gammaToBeta: Internal function: Transform gammas back to betas

In WALS: Weighted-Average Least Squares Model Averaging

Internal function: Transform gammas back to betas

Description

Transforms posterior means \hat{\gamma}_2 and variances corresponding to transformed auxiliary regressors Z_2 back to regression coefficients \hat{\beta} of original regressors X_1 and X_2.

Usage

gammaToBeta(
  posterior,
  y,
  Z1,
  Z2,
  Delta1,
  D2,
  sigma,
  Z1inv,
  method = "original",
  svdZ1
)

Arguments

posterior	Object returned from computePosterior.
y	Response y.
Z1	Transformed focus regressors Z_1.
Z2	Transformed auxiliary regressors Z_1.
Delta1	\Delta_1 or \bar{\Delta}_1.
D2	From semiorthogonalize, if postmult = FALSE was used, then D2 = \Delta_2 T \Lambda^{-1/2}, where T are the eigenvectors of \Xi and \Lambda the diagonal matrix containing the corresponding eigenvalues. If postmult = TRUE was used, then D2 = \Delta_2 T \Lambda^{-1/2} T^{\top} = \Delta_2 \Xi^{-1/2}.
sigma	Prespecified or estimated standard deviation of the error term.
Z1inv	(Z_{1}^{\top} Z_{1})^{-1}.
method	Character. \hat{\gamma}_1 is obtained from a linear regression of Z_1 on pseudo-responses y - Z_2 \hat{\gamma}_2. If method = original, then we use lm.fit to perform the linear regression, if method = "svd", then reuse the SVD of Z_1 in svdZ1 to perform the regression.
svdZ1	Optional, only needed if method = "svd". SVD of Z_1 computed using svd.

Details

The same transformations also work for GLMs, where we replace X_1, X_2, Z_1 and Z_2 with \bar{X}_1, \bar{X}_2, \bar{Z}_1 and \bar{Z}_2, respectively. Generally, we need to replace all variables with their corresponding "bar" version. Further, for GLMs sigma is always 1.

See \insertCitemagnus2016wals;textualWALS, \insertCitedeluca2018glm;textualWALS and \insertCitehuynhwals;textualWALS for the definitions of the variables.

References

\insertAllCited

WALS documentation built on June 22, 2024, 9:42 a.m.