# Purpose

This document demonstrates how to use std_selected() to compute the correct standardized solution of moderated regression.

# Setup the Environment

library(stdmod) # For computing the standardized moderation effect conveniently

# Load the Dataset

data(sleep_emo_con)
#>   case_id sleep_duration conscientiousness emotional_stability age gender
#> 1       1              6               3.6                 3.6  20 female
#> 2       2              4               3.8                 2.4  20 female
#> 3       3              7               4.3                 2.7  20 female

This data set has 500 cases of data. The variables are sleep duration, age, gender, and the scores from two personality scales, emotional stability and conscientiousness of the IPIP Big Five markers. Please refer to (citation to be added) for the detail of the data set.

The names of some variables are shortened for readability:

colnames(sleep_emo_con)[3:4] <- c("cons", "emot")
#>   case_id sleep_duration cons emot age gender
#> 1       1              6  3.6  3.6  20 female
#> 2       2              4  3.8  2.4  20 female
#> 3       3              7  4.3  2.7  20 female

# Moderated Regression

Suppose we are interested in predicting sleep duration by emotional stability, after controlling for gender and age. However, we suspect that the effect of emotional stability, if any, may be moderated by conscientiousness. Therefore, we conduct a moderated regression as follow:

lm_out <- lm(sleep_duration ~ age + gender + emot * cons,
data = sleep_emo_con)
summary(lm_out)
#>
#> Call:
#> lm(formula = sleep_duration ~ age + gender + emot * cons, data = sleep_emo_con)
#>
#> Residuals:
#>     Min      1Q  Median      3Q     Max
#> -6.0841 -0.7882  0.0089  0.9440  6.1189
#>
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)
#> (Intercept)  1.85154    1.35224   1.369  0.17155
#> age          0.01789    0.02133   0.838  0.40221
#> gendermale  -0.26127    0.16579  -1.576  0.11570
#> emot         1.32151    0.45039   2.934  0.00350 **
#> cons         1.20385    0.37062   3.248  0.00124 **
#> emot:cons   -0.33140    0.13273  -2.497  0.01286 *
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 1.384 on 494 degrees of freedom
#> Multiple R-squared:  0.0548, Adjusted R-squared:  0.04523
#> F-statistic: 5.728 on 5 and 494 DF,  p-value: 3.768e-05
plotmod(lm_out,
x = "emot",
w = "cons",
x_label = "Emotional Stability",
w_label = "Conscientiousness",
y_label = "Sleep Duration")

The results show that conscientiousness significantly moderates the effect of emotional stability on sleep duration.

# Standardized Moderation Effect

To get the correct standardized solution of the moderated regression, with the product term formed after standardization, we can use std_selected().

• The first argument is the regression output from lm().

• The argument to_center specifies variables to be mean centered.

• The argument to_scale specifies variables to be rescaled by their standard deviations after centering.

If we want to standardize or mean center all variables, we can use ~ . as a shortcut. Note that std_selected() will automatically skip categorical variables (i.e., factors or string variables in the regression model of lm()).

lm_stdall <- std_selected(lm_out,
to_center = ~ .,
to_scale  = ~ .)
summary(lm_stdall)
#>
#> Selected variable(s) are centered by mean and/or scaled by SD
#> - Variable(s) centered: sleep_duration age gender emot cons
#> - Variable(s) scaled: sleep_duration age gender emot cons
#>
#>                centered_by scaled_by                            Note
#> sleep_duration    6.776333 1.4168291 Standardized (mean = 0, SD = 1)
#> age              22.274000 2.9407857 Standardized (mean = 0, SD = 1)
#> gender                  NA        NA Nonnumeric
#> emot              2.713200 0.7629613 Standardized (mean = 0, SD = 1)
#> cons              3.343200 0.6068198 Standardized (mean = 0, SD = 1)
#>
#> Note:
#> - Categorical variables will not be centered or scaled even if requested.
#>
#> Call:
#> lm(formula = sleep_duration ~ age + gender + emot * cons, data = dat_mod)
#>
#> Residuals:
#>     Min      1Q  Median      3Q     Max
#> -4.2941 -0.5563  0.0063  0.6663  4.3187
#>
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)
#> (Intercept)  0.05492    0.04883   1.125  0.26124
#> age          0.03712    0.04428   0.838  0.40221
#> gendermale  -0.18440    0.11702  -1.576  0.11570
#> emot         0.11501    0.04493   2.560  0.01076 *
#> cons         0.13050    0.04517   2.889  0.00403 **
#> emot:cons   -0.10829    0.04337  -2.497  0.01286 *
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.9771 on 494 degrees of freedom
#> Multiple R-squared:  0.0548, Adjusted R-squared:  0.04523
#> F-statistic: 5.728 on 5 and 494 DF,  p-value: 3.768e-05
#>
#> Note:
#> - Estimates and their statistics are based on the data after mean-centering, scaling, or standardization.

The coefficient in this solution, -0.10829, can be interpreted as the change in the standardized effect of emotional stability for each one standard deviation increase of conscientiousness. Naturally, this can be called the standardized moderation effect of conscientiousness (Cheung, Cheung, Lau, Hui, & Vong, 2022).

The output of std_selected() can be passed to other functions that accept the output of lm(). This package also has a simple function, plotmod(), for generating a typical plot of the moderation effect:

plotmod(lm_stdall,
x = "emot",
w = "cons",
x_label = "Emotional Stability",
w_label = "Conscientiousness",
y_label = "Sleep Duration")

The function plotmod() also prints the conditional effects of the predictor (focal variable), emotional stability in this example.

# The Common (Incorrect) Standardized Solution

For comparison, this is the results of standardizing all variables, including the product term and the categorical variable.

library(lm.beta) # For generating the typical standardized solution
lm_beta <- lm.beta(lm_out)
summary(lm_beta)
#>
#> Call:
#> lm(formula = sleep_duration ~ age + gender + emot * cons, data = sleep_emo_con)
#>
#> Residuals:
#>     Min      1Q  Median      3Q     Max
#> -6.0841 -0.7882  0.0089  0.9440  6.1189
#>
#> Coefficients:
#>             Estimate Standardized Std. Error t value Pr(>|t|)
#> (Intercept)  1.85154           NA    1.35224   1.369  0.17155
#> age          0.01789      0.03712    0.02133   0.838  0.40221
#> gendermale  -0.26127     -0.06934    0.16579  -1.576  0.11570
#> emot         1.32151      0.71163    0.45039   2.934  0.00350 **
#> cons         1.20385      0.51560    0.37062   3.248  0.00124 **
#> emot:cons   -0.33140     -0.78201    0.13273  -2.497  0.01286 *
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 1.384 on 494 degrees of freedom
#> Multiple R-squared:  0.0548, Adjusted R-squared:  0.04523
#> F-statistic: 5.728 on 5 and 494 DF,  p-value: 3.768e-05

The coefficient of the standardized product term is -0.78201, which cannot be interpreted as the change in the standardized effect of emotional stability for each one standard deviation increase of conscientiousness because the product term is standardized and can no longer be interpreted as the product of two variables in the model.

# Improved Confidence Intervals

It has been shown (e.g., Yuan & Chan, 2011) that the standard errors of standardized regression coefficients computed just by standardizing the variables are biased, and consequently the confidence intervals are also invalid. The function std_selected_boot() is a wrapper of std_selected() that also forms the confidence interval of the regression coefficients when standardizing is conducted, using nonparametric bootstrapping as suggested by Cheung, Cheung, Lau, Hui, and Vong (2022).

We use the same example above that standardizes all variables except for categorical variables to illustrate this function. The argument nboot specifies the number of nonparametric bootstrap samples. The level of confidence is set by conf. The default is .95, denoting 95% confidence intervals. If this is the desired level, this argument can be omitted.

set.seed(649017)
lm_xwy_std_ci <- std_selected_boot(lm_out,
to_center = ~ .,
to_scale  = ~ .,
nboot = 2000)

If the default options are acceptable, the only additional argument is nboot.

summary(lm_xwy_std_ci)
#>
#> Selected variable(s) are centered by mean and/or scaled by SD
#> - Variable(s) centered: sleep_duration age gender emot cons
#> - Variable(s) scaled: sleep_duration age gender emot cons
#>
#>                centered_by scaled_by                            Note
#> sleep_duration    6.776333 1.4168291 Standardized (mean = 0, SD = 1)
#> age              22.274000 2.9407857 Standardized (mean = 0, SD = 1)
#> gender                  NA        NA Nonnumeric
#> emot              2.713200 0.7629613 Standardized (mean = 0, SD = 1)
#> cons              3.343200 0.6068198 Standardized (mean = 0, SD = 1)
#>
#> Note:
#> - Categorical variables will not be centered or scaled even if requested.
#> - Nonparametric bootstrapping 95% confidence intervals computed.
#> - The number of bootstrap samples is 2000.
#>
#> Call:
#> lm(formula = sleep_duration ~ age + gender + emot * cons, data = dat_mod)
#>
#> Residuals:
#>     Min      1Q  Median      3Q     Max
#> -4.2941 -0.5563  0.0063  0.6663  4.3187
#>
#> Coefficients:
#>              Estimate  CI Lower  CI Upper Std. Error t value Pr(>|t|)
#> (Intercept)  0.054919  0.002968  0.104340   0.048827   1.125  0.26124
#> age          0.037125 -0.036253  0.103590   0.044280   0.838  0.40221
#> gendermale  -0.184402 -0.438855  0.087551   0.117016  -1.576  0.11570
#> emot         0.115014  0.023642  0.202438   0.044927   2.560  0.01076 *
#> cons         0.130502  0.032359  0.224223   0.045167   2.889  0.00403 **
#> emot:cons   -0.108292 -0.203990 -0.009746   0.043374  -2.497  0.01286 *
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.9771 on 494 degrees of freedom
#> Multiple R-squared:  0.0548, Adjusted R-squared:  0.04523
#> F-statistic: 5.728 on 5 and 494 DF,  p-value: 3.768e-05
#>
#> Note:
#> - Estimates and their statistics are based on the data after mean-centering, scaling, or standardization.
#> - [CI Lower, CI Upper] are bootstrap percentile confidence intervals.
#> - Std. Error are not bootstrap SEs.

The standardized moderation effect is -0.1083, and the 95% nonparametric bootstrap confidence interval is -0.2040 to -0.0097.

Note: As a side product, the nonparametric bootstrap percentile confidence of the other coefficients are also reported. They can be used for other variables that are standardized in the same model, whether they are involved in the moderation or not.

# Further Information

vignette("plotmod") illustrates how to use plotmod() to plot a moderation effect. If variables are standardized by std_selected(), plotmod() can indicate this in the plot.

vignette("cond_effect") illustrates how to use cond_effect() to compute conditional effects, the effect of a predictor (focal variable) for selected levels of the moderator. cond_effect() support outputs from std_selected().

# Reference(s)

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.

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