Standardize Variables in a Regression Model
Source:R/std_selected.R, R/std_selected_boot.R
std_selected.RdStandardize, mean center, or scale by standard deviation selected variables in a regression model and refit the model
Usage
std_selected(lm_out, to_scale = NULL, to_center = NULL, to_standardize = NULL)
std_selected_boot(
lm_out,
to_scale = NULL,
to_center = NULL,
to_standardize = NULL,
conf = 0.95,
nboot = 100,
boot_args = NULL,
save_boot_est = TRUE,
full_output = FALSE,
do_boot = TRUE
)Arguments
- lm_out
The output from
lm().- to_scale
The terms to be rescaled by standard deviation, specified by a formula as in
lm(). For example, if the terms to be scaled arex1andx3, use~ x1 + x3. No need to specify the interaction term. To scale the outcome variable, list it on the right hand side as a predictor. Specify only the original variables. IfNULL, then no terms will be rescaled by their standard deviations. Variables that are not numeric will be ignored. Default isNULL.- to_center
The terms to be mean centered, specified by a formula as in
lm(). For example, if the terms to be centered isx1andx3, use~ x1 + x3. No need to specify the interaction term. To center the outcome variable, list it on the right hand side as a predictor. Specify only the original variables. IfNULL, then no term will be centered. Default isNULL.- to_standardize
The terms to be standardized, specified by a formula as in
lm(). For example, if the terms to be standardized isx1andx3, use~ x1 + x3. No need to specify the interaction term. To standardize the outcome variable, list it on the right hand side as a predictor. Specify only the original variables. This is a shortcut toto_centerandto_scale. Listing a variable into_standardizeis equivalent to listing this variable in bothto_centerandto_scale. Default isNULL.- conf
The level of confidence for the confidence interval. Default is .95.
- nboot
The number of bootstrap samples. Default is 100.
- boot_args
A named list of arguments to be passed to
boot::boot(). Default isNULL.- save_boot_est
If
TRUE, the default, the bootstrap estimates will be saved in the elementboot_estof the output.- full_output
Whether the full output from
boot::boot()is returned. Default isFALSE. IfTRUE, the full output fromboot::boot()will be saved in the elementboot_outof the output.- do_boot
Whether bootstrapping confidence intervals will be formed. Default is
TRUE. IfFALSE, all arguments related to bootstrapping will be ignored.
Value
The updated lm() output, with the class std_selected added. It will be
treated as a usual lm() object by most functions. These are the major
additional element in the list:
scaled_terms: If notNULL, a character vector of the variables scaled.centered_terms: If notNULL, a character vector of the variables mean-centered.scaled_by: A numeric vector of the scaling factors for all the variables in the model. The value is 1 for terms not scaled.centered_by: A numeric vector of the numbers used for centering for all the variables in the model. The value is 0 for terms not centered.std_selected_call: The original call.lm_out_call: The call inlm_out.
Like std_selected(), std_selected_boot() returns the updated lm()
output, with the class std_selected added. The output of std_selected_boot()
contain these additional elements in the list:
boot_ci: A data frame of the bootstrap confidence intervals of the regression coefficient.nboot: The number of bootstrap samples requested.conf: The level of confidence, in proportion.boot_est: A matrix of the bootstrap estimates of the regression coefficients. The number of rows equal tonboot, and the number of columns equal to the number of terms in the regression model.std_selected_boot_call: The call tostd_selected_boot().boot_out: If available, the original output fromboot::boot().
Details
std_selected() was originally developed to compute the standardized
moderation effect and the standardized coefficients for other predictors
given an lm() output (Cheung, Cheung, Lau, Hui, & Vong, 2022).
It has been extended such that users can specify
which variables in a regression model are to be mean-centered and/or
rescaled by
their standard deviations. If the model has one or more interaction terms,
they will be formed after the transformation, yielding the correct
standardized solution for a moderated regression model. Moreover,
categorical predictors will be automatically skipped in mean-centering
and rescaling.
Standardization is conducted when a variable is mean-centered and then rescaled by its standard deviation. Therefore, if the goal is to get the standardized solution of a moderated regression, users just instruct the function to standardize all non-categorical variables in the regression model.
std_selected_boot() is a wrapper of std_selected(). It calls
std_selected()
once
for each bootstrap sample, and then computes the nonparametric
bootstrap
percentile confidence intervals (Cheung, Cheung, Lau, Hui, & Vong, 2022).
If do_boot is FALSE, then the object it returns is identical to that
by std_selected().
This function intentionally does not have an argument for setting the seed
for
random number. Users are recommended to set the seed, e.g., using
set.seed()
before calling it, to ensure reproducibility.
Functions
std_selected(): The base function to center or scale selected variables in a regression modelstd_selected_boot(): A wrapper ofstd_selected()that forms nonparametric bootstrap confidence intervals.
References
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. doi:10.1037/hea0001188
Author
Shu Fai Cheung https://orcid.org/0000-0002-9871-9448
Examples
# Load a sample data set
dat <- test_x_1_w_1_v_1_cat1_n_500
head(dat)
#> dv iv mod v1 cat1
#> 1 4946.751 12.76737 96.85621 11.756899 gp1
#> 2 6635.081 14.89097 106.25696 11.371237 gp2
#> 3 6060.708 15.24101 97.85852 9.377471 gp2
#> 4 7240.781 16.65782 104.80266 10.508913 gp1
#> 5 5775.759 11.84448 95.85912 15.093480 gp3
#> 6 7725.783 16.31270 100.20561 3.442902 gp2
# Do a moderated regression by lm
lm_raw <- lm(dv ~ iv*mod + v1 + cat1, dat)
summary(lm_raw)
#>
#> Call:
#> lm(formula = dv ~ iv * mod + v1 + cat1, data = dat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2146.0 -431.9 -25.0 411.2 2309.3
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 308.767 4075.066 0.076 0.9396
#> iv 52.760 271.242 0.195 0.8459
#> mod 5.127 40.772 0.126 0.9000
#> v1 -12.760 10.174 -1.254 0.2104
#> cat1gp2 -158.673 71.834 -2.209 0.0276 *
#> cat1gp3 -43.166 75.283 -0.573 0.5666
#> iv:mod 3.416 2.709 1.261 0.2080
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 665 on 493 degrees of freedom
#> Multiple R-squared: 0.6352, Adjusted R-squared: 0.6307
#> F-statistic: 143 on 6 and 493 DF, p-value: < 2.2e-16
#>
# Mean center mod only
lm_cw <- std_selected(lm_raw, to_center = ~ mod)
summary(lm_cw)
#>
#> Call to std_selected():
#> std_selected(lm_out = lm_raw, to_center = ~mod)
#>
#> Selected variable(s) are centered by mean and/or scaled by SD
#> - Variable(s) centered: mod
#>
#> centered_by scaled_by Note
#> dv 0.000 1
#> iv 0.000 1
#> mod 100.395 1 Centered (mean = 0)
#> v1 0.000 1
#> cat1 NA NA Nonnumeric
#>
#> Note:
#> - Categorical variables will not be centered or scaled even if
#> requested.
#>
#> Call:
#> lm(formula = dv ~ iv * mod + v1 + cat1, data = dat_mod)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2146.0 -431.9 -25.0 411.2 2309.3
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 823.445 249.736 3.2973 0.001047 **
#> iv 395.692 14.684 26.9480 < 0.001 ***
#> mod 5.126 40.772 0.1257 0.899992
#> v1 -12.760 10.174 -1.2542 0.210365
#> cat1gp2 -158.673 71.834 -2.2089 0.027642 *
#> cat1gp3 -43.166 75.283 -0.5734 0.566645
#> iv:mod 3.416 2.709 1.2608 0.207990
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 665 on 493 degrees of freedom
#>
#> R-squared : 0.6352
#> Adjusted R-squared : 0.6307
#> ANOVA test of R-squared : F(6, 493) = 143.047, p < 0.001
#>
#> = Test the highest order term =
#> The highest order term : iv:mod
#> R-squared increase adding this term: 0.0012
#> F test of R-squared increase : F(1, 493) = 1.5895, p = 0.208
#>
#> 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()`.
#>
# Mean center mod and iv
lm_cwx <- std_selected(lm_raw, to_center = ~ mod + iv)
summary(lm_cwx)
#>
#> Call to std_selected():
#> std_selected(lm_out = lm_raw, to_center = ~mod + iv)
#>
#> Selected variable(s) are centered by mean and/or scaled by SD
#> - Variable(s) centered: mod iv
#>
#> centered_by scaled_by Note
#> dv 0.00000 1
#> iv 15.01576 1 Centered (mean = 0)
#> mod 100.39502 1 Centered (mean = 0)
#> v1 0.00000 1
#> cat1 NA NA Nonnumeric
#>
#> Note:
#> - Categorical variables will not be centered or scaled even if
#> requested.
#>
#> Call:
#> lm(formula = dv ~ iv * mod + v1 + cat1, data = dat_mod)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2146.0 -431.9 -25.0 411.2 2309.3
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 6765.069 115.857 58.3916 < 0.001 ***
#> iv 395.692 14.684 26.9480 < 0.001 ***
#> mod 56.418 5.941 9.4962 < 0.001 ***
#> v1 -12.760 10.174 -1.2542 0.21037
#> cat1gp2 -158.673 71.834 -2.2089 0.02764 *
#> cat1gp3 -43.166 75.283 -0.5734 0.56664
#> iv:mod 3.416 2.709 1.2608 0.20799
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 665 on 493 degrees of freedom
#>
#> R-squared : 0.6352
#> Adjusted R-squared : 0.6307
#> ANOVA test of R-squared : F(6, 493) = 143.047, p < 0.001
#>
#> = Test the highest order term =
#> The highest order term : iv:mod
#> R-squared increase adding this term: 0.0012
#> F test of R-squared increase : F(1, 493) = 1.5895, p = 0.208
#>
#> 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()`.
#>
# Standardize both mod and iv
lm_stdwx <- std_selected(lm_raw, to_scale = ~ mod + iv,
to_center = ~ mod + iv)
summary(lm_stdwx)
#>
#> Call to std_selected():
#> std_selected(lm_out = lm_raw, to_scale = ~mod + iv, to_center = ~mod +
#> iv)
#>
#> Selected variable(s) are centered by mean and/or scaled by SD
#> - Variable(s) centered: mod iv
#> - Variable(s) scaled: mod iv
#>
#> centered_by scaled_by Note
#> dv 0.00000 1.000000
#> iv 15.01576 2.039154 Standardized (mean = 0, SD = 1)
#> mod 100.39502 5.040823 Standardized (mean = 0, SD = 1)
#> v1 0.00000 1.000000
#> cat1 NA NA Nonnumeric
#>
#> Note:
#> - Categorical variables will not be centered or scaled even if
#> requested.
#>
#> Call:
#> lm(formula = dv ~ iv * mod + v1 + cat1, data = dat_mod)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2146.0 -431.9 -25.0 411.2 2309.3
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 6765.07 115.86 58.3916 < 0.001 ***
#> iv 806.88 29.94 26.9480 < 0.001 ***
#> mod 284.39 29.95 9.4962 < 0.001 ***
#> v1 -12.76 10.17 -1.2542 0.21037
#> cat1gp2 -158.67 71.83 -2.2089 0.02764 *
#> cat1gp3 -43.17 75.28 -0.5734 0.56664
#> iv:mod 35.11 27.85 1.2608 0.20799
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 665 on 493 degrees of freedom
#>
#> R-squared : 0.6352
#> Adjusted R-squared : 0.6307
#> ANOVA test of R-squared : F(6, 493) = 143.047, p < 0.001
#>
#> = Test the highest order term =
#> The highest order term : iv:mod
#> R-squared increase adding this term: 0.0012
#> F test of R-squared increase : F(1, 493) = 1.5895, p = 0.208
#>
#> 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()`.
#>
# Standardize all variables except for categorical variables.
# Interaction terms are formed after standardization.
lm_std <- std_selected(lm_raw, to_scale = ~ .,
to_center = ~ .)
summary(lm_std)
#>
#> Call to std_selected():
#> std_selected(lm_out = lm_raw, to_scale = ~., to_center = ~.)
#>
#> Selected variable(s) are centered by mean and/or scaled by SD
#> - Variable(s) centered: dv iv mod v1 cat1
#> - Variable(s) scaled: dv iv mod v1 cat1
#>
#> centered_by scaled_by Note
#> dv 6565.02965 1094.244465 Standardized (mean = 0, SD = 1)
#> iv 15.01576 2.039154 Standardized (mean = 0, SD = 1)
#> mod 100.39502 5.040823 Standardized (mean = 0, SD = 1)
#> v1 10.13884 2.938932 Standardized (mean = 0, SD = 1)
#> cat1 NA NA Nonnumeric
#>
#> Note:
#> - Categorical variables will not be centered or scaled even if
#> requested.
#>
#> Call:
#> lm(formula = dv ~ iv * mod + v1 + cat1, data = dat_mod)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -1.96117 -0.39474 -0.02285 0.37579 2.11040
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.0646 0.0483 1.3385 0.18136
#> iv 0.7374 0.0274 26.9480 < 0.001 ***
#> mod 0.2599 0.0274 9.4962 < 0.001 ***
#> v1 -0.0343 0.0273 -1.2542 0.21037
#> cat1gp2 -0.1450 0.0656 -2.2089 0.02764 *
#> cat1gp3 -0.0394 0.0688 -0.5734 0.56664
#> iv:mod 0.0321 0.0255 1.2608 0.20799
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 0.6077 on 493 degrees of freedom
#>
#> R-squared : 0.6352
#> Adjusted R-squared : 0.6307
#> ANOVA test of R-squared : F(6, 493) = 143.047, p < 0.001
#>
#> = Test the highest order term =
#> The highest order term : iv:mod
#> R-squared increase adding this term: 0.0012
#> F test of R-squared increase : F(1, 493) = 1.5895, p = 0.208
#>
#> 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()`.
#>
# Use to_standardize as a shortcut
lm_stdwx2 <- std_selected(lm_raw, to_standardize = ~ mod + iv)
# The results are the same
coef(lm_stdwx)
#> (Intercept) iv mod v1 cat1gp2 cat1gp3
#> 6765.06947 806.87814 284.39275 -12.75999 -158.67255 -43.16606
#> iv:mod
#> 35.11145
coef(lm_stdwx2)
#> (Intercept) iv mod v1 cat1gp2 cat1gp3
#> 6765.06947 806.87814 284.39275 -12.75999 -158.67255 -43.16606
#> iv:mod
#> 35.11145
all.equal(coef(lm_stdwx), coef(lm_stdwx2))
#> [1] TRUE
dat <- test_x_1_w_1_v_1_cat1_n_500
head(dat)
#> dv iv mod v1 cat1
#> 1 4946.751 12.76737 96.85621 11.756899 gp1
#> 2 6635.081 14.89097 106.25696 11.371237 gp2
#> 3 6060.708 15.24101 97.85852 9.377471 gp2
#> 4 7240.781 16.65782 104.80266 10.508913 gp1
#> 5 5775.759 11.84448 95.85912 15.093480 gp3
#> 6 7725.783 16.31270 100.20561 3.442902 gp2
# Do a moderated regression by lm
lm_raw <- lm(dv ~ iv*mod + v1 + cat1, dat)
summary(lm_raw)
#>
#> Call:
#> lm(formula = dv ~ iv * mod + v1 + cat1, data = dat)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2146.0 -431.9 -25.0 411.2 2309.3
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 308.767 4075.066 0.076 0.9396
#> iv 52.760 271.242 0.195 0.8459
#> mod 5.127 40.772 0.126 0.9000
#> v1 -12.760 10.174 -1.254 0.2104
#> cat1gp2 -158.673 71.834 -2.209 0.0276 *
#> cat1gp3 -43.166 75.283 -0.573 0.5666
#> iv:mod 3.416 2.709 1.261 0.2080
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 665 on 493 degrees of freedom
#> Multiple R-squared: 0.6352, Adjusted R-squared: 0.6307
#> F-statistic: 143 on 6 and 493 DF, p-value: < 2.2e-16
#>
# Standardize all variables as in std_selected above, and compute the
# nonparametric bootstrapping percentile confidence intervals.
set.seed(87053)
lm_std_boot <- std_selected_boot(lm_raw,
to_scale = ~ .,
to_center = ~ .,
conf = .95,
nboot = 100)
# In real analysis, nboot should be at least 2000.
summary(lm_std_boot)
#>
#> Call to std_selected_boot():
#> std_selected_boot(lm_out = lm_raw, to_scale = ~., to_center = ~.,
#> conf = 0.95, nboot = 100)
#>
#> Selected variable(s) are centered by mean and/or scaled by SD
#> - Variable(s) centered: dv iv mod v1 cat1
#> - Variable(s) scaled: dv iv mod v1 cat1
#>
#> centered_by scaled_by Note
#> dv 6565.02965 1094.244465 Standardized (mean = 0, SD = 1)
#> iv 15.01576 2.039154 Standardized (mean = 0, SD = 1)
#> mod 100.39502 5.040823 Standardized (mean = 0, SD = 1)
#> v1 10.13884 2.938932 Standardized (mean = 0, SD = 1)
#> cat1 NA NA Nonnumeric
#>
#> Note:
#> - Categorical variables will not be centered or scaled even if
#> requested.
#> - Nonparametric bootstrapping 95% confidence intervals computed.
#> - The number of bootstrap samples is 100.
#>
#> Call:
#> lm(formula = dv ~ iv * mod + v1 + cat1, data = dat_mod)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -1.96117 -0.39474 -0.02285 0.37579 2.11040
#>
#> Coefficients:
#> Estimate CI Lower CI Upper Std. Error t value Pr(>|t|)
#> (Intercept) 0.0646 -0.0301 0.1683 0.0483 1.3385 0.18136
#> iv 0.7374 0.7041 0.7948 0.0274 26.9480 < 0.001 ***
#> mod 0.2599 0.2129 0.3130 0.0274 9.4962 < 0.001 ***
#> v1 -0.0343 -0.0835 0.0392 0.0273 -1.2542 0.21037
#> cat1gp2 -0.1450 -0.3127 -0.0046 0.0656 -2.2089 0.02764 *
#> cat1gp3 -0.0394 -0.1845 0.1222 0.0688 -0.5734 0.56664
#> iv:mod 0.0321 -0.0103 0.0817 0.0255 1.2608 0.20799
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 0.6077 on 493 degrees of freedom
#>
#> R-squared : 0.6352
#> Adjusted R-squared : 0.6307
#> ANOVA test of R-squared : F(6, 493) = 143.047, p < 0.001
#>
#> = Test the highest order term =
#> The highest order term : iv:mod
#> R-squared increase adding this term: 0.0012
#> F test of R-squared increase : F(1, 493) = 1.5895, p = 0.208
#>
#> 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.
#>
# Use to_standardize as a shortcut
set.seed(87053)
lm_std_boot2 <- std_selected_boot(lm_raw,
to_standardize = ~ .,
conf = .95,
nboot = 100)
# The results are the same
confint(lm_std_boot)
#> 2.5 % 97.5 %
#> (Intercept) -0.03011698 0.16832106
#> iv 0.70408792 0.79477219
#> mod 0.21293423 0.31304112
#> v1 -0.08345938 0.03923478
#> cat1gp2 -0.31267483 -0.00464722
#> cat1gp3 -0.18447930 0.12223465
#> iv:mod -0.01033317 0.08169839
confint(lm_std_boot2)
#> 2.5 % 97.5 %
#> (Intercept) -0.03011698 0.16832106
#> iv 0.70408792 0.79477219
#> mod 0.21293423 0.31304112
#> v1 -0.08345938 0.03923478
#> cat1gp2 -0.31267483 -0.00464722
#> cat1gp3 -0.18447930 0.12223465
#> iv:mod -0.01033317 0.08169839
all.equal(confint(lm_std_boot), confint(lm_std_boot2))
#> [1] TRUE