In Gengrui-Zhang/R2spa: An R package for two-stage path analysis (2S-PA) to adjust for measurement errors

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

This vignette shows how to obtain grand standardized estimates for the latent variables for multi-group lavaan objects, using the R2spa::grandStandardizdSolution() function.

library(lavaan)
library(R2spa)

The example is from https://lavaan.ugent.be/tutorial/sem.html.

Single Group

For single groups, standardized solution can be obtained by first obtaining the latent variable covariance matrix:

$$ \begin{aligned} \bv \eta & = \bv \alpha + \bv \Gamma \bv X + \bv B \bv \eta + \bv \zeta \ (\bv I - \bv B) \bv \eta & = \bv \alpha + \bv \Gamma \bv X + \bv \zeta \ (\bv I - \bv B) \Var(\bv \eta) (\bv I - \bv B)^\top & = \Var(\bv \Gamma \bv X) + \bv \Psi \ \Var(\bv \eta) & = (\bv I - \bv B)^{-1} [\Var(\bv \Gamma \bv X) + \bv \Psi] {(\bv I - \bv B)^\top}^{-1} \end{aligned} $$

myModel <- '
   # latent variables
     ind60 =~ x1 + x2 + x3
     dem60 =~ y1 + y2 + y3 + y4
   # regressions
     dem60 ~ ind60
'
fit <- sem(model = myModel,
           data  = PoliticalDemocracy)
# Latent variable covariances
lavInspect(fit, "cov.lv")
# Using R2spa::veta()`
fit_est <- lavInspect(fit, "est")
R2spa:::veta(fit_est$beta, psi = fit_est$psi)

The standardized estimates in the $\bv B$ matrix are obtained as

$$ \bv B_s = \bv S_\eta^{-1} \bv B \bv S_\eta^{-1} $$

where $\bv S_\eta$ is a diagonal matrix containing the square root of the diagonal elements of $\Var(\bv \eta)$

S_eta <- diag(
  sqrt(diag(
    R2spa:::veta(fit_est$beta, psi = fit_est$psi)
  ))
)
solve(S_eta) %*% fit_est$beta %*% S_eta
# Compare to lavaan
standardizedSolution(fit)[8, ]

Multiple Groups

reg <- '
  # latent variable definitions
    visual =~ x1 + x2 + x3
    speed =~ x7 + x8 + x9

  # regressions
    visual ~ c(b1, b1) * speed
'
reg_fit <- sem(reg, data = HolzingerSwineford1939,
               group = "school",
               group.equal = c("loadings", "intercepts"))

Separate standardization by group

standardizedSolution(reg_fit, type = "std.lv") |>
  subset(subset = label == "b1")
# Compare to doing it by hand
reg_fit_est <- lavInspect(reg_fit, what = "est")
# Group 1:
S_eta1 <- diag(
  sqrt(diag(
    R2spa:::veta(reg_fit_est[[1]]$beta, psi = reg_fit_est[[1]]$psi)
  ))
)
solve(S_eta1) %*% reg_fit_est[[1]]$beta %*% S_eta1
# Group 2:
S_eta2 <- diag(
  sqrt(diag(
    R2spa:::veta(reg_fit_est[[2]]$beta, psi = reg_fit_est[[2]]$psi)
  ))
)
solve(S_eta2) %*% reg_fit_est[[2]]$beta %*% S_eta2

Grand standardization

For each group we have the group-specific covariance matrix $\Var(\bv \eta_g)$. The group-specific latent means are

$$\E(\bv \eta_g) = (\bv I - \bv B)^{-1}[\bv \alpha + \Gamma \E(\bv X)]$$

The grand mean is $\E(\bv \eta) = \sum_{g = 1}^G n_g \E(\bv \eta_g) / N$

The grand covariance matrix is:

$$\Var(\bv \eta) = \frac{1}{N} \sum_{g = 1}^G n_g \left{\Var(\bv \eta_g) + [\E(\bv \eta_g) - \E(\bv \eta)][\E(\bv \eta_g) - \E(\bv \eta)]^\top \right}$$

So we need to involve the mean as well

# Latent means
lavInspect(reg_fit, what = "mean.lv")  # lavaan
R2spa:::eeta(reg_fit_est[[1]]$beta, alpha = reg_fit_est[[1]]$alpha)
R2spa:::eeta(reg_fit_est[[2]]$beta, alpha = reg_fit_est[[2]]$alpha)
# Grand covariance
ns <- lavInspect(reg_fit, what = "nobs")
R2spa:::veta_grand(ns, beta_list = lapply(reg_fit_est, `[[`, "beta"),
                   psi_list = lapply(reg_fit_est, `[[`, "psi"),
                   alpha_list = lapply(reg_fit_est, `[[`, "alpha"))

The function grandStandardizdSolution() automates the computation of grand standardized coefficients, with the asymptotic standard error obtained using the delta method:

grandStandardizedSolution(reg_fit)

Gengrui-Zhang/R2spa documentation built on Sept. 6, 2024, 5:01 p.m.