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Beta-Select Demonstration: Regression by `lm()`"
In betaselectr: Betas-Select in Structural Equation Models and Linear Models
Introduction
This article demonstrates how to use
lm_betaselect() from the package
betaselectr
to standardize
selected variables in a model fitted
by lm() and forming confidence
intervals for the parameters.
Data and Model
The sample dataset from the package
betaselectr will be used for this
demonstration:
library(betaselectr)
head(data_test_mod_cat2)
#> dv iv mod cov1 cat1
#> 1 15.53 13.95 50.75 25.33 gp2
#> 2 17.69 15.07 49.67 20.96 gp1
#> 3 28.56 14.43 53.42 19.22 gp3
#> 4 25.00 11.22 42.55 20.18 gp2
#> 5 19.33 14.93 52.12 22.82 gp2
#> 6 20.62 10.22 39.36 18.41 gp1
This is the regression model, fitted by
lm():
lm_out <- lm(dv ~ iv * mod + cov1 + cat1,
data = data_test_mod_cat2)
The model has a moderator, mod, posited
to moderate the effect from iv to
med. The product term is iv:mod.
The variable cat1 is a categorical variable
with three groups: gp1, gp2, gp3.
These are the results:
summary(lm_out)
#>
#> Call:
#> lm(formula = dv ~ iv * mod + cov1 + cat1, data = data_test_mod_cat2)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -17.0892 -4.6312 0.0057 5.0186 18.7053
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 90.87211 34.04462 2.669 0.00803 **
#> iv -6.06932 2.33701 -2.597 0.00988 **
#> mod -1.61636 0.68840 -2.348 0.01954 *
#> cov1 0.09885 0.19433 0.509 0.61136
#> cat1gp2 1.71248 1.15064 1.488 0.13775
#> cat1gp3 2.47838 1.10562 2.242 0.02574 *
#> iv:mod 0.13230 0.04656 2.841 0.00481 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 6.743 on 293 degrees of freedom
#> Multiple R-squared: 0.1149, Adjusted R-squared: 0.09676
#> F-statistic: 6.338 on 6 and 293 DF, p-value: 2.759e-06
Problems With Standardization
One common type of standardized coefficients,
called "betas" in some programs, is
computed by standardizing all terms
in the model.
First, all variables in the model,
including the product term and dummy
variables, are computed:
data_test_mod_cat2_z <- data_test_mod_cat2
data_test_mod_cat2_z$iv_x_mod <- data_test_mod_cat2_z$iv *
data_test_mod_cat2_z$mod
data_test_mod_cat2_z$cat_gp2 <- as.numeric(data_test_mod_cat2_z$cat1 == "gp2")
data_test_mod_cat2_z$cat_gp3 <- as.numeric(data_test_mod_cat2_z$cat1 == "gp3")
head(data_test_mod_cat2_z)
#> dv iv mod cov1 cat1 iv_x_mod cat_gp2 cat_gp3
#> 1 15.53 13.95 50.75 25.33 gp2 707.9625 1 0
#> 2 17.69 15.07 49.67 20.96 gp1 748.5269 0 0
#> 3 28.56 14.43 53.42 19.22 gp3 770.8506 0 1
#> 4 25.00 11.22 42.55 20.18 gp2 477.4110 1 0
#> 5 19.33 14.93 52.12 22.82 gp2 778.1516 1 0
#> 6 20.62 10.22 39.36 18.41 gp1 402.2592 0 0
All the variables are then standardized:
data_test_mod_cat2_z <- data.frame(scale(data_test_mod_cat2_z[, -5]))
head(data_test_mod_cat2_z)
#> dv iv mod cov1 iv_x_mod cat_gp2
#> 1 -0.9458226 -0.44874323 0.23147783 2.553777460 -0.24816181 1.331109
#> 2 -0.6414005 0.13926755 -0.03378874 0.390649940 0.06187143 -0.748749
#> 3 0.8905756 -0.19673861 0.88727574 -0.470641109 0.23249121 -0.748749
#> 4 0.3888429 -1.88201951 -1.78258317 0.004553953 -2.01026427 1.331109
#> 5 -0.4102652 0.06576621 0.56797339 1.311340372 0.28829267 1.331109
#> 6 -0.2284575 -2.40702913 -2.56610202 -0.871586942 -2.58464861 -0.748749
#> cat_gp3
#> 1 -0.9401258
#> 2 -0.9401258
#> 3 1.0601418
#> 4 -0.9401258
#> 5 -0.9401258
#> 6 -0.9401258
The regression model is then fitted to the
standardized variables:
lm_std_common <- lm(dv ~ iv + mod + cov1 + cat_gp2 + cat_gp3 + iv_x_mod,
data = data_test_mod_cat2_z)
The "betas" commonly reported are the
coefficients in this model:
lm_std_common_summary <- summary(lm_std_common)
printCoefmat(lm_std_common_summary$coefficients,
digits = 5,
zap.ind = 1,
P.values = TRUE,
has.Pvalue = TRUE,
signif.stars = TRUE)
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.000000 0.054871 0.0000 1.000000
#> iv -1.629280 0.627360 -2.5970 0.009878 **
#> mod -0.927480 0.395006 -2.3480 0.019539 *
#> cov1 0.028140 0.055329 0.5087 0.611359
#> cat_gp2 0.116040 0.077970 1.4883 0.137753
#> cat_gp3 0.174620 0.077901 2.2416 0.025735 *
#> iv_x_mod 2.439510 0.858601 2.8413 0.004809 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
However, for this model, there are several
problems:
-
The product term is also
standardized (iv_x_mod, computed
using the standard deviations of
dv and iv:mod). This is inappropriate [@hayes_introduction_2022].
One simple but underused solution is
standardizing the variables before
forming the product term [@friedrich_defense_1982].
-
The confidence intervals are formed using
ordinary least squares (OLS), which does not
take into account the sampling variation
of the standardizers (the sample standard
deviations used in standardization) and
so the standard errors may be
biased [@yuan_biases_2011].
Although there
are situations in which the OLS
confidence and the nonparametric
percentile bootstrap confidences can be
similar (e.g., sample size is large
and the population values are not extreme),
it is recommended to use bootstrap
confidence intervals when computation
cost is low [@jones_computing_2013].
-
There are cases in which some variabLes
are measured by meaningful units and
do not need to be standardized. For
example, if cov1 is age measured by
year, then age is more
meaningful than "standardized age".
-
In regression models, categorical variables
are usually represented by dummy variables,
each of them having only two possible
values (0 or 1). It is not meaningful
to standardize the dummy variables
[@darlington_regression_2016].
Beta-Select by lm_betaselect()
The function lm_betaselect() can be used
to solve these problems by:
-
standardizing variables before product
terms are formed,
-
standardizing only variables for which
standardization can facilitate
interpretation, and
-
forming bootstrap confidence intervals
that take
into account selected standardization.
We call the coefficients computed by
this kind of standardization betas-select
($\beta{s}{Select}$, $\beta{Select}$
in singular form),
to differentiate them from coefficients
computed by standardizing all variables,
including product terms.
Estimates Only
Suppose we only need to
solve the first problem, standardizing all
numeric variables,
with the product
term computed after iv, mod, and dv
are standardized.
lm_beta_select <- lm_betaselect(dv ~ iv*mod + cov1 + cat1,
data = data_test_mod_cat2,
do_boot = FALSE)
The function lm_betaselect() can be
used as lm(), with applicable arguments
such as the model formula and data passed
to lm().
By default, all numeric variables will
be standardized before fitting the models.
Terms such as product terms are created
after standardization.
Moreover, categorical variables (factors and
string variables) will not be standardized.
Bootstrapping is done by default. In this
illustration, do_boot = FALSE is added
to disable it because we only want to
address the first problem. We will do bootstrapping when
addressing the issue with confidence intervals.
The summary() method can be used
ont the output of lm_betaselect():
summary(lm_beta_select)
#> Call to lm_betaselect():
#> betaselectr::lm_betaselect(formula = dv ~ iv * mod + cov1 + cat1,
#> data = data_test_mod_cat2, do_boot = FALSE)
#>
#> Variable(s) standardized: dv, iv, mod, cov1
#>
#> Call:
#> stats::lm(formula = dv ~ iv * mod + cov1 + cat1, data = betaselectr::std_data(data = data_test_mod_cat2,
#> to_standardize = c("dv", "iv", "mod", "cov1")))
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2.4085 -0.6527 0.0008 0.7073 2.6363
#>
#> Coefficients:
#> Estimate CI.Lower CI.Upper Std. Error t value Pr(>|t|)
#> (Intercept) -0.308 -0.576 -0.040 0.136 -2.259 0.02461 *
#> iv 0.140 0.021 0.258 0.060 2.324 0.02082 *
#> mod 0.196 0.078 0.315 0.060 3.264 0.00123 **
#> cov1 0.028 -0.081 0.137 0.055 0.509 0.61136
#> cat1gp2 0.241 -0.078 0.561 0.162 1.488 0.13775
#> cat1gp3 0.349 0.043 0.656 0.156 2.242 0.02574 *
#> iv:mod 0.145 0.044 0.245 0.051 2.841 0.00481 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.9504 on 293 degrees of freedom
#>
#> R-squared : 0.115
#> Adjusted R-squared : 0.097
#> ANOVA test of R-squared : F(6, 293) = 6.338, p < 0.001
#>
#> Note:
#> - Results *after* standardization are reported.
#> - Standard errors are least-squares standard errors.
#> - T values are computed by 'Estimate / Std. Error'.
#> - P-values are usual t-test p-values.
#> - Least squares standard errors, t values, p-values, and confidence
#> intervals (if reported) should not be used for coefficients involved
#> in standardization.
#> - Least squares 95.0% confidence interval reported.
Compared to the solution with the product
term standardized, the coefficient of
iv:mod changed substantially from
2.440 to
0.145. As shown by
@cheung_improving_2022, the coefficient
of standardized product term (iv:mod)
can be substantially different from the
properly standardized product term
(the product of standardized iv and
standardized mod).
Estimates and Bootstrap Confidence Interval
Suppose we want to address
both the first and the second problems,
with
-
the product term computed after iv,
mod, and dv are standardized, and
-
bootstrap confidence interval used.
We can call lm_betaselect() again, with
additional arguments
set:
lm_beta_select_boot <- lm_betaselect(dv ~ iv*mod + cov1 + cat1,
data = data_test_mod_cat2,
bootstrap = 5000,
iseed = 4567)
These are the additional arguments:
-
bootstrap: The number of bootstrap
samples to draw. Default is 100. It should
be set to 5000 or even 10000.
-
iseed: The seed for the random number
generator used in bootstrapping. Set
this to an integer to make the results
reproducible.
This is the output of summary()
summary(lm_beta_select_boot)
#> Call to lm_betaselect():
#> betaselectr::lm_betaselect(formula = dv ~ iv * mod + cov1 + cat1,
#> data = data_test_mod_cat2, bootstrap = 5000, iseed = 4567)
#>
#> Variable(s) standardized: dv, iv, mod, cov1
#>
#> Call:
#> stats::lm(formula = dv ~ iv * mod + cov1 + cat1, data = betaselectr::std_data(data = data_test_mod_cat2,
#> to_standardize = c("dv", "iv", "mod", "cov1")))
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2.4085 -0.6527 0.0008 0.7073 2.6363
#>
#> Coefficients:
#> Estimate CI.Lower CI.Upper Std. Error z value Pr(Boot)
#> (Intercept) -0.308 -0.536 -0.080 0.117 -2.636 0.0056 **
#> iv 0.140 0.009 0.268 0.066 2.109 0.0384 *
#> mod 0.196 0.075 0.317 0.061 3.208 0.0016 **
#> cov1 0.028 -0.075 0.131 0.052 0.537 0.5700
#> cat1gp2 0.241 -0.067 0.540 0.155 1.560 0.1276
#> cat1gp3 0.349 0.064 0.631 0.146 2.394 0.0152 *
#> iv:mod 0.145 0.059 0.228 0.043 3.356 0.0020 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.9504 on 293 degrees of freedom
#>
#> R-squared : 0.115
#> Adjusted R-squared : 0.097
#> ANOVA test of R-squared : F(6, 293) = 6.338, p < 0.001
#>
#> Note:
#> - Results *after* standardization are reported.
#> - Nonparametric bootstrapping conducted.
#> - The number of bootstrap samples is 5000.
#> - Standard errors are bootstrap standard errors.
#> - Z values are computed by 'Estimate / Std. Error'.
#> - The bootstrap p-values are asymmetric p-values by Asparouhov and
#> Muthén (2021).
#> - Percentile bootstrap 95.0% confidence interval reported.
By default, 95% percentile bootstrap
confidence intervals are printed
(CI.Lower and CI.Upper). The p-values
(Pr(Boot)) are asymmetric bootstrap
p-values [@asparouhov_bootstrap_2021].
Estimates and Bootstrap Confidence Intervals, With Only Selected Variables Standardized
Suppose we want to address also the
the third issue, and standardize only
some of the variables. This can be
done using either to_standardize
or not_to_standardize.
-
Use to_standardize when
the number of variables to standardize
is much fewer than number of the variables
not to standardize.
-
Use not_to_standardize
when the number of variables to standardize
is much more than the number of
variables not to standardize.
For example, suppose we only
need to standardize dv and
iv,
this is the call to do
this, setting
to_standardize to c("iv", "dv"):
lm_beta_select_boot_1 <- lm_betaselect(dv ~ iv*mod + cov1 + cat1,
data = data_test_mod_cat2,
to_standardize = c("dv", "iv"),
bootstrap = 5000,
iseed = 4567)
If we want to standardize all
variables except for mod
and cov1, we can use
this call, and set
not_to_standardize to c("mod", "cov1"):
lm_beta_select_boot_2 <- lm_betaselect(dv ~ iv*mod + cov1 + cat1,
data = data_test_mod_cat2,
not_to_standardize = c("mod", "cov1"),
bootstrap = 5000,
iseed = 4567)
The results of these calls are identical,
and only those of the first version are
printed:
summary(lm_beta_select_boot_1)
#> Call to lm_betaselect():
#> betaselectr::lm_betaselect(formula = dv ~ iv * mod + cov1 + cat1,
#> data = data_test_mod_cat2, to_standardize = c("dv", "iv"),
#> bootstrap = 5000, iseed = 4567)
#>
#> Variable(s) standardized: dv, iv
#>
#> Call:
#> stats::lm(formula = dv ~ iv * mod + cov1 + cat1, data = betaselectr::std_data(data = data_test_mod_cat2,
#> to_standardize = c("dv", "iv")))
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -2.4085 -0.6527 0.0008 0.7073 2.6363
#>
#> Coefficients:
#> Estimate CI.Lower CI.Upper Std. Error z value Pr(Boot)
#> (Intercept) -2.991 -4.769 -1.196 0.899 -3.326 0.0024 **
#> iv -1.629 -2.667 -0.573 0.539 -3.021 0.0036 **
#> mod 0.048 0.019 0.078 0.015 3.199 0.0016 **
#> cov1 0.014 -0.037 0.066 0.026 0.533 0.5700
#> cat1gp2 0.241 -0.067 0.540 0.155 1.560 0.1276
#> cat1gp3 0.349 0.064 0.631 0.146 2.394 0.0152 *
#> iv:mod 0.036 0.015 0.056 0.011 3.366 0.0020 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.9504 on 293 degrees of freedom
#>
#> R-squared : 0.115
#> Adjusted R-squared : 0.097
#> ANOVA test of R-squared : F(6, 293) = 6.338, p < 0.001
#>
#> Note:
#> - Results *after* standardization are reported.
#> - Nonparametric bootstrapping conducted.
#> - The number of bootstrap samples is 5000.
#> - Standard errors are bootstrap standard errors.
#> - Z values are computed by 'Estimate / Std. Error'.
#> - The bootstrap p-values are asymmetric p-values by Asparouhov and
#> Muthén (2021).
#> - Percentile bootstrap 95.0% confidence interval reported.
For betas-select, researchers need
to state which variables
are standardized and which are not.
This can be done in table notes.
Categorical Variables
When calling lm_betaselect(),
categorical variables (factors and
string variables) will never be standardized.
In the example above, the coefficients
of the two dummy variables when both
the dummy variables and the outcome
variables are standardized are
0.116 and
0.175:
printCoefmat(lm_std_common_summary$coefficients[5:6, ],
digits = 5,
zap.ind = 1,
P.values = TRUE,
has.Pvalue = TRUE,
signif.stars = TRUE)
#> Estimate Std. Error t value Pr(>|t|)
#> cat_gp2 0.116041 0.077970 1.4883 0.13775
#> cat_gp3 0.174623 0.077901 2.2416 0.02574 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
These two values are not interpretable
because it does not make sense to talk
about a "one-SD change" in the dummy variables.
The betas-Select of the dummy variables,
with only the outcome variable standardized,
are
0.241 and
0.349.
lm_beta_select_boot_summary <- summary(lm_beta_select_boot)
printCoefmat(lm_beta_select_boot_summary$coefficients[5:6, ],
digits = 5,
zap.ind = 1,
P.values = TRUE,
has.Pvalue = TRUE,
signif.stars = TRUE)
#> Estimate CI.Lower CI.Upper Std. Error z value Pr(Boot)
#> cat1gp2 0.241350 -0.067312 0.540477 0.154668 1.5604 0.1276
#> cat1gp3 0.349290 0.064136 0.630539 0.145927 2.3936 0.0152 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
They
represent the difference between gp2
and gp3 from the reference group,
gp1, on the standardized outcome variable.
That is, their meanings are the same before
and after standardization. The only difference
is in the unit of the outcome variable.
Conclusion
In regression analysis, there
are situations in which standardizing
all variables is not appropriate, or
when standardization needs to be done
before forming product terms. We are
not aware of tools that can do appropriate
standardization and form confidence
intervals that takes into account the
selective standardization. By promoting
the use of betas-select using
lm_betaselect(), we hope to make it
easier for researchers to do appropriate
standardization when reporting regression
results.
References
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Introduction
This article demonstrates how to use
lm_betaselect() from the package
betaselectr
to standardize
selected variables in a model fitted
by lm() and forming confidence
intervals for the parameters.
Data and Model
The sample dataset from the package
betaselectr will be used for this
demonstration:
library(betaselectr) head(data_test_mod_cat2) #> dv iv mod cov1 cat1 #> 1 15.53 13.95 50.75 25.33 gp2 #> 2 17.69 15.07 49.67 20.96 gp1 #> 3 28.56 14.43 53.42 19.22 gp3 #> 4 25.00 11.22 42.55 20.18 gp2 #> 5 19.33 14.93 52.12 22.82 gp2 #> 6 20.62 10.22 39.36 18.41 gp1
This is the regression model, fitted by
lm():
lm_out <- lm(dv ~ iv * mod + cov1 + cat1, data = data_test_mod_cat2)
The model has a moderator, mod, posited
to moderate the effect from iv to
med. The product term is iv:mod.
The variable cat1 is a categorical variable
with three groups: gp1, gp2, gp3.
These are the results:
summary(lm_out) #> #> Call: #> lm(formula = dv ~ iv * mod + cov1 + cat1, data = data_test_mod_cat2) #> #> Residuals: #> Min 1Q Median 3Q Max #> -17.0892 -4.6312 0.0057 5.0186 18.7053 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 90.87211 34.04462 2.669 0.00803 ** #> iv -6.06932 2.33701 -2.597 0.00988 ** #> mod -1.61636 0.68840 -2.348 0.01954 * #> cov1 0.09885 0.19433 0.509 0.61136 #> cat1gp2 1.71248 1.15064 1.488 0.13775 #> cat1gp3 2.47838 1.10562 2.242 0.02574 * #> iv:mod 0.13230 0.04656 2.841 0.00481 ** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 6.743 on 293 degrees of freedom #> Multiple R-squared: 0.1149, Adjusted R-squared: 0.09676 #> F-statistic: 6.338 on 6 and 293 DF, p-value: 2.759e-06
Problems With Standardization
One common type of standardized coefficients, called "betas" in some programs, is computed by standardizing all terms in the model.
First, all variables in the model, including the product term and dummy variables, are computed:
data_test_mod_cat2_z <- data_test_mod_cat2 data_test_mod_cat2_z$iv_x_mod <- data_test_mod_cat2_z$iv * data_test_mod_cat2_z$mod data_test_mod_cat2_z$cat_gp2 <- as.numeric(data_test_mod_cat2_z$cat1 == "gp2") data_test_mod_cat2_z$cat_gp3 <- as.numeric(data_test_mod_cat2_z$cat1 == "gp3") head(data_test_mod_cat2_z) #> dv iv mod cov1 cat1 iv_x_mod cat_gp2 cat_gp3 #> 1 15.53 13.95 50.75 25.33 gp2 707.9625 1 0 #> 2 17.69 15.07 49.67 20.96 gp1 748.5269 0 0 #> 3 28.56 14.43 53.42 19.22 gp3 770.8506 0 1 #> 4 25.00 11.22 42.55 20.18 gp2 477.4110 1 0 #> 5 19.33 14.93 52.12 22.82 gp2 778.1516 1 0 #> 6 20.62 10.22 39.36 18.41 gp1 402.2592 0 0
All the variables are then standardized:
data_test_mod_cat2_z <- data.frame(scale(data_test_mod_cat2_z[, -5])) head(data_test_mod_cat2_z) #> dv iv mod cov1 iv_x_mod cat_gp2 #> 1 -0.9458226 -0.44874323 0.23147783 2.553777460 -0.24816181 1.331109 #> 2 -0.6414005 0.13926755 -0.03378874 0.390649940 0.06187143 -0.748749 #> 3 0.8905756 -0.19673861 0.88727574 -0.470641109 0.23249121 -0.748749 #> 4 0.3888429 -1.88201951 -1.78258317 0.004553953 -2.01026427 1.331109 #> 5 -0.4102652 0.06576621 0.56797339 1.311340372 0.28829267 1.331109 #> 6 -0.2284575 -2.40702913 -2.56610202 -0.871586942 -2.58464861 -0.748749 #> cat_gp3 #> 1 -0.9401258 #> 2 -0.9401258 #> 3 1.0601418 #> 4 -0.9401258 #> 5 -0.9401258 #> 6 -0.9401258
The regression model is then fitted to the standardized variables:
lm_std_common <- lm(dv ~ iv + mod + cov1 + cat_gp2 + cat_gp3 + iv_x_mod, data = data_test_mod_cat2_z)
The "betas" commonly reported are the coefficients in this model:
lm_std_common_summary <- summary(lm_std_common) printCoefmat(lm_std_common_summary$coefficients, digits = 5, zap.ind = 1, P.values = TRUE, has.Pvalue = TRUE, signif.stars = TRUE) #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 0.000000 0.054871 0.0000 1.000000 #> iv -1.629280 0.627360 -2.5970 0.009878 ** #> mod -0.927480 0.395006 -2.3480 0.019539 * #> cov1 0.028140 0.055329 0.5087 0.611359 #> cat_gp2 0.116040 0.077970 1.4883 0.137753 #> cat_gp3 0.174620 0.077901 2.2416 0.025735 * #> iv_x_mod 2.439510 0.858601 2.8413 0.004809 ** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
However, for this model, there are several problems:
-
The product term is also standardized (
iv_x_mod, computed using the standard deviations ofdvandiv:mod). This is inappropriate [@hayes_introduction_2022]. One simple but underused solution is standardizing the variables before forming the product term [@friedrich_defense_1982]. -
The confidence intervals are formed using ordinary least squares (OLS), which does not take into account the sampling variation of the standardizers (the sample standard deviations used in standardization) and so the standard errors may be biased [@yuan_biases_2011]. Although there are situations in which the OLS confidence and the nonparametric percentile bootstrap confidences can be similar (e.g., sample size is large and the population values are not extreme), it is recommended to use bootstrap confidence intervals when computation cost is low [@jones_computing_2013].
-
There are cases in which some variabLes are measured by meaningful units and do not need to be standardized. For example, if
cov1is age measured by year, then age is more meaningful than "standardized age". -
In regression models, categorical variables are usually represented by dummy variables, each of them having only two possible values (0 or 1). It is not meaningful to standardize the dummy variables [@darlington_regression_2016].
Beta-Select by lm_betaselect()
The function lm_betaselect() can be used
to solve these problems by:
-
standardizing variables before product terms are formed,
-
standardizing only variables for which standardization can facilitate interpretation, and
-
forming bootstrap confidence intervals that take into account selected standardization.
We call the coefficients computed by this kind of standardization betas-select ($\beta{s}{Select}$, $\beta{Select}$ in singular form), to differentiate them from coefficients computed by standardizing all variables, including product terms.
Estimates Only
Suppose we only need to
solve the first problem, standardizing all
numeric variables,
with the product
term computed after iv, mod, and dv
are standardized.
lm_beta_select <- lm_betaselect(dv ~ iv*mod + cov1 + cat1, data = data_test_mod_cat2, do_boot = FALSE)
The function lm_betaselect() can be
used as lm(), with applicable arguments
such as the model formula and data passed
to lm().
By default, all numeric variables will be standardized before fitting the models. Terms such as product terms are created after standardization.
Moreover, categorical variables (factors and string variables) will not be standardized.
Bootstrapping is done by default. In this
illustration, do_boot = FALSE is added
to disable it because we only want to
address the first problem. We will do bootstrapping when
addressing the issue with confidence intervals.
The summary() method can be used
ont the output of lm_betaselect():
summary(lm_beta_select) #> Call to lm_betaselect(): #> betaselectr::lm_betaselect(formula = dv ~ iv * mod + cov1 + cat1, #> data = data_test_mod_cat2, do_boot = FALSE) #> #> Variable(s) standardized: dv, iv, mod, cov1 #> #> Call: #> stats::lm(formula = dv ~ iv * mod + cov1 + cat1, data = betaselectr::std_data(data = data_test_mod_cat2, #> to_standardize = c("dv", "iv", "mod", "cov1"))) #> #> Residuals: #> Min 1Q Median 3Q Max #> -2.4085 -0.6527 0.0008 0.7073 2.6363 #> #> Coefficients: #> Estimate CI.Lower CI.Upper Std. Error t value Pr(>|t|) #> (Intercept) -0.308 -0.576 -0.040 0.136 -2.259 0.02461 * #> iv 0.140 0.021 0.258 0.060 2.324 0.02082 * #> mod 0.196 0.078 0.315 0.060 3.264 0.00123 ** #> cov1 0.028 -0.081 0.137 0.055 0.509 0.61136 #> cat1gp2 0.241 -0.078 0.561 0.162 1.488 0.13775 #> cat1gp3 0.349 0.043 0.656 0.156 2.242 0.02574 * #> iv:mod 0.145 0.044 0.245 0.051 2.841 0.00481 ** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 0.9504 on 293 degrees of freedom #> #> R-squared : 0.115 #> Adjusted R-squared : 0.097 #> ANOVA test of R-squared : F(6, 293) = 6.338, p < 0.001 #> #> Note: #> - Results *after* standardization are reported. #> - Standard errors are least-squares standard errors. #> - T values are computed by 'Estimate / Std. Error'. #> - P-values are usual t-test p-values. #> - Least squares standard errors, t values, p-values, and confidence #> intervals (if reported) should not be used for coefficients involved #> in standardization. #> - Least squares 95.0% confidence interval reported.
Compared to the solution with the product
term standardized, the coefficient of
iv:mod changed substantially from
2.440 to
0.145. As shown by
@cheung_improving_2022, the coefficient
of standardized product term (iv:mod)
can be substantially different from the
properly standardized product term
(the product of standardized iv and
standardized mod).
Estimates and Bootstrap Confidence Interval
Suppose we want to address both the first and the second problems, with
-
the product term computed after
iv,mod, anddvare standardized, and -
bootstrap confidence interval used.
We can call lm_betaselect() again, with
additional arguments
set:
lm_beta_select_boot <- lm_betaselect(dv ~ iv*mod + cov1 + cat1, data = data_test_mod_cat2, bootstrap = 5000, iseed = 4567)
These are the additional arguments:
-
bootstrap: The number of bootstrap samples to draw. Default is 100. It should be set to 5000 or even 10000. -
iseed: The seed for the random number generator used in bootstrapping. Set this to an integer to make the results reproducible.
This is the output of summary()
summary(lm_beta_select_boot) #> Call to lm_betaselect(): #> betaselectr::lm_betaselect(formula = dv ~ iv * mod + cov1 + cat1, #> data = data_test_mod_cat2, bootstrap = 5000, iseed = 4567) #> #> Variable(s) standardized: dv, iv, mod, cov1 #> #> Call: #> stats::lm(formula = dv ~ iv * mod + cov1 + cat1, data = betaselectr::std_data(data = data_test_mod_cat2, #> to_standardize = c("dv", "iv", "mod", "cov1"))) #> #> Residuals: #> Min 1Q Median 3Q Max #> -2.4085 -0.6527 0.0008 0.7073 2.6363 #> #> Coefficients: #> Estimate CI.Lower CI.Upper Std. Error z value Pr(Boot) #> (Intercept) -0.308 -0.536 -0.080 0.117 -2.636 0.0056 ** #> iv 0.140 0.009 0.268 0.066 2.109 0.0384 * #> mod 0.196 0.075 0.317 0.061 3.208 0.0016 ** #> cov1 0.028 -0.075 0.131 0.052 0.537 0.5700 #> cat1gp2 0.241 -0.067 0.540 0.155 1.560 0.1276 #> cat1gp3 0.349 0.064 0.631 0.146 2.394 0.0152 * #> iv:mod 0.145 0.059 0.228 0.043 3.356 0.0020 ** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 0.9504 on 293 degrees of freedom #> #> R-squared : 0.115 #> Adjusted R-squared : 0.097 #> ANOVA test of R-squared : F(6, 293) = 6.338, p < 0.001 #> #> Note: #> - Results *after* standardization are reported. #> - Nonparametric bootstrapping conducted. #> - The number of bootstrap samples is 5000. #> - Standard errors are bootstrap standard errors. #> - Z values are computed by 'Estimate / Std. Error'. #> - The bootstrap p-values are asymmetric p-values by Asparouhov and #> Muthén (2021). #> - Percentile bootstrap 95.0% confidence interval reported.
By default, 95% percentile bootstrap
confidence intervals are printed
(CI.Lower and CI.Upper). The p-values
(Pr(Boot)) are asymmetric bootstrap
p-values [@asparouhov_bootstrap_2021].
Estimates and Bootstrap Confidence Intervals, With Only Selected Variables Standardized
Suppose we want to address also the
the third issue, and standardize only
some of the variables. This can be
done using either to_standardize
or not_to_standardize.
-
Use
to_standardizewhen the number of variables to standardize is much fewer than number of the variables not to standardize. -
Use
not_to_standardizewhen the number of variables to standardize is much more than the number of variables not to standardize.
For example, suppose we only
need to standardize dv and
iv,
this is the call to do
this, setting
to_standardize to c("iv", "dv"):
lm_beta_select_boot_1 <- lm_betaselect(dv ~ iv*mod + cov1 + cat1, data = data_test_mod_cat2, to_standardize = c("dv", "iv"), bootstrap = 5000, iseed = 4567)
If we want to standardize all
variables except for mod
and cov1, we can use
this call, and set
not_to_standardize to c("mod", "cov1"):
lm_beta_select_boot_2 <- lm_betaselect(dv ~ iv*mod + cov1 + cat1, data = data_test_mod_cat2, not_to_standardize = c("mod", "cov1"), bootstrap = 5000, iseed = 4567)
The results of these calls are identical, and only those of the first version are printed:
summary(lm_beta_select_boot_1) #> Call to lm_betaselect(): #> betaselectr::lm_betaselect(formula = dv ~ iv * mod + cov1 + cat1, #> data = data_test_mod_cat2, to_standardize = c("dv", "iv"), #> bootstrap = 5000, iseed = 4567) #> #> Variable(s) standardized: dv, iv #> #> Call: #> stats::lm(formula = dv ~ iv * mod + cov1 + cat1, data = betaselectr::std_data(data = data_test_mod_cat2, #> to_standardize = c("dv", "iv"))) #> #> Residuals: #> Min 1Q Median 3Q Max #> -2.4085 -0.6527 0.0008 0.7073 2.6363 #> #> Coefficients: #> Estimate CI.Lower CI.Upper Std. Error z value Pr(Boot) #> (Intercept) -2.991 -4.769 -1.196 0.899 -3.326 0.0024 ** #> iv -1.629 -2.667 -0.573 0.539 -3.021 0.0036 ** #> mod 0.048 0.019 0.078 0.015 3.199 0.0016 ** #> cov1 0.014 -0.037 0.066 0.026 0.533 0.5700 #> cat1gp2 0.241 -0.067 0.540 0.155 1.560 0.1276 #> cat1gp3 0.349 0.064 0.631 0.146 2.394 0.0152 * #> iv:mod 0.036 0.015 0.056 0.011 3.366 0.0020 ** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 0.9504 on 293 degrees of freedom #> #> R-squared : 0.115 #> Adjusted R-squared : 0.097 #> ANOVA test of R-squared : F(6, 293) = 6.338, p < 0.001 #> #> Note: #> - Results *after* standardization are reported. #> - Nonparametric bootstrapping conducted. #> - The number of bootstrap samples is 5000. #> - Standard errors are bootstrap standard errors. #> - Z values are computed by 'Estimate / Std. Error'. #> - The bootstrap p-values are asymmetric p-values by Asparouhov and #> Muthén (2021). #> - Percentile bootstrap 95.0% confidence interval reported.
For betas-select, researchers need to state which variables are standardized and which are not. This can be done in table notes.
Categorical Variables
When calling lm_betaselect(),
categorical variables (factors and
string variables) will never be standardized.
In the example above, the coefficients of the two dummy variables when both the dummy variables and the outcome variables are standardized are 0.116 and 0.175:
printCoefmat(lm_std_common_summary$coefficients[5:6, ], digits = 5, zap.ind = 1, P.values = TRUE, has.Pvalue = TRUE, signif.stars = TRUE) #> Estimate Std. Error t value Pr(>|t|) #> cat_gp2 0.116041 0.077970 1.4883 0.13775 #> cat_gp3 0.174623 0.077901 2.2416 0.02574 * #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
These two values are not interpretable because it does not make sense to talk about a "one-SD change" in the dummy variables.
The betas-Select of the dummy variables, with only the outcome variable standardized, are 0.241 and 0.349.
lm_beta_select_boot_summary <- summary(lm_beta_select_boot) printCoefmat(lm_beta_select_boot_summary$coefficients[5:6, ], digits = 5, zap.ind = 1, P.values = TRUE, has.Pvalue = TRUE, signif.stars = TRUE) #> Estimate CI.Lower CI.Upper Std. Error z value Pr(Boot) #> cat1gp2 0.241350 -0.067312 0.540477 0.154668 1.5604 0.1276 #> cat1gp3 0.349290 0.064136 0.630539 0.145927 2.3936 0.0152 * #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
They
represent the difference between gp2
and gp3 from the reference group,
gp1, on the standardized outcome variable.
That is, their meanings are the same before
and after standardization. The only difference
is in the unit of the outcome variable.
Conclusion
In regression analysis, there
are situations in which standardizing
all variables is not appropriate, or
when standardization needs to be done
before forming product terms. We are
not aware of tools that can do appropriate
standardization and form confidence
intervals that takes into account the
selective standardization. By promoting
the use of betas-select using
lm_betaselect(), we hope to make it
easier for researchers to do appropriate
standardization when reporting regression
results.
References
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