# Standardized Moderation Effect by std_selected()

#### Shu Fai Cheung and David Weng Ngai Vong

#### 2024-03-26

Source:`vignettes/moderation.Rmd`

`moderation.Rmd`

## Purpose

This document demonstrates how to use `std_selected()`

from the `stdmod`

package to compute the correct standardized
solution of moderated regression. More about this package can be found
in `vignette("stdmod", package = "stdmod")`

or at https://sfcheung.github.io/stdmod/.

## Load the Dataset

```
data(sleep_emo_con)
head(sleep_emo_con, 3)
#> 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:

## 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.In

`stdmod`

0.2.6.3, the argument`to_standardize`

was introduced as a shortcut. Listing a variable in`to_standardize`

is equivalent to listing it in`to_center`

and`to_scale`

.

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_standardize = ~ .)
```

Before 0.2.6.3, to standardize all variables except for categorical
variables, we need to use both `to_center = ~ .`

and
`to_scale = ~ .`

. Since 0.2.6.3, we can just use
`to_standardize = ~ .`

, as shown above. If
`to_standardize = ~ .`

does not work, just use
`to_center`

and `to_scale`

as shown below:

```
lm_stdall <- std_selected(lm_out,
to_center = ~ .,
to_scale = ~ .)
```

A summary of the results of `std_selected()`

can be
generated by `summary()`

:

```
summary(lm_stdall)
#>
#> Call to std_selected():
#> std_selected(lm_out = lm_out, to_standardize = ~.)
#>
#> 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) 0.0549 0.0488 1.1248 0.26124
#> age 0.0371 0.0443 0.8384 0.40221
#> gendermale -0.1844 0.1170 -1.5759 0.11570
#> emot 0.1150 0.0449 2.5600 0.01076 *
#> cons 0.1305 0.0452 2.8893 0.00403 **
#> emot:cons -0.1083 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.
#> - One or more variables are scaled by SD or standardized. OLS standard
#> errors and confidence intervals may be biased for their coefficients.
#> Please use `std_selected_boot()`.
```

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
packageVersion("lm.beta")
#> [1] '1.7.2'
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_standardize = ~ .,
nboot = 2000)
```

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

.

```
summary(lm_xwy_std_ci)
#>
#> Call to std_selected_boot():
#> std_selected_boot(lm_out = lm_out, to_scale = ~., to_center = ~.,
#> nboot = 2000)
#>
#> 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) 0.0549 0.0030 0.1043 0.0488 1.1248 0.26124
#> age 0.0371 -0.0363 0.1036 0.0443 0.8384 0.40221
#> gendermale -0.1844 -0.4389 0.0876 0.1170 -1.5759 0.11570
#> emot 0.1150 0.0236 0.2024 0.0449 2.5600 0.01076 *
#> cons 0.1305 0.0324 0.2242 0.0452 2.8893 0.00403 **
#> emot:cons -0.1083 -0.2040 -0.0097 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 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", package = "stdmod")`

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", package = "stdmod")`

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()`

supports 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