustrated edition). The Guilford Press.

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

Hayes, A. F. (2022). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach (Third edition). The Guilford Press.

Jones, J. A., & Waller, N. G. (2013). Computing confidence intervals for standardized regression coefficients. Psychological Methods, 18(4), 435–453. https://doi.org/10.1037/a0033269

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

2024-11-08