From ee57b1bc49cd6873eb3904d827f3c228d5744a1e Mon Sep 17 00:00:00 2001 From: "Hok Chio (Mark) Lai" Date: Sat, 18 Feb 2023 16:20:42 -0800 Subject: [PATCH 001/104] Draft vignette of some trial results --- vignettes/corrected-se.Rmd | 196 +++++++++++++++++++++++++++++++++++++ 1 file changed, 196 insertions(+) create mode 100644 vignettes/corrected-se.Rmd diff --git a/vignettes/corrected-se.Rmd b/vignettes/corrected-se.Rmd new file mode 100644 index 0000000..1d1acab --- /dev/null +++ b/vignettes/corrected-se.Rmd @@ -0,0 +1,196 @@ +--- +title: "Corrected Standard Errors" +output: rmarkdown::html_vignette +vignette: > + %\VignetteIndexEntry{corrected-se} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteEncoding{UTF-8} +header-includes: + - \DeclareMathOperator{\Var}{\mathrm{Var}} + - \DeclareMathOperator{\E}{\mathrm{E}} + - \newcommand{\bv}[1]{\boldsymbol{\mathbf{#1}}} %APA-consistent bold +--- + +```{r, include = FALSE} +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. + +```{r load-pkg, message = FALSE} +library(lavaan) +library(R2spa) +library(numDeriv) +library(boot) +``` + +```{r} +model <- ' + # latent variable definitions + ind60 =~ x1 + x2 + x3 + dem60 =~ y1 + y2 + y3 + y4 + + # regressions + dem60 ~ ind60 +' +``` + +```{r} +cfa_ind60 <- cfa("ind60 =~ x1 + x2 + x3", data = PoliticalDemocracy, + std.lv = TRUE) +vc1 <- vcov(cfa_ind60) +# Function for error variance of factor score +compute_ev_fs <- function(x, lam_idx, th_idx) { + lam <- x[lam_idx] + th <- diag(x[th_idx]) + covy <- tcrossprod(lam) + th + ginvcovy <- MASS::ginv(covy) + tlam_invcov <- crossprod(lam, ginvcovy) + tlam_invcov %*% th %*% t(tlam_invcov) +} +# compute_ev_fs(coef(cfa_ind60)) +coef_ind60 <- coef(cfa_ind60) +g_ev_fs1 <- grad(compute_ev_fs, coef(cfa_ind60), + lam_idx = 1:3, th_idx = 4:6) # gradient +# Delta method variance +v_ev_fs1 <- crossprod(g_ev_fs1, vcov(cfa_ind60)) %*% g_ev_fs1 + +cfa_dem60 <- cfa("dem60 =~ y1 + y2 + y3 + y4", + data = PoliticalDemocracy, std.lv = TRUE) +vc2 <- vcov(cfa_dem60) +# compute_ev_fs(coef(cfa_ind60)) +g_ev_fs2 <- grad(compute_ev_fs, coef(cfa_dem60), + lam_idx = 1:4, th_idx = 5:8) # gradient +# Delta method variance +v_ev_fs2 <- crossprod(g_ev_fs2, vcov(cfa_dem60)) %*% g_ev_fs2 +fs_dat <- data.frame( + fs_ind60 = as.numeric(lavPredict(cfa_ind60)), + fs_dem60 = as.numeric(lavPredict(cfa_dem60)) +) +se_fs <- list( + ind60 = sqrt(compute_ev_fs(coef(cfa_ind60), lam_idx = 1:3, th_idx = 4:6)), + dem60 = sqrt(compute_ev_fs(coef(cfa_dem60), lam_idx = 1:4, th_idx = 5:8)) +) +vc_ev_fs <- diag(c(v_ev_fs1, v_ev_fs2)) +``` + +```{r} +# 2S-PA +tspa_fit <- tspa(model = "dem60 ~ ind60", + data = fs_dat, + se = se_fs) +# Estimate as function of constraints +tspa_est <- function(se) { + coef(tspa(model = "dem60 ~ ind60", + data = fs_dat, + se = list(ind60 = se[1], dem60 = se[2]))) +} +tspa_est(unlist(se_fs)) +# Jacobian +jac <- jacobian(tspa_est, x = unlist(se_fs)) +# Corrected covariance matrix +vcov(tspa_fit) + jac %*% vc_ev_fs %*% t(jac) +``` + +Bootstrap Standard Errors + +```{r, eval = FALSE} +run_tspa <- function(df, inds) { + fs_ind60 <- get_fs(data = df[inds, ], model = "ind60 =~ x1 + x2 + x3", + se = "none", test = "none") + fs_dem60 <- get_fs(data = df[inds, ], model = "dem60 =~ y1 + y2 + y3 + y4", + se = "none", test = "none") + fs_dat <- cbind(fs_ind60, fs_dem60) + tspa_fit <- tspa(model = "dem60 ~ ind60", + data = fs_dat, + se = list(ind60 = fs_ind60[1, 2], + dem60 = fs_dem60[1, 2]), + test = "none") + coef(tspa_fit) +} +boo <- boot(PoliticalDemocracy, statistic = run_tspa, R = 999) +``` + +Use joint measurement model + +```{r} +cfa_joint <- cfa("ind60 =~ x1 + x2 + x3 + dem60 =~ y1 + y2 + y3 + y4", + data = PoliticalDemocracy) +compute_pars_joint <- function(x) { + lam <- matrix(0, nrow = 7, ncol = 2) + lam[1:3, 1] <- c(1, x[1:2]) + lam[4:7, 2] <- c(1, x[3:5]) + th <- diag(x[6:12]) + psi <- lav_matrix_lower2full(matrix(c(x[c(13, 15, 14)]))) + covy <- lam %*% psi %*% t(lam) + th + ginvcovy <- MASS::ginv(covy) + tlam_invcov <- crossprod(lam, ginvcovy) + a_mat <- psi %*% tlam_invcov + fsA <- a_mat %*% lam + av_efs <- a_mat %*% th %*% t(a_mat) + c(fsA, av_efs[lower.tri(av_efs, diag = TRUE)]) +} +# Measurement parameters +pars_joint <- compute_pars_joint(coef(cfa_joint)) +g_joint <- jacobian(compute_pars_joint, coef(cfa_joint)) +# Delta method variance +v_pars_joint <- g_joint %*% vcov(cfa_joint) %*% t(g_joint) +# Factor score +fs_joint <- get_fs(PoliticalDemocracy, + model = "ind60 =~ x1 + x2 + x3 + dem60 =~ y1 + y2 + y3 + y4") +``` + +```{r} +tspa_fit2 <- tspa(model = "dem60 ~ ind60", + data = fs_joint, + vc = attr(fs_joint, "av_efs"), + cross_loadings = attr(fs_joint, "fsA")) +# Estimate as function of constraints +tspa_est <- function(pars) { + fsA <- matrix(pars[1:4], nrow = 2, ncol = 2) + av_efs <- lav_matrix_lower2full(pars[5:7]) + dimnames(av_efs) <- list(c("ind60", "dem60"), c("ind60", "dem60")) + coef(tspa(model = "dem60 ~ ind60", data = fs_joint, + vc = av_efs, cross_loadings = fsA)) +} +tspa_est(pars_joint) +# Jacobian +jac <- jacobian(tspa_est, x = pars_joint) +# Corrected covariance matrix +vcov(tspa_fit2) + jac %*% v_pars_joint %*% t(jac) +``` + +Compare to joint model + +```{r} +sem_fit <- sem(model, data = PoliticalDemocracy) +vcov(sem_fit)[c(6, 14, 15), c(6, 14, 15)] +``` + +Bootstrap Standard Errors + +```{r, eval = FALSE} +run_tspa2 <- function(df, inds) { + fs_joint <- get_fs(df[inds, ], + model = "ind60 =~ x1 + x2 + x3 + dem60 =~ y1 + y2 + y3 + y4", + se = "none", test = "none") + tspa_fit2 <- tspa(model = "dem60 ~ ind60", + data = fs_joint, + vc = attr(fs_joint, "av_efs"), + cross_loadings = attr(fs_joint, "fsA"), + test = "none") + coef(tspa_fit2) +} +boo <- boot(PoliticalDemocracy, statistic = run_tspa2, R = 999) +run_sem2 <- function(df, inds) { + sem_fit <- sem(model, data = df[inds, ]) + coef(sem_fit)[c(6, 14, 15)] +} +boo2 <- boot(PoliticalDemocracy, statistic = run_sem2, R = 999) +``` From ae49e4d08b4082a0296e8fe3bed89658670d9966 Mon Sep 17 00:00:00 2001 From: "Hok Chio (Mark) Lai" Date: Sun, 19 Feb 2023 09:38:17 -0800 Subject: [PATCH 002/104] Add Bartlett score --- vignettes/corrected-se.Rmd | 62 ++++++++++++++++++++++++++++++++++++-- 1 file changed, 60 insertions(+), 2 deletions(-) diff --git a/vignettes/corrected-se.Rmd b/vignettes/corrected-se.Rmd index 1d1acab..ce5cb72 100644 --- a/vignettes/corrected-se.Rmd +++ b/vignettes/corrected-se.Rmd @@ -165,6 +165,51 @@ jac <- jacobian(tspa_est, x = pars_joint) vcov(tspa_fit2) + jac %*% v_pars_joint %*% t(jac) ``` +Bartlett's Method + +```{r} +compute_parsb_joint <- function(x) { + lam <- matrix(0, nrow = 7, ncol = 2) + lam[1:3, 1] <- c(1, x[1:2]) + lam[4:7, 2] <- c(1, x[3:5]) + th <- diag(x[6:12]) + ginvth <- MASS::ginv(th) + tlam_invth <- crossprod(lam, ginvth) + a_mat <- solve(tlam_invth %*% lam, tlam_invth) + av_efs <- a_mat %*% th %*% t(a_mat) + diag(av_efs) +} +# Measurement parameters +parsb_joint <- compute_parsb_joint(coef(cfa_joint)) +gb_joint <- jacobian(compute_parsb_joint, coef(cfa_joint)) +# Delta method variance +v_parsb_joint <- gb_joint %*% vcov(cfa_joint) %*% t(gb_joint) +# Factor score +fsb_joint <- get_fs(PoliticalDemocracy, + model = "ind60 =~ x1 + x2 + x3 + dem60 =~ y1 + y2 + y3 + y4", + method = "Bartlett") +``` + +```{r} +tspa_fit2b <- tspa(model = "dem60 ~ ind60", + data = fsb_joint, + vc = attr(fsb_joint, "av_efs"), + cross_loadings = diag(2)) +# Estimate as function of constraints +tspa_estb <- function(vc) { + av_efs <- diag(vc) + dimnames(av_efs) <- list(c("ind60", "dem60"), c("ind60", "dem60")) + coef(tspa(model = "dem60 ~ ind60", data = fsb_joint, + vc = av_efs, cross_loadings = diag(2))) +} +tspa_estb(parsb_joint) +# Jacobian +jac <- jacobian(tspa_estb, x = parsb_joint) +# Corrected covariance matrix +vcov(tspa_fit2b) + jac %*% v_parsb_joint %*% t(jac) +``` + Compare to joint model ```{r} @@ -187,10 +232,23 @@ run_tspa2 <- function(df, inds) { test = "none") coef(tspa_fit2) } -boo <- boot(PoliticalDemocracy, statistic = run_tspa2, R = 999) +boo <- boot(PoliticalDemocracy, statistic = run_tspa2, R = 4999) +run_tspa2b <- function(df, inds) { + fsb_joint <- get_fs(df[inds, ], + model = "ind60 =~ x1 + x2 + x3 + dem60 =~ y1 + y2 + y3 + y4", + se = "none", test = "none", method = "Bartlett") + tspa_fit2b <- tspa(model = "dem60 ~ ind60", + data = fsb_joint, + vc = attr(fsb_joint, "av_efs"), + cross_loadings = diag(2), + test = "none") + coef(tspa_fit2b) +} +boo2b <- boot(PoliticalDemocracy, statistic = run_tspa2b, R = 4999) run_sem2 <- function(df, inds) { sem_fit <- sem(model, data = df[inds, ]) coef(sem_fit)[c(6, 14, 15)] } -boo2 <- boot(PoliticalDemocracy, statistic = run_sem2, R = 999) +boo2j <- boot(PoliticalDemocracy, statistic = run_sem2, R = 4999) ``` From f7812458a369b30032506982a6f77c841b042e90 Mon Sep 17 00:00:00 2001 From: "Hok Chio (Mark) Lai" Date: Mon, 27 Feb 2023 12:24:17 -0800 Subject: [PATCH 003/104] Update to allow reinitiating pull request --- R/get_fscore.R | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/R/get_fscore.R b/R/get_fscore.R index 3d0ee16..b9ba330 100755 --- a/R/get_fscore.R +++ b/R/get_fscore.R @@ -83,7 +83,7 @@ augment_fs <- function(est, fs, fs_se) { if (is.vector(fs_se) || nrow(fs_se) != 1) { fs_se <- t(as.matrix(fs_se)) } - psi <- est$psi + # psi <- est$psi # fs_rho <- 1 - fs_se^2 / diag(psi) colnames(fs) <- paste0("fs_", colnames(fs)) colnames(fs_se) <- paste0("fs_", colnames(fs_se), "_se") From 2eb581b7c5706cf8bbde71a904a6d480bb42cabb Mon Sep 17 00:00:00 2001 From: "Hok Chio (Mark) Lai" Date: Tue, 28 Feb 2023 18:55:34 -0800 Subject: [PATCH 004/104] Force matrix to be PD #40 --- R/get_fscore.R | 2 ++ 1 file changed, 2 insertions(+) diff --git a/R/get_fscore.R b/R/get_fscore.R index 9ae01f3..5f43552 100755 --- a/R/get_fscore.R +++ b/R/get_fscore.R @@ -100,6 +100,7 @@ augment_fs <- function(est, fs, fs_ev) { # fs_rho <- 1 - fs_se^2 / diag(psi) # colnames(fs_rho) <- paste0("fs_", colnames(fs_rho), "_rel") fs_se <- t(as.matrix(sqrt(diag(fs_ev)))) + # fs_se[is.nan(fs_se)] <- 0 colnames(fs) <- paste0("fs_", colnames(fs)) colnames(fs_se) <- paste0("fs_", colnames(fs_se), "_se") num_lvs <- ncol(fs_ev) @@ -177,6 +178,7 @@ compute_fscore <- function(y, lambda, theta, psi, method <- match.arg(method) if (is.null(nu)) nu <- colMeans(y) if (is.null(alpha)) alpha <- rep(0, ncol(as.matrix(lambda))) + diag(theta) <- pmax(diag(theta), 0) # force PD Theta covy <- lambda %*% psi %*% t(lambda) + theta meany <- lambda %*% alpha + nu y1c <- t(as.matrix(y)) - as.vector(meany) From e6a3253ce20c43417fe3e975d851037cb004f653 Mon Sep 17 00:00:00 2001 From: "Hok Chio (Mark) Lai" Date: Tue, 28 Feb 2023 20:14:53 -0800 Subject: [PATCH 005/104] Update some formatting suggestions --- R/tspa_plot.R | 28 ++++++++++++++-------------- 1 file changed, 14 insertions(+), 14 deletions(-) diff --git a/R/tspa_plot.R b/R/tspa_plot.R index 2ed235f..4625626 100644 --- a/R/tspa_plot.R +++ b/R/tspa_plot.R @@ -7,9 +7,9 @@ #' @param title Character. A default or user-defined name or of the title of scatterplot. #' @param label_x Character. A default or user-defined name of the x-aix of scatterplot. #' @param label_y Character. A default or user-defined name of the y-aix of scatterplot. +#' @param abbreviation Logic input. If 'False' is indicated #' @param ... Additional arguments passed to \code{\link[graphics]{plot}}. See #' \code{\link[graphics]{plot}} for a list. -#' @param abbreviation Logic input. If 'False' is indicated #' #' @return A scatterplot between factor scores, and a residual plot. #' @@ -34,13 +34,13 @@ #' se = list(ind60 = 0.1213615, dem60 = 0.6756472)) #' tspa_plot(tspa_fit) -tspa_plot <- function (tspa_fit, - ask = FALSE, - title = NULL, - label_x = NULL, - label_y = NULL, - abbreviation = TRUE, - ...) { +tspa_plot <- function(tspa_fit, + ask = FALSE, + title = NULL, + label_x = NULL, + label_y = NULL, + abbreviation = TRUE, + ...) { fit_data <- parameterestimates(tspa_fit) latent_scores <- lavInspect(tspa_fit, what = "data") @@ -65,8 +65,8 @@ tspa_plot <- function (tspa_fit, invisible(readline(prompt = "Hit to see next plot: ")) } - plot(latent_scores[[g]][ ,paste0("fs_", latent_iv[i])], - latent_scores[[g]][ ,paste0("fs_", latent_dv[i])], + plot(latent_scores[[g]][ , paste0("fs_", latent_iv[i])], + latent_scores[[g]][ , paste0("fs_", latent_dv[i])], ylab = ifelse(is.null(label_y), paste0("fs_", latent_dv[i]), ifelse(length(label_y) > 1, label_y[i], label_y)), xlab = ifelse(is.null(label_x), paste0("fs_", latent_iv[i]), @@ -99,8 +99,8 @@ tspa_plot <- function (tspa_fit, } else { latent_names <- colnames(latent_scores) df_latent_scores <- data.frame(latent_scores) - latent_dv <- c(t(fit_data[which(fit_data$op == "~"),]["lhs"])) - latent_iv <- c(t(fit_data[which(fit_data$op == "~"),]["rhs"])) + latent_dv <- c(t(fit_data[which(fit_data$op == "~"), ]["lhs"])) + latent_iv <- c(t(fit_data[which(fit_data$op == "~"), ]["rhs"])) for (i in seq(length(latent_dv))) { latent_model <- lm(as.numeric(t(df_latent_scores[paste0("fs_", latent_dv[i])])) ~ @@ -111,8 +111,8 @@ tspa_plot <- function (tspa_fit, invisible(readline(prompt = "Hit to see next plot: ")) } - plot(latent_scores[ ,paste0("fs_", latent_iv[i])], - latent_scores[ ,paste0("fs_", latent_dv[i])], + plot(latent_scores[ , paste0("fs_", latent_iv[i])], + latent_scores[ , paste0("fs_", latent_dv[i])], ylab = ifelse(is.null(label_y), paste0("fs_", latent_dv[i]), ifelse(length(label_y) > 1, label_y[i], label_y)), xlab = ifelse(is.null(label_x), paste0("fs_", latent_iv[i]), From a4f42de28cdf801cb227098c1c6c575616ce0750 Mon Sep 17 00:00:00 2001 From: "Hok Chio (Mark) Lai" Date: Tue, 14 Mar 2023 18:07:08 -0700 Subject: [PATCH 006/104] Minor update to vignette --- vignettes/corrected-se.Rmd | 34 ++++++++++++++++++++++++++++++---- 1 file changed, 30 insertions(+), 4 deletions(-) diff --git a/vignettes/corrected-se.Rmd b/vignettes/corrected-se.Rmd index ce5cb72..866bb5d 100644 --- a/vignettes/corrected-se.Rmd +++ b/vignettes/corrected-se.Rmd @@ -79,13 +79,13 @@ vc_ev_fs <- diag(c(v_ev_fs1, v_ev_fs2)) ```{r} # 2S-PA -tspa_fit <- tspa(model = "dem60 ~ ind60", - data = fs_dat, +tspa_fit <- tspa(model = "dem60 ~ ind60", + data = fs_dat, se = se_fs) # Estimate as function of constraints tspa_est <- function(se) { - coef(tspa(model = "dem60 ~ ind60", - data = fs_dat, + coef(tspa(model = "dem60 ~ ind60", + data = fs_dat, se = list(ind60 = se[1], dem60 = se[2]))) } tspa_est(unlist(se_fs)) @@ -95,6 +95,32 @@ jac <- jacobian(tspa_est, x = unlist(se_fs)) vcov(tspa_fit) + jac %*% vc_ev_fs %*% t(jac) ``` +Standardized solution + +```{r} +tspa_est_std <- function(se) { + tfit <- tspa(model = "dem60 ~ ind60", + data = fs_dat, + se = list(ind60 = se[1], dem60 = se[2])) + standardizedSolution(tfit)$est.std[5] +} +tspa_est_std(unlist(se_fs)) +# Jacobian +jac_std <- jacobian(tspa_est_std, x = unlist(se_fs)) +# Corrected standard error +sqrt( + standardizedSolution(tspa_fit)[5, "se"]^2 + + jac_std %*% vc_ev_fs %*% t(jac_std) +) +``` + +Compare to joint model + +```{r} +sem_fit <- sem(model, data = PoliticalDemocracy) +standardizedSolution(sem_fit)[8, "se"] +``` + Bootstrap Standard Errors ```{r, eval = FALSE} From 0e1a16ae46c12d7a4622589e69952ba665e79198 Mon Sep 17 00:00:00 2001 From: Jimmy Zhang Date: Sun, 26 Mar 2023 22:35:46 -0700 Subject: [PATCH 007/104] Commit changes --- R/tspa_plot.R | 26 ++++++++++++-------- man/tspa_plot.Rd | 19 ++++++++------- vignettes/tspa-plot-vignette.Rmd | 42 ++++++++++---------------------- vignettes/tspa-vignette.Rmd | 39 ++--------------------------- 4 files changed, 41 insertions(+), 85 deletions(-) diff --git a/R/tspa_plot.R b/R/tspa_plot.R index 2ed235f..08eccb7 100644 --- a/R/tspa_plot.R +++ b/R/tspa_plot.R @@ -2,20 +2,21 @@ #' #' @param tspa_fit An object of class \code{lavaan}, #' representing the output generated from `tspa()` function. -#' @param ask Logic input. If 'True' is indicated, the user will be asked before +#' @param ask Logic input. If 'TRUE' is indicated, the user will be asked before #' before each plot is generated. The default setting is 'False'. -#' @param title Character. A default or user-defined name or of the title of scatterplot. -#' @param label_x Character. A default or user-defined name of the x-aix of scatterplot. -#' @param label_y Character. A default or user-defined name of the y-aix of scatterplot. +#' @param title Character. Set the name of scatter plot. The default value is "Scatterplot". +#' @param label_x Character. Set the name of the x-axis. The default value is "fs_" followed by variable names. +#' @param label_y Character. Set the name of the y-axis. The default value is "fs_" followed by variable names. #' @param ... Additional arguments passed to \code{\link[graphics]{plot}}. See #' \code{\link[graphics]{plot}} for a list. -#' @param abbreviation Logic input. If 'False' is indicated +#' @param abbreviation Logic input. If 'FALSE' is indicated #' #' @return A scatterplot between factor scores, and a residual plot. #' #' @export #' #' @examples +#' library(lavaan) #' model <- ' #' # latent variable definitions #' ind60 =~ x1 + x2 + x3 @@ -24,13 +25,13 @@ #' dem60 ~ ind60 #' ' #' fs_dat_ind60 <- get_fs(data = PoliticalDemocracy, -#' model = "ind60 =~ x1 + x2 + x3") +#' model = "ind60 =~ x1 + x2 + x3") #' fs_dat_dem60 <- get_fs(data = PoliticalDemocracy, #' model = "dem60 =~ y1 + y2 + y3 + y4") #' fs_dat <- cbind(fs_dat_ind60, fs_dat_dem60) #' #' tspa_fit <- tspa(model = "dem60 ~ ind60", -#' data = fs_dat, +#' data = fs_dat, #' se = list(ind60 = 0.1213615, dem60 = 0.6756472)) #' tspa_plot(tspa_fit) @@ -42,11 +43,16 @@ tspa_plot <- function (tspa_fit, abbreviation = TRUE, ...) { - fit_data <- parameterestimates(tspa_fit) + # Check if the model input is a tspa fit + if (is.null(attributes(tspa_fit)$tspaModel)) { + stop("tspa_plot() function only supports outputs from tspa()") + } + + fit_data <- lavaan::parameterestimates(tspa_fit) latent_scores <- lavInspect(tspa_fit, what = "data") if (is.list(latent_scores)) { - latent_names <- colnames(latent_scores[[1]]) + # latent_names <- colnames(latent_scores[[1]]) df_latent_scores <- lapply(latent_scores, data.frame) ifelse(abbreviation == TRUE, g_names <- abbreviate(names(df_latent_scores)), @@ -97,7 +103,7 @@ tspa_plot <- function (tspa_fit, } } else { - latent_names <- colnames(latent_scores) + # latent_names <- colnames(latent_scores) df_latent_scores <- data.frame(latent_scores) latent_dv <- c(t(fit_data[which(fit_data$op == "~"),]["lhs"])) latent_iv <- c(t(fit_data[which(fit_data$op == "~"),]["rhs"])) diff --git a/man/tspa_plot.Rd b/man/tspa_plot.Rd index f2c1736..6a3eb02 100644 --- a/man/tspa_plot.Rd +++ b/man/tspa_plot.Rd @@ -8,8 +8,8 @@ tspa_plot( tspa_fit, ask = FALSE, title = NULL, - xlab = NULL, - ylab = NULL, + label_x = NULL, + label_y = NULL, abbreviation = TRUE, ... ) @@ -18,16 +18,16 @@ tspa_plot( \item{tspa_fit}{An object of class \code{lavaan}, representing the output generated from \code{tspa()} function.} -\item{ask}{Logic input. If 'True' is indicated, the user will be asked before +\item{ask}{Logic input. If 'TRUE' is indicated, the user will be asked before before each plot is generated. The default setting is 'False'.} -\item{title}{Character. A default or user-defined name or of the title of scatterplot.} +\item{title}{Character. Set the name of scatter plot. The default value is "Scatterplot".} -\item{xlab}{Character. A default or user-defined name of the x-aix of scatterplot.} +\item{label_x}{Character. Set the name of the x-axis. The default value is "fs_" followed by variable names.} -\item{ylab}{Character. A default or user-defined name of the y-aix of scatterplot.} +\item{label_y}{Character. Set the name of the y-axis. The default value is "fs_" followed by variable names.} -\item{abbreviation}{Logic input. If 'False' is indicated} +\item{abbreviation}{Logic input. If 'FALSE' is indicated} \item{...}{Additional arguments passed to \code{\link[graphics]{plot}}. See \code{\link[graphics]{plot}} for a list.} @@ -39,6 +39,7 @@ A scatterplot between factor scores, and a residual plot. Diagnostic plots of fitted 2S-PA model } \examples{ +library(lavaan) model <- ' # latent variable definitions ind60 =~ x1 + x2 + x3 @@ -47,13 +48,13 @@ dem60 =~ y1 + a*y2 + b*y3 + c*y4 dem60 ~ ind60 ' fs_dat_ind60 <- get_fs(data = PoliticalDemocracy, -model = "ind60 =~ x1 + x2 + x3") + model = "ind60 =~ x1 + x2 + x3") fs_dat_dem60 <- get_fs(data = PoliticalDemocracy, model = "dem60 =~ y1 + y2 + y3 + y4") fs_dat <- cbind(fs_dat_ind60, fs_dat_dem60) tspa_fit <- tspa(model = "dem60 ~ ind60", -data = fs_dat, + data = fs_dat, se = list(ind60 = 0.1213615, dem60 = 0.6756472)) tspa_plot(tspa_fit) } diff --git a/vignettes/tspa-plot-vignette.Rmd b/vignettes/tspa-plot-vignette.Rmd index 7148a91..38f5360 100644 --- a/vignettes/tspa-plot-vignette.Rmd +++ b/vignettes/tspa-plot-vignette.Rmd @@ -1,10 +1,14 @@ --- -title: "tspa-plot-vignette" +title: "tspa-diagnostic-plot-vignette" author: "Jimmy Zhang" -date: "2023-02-16" -output: - html_document: default - pdf_document: default +date: 'r Sys.Date()' +output: + rmarkdown::html_vignette: + toc: true +vignette: > + %\VignetteIndexEntry{2spa_vignette} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteEncoding{UTF-8} --- ```{r setup, include=FALSE} @@ -18,7 +22,6 @@ Load packages ```{r load-pkg, message = FALSE} library(lavaan) library(R2spa) -library(semdplot) ``` ## Single group, single predictor @@ -47,6 +50,7 @@ tspa_fit_1 <- tspa(model = "dem60 ~ ind60", We recommend researchers to examine the diagnostic plot of factor scores in the model. They can use the function `tspa_plot()` to obtain two plots: (a) a scatter plot between factors, (b) a residual plot between factors. Features of `tspa_plot()`: + - The function is able to pass arguments to the base R `plot()` function. - Users can define title of the scatter plot and label names of axis. - Abbreviation argument allows users to choose whether using abbreviated group names. @@ -61,12 +65,6 @@ tspa_plot(tspa_fit_1, ask = TRUE) ``` -```{r example using casewise_residuals function} -casewise_residuals(tspa_fit_1) -``` - -It seems that `casewise_residuals` function does not support tspa model fits because they have latent variables. - ## Single group, multiple predictors ```{r} @@ -96,7 +94,7 @@ fs_dat_dem65 <- get_fs(data = PoliticalDemocracy, model = "dem65 =~ y5 + y6 + y7 + y8") fs_dat <- cbind(fs_dat_ind60, fs_dat_dem60, fs_dat_dem65) -tspa_3var_fit <- tspa(model = "dem60 ~ ind60 +tspa_fit_2 <- tspa(model = "dem60 ~ ind60 dem65 ~ ind60 + dem60", data = fs_dat, se = list(ind60 = 0.1213615, dem60 = 0.6756472, @@ -105,14 +103,14 @@ tspa_3var_fit <- tspa(model = "dem60 ~ ind60 ```{r example_single_group_multiple_factors} # Title, xlab, and ylab each with same names. -tspa_plot(tspa_3var_fit, +tspa_plot(tspa_fit_2, ps = 10, col = "darkgray", cex.lab = 1.2, cex.axis = 1) # Title, xlab, and ylab each with separate names. -tspa_plot(tspa_3var_fit, +tspa_plot(tspa_fit_2, ps = 10, col = "darkgray", cex.lab = 1.2, @@ -158,14 +156,12 @@ tspa_fit_3 <- tspa(model = "visual ~ speed", ``` ```{r example_multiple_group_single_factor} -par(mar = c(4,4,4,4)) tspa_plot(tspa_fit_3, ps = 10, col = "darkgray", cex.lab = 1.2, cex.axis = 1) -par(mar = c(4,4,4,4)) tspa_plot(tspa_fit_3, ps = 10, col = "darkgray", @@ -174,18 +170,6 @@ tspa_plot(tspa_fit_3, abbreviation = FALSE) ``` -```{r} -fs_sem3 <- sem(model = "fs_visual ~ fs_speed", - data = fs_hs) - -fit_res3 <- casewise_residuals(fs_sem3) -p <- plot(fit_res3, - x_names = c("fs_speed"), - y_names = c("fs_visual")) - -plot(p) -``` - diff --git a/vignettes/tspa-vignette.Rmd b/vignettes/tspa-vignette.Rmd index 4a01e91..09896fb 100644 --- a/vignettes/tspa-vignette.Rmd +++ b/vignettes/tspa-vignette.Rmd @@ -1,6 +1,6 @@ --- title: "Two-Stage Path Analysis (2S-PA) Model Examples" -author: "Yixiao Li, Jimmy Zhang" +author: "Yixiao Li" date: "`r Sys.Date()`" output: rmarkdown::html_vignette: @@ -63,19 +63,6 @@ To view the Two-Stage Path Analysis model, use `attributes([model name])$tspaMod cat(attributes(tspa_fit)$tspaModel) ``` -We recommend researchers to examine the diagnostic plot of factor scores in the model. They can use the function `tspa_plot()` to obtain two plots: (a) a scatter plot between factors, (b) a residual plot between factors. - -```{r} -# par(mar = c(2,2,3,2)) -tspa_plot(tspa_fit, - ps = 10, - col = "darkgray", - cex.lab = 1.2, - cex.axis = 1, - abbreviation = FALSE, - ask = FALSE) -``` - ## Single group, multiple predictors ```{r} @@ -115,7 +102,7 @@ To build a Two-Stage Path Analysis model, simply call the `tspa()` function with [^2]: The text inside `[]` should be replaced with actual variable and model name. ```{r} -tspa_fit_2 <- tspa(model = "dem60 ~ ind60 +tspa_3var_fit <- tspa(model = "dem60 ~ ind60 dem65 ~ ind60 + dem60", data = fs_dat, se = list(ind60 = 0.1213615, dem60 = 0.6756472, @@ -123,17 +110,6 @@ tspa_fit_2 <- tspa(model = "dem60 ~ ind60 cat(attributes(tspa_3var_fit)$tspaModel) ``` -Diagnotisc plot: - -```{r} -tspa_plot(tspa_fit_2, - ps = 10, - col = "darkgray", - cex.lab = 1.2, - cex.axis = 1, - mar = c(0.1, 0.05, 0.05, 0.1)) -``` - ## Multigroup, single predictor ```{r} @@ -187,14 +163,3 @@ To view the Two-Stage Path Analysis model for multigroup, use `attributes([model ```{r} cat(attributes(tspa_fit)$tspaModel) ``` - -Diagnotisc plot: - -```{r} -par(mar = c(4,4,4,4)) -tspa_plot(tspa_fit, - ps = 10, - col = "darkgray", - cex.lab = 1.2, - cex.axis = 1) -``` From 76927ef0fb64b81a929913c2c35063207094215b Mon Sep 17 00:00:00 2001 From: Jimmy Zhang Date: Mon, 3 Apr 2023 10:20:52 -0700 Subject: [PATCH 008/104] commit changes --- R/tspa_plot_revised.R | 44 ++++++++++++++++++++++++++++++++ vignettes/tspa-plot-vignette.Rmd | 40 ++++++++++++++++++++++++++++- 2 files changed, 83 insertions(+), 1 deletion(-) create mode 100644 R/tspa_plot_revised.R diff --git a/R/tspa_plot_revised.R b/R/tspa_plot_revised.R new file mode 100644 index 0000000..90f20fd --- /dev/null +++ b/R/tspa_plot_revised.R @@ -0,0 +1,44 @@ +tspa_plot <- function(tspa_fit, + ask = FALSE, + title = NULL, + label_x = NULL, + label_y = NULL, + abbreviation = TRUE, + ...) { + + fit_pars <- lavaan::parameterestimates(tspa_fit) # parameter estimates + fscores_func <- lavaan::lavInspect(tspa_fit, what = "data") # factor scores from `get_fs` + fscores_est <- lavaan::lavPredict(tspa_fit) # factor scores from estimation using `lavPredict` + + # determine type of scatterplot + if (is.null(fscore_type)) { + fscores <- fscores_est + } else { + fscores <- fscore_func + } + + # determine if this is a multi-group sample + if (is.list(fs_scores)) { + df_latent_scores <- lapply(fscores, data.frame) + ifelse(abbreviation == TRUE, + g_names <- abbreviate(names(df_latent_scores)), + g_names <- names(df_latent_scores)) + latent_dv <- c(t(na.omit(fit_pars[1:(nrow(fit_pars)/length(fscores)), ][which(fit_pars$op == "~"),]["lhs"]))) + latent_iv <- c(t(na.omit(fit_pars[1:(nrow(fit_pars)/length(fscores)), ][which(fit_pars$op == "~"),]["rhs"]))) + } else { + df_latent_scores <- data.frame(fscores) + latent_dv <- c(t(fit_pars[which(fit_pars$op == "~"), ]["lhs"])) + latent_iv <- c(t(fit_pars[which(fit_pars$op == "~"), ]["rhs"])) + } + + + + + + + + + + + +} diff --git a/vignettes/tspa-plot-vignette.Rmd b/vignettes/tspa-plot-vignette.Rmd index 38f5360..482d196 100644 --- a/vignettes/tspa-plot-vignette.Rmd +++ b/vignettes/tspa-plot-vignette.Rmd @@ -65,6 +65,26 @@ tspa_plot(tspa_fit_1, ask = TRUE) ``` +### Using estimates from `tspa()` function + +```{r} +latent_estimates <- data.frame(lavPredict(tspa_fit_1)) +plot(latent_estimates$ind60, + latent_estimates$dem60, + main="Scatterplot", + xlab="ind60", + ylab="dem60", + pch=16, + ps=1.25, + cex.lab=1.2, + cex.axis=1.2) +slope <- coef(tspa_fit_1)["dem60~ind60"] +abline(a = 0, b = slope, lwd=2) + +parameterestimates(tspa_fit_1) +``` + + ## Single group, multiple predictors ```{r} @@ -120,6 +140,24 @@ tspa_plot(tspa_fit_2, label_y = c("factor_a", "factor_b", "factor_c")) ``` +### Using estimates from `tspa()` function + +```{r} +# dem60 and ind60 +latent_estimates <- data.frame(lavPredict(tspa_fit_2)) +plot(latent_estimates$ind60, + latent_estimates$dem60, + main="Scatterplot", + xlab="ind60", + ylab="dem60", + pch=16, + ps=1.25, + cex.lab=1.2, + cex.axis=1.2) +slope <- coef(tspa_fit_2)["dem60~ind60"] +abline(a = 0, b = slope, lwd=2) +``` + ## Multigroup, single predictor ```{r} @@ -143,7 +181,7 @@ fs_dat_visual <- get_fs(data = HolzingerSwineford1939, group = "school") fs_dat_speed <- get_fs(data = HolzingerSwineford1939, model = "speed =~ x7 + x8 + x9", - group = "school") + group = "school") fs_hs <- cbind(fs_dat_visual, fs_dat_speed) tspa_fit_3 <- tspa(model = "visual ~ speed", From 6bb1b3ee3a520c53cb546e294957a0cb751b7e0e Mon Sep 17 00:00:00 2001 From: "Hok Chio (Mark) Lai" Date: Mon, 24 Apr 2023 23:42:36 -0700 Subject: [PATCH 009/104] Move out func for computing the scoring matrix --- R/get_fscore.R | 38 ++++++++++++++++++++++++++++++++------ 1 file changed, 32 insertions(+), 6 deletions(-) diff --git a/R/get_fscore.R b/R/get_fscore.R index 4ed8a44..648aa33 100755 --- a/R/get_fscore.R +++ b/R/get_fscore.R @@ -172,14 +172,10 @@ compute_fscore <- function(y, lambda, theta, psi, stop("input of psi (latent covariance) is needed for regression scores") } # Regression score - ginvcovy <- MASS::ginv(covy) - tlam_invcov <- crossprod(lambda, ginvcovy) - a_mat <- psi %*% tlam_invcov + a_mat <- compute_a_reg(lambda, psi, theta) } else if (method == "Bartlett") { # Bartlett score - ginvth <- MASS::ginv(theta) - tlam_invth <- crossprod(lambda, ginvth) - a_mat <- solve(tlam_invth %*% lambda, tlam_invth) + a_mat <- compute_a_bartlett(lambda, psi, theta) } fs <- t(a_mat %*% y1c + as.vector(alpha)) if (acov) { @@ -201,3 +197,33 @@ compute_fscore <- function(y, lambda, theta, psi, } fs } + +compute_a <- function(par, lavobj, method = c("regression", "Bartlett")) { + method = match.arg(method) + free <- lavInspect(lavobj, what = "free") + free_list <- lapply(free, FUN = \(x) x[which(x > 0)]) + mat <- lavInspect(lavobj, what = "est") + for (l in seq_along(free_list)) { + for (i in free_list[[l]]) { + mat[[l]][which(free[[l]] == i)] <- par[i] + } + } + if (method == "regression") { + return(do.call(compute_a_reg, args = mat[c("lambda", "psi", "theta")])) + } else if (method == "Bartlett") { + return(do.call(compute_a_bartlett, args = mat[c("lambda", "psi", "theta")])) + } +} + +compute_a_reg <- function(lambda, psi, theta) { + covy <- lambda %*% psi %*% t(lambda) + theta + ginvcovy <- MASS::ginv(covy) + tlam_invcov <- crossprod(lambda, ginvcovy) + psi %*% tlam_invcov +} + +compute_a_bartlett <- function(lambda, psi, theta) { + ginvth <- MASS::ginv(theta) + tlam_invth <- crossprod(lambda, ginvth) + solve(tlam_invth %*% lambda, tlam_invth) +} From 035741857a213fff259a47ad4ee897a2a624fd44 Mon Sep 17 00:00:00 2001 From: "Hok Chio (Mark) Lai" Date: Tue, 25 Apr 2023 09:44:48 -0700 Subject: [PATCH 010/104] Force alpha as matrix in `compute_fscore()` --- R/get_fscore.R | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/R/get_fscore.R b/R/get_fscore.R index 648aa33..2c42ded 100755 --- a/R/get_fscore.R +++ b/R/get_fscore.R @@ -163,7 +163,7 @@ compute_fscore <- function(y, lambda, theta, psi, fs_matrices = FALSE) { method <- match.arg(method) if (is.null(nu)) nu <- colMeans(y) - if (is.null(alpha)) alpha <- rep(0, ncol(as.matrix(lambda))) + if (is.null(alpha)) alpha <- matrix(0, ncol = ncol(as.matrix(lambda))) covy <- lambda %*% psi %*% t(lambda) + theta meany <- lambda %*% alpha + nu y1c <- t(as.matrix(y)) - as.vector(meany) From d56d0ac1e8ecb793d4f6ba326b46139a9af8b832 Mon Sep 17 00:00:00 2001 From: "Hok Chio (Mark) Lai" Date: Tue, 25 Apr 2023 10:57:15 -0700 Subject: [PATCH 011/104] Add scoring matrix as attribute - fix bug in previous commit --- R/get_fscore.R | 8 ++++++-- 1 file changed, 6 insertions(+), 2 deletions(-) diff --git a/R/get_fscore.R b/R/get_fscore.R index 2c42ded..a122b3f 100755 --- a/R/get_fscore.R +++ b/R/get_fscore.R @@ -68,7 +68,7 @@ get_fs <- function(data, model = NULL, group = NULL, if (is.null(group)) { prepare_fs_dat(y, est) } else { - fs_lst <- av_efs_lst <- fsA_lst <- fsb_lst <- + fs_lst <- av_efs_lst <- fsA_lst <- fsb_lst <- fsS_lst <- setNames(vector("list", length = length(est)), fit@Data@group.label) for (i in seq_along(fs_lst)) { @@ -77,10 +77,12 @@ get_fs <- function(data, model = NULL, group = NULL, av_efs_lst[[i]] <- attr(fs_lst[[i]], "av_efs") fsA_lst[[i]] <- attr(fs_lst[[i]], "fsA") fsb_lst[[i]] <- attr(fs_lst[[i]], "fsb") + fsS_lst[[i]] <- attr(fs_lst[[i]], "scoring_matrix") } attr(fs_lst, which = "av_efs") <- av_efs_lst attr(fs_lst, which = "fsA") <- fsA_lst attr(fs_lst, which = "fsb") <- fsb_lst + attr(fs_lst, which = "scoring_matrix") <- fsS_lst fs_lst } } @@ -113,6 +115,7 @@ augment_fs <- function(est, fs, fs_ev) { attr(fs_dat, "av_efs") <- fs_ev attr(fs_dat, "fsA") <- fsA attr(fs_dat, "fsb") <- attr(fs, "fsb") + attr(fs_dat, "scoring_matrix") <- attr(fs, "scoring_matrix") fs_dat } @@ -163,7 +166,7 @@ compute_fscore <- function(y, lambda, theta, psi, fs_matrices = FALSE) { method <- match.arg(method) if (is.null(nu)) nu <- colMeans(y) - if (is.null(alpha)) alpha <- matrix(0, ncol = ncol(as.matrix(lambda))) + if (is.null(alpha)) alpha <- matrix(0, nrow = ncol(as.matrix(lambda))) covy <- lambda %*% psi %*% t(lambda) + theta meany <- lambda %*% alpha + nu y1c <- t(as.matrix(y)) - as.vector(meany) @@ -187,6 +190,7 @@ compute_fscore <- function(y, lambda, theta, psi, unclass(dir_minus * (psi - a_mat %*% covy %*% t(a_mat))) } if (fs_matrices) { + attr(fs, "scoring_matrix") <- a_mat fsA <- unclass(a_mat %*% lambda) attr(fs, "fsA") <- fsA attr(fs, "fsb") <- alpha - fsA %*% alpha From b49aff5811e32e578a607c69134ec863ac714227 Mon Sep 17 00:00:00 2001 From: "Hok Chio (Mark) Lai" Date: Tue, 25 Apr 2023 10:57:45 -0700 Subject: [PATCH 012/104] Draft vignette --- vignettes/correction-error.Rmd | 136 +++++++++++++++++++++++++++++ vignettes/sim_correction-error.RDS | Bin 0 -> 3906825 bytes 2 files changed, 136 insertions(+) create mode 100644 vignettes/correction-error.Rmd create mode 100644 vignettes/sim_correction-error.RDS diff --git a/vignettes/correction-error.Rmd b/vignettes/correction-error.Rmd new file mode 100644 index 0000000..0ad6baf --- /dev/null +++ b/vignettes/correction-error.Rmd @@ -0,0 +1,136 @@ +--- +title: "Correction to Measurement Error" +author: "Mark Lai" +date: "`r Sys.Date()`" +output: rmarkdown::html_vignette +vignette: > + %\VignetteIndexEntry{correction-error} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteEncoding{UTF-8} +--- + +```{r setup, include = FALSE} +knitr::opts_chunk$set( + collapse = TRUE, + comment = "#>" +) +``` + +When computing the reliability or the standard errors of factor scores, we typically condition on the estimates of the model parameters, aka assuming that they are known. However, when sample size is small, ignoring those can be problematic as the model parameters are imprecise. This vignette provides details on how to obtain more realistic estimates of standard errors of factor scores. + +# Unidimensional Case + +Consider the conditional variance of the factor scores, $Var(\tilde \eta \mid \eta)$, with $\tilde \eta = \boldsymbol{a}^\top \boldsymbol{y}$. From the factor model, we have + +$$Var(\tilde \eta \mid \eta) = Var(\boldsymbol{a}^\top \boldsymbol{\lambda} \eta + \boldsymbol{a}^\top \boldsymbol{\varepsilon} \mid \eta) = \eta^2 Var(\boldsymbol{a}^\top \boldsymbol{\lambda}) + Var(\boldsymbol{a}^\top \boldsymbol{\varepsilon}).$$ + +The second term is the error component. Here is a small simulation: + +```{r} +#' Load package and set seed +library(lavaan) +library(R2spa) +set.seed(191254) + +#' Fixed conditions +num_obs <- 100 +lambda <- c(.3, .5, .7, .9) +theta <- c(.5, .4, .5, .7) + +#' Simulate true score, and standardized +eta <- rnorm(num_obs) +eta <- (eta - mean(eta)) +eta <- eta / sqrt(mean(eta^2)) + +#' Function for simulating y +simy <- function(eta, lambda) { + t(tcrossprod(lambda, eta) + + rnorm(length(lambda) * length(eta), sd = sqrt(theta))) +} +``` + +```{r, eval = FALSE} +#' Simulation +nsim <- 2500 +tfsy_sim <- fsy_sim <- matrix(NA, nrow = num_obs, ncol = nsim) +# Also save the scoring matrix +a_sim <- matrix(NA, nrow = length(lambda), ncol = nsim) +for (i in seq_len(nsim)) { + y <- simy(eta = eta, lambda = lambda) + tfsy_sim[, i] <- R2spa::compute_fscore( + y, lambda = lambda, theta = diag(theta), psi = matrix(1) + ) + fsy <- R2spa::get_fs(y, std.lv = TRUE) + fsy_sim[, i] <- fsy[, 1] + a_sim[, i] <- attr(fsy, which = "scoring_matrix") +} +``` + +```{r echo = FALSE, eval = FALSE} +saveRDS(list(tfsy_sim, fsy_sim, a_sim), file = "sim_correction-error.RDS") +``` + +```{r echo = FALSE, include = FALSE} +tfsy_sim <- readRDS("sim_correction-error.RDS")[[1]] +fsy_sim <- readRDS("sim_correction-error.RDS")[[2]] +a_sim <- readRDS("sim_correction-error.RDS")[[3]] +``` + +```{r} +#' Average conditional variance +# a known +apply(tfsy_sim, 1, var) |> mean() +# compare to theoretical value +true_a <- R2spa:::compute_a_reg(lambda, psi = matrix(1), theta = diag(theta)) +true_a %*% diag(theta) %*% t(true_a) +# a estimated +apply(fsy_sim, 1, var) |> mean() +``` + +The standard errors are larger when using sample estimates of weights. + +When $\boldsymbol{a}$ is known, the error variance is simply $\boldsymbol{a}^\top \boldsymbol{\Theta} \boldsymbol{a}$. However, when $\boldsymbol{a}$ is unknown and estimated as $\tilde{\boldsymbol{a}}$ from the data, + +$$Var(\tilde{\boldsymbol{a}}^\top \boldsymbol{\varepsilon}) = E[Var(\tilde{\boldsymbol{a}}^\top \boldsymbol{\varepsilon} \mid \tilde{\boldsymbol{a}})] + Var[E(\tilde{\boldsymbol{a}}^\top \boldsymbol{\varepsilon} \mid \tilde{\boldsymbol{a}})].$$ + +Under the assumption of $E(\boldsymbol{\varepsilon})$ = $\boldsymbol{0}$, the second term is 0. The first term is an expectation of a quadratic form, + +$$E(\tilde{\boldsymbol{a}}^\top \boldsymbol{\Theta} \tilde{\boldsymbol{a}}) = E(\tilde{\boldsymbol{a}}^\top) \boldsymbol{\Theta} E(\tilde{\boldsymbol{a}}) + \mathrm{tr}(\boldsymbol{\Theta} \boldsymbol{V}_{\tilde{\boldsymbol{a}}})$$ + +where $\boldsymbol{V}_{\tilde{\boldsymbol{a}}}$ is the covariance matrix of $\tilde{\boldsymbol{a}}$. We can see this using the simulated data + +```{r} +correct_fac <- sum(diag(diag(theta) %*% cov(t(a_sim)))) +true_a %*% diag(theta) %*% t(true_a) + correct_fac +``` + +## Correction Factor + +We can approximate $\boldsymbol{V}_{\tilde{\boldsymbol{a}}}$ using the [delta method](https://en.wikipedia.org/wiki/Delta_method), and the other terms in the above using sample estimates + +$$\tilde{\boldsymbol{a}}^\top \hat{\boldsymbol{\Theta}} \tilde{\boldsymbol{a}} + \mathrm{tr}(\hat{\boldsymbol{\Theta}} \hat{\boldsymbol{V}}_{\tilde{\boldsymbol{a}}}).$$ + +The delta method estimate, $\hat{\boldsymbol{V}}_{\tilde{\boldsymbol{a}}}$, is + +$$\boldsymbol{J}_{\tilde{\boldsymbol{a}}}(\boldsymbol{\theta})^\top \hat{\boldsymbol{V}}_{\tilde{\boldsymbol{\theta}}} \boldsymbol{J}_{\tilde{\boldsymbol{a}}}(\boldsymbol{\theta}),$$ + +where $\boldsymbol{J}_{\tilde{\boldsymbol{a}}}(\boldsymbol{\theta})$ is the Jacobian matrix, and $\hat{\boldsymbol{V}}_{\tilde{\boldsymbol{\theta}}}$ is the sampling variance of the sample estimator of $\boldsymbol{\theta}$. The Jacobian can be approximated using finite difference with `lavaan::lav_func_jacobian_complex()`. + +```{r} +y <- simy(eta = eta, lambda = lambda) +cfa_fit <- cfa("f =~ y1 + y2 + y3 + y4", + data = setNames(data.frame(y), paste0("y", 1:4)), + std.lv = TRUE) +# Jacobian +R2spa:::compute_a(coef(cfa_fit), lavobj = cfa_fit) +jac_a <- lavaan::lav_func_jacobian_complex( + \(x) R2spa:::compute_a(x, lavobj = cfa_fit), + coef(cfa_fit) +) +sum(diag(lavInspect(cfa_fit, what = "est")$theta %*% + jac_a %*% vcov(cfa_fit) %*% t(jac_a))) +fsy <- R2spa::get_fs(y, std.lv = TRUE) +attr(fsy, which = "av_efs") +``` + +# Multidimensional Case diff --git a/vignettes/sim_correction-error.RDS b/vignettes/sim_correction-error.RDS new file mode 100644 index 0000000000000000000000000000000000000000..1c1551a52143208d3caab52615221e6c466daffb GIT binary patch literal 3906825 zcmV(#K;*w4iwFP!00000100%pG?efA$88~@jmVZHB~l4V`CL-gR6-O&B_x$lQTCKb z2w6(9Wedr^U-o_9_ubeTGxqi8`}=3kb7sz&XP*1M?)UrkI%n=OP*6}%Q_xWFyzbh$ 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