How lav_betaselect() Works
Source:vignettes/articles/lav_betaselect_technical.Rmd
lav_betaselect_technical.RmdGoal
This technical appendix describes how a \(\beta_{Select}\) is computed in
lav_betaselect() from the package betaselectr.
Beta-Select (\(\beta_{Select}\))
Unlike multiple linear regression and generalized linear model, refitting a structural equation model with some variables standardized is not a suitable approach because (a) the standardization may not be consistent with the model being fitted (the sample standard deviation and the model implied standard deviation may be different) (b) doing so does not allow the delta method to take into account the sampling error in the standard deviations. Therefore, \(\beta{s}_{Select}\) in structural equation model are computed as function of model parameters.
A Predictor Not Involved In a Product Term
Let’s consider a predictor not involved in a product term.
This is an example, with the effect of \(x_1\) not moderated and \(x_1\) also not a moderator:
\[ y = B_0 + B_1x_1 + B_2x_2 + B_3w + B_4x_2w + e \]
The general form of the standardized coefficient of \(x_1\) is:
\[ \beta_{Select} = B_1\frac{SD_{x_1}}{SD_y} \]
where \(SD_{x_1}\) and \(SD_y\) are the standard deviations of \(x_1\) and \(y\), respectively.
If only \(x_1\) is standardized, then
\[ \beta_{Select} = B_1{SD_{x_1}} \]
If only \(y\) is standardized, then
\[ \beta_{Select} = B_1\frac{1}{SD_y} \]
The function lav_betaselect() will compute the \(\beta_{Select}\) of a predictor based on
the variables being standardized.
A Product Term
\(B_4x_2w\) in the following model is a product term (interaction term), representint the moderation effect of \(w\) on the effect of \(x_2\) on \(y\):
\[ y = B_0 + B_1x_1 + B_2x_2 + B_3w + B_4x_2w + e \]
The general form of the standardized coefficient of the product term \(B_4x_2w\) is:
\[ \beta_{Select} = B_4\frac{SD_{x_2}SD_{w}}{SD_y} \]
where \(SD_{x_2}\), \(SD_{w}\), and \(SD_y\) are the standard deviations of \(x_2\), \(w\), and \(y\), respectively.
If only \(x_2\) is standardized, then
\[ \beta_{Select} = B_4SD_{x_2} \]
If only \(y\) is standardized, then
\[ \beta_{Select} = B_4\frac{1}{SD_y} \]
If only \(w\) is standardized, then
\[ \beta_{Select} = B_4SD_{w} \]
If only \(x_2\) and \(w\) are standardized, then
\[ \beta_{Select} = B_4SD_{x_2}SD_{w} \]
If only \(y\) and \(w\) are standardized, then
\[ \beta_{Select} = B_4\frac{SD_w}{SD_y} \]
If only \(y\) and \(x_2\) are standardized, then
\[ \beta_{Select} = B_4\frac{SD_{x_2}}{SD_y} \]
The function lav_betaselect() will compute the \(\beta_{Select}\) of a predictor based on
the variables being standardized.
A Predictor Involved In a Product Term
Let’s consider a predictor involved in a product term.
In the following model, the effect of \(x_2\) is moderated by \(w\), and the conditional effect of \(x_2\) when \(w = 0\) is given by \(B_2\):
\[ y = B_0 + B_1x_1 + B_2x_2 + B_3w + B_4x_2w + e \]
The general form of the standardized coefficient of \(x_2\) is:
\[ \beta_{Select} = (B_2 + B_4M_w)\frac{SD_{x_2}}{SD_y} \]
where \(SD_{x_2}\) and \(SD_y\) are the standard deviations of \(x_2\) and \(y\), respectively, and \(M_w\) is the mean of \(w\).
If only \(x_2\) is standardized, then
\[ \beta_{Select} = (B_2 + B_4M_w)SD_{x_2} \]
If only \(y\) is standardized, then
\[ \beta_{Select} = (B_2 + B_4M_w)\frac{1}{SD_y} \]
The function lav_betaselect() will compute the \(\beta_{Select}\) based on the variables
being standardized.
Standard Error, \(p\)-Values, and Confidence Interval
The Delta Method
If the delta method (Rao, 1973) is used
to compute the standard error of a \(\beta_{Select}\), the \(\beta_{Select}\) will be treated as a
function of the model parameters, and the point estimates and the
variance-covariance matrix of these estimates returned by
lavaan() will be used to derive the asymptotical standard
error of the \(\beta_{Select}\). The
\(p\)-value and the confidence interval
will then be computed using the standard normal distribution.
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, and the \(\beta_{Select}\) will be computed in each sample using one of the formulas above. The standard error is the standard deviation of the \(R\) bootstrap estimates of the \(\beta_{Select}\). 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
For a structural equation model, the model implied standard deviations, instead of the sample standard deviations, are used. This ensures that the standardization conducted is consistent with the model being fitted.
For example, if a multigroup model is fitted and equality constraints are imposed on the standard deviations, then the model implied common standard deviation, which can be different from the full sample standard deviation, will be used in the standardization.
Moreover, if missing data is present and method such as maximum likelihood (ML) is used, then this method also allows using the ML estimates of the standard deviations, instead of listwise or pairwise estimates of them.