This vignette illustrates how to use `std_selected()`

, the
main function from the `stdmod`

package. More about this
package can be found in
`vignette("stdmod", package = "stdmod")`

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

`std_selected()`

to:get the correct standardized regression coefficients of a moderated regression model, and

form the valid confidence intervals of the standardized regression coefficients using nonparametric bootstrapping that takes into account the sampling variation due to standardization.

```
library(stdmod)
dat <- sleep_emo_con
head(dat, 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
```

This dataset has 500 cases, with sleep duration (measured in average
hours), conscientiousness, emotional stability, age, and gender (a
`"female"`

and `"male"`

).

The names of some variables are shortened for readability:

Suppose this is the moderated regression model:

Dependent variable (Outcome Variable): sleep duration (

`sleep`

)Independent variable (Predictor / Focal Variable): emotional stability (

`emot`

)Moderator: conscientiousness (

`cons`

)Control variables:

`age`

and`gender`

`lm()`

can be used to fit this model:

```
lm_out <- lm(sleep ~ age + gender + emot * cons,
dat = dat)
summary(lm_out)
#>
#> Call:
#> lm(formula = sleep ~ age + gender + emot * cons, data = dat)
#>
#> 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 unstandardized moderation effect is significant, B = -0.3314. For each one unit increase of conscientiousness score, the effect of emotional stability decreases by 0.3314.

Suppose we want to find the correct standardized solution for the
moderated regression, that is, all variables except for categorical
variables are standardized. In a moderated regression model, the product
term should be formed *after* standardization.

Instead of doing the standardization ourselves before calling
`lm()`

, we can pass the `lm()`

output to
`std_selected()`

, and use `~ .`

for the arguments
`to_scale`

and `to_center`

.

Since 0.2.6.3, `to_standardize`

can be used as a
shortcut:

```
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 age gender emot cons
#> - Variable(s) scaled: sleep age gender emot cons
#>
#> centered_by scaled_by Note
#> sleep 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 ~ 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()`.
```

In this example, the coefficient of the product term, which naturally
can be called the **standardized moderation effect**, is
significant, B = -0.1083. For each one *standard deviation*
increase of conscientiousness score, the **standardized
effect** of emotional stability decreases by 0.1083.

Standardization is equivalent to centering by mean and then scaling
by (dividing by) standard deviation. The argument `to_center`

specifies the variables to be centered by their means, and the argument
`to_scale`

specifies the variables to be scaled by their
standard deviations. The formula interface of `lm()`

is used
in these two arguments, with the variables on the right hand side being
the variables to be centered and/or scaled.

The “`.`

” on the right hand side represents all variables
in the model, including the outcome variable (sleep duration in this
example).

`std_selected()`

will also skip categorical variables
automatically skipped because standardizing them will make their
coefficients not easy to interpret.

Since 0.2.6.3, `to_standardize`

is added as a shortcut.
Listing a variable on `to_standardize`

is equivalent to
listing this variable on both `to_center`

and
`to_scale`

.

Using `std_selected`

minimizes impact on the workflow. Do
regression as usual. Get the correct standardized coefficients only when
we need to interpret them.

There is one problem with standardized coefficients. The confidence intervals based on ordinary least squares (OLS) fitted to the standardized variables do not take into account the sampling variation of the sample means and standard deviations (Yuan & Chan, 2011). Cheung, Cheung, Lau, Hui, and Vong (2022) suggest using nonparametric bootstrapping, with standardization conducted in each bootstrap sample.

This can be done by `std_selected_boot()`

, a wrapper of
`std_selected()`

:

```
set.seed(870432)
lm_stdall_boot <- std_selected_boot(lm_out,
to_scale = ~ .,
to_center = ~ .,
nboot = 5000)
```

Since 0.2.6.3, `to_standardize`

can be used as a
shortcut:

The minimum additional argument is `nboot`

, the number of
bootstrap samples.

```
summary(lm_stdall_boot)
#>
#> Call to std_selected_boot():
#> std_selected_boot(lm_out = lm_out, to_scale = ~., to_center = ~.,
#> nboot = 5000)
#>
#> Selected variable(s) are centered by mean and/or scaled by SD
#> - Variable(s) centered: sleep age gender emot cons
#> - Variable(s) scaled: sleep age gender emot cons
#>
#> centered_by scaled_by Note
#> sleep 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 5000.
#>
#> Call:
#> lm(formula = sleep ~ 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 0.0072 0.1045 0.0488 1.1248 0.26124
#> age 0.0371 -0.0347 0.1072 0.0443 0.8384 0.40221
#> gendermale -0.1844 -0.4392 0.0783 0.1170 -1.5759 0.11570
#> emot 0.1150 0.0291 0.2012 0.0449 2.5600 0.01076 *
#> cons 0.1305 0.0288 0.2265 0.0452 2.8893 0.00403 **
#> emot:cons -0.1083 -0.2043 -0.0090 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 output is similar to that of `std_selected()`

, with
additional information on the bootstrapping process.

The 95% bootstrap percentile confidence interval of the standardized moderation effect is -0.2043 to -0.0090.

`std_selected()`

and `std_selected_boot()`

can
also be used to standardize only selected variables. There are cases in
which we do not want to standardize some continuous variables because
they are measured on interpretable units, such as hours.

Suppose we want to standardize only emotional stability and
conscientiousness, and do not standardize sleep duration. We just list
`emot`

and `cons`

on `to_center`

and
`to_scale`

:

Since 0.2.6.3, `to_standardize`

can be used a
shortuct:

```
summary(lm_std1)
#>
#> Call to std_selected():
#> std_selected(lm_out = lm_out, to_standardize = ~emot + cons)
#>
#> Selected variable(s) are centered by mean and/or scaled by SD
#> - Variable(s) centered: emot cons
#> - Variable(s) scaled: emot cons
#>
#> centered_by scaled_by Note
#> sleep 0.0000 1.0000000
#> age 0.0000 1.0000000
#> gender NA NA Nonnumeric
#> emot 2.7132 0.7629613 Standardized (mean = 0, SD = 1)
#> cons 3.3432 0.6068198 Standardized (mean = 0, SD = 1)
#>
#> Note:
#> - Categorical variables will not be centered or scaled even if
#> requested.
#>
#> Call:
#> lm(formula = sleep ~ age + gender + emot * cons, data = dat_mod)
#>
#> 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) 6.4557 0.4783 13.4979 < 0.001 ***
#> age 0.0179 0.0213 0.8384 0.40221
#> gendermale -0.2613 0.1658 -1.5759 0.11570
#> emot 0.1630 0.0637 2.5600 0.01076 *
#> cons 0.1849 0.0640 2.8893 0.00403 **
#> emot:cons -0.1534 0.0615 -2.4967 0.01286 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 1.384 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 *partially* standardized moderation effect is -0.1534. For
each one *standard deviation* increase of conscientiousness
score, the *partially* standardized effect of emotional stability
decreases by 0.1534.

The function `std_selected_boot()`

can also be used to
form the nonparametric bootstrap confidence interval when only some of
the variables are standardized:

```
set.seed(870432)
lm_std1_boot <- std_selected_boot(lm_out,
to_scale = ~ emot + cons,
to_center = ~ emot + cons,
nboot = 5000)
```

Since 0.2.6.3, `to_standardize`

can be used as a
shortcut:

Again, the only additional argument is `nboot`

.

```
summary(lm_std1_boot)
#>
#> Call to std_selected_boot():
#> std_selected_boot(lm_out = lm_out, to_scale = ~emot + cons, to_center = ~emot +
#> cons, nboot = 5000)
#>
#> Selected variable(s) are centered by mean and/or scaled by SD
#> - Variable(s) centered: emot cons
#> - Variable(s) scaled: emot cons
#>
#> centered_by scaled_by Note
#> sleep 0.0000 1.0000000
#> age 0.0000 1.0000000
#> gender NA NA Nonnumeric
#> emot 2.7132 0.7629613 Standardized (mean = 0, SD = 1)
#> cons 3.3432 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 5000.
#>
#> Call:
#> lm(formula = sleep ~ age + gender + emot * cons, data = dat_mod)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -6.0841 -0.7882 0.0089 0.9440 6.1189
#>
#> Coefficients:
#> Estimate CI Lower CI Upper Std. Error t value Pr(>|t|)
#> (Intercept) 6.4557 5.6487 7.2735 0.4783 13.4979 < 0.001 ***
#> age 0.0179 -0.0184 0.0544 0.0213 0.8384 0.40221
#> gendermale -0.2613 -0.6233 0.1105 0.1658 -1.5759 0.11570
#> emot 0.1630 0.0405 0.2893 0.0637 2.5600 0.01076 *
#> cons 0.1849 0.0415 0.3229 0.0640 2.8893 0.00403 **
#> emot:cons -0.1534 -0.2915 -0.0124 0.0615 -2.4967 0.01286 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 1.384 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 95% bootstrap percentile confidence interval of the partially standardized moderation effect is -0.2915 to -0.0124.

A more detailed illustration can be found at
`vignette("moderation", package = "stdmod")`

.

`vignette("std_selected", package = "stdmod")`

illustrates
how `std_selected()`

can be used to form nonparametric
bootstrap percentile confidence interval for standardized regression
coefficients (“betas”) for regression models without a product term.

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 request is large.

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