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On this page, we describe the methods that are used for decomposing the effects on parameter estimations in terms of different estimates of bias with respect to standard errors.
First, we need to compute what the true'' standard errors are with respect to each parameter. We have generated a sampling distribution for each parameter in our study, and so can use this sample distribution to compute anempirical standard error.’’ The empirical standard error is simply the standard deviation of the sampling distribution of the parameter (in that specific condition) (Bandalos & Gagné, 2012). Which we will call \(SE(\hat{\theta}_i)\), the empirical standard error of the \(i\)-th parameter.
The bias of standard errors is formally computed as
\[ Bias\left(SE(\hat{\theta}_i)\right) = \sum_{j=1}^{n_r}\left(\frac{\widehat{SE}(\hat{\theta}_i)_j- SE(\hat{\theta}_i)}{SE(\hat{\theta}_i)}\right)/n_r\times 100 \] where, \(\widehat{SE}(\hat{\theta}_i)_j\) is the estimated standard error of the \(i\)-th parameter in the \(j\)-th replication. We will follow the suggestion of Hoogland and Boomsma (1998) that bias \(\leq \mid 5\%\mid\) is considered an ``acceptable’’ level of bias.
Bias will be estimated for
Another aspect of the standard error estimates that was of interest was the variability in standard error estimates between estimation methods. Meaning, which estimation method was least variable relative to the other methods. We compute an efficiency ratio (or relative efficiency) between estimation methods (m = MLR, u = ULSMV, and w = WLSMV). \[ RE = \sqrt{\left(\frac{\sum_{\forall j}(\widehat{SE}(\hat{\theta}_i)_{mj}- SE(\hat{\theta}_i))^2}{\sum_{\forall j}(\widehat{SE}(\hat{\theta}_i)_{uj}- SE(\hat{\theta}_i))^2}\right)} \] where RE = relative efficiency, and the ratio was computed for all three pairwise comparisons of (m, u, w).
Here, we estimated the bias estimates.
First, we set up some functions to compute the values of interest.
#compute RB
# p = parameter estimate of interest
# pt = true value of parameter of interest
est_relative_bias <- function(data, p, pt){
nr <- nrow(data)
data[, pt] <- as.numeric(data[,pt, drop=T])
rb <- sum((data[, p] - data[, pt])/data[,pt], na.rm = T)/nr*100
names(rb) <- 'RB'
return(rb)
}
# compute RMSE components
est_rmse <- function(data, p, pt){
nr <- nrow(data)
data[, pt] <- as.numeric(data[,pt, drop=T])
est_a <- mean(data[,p, drop=T], na.rm = T)
bias <- (est_a - data[,pt, drop=T][1])**2
sv <- sum((data[, p, drop=T] - est_a)**2, na.rm=T)/nr
rmse <- bias + sv
out <- c(rmse, bias, sv)
names(out) <- c('RMSE', 'Bias', 'SampVar')
return(out)
}
# compute estimated relative efficiency of
# parameter est between two estimation methods
est_relative_efficiency <- function(data, p, pt, est1, est2){
dat1 <- filter(data, Estimator == est1)
dat2 <- filter(data, Estimator == est2)
dat1[,pt] <- as.numeric(dat1[,pt, drop=T])
dat2[,pt] <- as.numeric(dat2[,pt, drop=T])
re <- sqrt(sum( (dat1[, p, drop=T] - dat1[, pt, drop=T])**2, na.rm=T)/sum( (dat2[, p, drop=T] - dat2[, pt, drop=T])**2, na.rm=T))
names(re) <- 'RE'
return(re)
}Next, loop through the desired results to get the estimates of bias of interest. For more details on the naming of variables for the true values, see the Data Cleaning page.
source(paste0(getwd(),"/code/load_packages.R"))
source(paste0(getwd(),"/code/get_data.R"))
# take out non-converged/inadmissible cases
sim_results <- filter(sim_results, Converge==1, Admissible==1)
# set up vectors of variable names
pvec <- c(paste0('selambda1',1:6), paste0('selambda2',6:10), 'sepsiW12','sepsiB1', 'sepsiB2', 'sepsiB12', paste0('sethetaB',1:10))
# Compute empirical standard errors
sim_results <- sim_results %>%
group_by(Condition, Estimator) %>%
mutate(EmpSElambda11 = sd(lambda11), EmpSElambda12 = sd(lambda12),
EmpSElambda13 = sd(lambda13), EmpSElambda14 = sd(lambda14),
EmpSElambda15 = sd(lambda15), EmpSElambda16 = sd(lambda16),
EmpSElambda26 = sd(lambda26), EmpSElambda27 = sd(lambda27),
EmpSElambda28 = sd(lambda28), EmpSElambda29 = sd(lambda29),
EmpSElambda210 = sd(lambda210),
EmpSEpsiW12 = sd(psiW12), EmpSEpsiB1 = sd(psiB1),
EmpSEpsiB2 = sd(psiB2), EmpSEpsiB12 = sd(psiB12),
EmpSEthetaB1 = sd(thetaB1), EmpSEthetaB2 = sd(thetaB2),
EmpSEthetaB3 = sd(thetaB3), EmpSEthetaB4 = sd(thetaB4),
EmpSEthetaB5 = sd(thetaB5), EmpSEthetaB6 = sd(thetaB6),
EmpSEthetaB7 = sd(thetaB7), EmpSEthetaB8 = sd(thetaB8),
EmpSEthetaB9 = sd(thetaB9), EmpSEthetaB10 = sd(thetaB10))
# vector of empirical SE names
# set up vectors of variable names
ptvec <- c(paste0('EmpSElambda1',1:6), paste0('EmpSElambda2',6:10), 'EmpSEpsiW12','EmpSEpsiB1', 'EmpSEpsiB2', 'EmpSEpsiB12', paste0('EmpSEthetaB',1:10))
# iterators - conditions
N1 <- unique(sim_results$ss_l1)
N2 <- unique(sim_results$ss_l2)
ICC_LV <- unique(sim_results$icc_lv)
ICC_OV <- unique(sim_results$icc_ov)
EST <- unique(sim_results$Estimator)
# results matrix
result <- data.frame(matrix(nrow=length(N1)*length(N2)*length(ICC_LV)*length(ICC_OV)*length(pvec)*length(EST), ncol=(17)))
colnames(result) <- c('N1', 'N2', 'ICC_LV' ,'ICC_OV', 'Variable', 'Estimator','EmpSE', 'RB', 'RMSE', 'Bias', 'SampVar', 'muRE', 'mwRE', 'uwRE', 'nRep', 'estMean', 'estSD')
j <- 1 # row id
for(n1 in N1){
for(n2 in N2){
for(iccl in ICC_LV){
for(icco in ICC_OV){
for(i in 1:length(pvec)){
dat <- filter(sim_results, ss_l1 == n1, ss_l2==n2,
icc_lv==iccl, icc_ov==icco)
result[j:(j+2), 1:5] <- matrix(rep(c(n1,n2, iccl,icco, pvec[i]),3),
ncol=5, byrow = T)
# compute bias by each estimation method
k <- 0
for(est in EST){
sdat <- filter(dat, Estimator==est)
result[j+k, 6] <- est
result[j+k, 7] <- sdat[1, ptvec[i], drop=T]
result[j+k, 8] <- est_relative_bias(sdat, pvec[i], ptvec[i])
result[j+k, 9:11] <- est_rmse(sdat, pvec[i], ptvec[i])
result[j+k, 15] <- nrow(sdat) # number of converged replications
result[j+k, 16] <- mean(sdat[, pvec[i], drop=T])
result[j+k, 17] <- sd(sdat[, pvec[i], drop=T])
k<-k+1
}
# Compute Relative Efficiency
# MLR vs. ULSMV
result[j:(j+2), 12] <- est_relative_efficiency(dat, pvec[i], ptvec[i],
'MLR','ULSMV')
# MLR vs. WLSMV
result[j:(j+2), 13] <- est_relative_efficiency(dat, pvec[i], ptvec[i],
'MLR','WLSMV')
# ULSMV vs. WLSMV
result[j:(j+2), 14] <- est_relative_efficiency(dat, pvec[i], ptvec[i],
'ULSMV','WLSMV')
# update row
j <- j+3
}
}
}
}
}So, there are a lot of results that could be reported from this matrix of results. We have saved these results and these estimates are included in the accompanying Shiny app for more detailed exploration by those interested. Here, we describe a subset of the results that we feel are most important.
First, we will plot estimates (boxplots) to show how these estimates changed across conditions. To summarize the results we will average over the parameters that only differ y indices. Meaning we will describe the “average factor loading bias” by reporting the average bias for factor loadings. Additionally, different conditions resultedin different “sample sizes.” By this we mean the number of uses replications. The different number of cases per condition was accounted for by creating a “weight” variable for each row of the result object. This meant that conditions that had more usable replications counted more towards to averages reported (or count as much as if we averaged over the individual replications).
result$wi <- result$nRep/500
# 500 is the max number of replications per cell
# Save Results
write_csv(result, path=paste0(w.d, "/data/results_bias_se.csv"))Bandalos, D. L., & Gagné, P. (2012). Simulation methods in structural equation modeling. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (pp. 92–108). New York, NY: Guilford Press.
Hoogland, J. J., & Boomsma, A. (1998). Robustness studies in covariance structure modeling: An overview and a meta-analysis. Sociological Methods & Research, 26(3), 329-367.
sessionInfo()R version 3.6.3 (2020-02-29)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 18362)
Matrix products: default
locale:
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[2] LC_CTYPE=English_United States.1252
[3] LC_MONETARY=English_United States.1252
[4] LC_NUMERIC=C
[5] LC_TIME=English_United States.1252
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] workflowr_1.6.2
loaded via a namespace (and not attached):
[1] Rcpp_1.0.4.6 rprojroot_1.3-2 digest_0.6.25 later_1.0.0
[5] R6_2.4.1 backports_1.1.7 git2r_0.27.1 magrittr_1.5
[9] evaluate_0.14 stringi_1.4.6 rlang_0.4.6 fs_1.4.1
[13] promises_1.1.0 rmarkdown_2.1 tools_3.6.3 stringr_1.4.0
[17] glue_1.4.1 httpuv_1.5.2 xfun_0.14 yaml_2.2.1
[21] compiler_3.6.3 htmltools_0.4.0 knitr_1.28