Nothing
Standardized Moderation Effect by std_selected()"
In stdmod: Standardized Moderation Effect and Its Confidence Interval
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/.
Setup the Environment
library(stdmod) # For computing the standardized moderation effect conveniently
Load the Dataset
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 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")
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:
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")
plot of chunk mod_reg
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")
plot of chunk mod_reg_stdall
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] '1.7.3'
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_standardize = ~., 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
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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/.
Setup the Environment
library(stdmod) # For computing the standardized moderation effect conveniently
Load the Dataset
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 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") 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:
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")
plot of chunk mod_reg
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_centerspecifies variables to be mean centered. -
The argument
to_scalespecifies variables to be rescaled by their standard deviations after centering. -
In
stdmod0.2.6.3, the argumentto_standardizewas introduced as a shortcut. Listing a variable into_standardizeis equivalent to listing it into_centerandto_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")
plot of chunk mod_reg_stdall
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] '1.7.3' 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_standardize = ~., 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
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