How lm_betaselect() and glm_betaselect() Work
In betaselectr: Betas-Select in Structural Equation Models and Linear Models

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Goal

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 [@rao_large_1973; @pesigan_betadelta_2023] 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_introduction_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_bootstrap_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_betadelta_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.