How lm_betaselect() and glm_betaselect() Work
Source:vignettes/articles/lm_betaselect_technical.Rmd
lm_betaselect_technical.RmdGoal
This technical appendix describes how a \(\beta_{Select}\) is computed in
lm_betaselect() and glm_betaselect() from the
package betaselectr.
Beta-Select (\(\beta_{Select}\))
Suppose this is the linear regression model:
\[ y = B_0 + B_1x_1 + B_2x_2 + B_3w + B_4x_2w + e \]
If only some of the variables are selected to be standardized, then only the two functions will simply standardize the selected variables using sample means and SDs, and refit the model.
For example, if only \(y\) and \(x_2\) are standardized, then both
lm_betaselect() and glm_betaselect() will
standardize \(y\) and \(x_2\), and then fit the model as usual. The
coefficients in the resulting model is then the \(\beta{s}_{Select}\) requested.
For a model to be fitted by glm(), such as a logistic
regression model, the outcome variable should not be standardized.
Standard Error, \(p\)-Values, and Confidence Interval
Although formulas for delta method standard errors (Pesigan et al., 2023; Rao, 1973) for standardized coefficients in multiple regression are available, they assumes that all variables are standardized. To our knowledge, formulas are not yet available for coefficients with only selected variables standardized, and for the coefficients of product terms. Therefore, for now, only nonparametric bootstrapping is supported.
Nonparametric Bootstrapping
If nonparametric bootstrapping (Efron &
Tibshirani, 1993) is used to compute the standard error of a
\(\beta_{Select}\), then \(R\) bootstrap samples will be drawn,
selected variables standardized, and then the model is fitted using
lm() or glm(). The standard error is the
standard deviation of the \(R\)
bootstrap estimates of the regression model. The \(p\)-value is computed using the method
proposed by Asparouhov & Muthén
(2021). The confidence interval can be formed by either the
percentile method (the default) or the bias-corrected method.
Miscellaneous
If missing data is present, listwise deletion will be used, using only the variables in the model, to determin the cases to be used for computing the means and standard deviations for the standardization.
If all variables are to be standardized and no higher order terms
such as product terms are present, then existing methods, such as those
available in Pesigan et al. (2023), can
also be used. The package betaselectr is for cases in which
only some of the variables are to be standardized and/or the model has
one or more product term.