---
title: "Mean Center and Standardize Selected Variable by std_selected()"
author: "Shu Fai Cheung and David Weng Ngai Vong"
date: "2026-01-04"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Mean Center and Standardize Selected Variable by std_selected()}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---



# Purpose

Instead of standardizing all variables, even variables that (a) are categorical
and should not be standardized, or (b) measured on meaningful unites and do
not need to be standardized, `std_selected()` from the package
`stdmod` allows users to have more
control on how standardization is to be conducted.

A moderated regression model is used as an example but it can also be used for
regression models without interaction terms.

More about this package can be found
in `vignette("stdmod", package = "stdmod")`
or at [https://sfcheung.github.io/stdmod/](https://sfcheung.github.io/stdmod/).

# Setup the Environment


``` r
library(stdmod)
```

# Load the Dataset


``` r
data(sleep_emo_con)
head(sleep_emo_con, 3)
#> # A tibble: 3 × 6
#>   case_id sleep_duration conscientiousness emotional_stability   age gender
#>     <int>          <dbl>             <dbl>               <dbl> <dbl> <chr> 
#> 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 included) for the detail of the data set.

The names of some variables are shortened for readability:


``` r
colnames(sleep_emo_con)[3:4] <- c("cons", "emot")
head(sleep_emo_con, 3)
#> # A tibble: 3 × 6
#>   case_id sleep_duration  cons  emot   age gender
#>     <int>          <dbl> <dbl> <dbl> <dbl> <chr> 
#> 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:


``` r
lm_raw <- lm(sleep_duration ~ age + gender + emot * cons,
             data = sleep_emo_con)
summary(lm_raw)
#> 
#> 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
```

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

This package has a simple function, `plotmod()`, for generating a typical plot
of the moderation effect:


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

<div class="figure" style="text-align: center">
<img src="std_selected_lm_raw_plot-1.png" alt="plot of chunk std_selected_lm_raw_plot"  />
<p class="caption">plot of chunk std_selected_lm_raw_plot</p>
</div>

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

# Mean Center the Moderator

To know the effect of emotional stability when conscientiousness is equal to its
mean, we can center conscientiousness by its mean in the data and redo the
moderated regression. Instead of creating the new variable and rerun the
regression, we can pass the `lm()` output to `std_selected()` and specify the
variables to be mean centered:



``` r
lm_w_centered <- std_selected(lm_raw,
                              to_center = ~ cons)
printCoefmat(summary(lm_w_centered)$coefficients, digits = 3)
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   5.8763     0.5170   11.37   <2e-16 ***
#> age           0.0179     0.0213    0.84   0.4022    
#> gendermale   -0.2613     0.1658   -1.58   0.1157    
#> emot          0.2136     0.0834    2.56   0.0108 *  
#> cons          1.2039     0.3706    3.25   0.0012 ** 
#> emot:cons    -0.3314     0.1327   -2.50   0.0129 *  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

The argument for meaning centering is `to_center`. The variable is specified
in the formula form, placing them on the right hand side of the formula.

In this example, when conscientiousness
is at mean level, the effect of emotional stability is
0.2136.

# Mean Center The Moderator and the Focal Variable

This example demonstrates centering more than one variable. In the following
model, both emotional stability and conscientiousness are centered. They are
placed after `~` and joined by `+`.


``` r
lm_xw_centered <- std_selected(lm_raw,
                               to_center = ~ emot + cons)
printCoefmat(summary(lm_xw_centered)$coefficients, digits = 3)
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   6.4557     0.4783   13.50   <2e-16 ***
#> age           0.0179     0.0213    0.84    0.402    
#> gendermale   -0.2613     0.1658   -1.58    0.116    
#> emot          0.2136     0.0834    2.56    0.011 *  
#> cons          0.3047     0.1055    2.89    0.004 ** 
#> emot:cons    -0.3314     0.1327   -2.50    0.013 *  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

# Standardize The Moderator and The Focal Variable

To standardize a variable we first mean center it and then scale it by its
standard deviation. Scaling is done by listing the variable on `to_scale`.
The input format is identical to that of `to_center`.

```r
lm_xw_std <- std_selected(lm_raw,
                          to_center = ~ emot + cons,
                          to_scale  = ~ emot + cons)
```

Since 0.2.6.3 of `stdmod`, `to_standardize` can be used as a shortcut.
Listing a variable on `to_standardize` is equivalent to listing it
on both `to_center` and `to_scale`. Therefore, the following
call can also be used:


``` r
lm_xw_std <- std_selected(lm_raw,
                          to_standardize = ~ emot + cons)
```


``` r
printCoefmat(summary(lm_xw_std)$coefficients, digits = 3)
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   6.4557     0.4783   13.50   <2e-16 ***
#> age           0.0179     0.0213    0.84    0.402    
#> gendermale   -0.2613     0.1658   -1.58    0.116    
#> emot          0.1630     0.0637    2.56    0.011 *  
#> cons          0.1849     0.0640    2.89    0.004 ** 
#> emot:cons    -0.1534     0.0615   -2.50    0.013 *  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

In this example, when conscientiousness
is at mean level, for each one standard deviation increase of
emotional stability, the predicted sleep duration increases by
0.1630 hour.


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

<div class="figure" style="text-align: center">
<img src="std_selected_lm_xw_std_plot-1.png" alt="plot of chunk std_selected_lm_xw_std_plot"  />
<p class="caption">plot of chunk std_selected_lm_xw_std_plot</p>
</div>

The function `plotmod()` automatically checks whether a variable is
standardized. If yes, it will report this in the plot as table note on the
bottom.

The pattern of
the plot does not change. However, the conditional effects reported
in the graph are now
based on the model with emotional stability and conscientiousness
standardized.

# Standardize The Moderator, The Focal Variable, and the Outcome Variable

We can also mean center or standardize the outcome variable (dependent
variable). We
just add the variable to the right hand side of `~` in `to_center` and
`to_scale` as appropriate.

```r
lm_xwy_std <- std_selected(lm_raw,
                           to_center = ~ emot + cons + sleep_duration,
                           to_scale  = ~ emot + cons + sleep_duration)
```

Since 0.2.6.3, `to_standardize` can be used as a shortcut:


``` r
lm_xwy_std <- std_selected(lm_raw,
                           to_standardize = ~ emot + cons + sleep_duration)
printCoefmat(summary(lm_xwy_std)$coefficients, digits = 3)
#>             Estimate Std. Error t value Pr(>|t|)   
#> (Intercept)  -0.2263     0.3376   -0.67    0.503   
#> age           0.0126     0.0151    0.84    0.402   
#> gendermale   -0.1844     0.1170   -1.58    0.116   
#> emot          0.1150     0.0449    2.56    0.011 * 
#> cons          0.1305     0.0452    2.89    0.004 **
#> emot:cons    -0.1083     0.0434   -2.50    0.013 * 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

In this example, when conscientiousness
is at mean level, the standardized moderation effect of
emotional stability on sleep duration is
0.1150.


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

<div class="figure" style="text-align: center">
<img src="std_selected_lm_xwy_std_plot-1.png" alt="plot of chunk std_selected_lm_xwy_std_plot"  />
<p class="caption">plot of chunk std_selected_lm_xwy_std_plot</p>
</div>

Again, the pattern of
the plot does not change, but the conditional effects reported
in the graph are now
based on the model with emotional stability, conscientiousness,
and sleep duration standardized.

# Standardize All Variables

If we want to standardize all variables except for categorical variables, if any,
we can use `~ .` as a shortcut. `std_selected()` will automatically
skip categorical variables (i.e., factors or string variables in the
regression model of `lm()`).

```r
lm_all_std <- std_selected(lm_raw,
                           to_center = ~ .,
                           to_scale  = ~ .)
```

Since 0.2.6.3, `to_standardize` can be used as a shortcut:


``` r
lm_all_std <- std_selected(lm_raw,
                           to_standardize = ~ .)
printCoefmat(summary(lm_all_std)$coefficients, digits = 3)
#>             Estimate Std. Error t value Pr(>|t|)   
#> (Intercept)   0.0549     0.0488    1.12    0.261   
#> age           0.0371     0.0443    0.84    0.402   
#> gendermale   -0.1844     0.1170   -1.58    0.116   
#> emot          0.1150     0.0449    2.56    0.011 * 
#> cons          0.1305     0.0452    2.89    0.004 **
#> emot:cons    -0.1083     0.0434   -2.50    0.013 * 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

# The Usual Standardized Solution

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


``` r
library(lm.beta) # For generating the typical standardized solution
packageVersion("lm.beta")
#> [1] '1.7.3'
lm_usual_std <- lm.beta(lm_raw)
printCoefmat(summary(lm_usual_std)$coefficients, digits = 3)
#>             Estimate Standardized Std. Error t value Pr(>|t|)   
#> (Intercept)   1.8515           NA     1.3522    1.37   0.1715   
#> age           0.0179       0.0371     0.0213    0.84   0.4022   
#> gendermale   -0.2613      -0.0693     0.1658   -1.58   0.1157   
#> emot          1.3215       0.7116     0.4504    2.93   0.0035 **
#> cons          1.2039       0.5156     0.3706    3.25   0.0012 **
#> emot:cons    -0.3314      -0.7820     0.1327   -2.50   0.0129 * 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

In moderated regression, the coefficient of *standardized* product term,
-0.7820,
is not interpretable. The coefficient of *standardized* gender,
-0.0693, is also
difficult to interpret.

# Improved Confidence Interval For "Betas"

It has been shown (e.g., [Yuan & Chan, 2011](https://doi.org/10.1007/s11336-011-9224-6)) that the standard errors of
standardized regression coefficients (betas) 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 standardization
is conducted,
using nonparametric bootstrapping as suggested by
[Cheung, Cheung, Lau, Hui, and Vong (2022)](https://doi.org/10.1037/hea0001188).

We use the same example above that standardizes emotional stability,
conscientiousness, and sleep duration, to illustrate this function.
The argument `nboot` specifies the number of nonparametric bootstrap samples.
The level of confidence set by `conf`. The default is .95, denoting 95%
confidence intervals. If this is the desired level, this argument can be omitted.




``` r
set.seed(58702)
lm_xwy_std_ci <- std_selected_boot(lm_raw,
                    to_standardize = ~ emot + cons + sleep_duration,
                    nboot = 2000)
```


``` r
summary(lm_xwy_std_ci)
#> 
#> Call to std_selected_boot():
#> std_selected_boot(lm_out = lm_raw, to_standardize = ~emot + cons + 
#>     sleep_duration, nboot = 2000)
#> 
#> Selected variable(s) are centered by mean and/or scaled by SD
#> - Variable(s) centered: emot cons sleep_duration
#> - Variable(s) scaled: emot cons sleep_duration
#> 
#>                centered_by scaled_by                            Note
#> sleep_duration    6.776333 1.4168291 Standardized (mean = 0, SD = 1)
#> age               0.000000 1.0000000                                
#> 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)  -0.2263  -0.8315   0.3503     0.3376 -0.6703  0.50298   
#> age           0.0126  -0.0128   0.0397     0.0151  0.8384  0.40221   
#> gendermale   -0.1844  -0.4484   0.0723     0.1170 -1.5759  0.11570   
#> emot          0.1150   0.0256   0.2001     0.0449  2.5600  0.01076 * 
#> cons          0.1305   0.0289   0.2323     0.0452  2.8893  0.00403 **
#> emot:cons    -0.1083  -0.2005  -0.0077     0.0434 -2.4967  0.01286 * 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 0.9771 on 494 degrees of freedom
#> 
#> R-squared                : 0.0548
#> Adjusted R-squared       : 0.0452
#> ANOVA test of R-squared  : F(5, 494) = 5.7277, p < 0.001
#> 
#> = Test the highest order term =
#> The highest order term             : emot:cons
#> R-squared increase adding this term: 0.0119
#> F test of R-squared increase       : F(1, 494) = 6.2335, p = 0.013
#> 
#> 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 percentile confidence interval is
-0.2005 to
-0.0077.

Note: As a side product, the nonparametric bootstrap 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

Further information on the functions can be found in their help pages
(`std_selected()` and `std_selected_boot()`). For example, parallel computation
can be used when doing bootstrapping, if the number of bootstrapping samples
requested is large.

# Reference

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