Last updated: 2020-03-31

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rm(list=ls())
source(paste0(getwd(),"/code/load_packages.R"))
#source(paste0(getwd(),"/code/get_data.R"))
sessionInfo()
R version 3.6.1 (2019-07-05)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 18362)

Matrix products: default

locale:
[1] LC_COLLATE=English_United States.1252 
[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] xtable_1.8-4          kableExtra_1.1.0      MplusAutomation_0.7-3
 [4] data.table_1.12.6     patchwork_1.0.0       forcats_0.4.0        
 [7] stringr_1.4.0         dplyr_0.8.3           purrr_0.3.3          
[10] readr_1.3.1           tidyr_1.0.0           tibble_2.1.3         
[13] ggplot2_3.2.1         tidyverse_1.3.0      

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.3        lubridate_1.7.4   lattice_0.20-38   assertthat_0.2.1 
 [5] zeallot_0.1.0     rprojroot_1.3-2   digest_0.6.23     R6_2.4.1         
 [9] cellranger_1.1.0  plyr_1.8.4        backports_1.1.5   reprex_0.3.0     
[13] evaluate_0.14     coda_0.19-3       httr_1.4.1        pillar_1.4.2     
[17] rlang_0.4.2       lazyeval_0.2.2    readxl_1.3.1      rstudioapi_0.10  
[21] texreg_1.36.23    rmarkdown_1.18    gsubfn_0.7        proto_1.0.0      
[25] webshot_0.5.2     pander_0.6.3      munsell_0.5.0     broom_0.5.2      
[29] compiler_3.6.1    httpuv_1.5.2      modelr_0.1.5      xfun_0.11        
[33] pkgconfig_2.0.3   htmltools_0.4.0   tidyselect_0.2.5  workflowr_1.5.0  
[37] viridisLite_0.3.0 crayon_1.3.4      dbplyr_1.4.2      withr_2.1.2      
[41] later_1.0.0       grid_3.6.1        nlme_3.1-140      jsonlite_1.6     
[45] gtable_0.3.0      lifecycle_0.1.0   DBI_1.0.0         git2r_0.26.1     
[49] magrittr_1.5      scales_1.1.0      cli_1.1.0         stringi_1.4.3    
[53] fs_1.3.1          promises_1.1.0    xml2_1.2.2        generics_0.0.2   
[57] vctrs_0.2.0       boot_1.3-22       tools_3.6.1       glue_1.3.1       
[61] hms_0.5.2         parallel_3.6.1    yaml_2.2.0        colorspace_1.4-1 
[65] rvest_0.3.5       knitr_1.26        haven_2.2.0      
# general options
theme_set(theme_bw())
options(digits=3)
# set up vectors of variable names
pvec <- c(paste0('lambda1',1:6), paste0('lambda2',6:10), 'psiW12','psiB1', 'psiB2', 'psiB12', paste0('thetaB',1:10), 'icc_lv1_est', 'icc_lv2_est', paste0('icc_ov',1:10,'_est'))
# stored "true" values of parameters by each condition
ptvec <- c(rep('lambdaT',11), 'psiW12T', 'psiB1T', 'psiB2T', 'psiB12T', rep("thetaBT", 10), rep('icc_lv',2), rep('icc_ov',10))

result <- read_csv(paste0(w.d, "/data/results_bias_est.csv"))
Parsed with column specification:
cols(
  N1 = col_double(),
  N2 = col_double(),
  ICC_LV = col_double(),
  ICC_OV = col_double(),
  Variable = col_character(),
  Estimator = col_character(),
  TrueValue = col_double(),
  RB = col_double(),
  RMSE = col_double(),
  Bias = col_double(),
  SampVar = col_double(),
  muRE = col_double(),
  mwRE = col_double(),
  uwRE = col_double(),
  nRep = col_double(),
  estMean = col_double(),
  estSD = col_double()
)
# Set conditions levels as categorical values
result <- result %>%
  mutate(N1 = factor(N1, c("5", "10", "30")),
         N2 = factor(N2, c("30", "50", "100", "200")),
         ICC_OV = factor(ICC_OV, c("0.1","0.3", "0.5")),
         ICC_LV = factor(ICC_LV, c("0.1", "0.5")),
         wi = nRep/500)

Summarizing Results

First, we will plot estimates (botxplots) 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).

*Click here for more details

Level-2 Factor Covariance

sdat <- filter(result, Variable %in% c("psiB12"))
sdat <- sdat %>%
  mutate(TrueValue = factor(TrueValue))

TRUEVALUE <- as.numeric(levels(sdat$TrueValue))

p1 <- ggplot(sdat, aes(y=estMean, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept = TRUEVALUE, color="red")+
  labs(y="Average Level-2 Factor Covariance")

p2 <- ggplot(sdat, aes(y=estSD, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="SD of Level-2 Factor Covariance")

p3 <- ggplot(sdat, aes(y=RB, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept=-10, color="red", linetype="dashed")+
  geom_hline(yintercept=10, color="red", linetype="dashed")+
  labs(y="Relative Bias")

p4 <- ggplot(sdat, aes(y=RMSE, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Root Mean Square Error")

p5 <- ggplot(sdat, aes(y=Bias, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Sqaured Bias")

p6 <- ggplot(sdat, aes(y=SampVar, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Sampling Variance of Estimates")


p <- (p1 + p2 + p3)/(p4 + p5 + p6) + 
  plot_annotation(title="Summarizing bias indices of LEVEL-2 FACTOR COVARIANCE")
p

Single Condition Breakdown

Estimation Method

ggplot(sdat, aes(y=estMean, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept = TRUEVALUE, color="red")+
  labs(y="Average Level-2 Factor Covariance",
       title="LEVEL-2 FACTOR COVARIANCE by Estimation Method",
       subtitle="Parameter Estimates")+
  facet_wrap(.~Estimator)

ggplot(sdat, aes(y=estSD, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="SD of Level-2 Factor Covariances",
       title="LEVEL-2 FACTOR COVARIANCE by Estimation Method",
       subtitle="Standard Deviation of Estimates")+
  facet_wrap(.~Estimator)

ggplot(sdat, aes(y=RB, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept=-10, color="red", linetype="dashed")+
  geom_hline(yintercept=10, color="red", linetype="dashed")+
  labs(y="Relative Bias",
       title="LEVEL-2 FACTOR COVARIANCE by Estimation Method",
       subtitle="Relative Bias of Estimates")+
  facet_wrap(.~Estimator)

ggplot(sdat, aes(y=RMSE, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Root Mean Square Error",
       title="LEVEL-2 FACTOR COVARIANCE by Estimation Method",
       subtitle="Root Mean Square Error of Estimates")+
  facet_wrap(.~Estimator)

ggplot(sdat, aes(y=Bias, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Sqaured Bias",
       title="LEVEL-2 FACTOR COVARIANCE by Estimation Method",
       subtitle="Squared Bias of Estiamtes")+
  facet_wrap(.~Estimator)

ggplot(sdat, aes(y=SampVar, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Sampling Variance",
       title="LEVEL-2 FACTOR COVARIANCE by Estimation Method",
       subtitle="Sampling Variance of Estimates")+
  facet_wrap(.~Estimator)

c <- sdat %>%
  group_by(Estimator, TrueValue) %>%
  summarise(est = weighted.mean(estMean, wi),
            RB = weighted.mean(RB, wi),
            RMSE = weighted.mean(RMSE, wi),
            Bias = weighted.mean(Bias, wi),
            SampVar =weighted.mean(SampVar, wi))

kable(c, format='html', digits=3,
      caption="Summary Indices of LEVEL-2 FACTOR COVARIANCE by Estimation Method") %>%
  kable_styling(full_width = T)
Summary Indices of LEVEL-2 FACTOR COVARIANCE by Estimation Method
Estimator TrueValue est RB RMSE Bias SampVar
MLR 0.0333 0.032 -4.29 0.006 0.000 0.006
MLR 0.3 0.295 -1.66 0.041 0.000 0.041
ULSMV 0.0333 0.032 -4.57 0.009 0.000 0.009
ULSMV 0.3 0.318 5.91 0.050 0.001 0.049
WLSMV 0.0333 0.026 -21.04 0.004 0.000 0.004
WLSMV 0.3 0.271 -9.62 0.035 0.001 0.034

Level-2 Sample Size

ggplot(sdat, aes(y=estMean, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept = TRUEVALUE, color="red")+
  labs(y="Average Level-2 Factor Covariance",
       title="LEVEL-2 FACTOR COVARIANCE by Level-2 Sample Size",
       subtitle="Parameter Estimates")+
  facet_wrap(.~N2)

ggplot(sdat, aes(y=estSD, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="SD of Level-2 Factor Covariances",
       title="LEVEL-2 FACTOR COVARIANCE by Level-2 Sample Size",
       subtitle="Standard Deviation of Parameter Estimates")+
  facet_wrap(.~N2)

ggplot(sdat, aes(y=RB, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept=-10, color="red", linetype="dashed")+
  geom_hline(yintercept=10, color="red", linetype="dashed")+
  labs(y="Relative Bias",
       title="LEVEL-2 FACTOR COVARIANCE by Level-2 Sample Size",
       subtitle="Relative Bias Parameter Estimates")+
  facet_wrap(.~N2)

ggplot(sdat, aes(y=RMSE, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Root Mean Square Error",
       title="LEVEL-2 FACTOR COVARIANCE by Level-2 Sample Size",
       subtitle="Root Mean Square Error")+
  facet_wrap(.~N2)

ggplot(sdat, aes(y=Bias, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Sqaured Bias",
       title="LEVEL-2 FACTOR COVARIANCE by Level-2 Sample Size",
       subtitle="Squared Bias of Parameter Estimates")+
  facet_wrap(.~N2)

ggplot(sdat, aes(y=SampVar, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Sampling Variance of Estimates",
       title="LEVEL-2 FACTOR COVARIANCE by Level-2 Sample Size",
       subtitle="Sampling Variance of Parameter Estimates")+
  facet_wrap(.~N2)

c <- sdat %>%
  group_by(N2, TrueValue) %>%
  summarise(est = weighted.mean(estMean, wi),
            RB = weighted.mean(RB, wi),
            RMSE = weighted.mean(RMSE, wi),
            Bias = weighted.mean(Bias, wi),
            SampVar =weighted.mean(SampVar, wi))

kable(c, format='html', digits=3, 
      caption="Summary Indices of LEVEL-2 FACTOR COVARIANCE by Level-2 Sample Size") %>%
  kable_styling(full_width = T)
Summary Indices of LEVEL-2 FACTOR COVARIANCE by Level-2 Sample Size
N2 TrueValue est RB RMSE Bias SampVar
30 0.0333 0.027 -18.93 0.018 0.000 0.017
30 0.3 0.292 -2.64 0.096 0.002 0.094
50 0.0333 0.027 -18.70 0.008 0.000 0.008
50 0.3 0.297 -1.09 0.050 0.001 0.049
100 0.0333 0.031 -6.27 0.004 0.000 0.004
100 0.3 0.295 -1.83 0.024 0.000 0.024
200 0.0333 0.033 -1.59 0.002 0.000 0.002
200 0.3 0.297 -1.13 0.012 0.000 0.012

Level-1 Sample Size

ggplot(sdat, aes(y=estMean, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept = TRUEVALUE, color="red")+
  labs(y="Average Level-2 Factor Covariance",
       title="LEVEL-2 FACTOR COVARIANCE by Level-1",
       subtitle="Parameter Estimates")+
  facet_wrap(.~N1)

ggplot(sdat, aes(y=estSD, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="SD of Level-2 Factor Covariances",
       title="LEVEL-2 FACTOR COVARIANCE by Level-1 Sample Size",
       subtitle="Standard Deviation of Parameter Estimates")+
  facet_wrap(.~N1)

ggplot(sdat, aes(y=RB, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept=-10, color="red", linetype="dashed")+
  geom_hline(yintercept=10, color="red", linetype="dashed")+
  labs(y="Relative Bias",
       title="LEVEL-2 FACTOR COVARIANCE by Level-1 Sample Size",
       subtitle="Relative Bias of Parameter Estimates")+
  facet_wrap(.~N1)

ggplot(sdat, aes(y=RMSE, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Root Mean Square Error",
       title="LEVEL-2 FACTOR COVARIANCE by Level-1 Sample Size",
       subtitle="Root Mean Square Error")+
  facet_wrap(.~N1)

ggplot(sdat, aes(y=Bias, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Sqaured Bias",
       title="LEVEL-2 FACTOR COVARIANCE by Level-1 Sample Size",
       subtitle="Squared Bias of Parameter Estimates")+
  facet_wrap(.~N1)

ggplot(sdat, aes(y=SampVar, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Sampling Variance of Estimates",
       title="LEVEL-2 FACTOR COVARIANCE by Level-1 Sample Size",
       subtitle="Sampling Variance of Parameter Estimates")+
  facet_wrap(.~N1)

c <- sdat %>%
  group_by(N1, TrueValue) %>%
  summarise(est = weighted.mean(estMean, wi),
            RB = weighted.mean(RB, wi),
            RMSE = weighted.mean(RMSE, wi),
            Bias = weighted.mean(Bias, wi),
            SampVar =weighted.mean(SampVar, wi))

kable(c, format='html', digits=3,
      caption="Summary Indices of LEVEL-2 FACTOR COVARIANCE  by Level-1 Sample Size") %>%
  kable_styling(full_width = T)
Summary Indices of LEVEL-2 FACTOR COVARIANCE by Level-1 Sample Size
N1 TrueValue est RB RMSE Bias SampVar
5 0.0333 0.029 -13.14 0.012 0.000 0.012
5 0.3 0.298 -0.65 0.056 0.001 0.055
10 0.0333 0.030 -8.55 0.006 0.000 0.006
10 0.3 0.294 -1.86 0.040 0.001 0.039
30 0.0333 0.031 -8.31 0.004 0.000 0.004
30 0.3 0.293 -2.20 0.033 0.000 0.033

ICC Observed Variables

ggplot(sdat, aes(y=estMean, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept = TRUEVALUE, color="red")+
  labs(y="Average Level-2 Factor Covariance",
       title="LEVEL-2 FACTOR COVARIANCE by ICC of Observed Variables",
       subtitle="Parameter Estimates")+
  facet_wrap(.~ICC_OV)

ggplot(sdat, aes(y=estSD, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="SD of Level-2 Factor Covariances",
       title="LEVEL-2 FACTOR COVARIANCE by ICC of Observed Variables",
       subtitle="Standard Deviation of Parameter Estimates")+
  facet_wrap(.~ICC_OV)

ggplot(sdat, aes(y=RB, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept=-10, color="red", linetype="dashed")+
  geom_hline(yintercept=10, color="red", linetype="dashed")+
  labs(y="Relative Bias",
       title="LEVEL-2 FACTOR COVARIANCE by ICC of Observed Variables",
       subtitle="Relative Bias of Parameter Estimates")+
  facet_wrap(.~ICC_OV)

ggplot(sdat, aes(y=RMSE, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Root Mean Square Error",
       title="LEVEL-2 FACTOR COVARIANCE by ICC of Observed Variables",
       subtitle="Root Mean Square Error of Parameter Estimates")+
  facet_wrap(.~ICC_OV)

ggplot(sdat, aes(y=Bias, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Sqaured Bias",
       title="LEVEL-2 FACTOR COVARIANCE by ICC of Observed Variables",
       subtitle="Squared Bias of Parameter Estimates")+
  facet_wrap(.~ICC_OV)

ggplot(sdat, aes(y=SampVar, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Sampling Variance of Estimates",
       title="LEVEL-2 FACTOR COVARIANCE by ICC of Observed Variables",
       subtitle="Sampling Variance of Parameter Estimates")+
  facet_wrap(.~ICC_OV)

c <- sdat %>%
  group_by(ICC_OV, TrueValue) %>%
  summarise(est = weighted.mean(estMean, wi),
            RB = weighted.mean(RB, wi),
            RMSE = weighted.mean(RMSE, wi),
            Bias = weighted.mean(Bias, wi),
            SampVar =weighted.mean(SampVar, wi))

kable(c, format='html', digits=3, caption="Summary Indices of LEVEL-2 FACTOR COVARIANCE by ICC of Observed Variables") %>%
  kable_styling(full_width = T)
Summary Indices of LEVEL-2 FACTOR COVARIANCE by ICC of Observed Variables
ICC_OV TrueValue est RB RMSE Bias SampVar
0.1 0.0333 0.032 -4.14 0.002 0.000 0.002
0.1 0.3 0.295 -1.75 0.021 0.000 0.021
0.3 0.0333 0.032 -2.74 0.005 0.000 0.005
0.3 0.3 0.293 -2.19 0.037 0.001 0.036
0.5 0.0333 0.025 -25.62 0.015 0.000 0.015
0.5 0.3 0.297 -1.01 0.061 0.001 0.059

ICC Latent Variables

ggplot(sdat, aes(y=estMean, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept = TRUEVALUE, color="red")+
  labs(y="Average Level-2 Factor Covariance",
       title="LEVEL-2 FACTOR COVARIANCE by ICC of Latent Variables",
       subtitle="Parameter Estimates")+
  facet_wrap(.~ICC_LV)

ggplot(sdat, aes(y=estSD, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="SD of Level-2 Factor Covariances",
       title="LEVEL-2 FACTOR COVARIANCE by ICC of Latent Variables",
       subtitle="Standard Deviation of Parameter Estimates")+
  facet_wrap(.~ICC_LV)

ggplot(sdat, aes(y=RB, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept=-10, color="red", linetype="dashed")+
  geom_hline(yintercept=10, color="red", linetype="dashed")+
  labs(y="Relative Bias",
       title="LEVEL-2 FACTOR COVARIANCE by ICC of Latent Variables",
       subtitle="Relative Bias of Parameter Estimates")+
  facet_wrap(.~ICC_LV)

ggplot(sdat, aes(y=RMSE, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Root Mean Square Error",
       title="LEVEL-2 FACTOR COVARIANCE by ICC of Latent Variables",
       subtitle="Root Mean Square Error of Parameter Estimates")+
  facet_wrap(.~ICC_LV)

ggplot(sdat, aes(y=Bias, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Sqaured Bias",
       title="LEVEL-2 FACTOR COVARIANCE by ICC of Latent Variables",
       subtitle="Squared Bias of Parameter Estimates")+
  facet_wrap(.~ICC_LV)

ggplot(sdat, aes(y=SampVar, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Sampling Variance of Estimates",
       title="LEVEL-2 FACTOR COVARIANCE by ICC of Latent Variables",
       subtitle="Sampling Variance of Parameter Estimates")+
  facet_wrap(.~ICC_LV)

c <- sdat %>%
  group_by(ICC_LV, TrueValue) %>%
  summarise(est = weighted.mean(estMean, wi),
            RB = weighted.mean(RB, wi),
            RMSE = weighted.mean(RMSE, wi),
            Bias = weighted.mean(Bias, wi),
            SampVar =weighted.mean(SampVar, wi))

kable(c, format='html', digits=3,
      caption="Summary Indices of LEVEL-2 FACTOR COVARIANCE by ICC of Latent Variables") %>%
  kable_styling(full_width = T)
Summary Indices of LEVEL-2 FACTOR COVARIANCE by ICC of Latent Variables
ICC_LV TrueValue est RB RMSE Bias SampVar
0.1 0.0333 0.030 -9.65 0.007 0.000 0.006
0.5 0.3 0.295 -1.62 0.042 0.001 0.042

Loadings by Estimation Method and Sample Sizes

Estimation Method & Level-2 Sample Size

ggplot(sdat, aes(y=estMean, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept = TRUEVALUE, color="red")+
  labs(y="Average Level-2 Factor Covariance")+
  facet_grid(N2~Estimator)

ggplot(sdat, aes(y=RB, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept=-10, color="red", linetype="dashed")+
  geom_hline(yintercept=10, color="red", linetype="dashed")+
  labs(y="Relative Bias")+
  facet_grid(N2~Estimator)

ggplot(sdat, aes(y=RMSE, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Root Mean Square Error")+
  facet_grid(N2~Estimator)

c <- sdat %>%
  group_by(Estimator, N2, TrueValue) %>%
  summarise(est = weighted.mean(estMean, wi),
            RB = weighted.mean(RB, wi),
            RMSE = weighted.mean(RMSE, wi),
            Bias = weighted.mean(Bias, wi),
            SampVar =weighted.mean(SampVar, wi))

c1 <- cbind(c[ c$Estimator == 'MLR', c( 'N2','TrueValue', 'est', 'RB', 'RMSE')], 
           c[ c$Estimator == 'ULSMV', c('est', 'RB', 'RMSE')], 
           c[ c$Estimator == 'WLSMV', c('est', 'RB', 'RMSE')])
colnames(c1) <- c('N2','True Value',  rep(c('est', 'RB', 'RMSE'), 3))

kable(c1, format='html', digits=3, row.names = F) %>%
  kable_styling(full_width = T) %>%
  add_header_above(c(' '=2, 'MLR'=3, 'ULSMV'=3, 'WLSMV'=3))
MLR
ULSMV
WLSMV
N2 True Value est RB RMSE est RB RMSE est RB RMSE
30 0.0333 0.030 -9.580 0.015 0.031 -6.413 0.027 0.018 -46.43 0.010
30 0.3 0.292 -2.788 0.089 0.344 14.675 0.123 0.236 -21.49 0.076
50 0.0333 0.030 -8.497 0.009 0.028 -14.999 0.010 0.022 -34.92 0.007
50 0.3 0.297 -1.008 0.047 0.326 8.504 0.058 0.265 -11.65 0.044
100 0.0333 0.033 -2.047 0.004 0.033 -1.441 0.004 0.028 -15.72 0.003
100 0.3 0.294 -1.859 0.024 0.307 2.461 0.026 0.281 -6.37 0.023
200 0.0333 0.033 0.069 0.002 0.034 0.654 0.002 0.031 -5.53 0.002
200 0.3 0.297 -1.118 0.012 0.302 0.620 0.013 0.291 -2.94 0.012

Estimation Method & Level-1 Sample Size

ggplot(sdat, aes(y=estMean, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept = TRUEVALUE, color="red")+
  labs(y="Average Level-2 Factor Covariance")+
  facet_grid(N1~Estimator)

ggplot(sdat, aes(y=RB, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept=-10, color="red", linetype="dashed")+
  geom_hline(yintercept=10, color="red", linetype="dashed")+
  labs(y="Relative Bias")+
  facet_grid(N1~Estimator)

ggplot(sdat, aes(y=RMSE, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Root Mean Square Error")+
  facet_grid(N1~Estimator)

c <- sdat %>%
  group_by(Estimator, N1, TrueValue) %>%
  summarise(est = weighted.mean(estMean, wi),
            RB = weighted.mean(RB, wi),
            RMSE = weighted.mean(RMSE, wi),
            Bias = weighted.mean(Bias, wi),
            SampVar =weighted.mean(SampVar, wi))

c1 <- cbind(c[ c$Estimator == 'MLR', c( 'N1','TrueValue', 'est', 'RB', 'RMSE')], 
           c[ c$Estimator == 'ULSMV', c('est', 'RB', 'RMSE')], 
           c[ c$Estimator == 'WLSMV', c('est', 'RB', 'RMSE')])
colnames(c1) <- c('N1','True Value',  rep(c('est', 'RB', 'RMSE'), 3))

kable(c1, format='html', digits=3, row.names = F) %>%
  kable_styling(full_width = T) %>%
  add_header_above(c(' '=2, 'MLR'=3, 'ULSMV'=3, 'WLSMV'=3))
MLR
ULSMV
WLSMV
N1 True Value est RB RMSE est RB RMSE est RB RMSE
5 0.0333 0.031 -5.95 0.011 0.031 -7.99 0.018 0.024 -28.50 0.007
5 0.3 0.296 -1.20 0.053 0.329 9.77 0.070 0.267 -10.91 0.044
10 0.0333 0.032 -3.46 0.006 0.032 -2.93 0.007 0.027 -20.09 0.004
10 0.3 0.295 -1.62 0.039 0.317 5.70 0.048 0.269 -10.25 0.033
30 0.0333 0.032 -3.77 0.004 0.032 -3.84 0.004 0.027 -17.51 0.003
30 0.3 0.294 -2.09 0.033 0.309 3.10 0.037 0.276 -8.03 0.031

Estimation Method, Level-2 Sample Size & Level-1 Sample Size

ggplot(sdat, aes(y=estMean, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept = TRUEVALUE, color="red")+
  labs(y="Average Level-12 Factor Covariance")+
  facet_grid(N2+N1~Estimator)

ggplot(sdat, aes(y=RB, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  geom_hline(yintercept=-10, color="red", linetype="dashed")+
  geom_hline(yintercept=10, color="red", linetype="dashed")+
  labs(y="Relative Bias")+
  facet_grid(N2+N1~Estimator)

ggplot(sdat, aes(y=RMSE, x=TrueValue, group=TrueValue))+
  geom_boxplot()+
  labs(y="Root Mean Square Error")+
  facet_grid(N2+N1~Estimator)

c <- sdat %>%
  group_by(Estimator, N2, N1, TrueValue) %>%
  summarise(est = weighted.mean(estMean, wi),
            RB = weighted.mean(RB, wi),
            RMSE = weighted.mean(RMSE, wi),
            Bias = weighted.mean(Bias, wi),
            SampVar =weighted.mean(SampVar, wi))

c1 <- cbind(c[ c$Estimator == 'MLR', c( 'N2','N1', 'TrueValue', 'est', 'RB', 'RMSE')], 
           c[ c$Estimator == 'ULSMV', c('est', 'RB', 'RMSE')], 
           c[ c$Estimator == 'WLSMV', c('est', 'RB', 'RMSE')])
colnames(c1) <- c('N2','N1', 'True Value', rep(c('est', 'RB', 'RMSE'), 3))

kable(c1, format='html', digits=3, row.names = F) %>%
  kable_styling(full_width = T) %>%
  add_header_above(c(' '=3, 'MLR'=3, 'ULSMV'=3, 'WLSMV'=3))
MLR
ULSMV
WLSMV
N2 N1 True Value est RB RMSE est RB RMSE est RB RMSE
30 5 0.0333 0.026 -20.880 0.029 0.032 -3.344 0.090 0.007 -78.95 0.024
30 5 0.3 0.304 1.447 0.128 0.385 28.440 0.190 0.232 -22.65 0.096
30 10 0.0333 0.032 -2.680 0.014 0.033 -2.048 0.022 0.019 -44.10 0.009
30 10 0.3 0.288 -3.979 0.084 0.338 12.656 0.118 0.226 -24.61 0.069
30 30 0.0333 0.031 -7.696 0.008 0.030 -10.356 0.009 0.020 -39.12 0.006
30 30 0.3 0.286 -4.822 0.066 0.321 6.997 0.082 0.246 -17.89 0.067
50 5 0.0333 0.029 -11.997 0.015 0.019 -42.178 0.022 0.010 -69.17 0.014
50 5 0.3 0.294 -2.001 0.061 0.333 10.853 0.080 0.252 -16.14 0.057
50 10 0.0333 0.032 -3.079 0.007 0.032 -3.936 0.008 0.025 -26.35 0.006
50 10 0.3 0.295 -1.657 0.047 0.326 8.675 0.059 0.261 -12.96 0.042
50 30 0.0333 0.030 -10.696 0.005 0.030 -10.496 0.006 0.024 -28.17 0.005
50 30 0.3 0.301 0.410 0.037 0.320 6.686 0.042 0.279 -7.16 0.036
100 5 0.0333 0.033 -0.521 0.006 0.033 0.247 0.007 0.027 -20.34 0.006
100 5 0.3 0.292 -2.534 0.031 0.315 5.054 0.034 0.278 -7.37 0.030
100 10 0.0333 0.032 -3.859 0.003 0.032 -4.584 0.003 0.027 -18.14 0.003
100 10 0.3 0.297 -0.842 0.022 0.308 2.563 0.025 0.283 -5.79 0.022
100 30 0.0333 0.033 -1.537 0.002 0.033 0.296 0.002 0.030 -10.43 0.002
100 30 0.3 0.293 -2.262 0.019 0.301 0.353 0.021 0.282 -6.11 0.019
200 5 0.0333 0.034 1.155 0.003 0.033 0.407 0.003 0.030 -8.80 0.003
200 5 0.3 0.296 -1.167 0.015 0.304 1.343 0.017 0.291 -2.85 0.015
200 10 0.0333 0.032 -3.825 0.002 0.033 -1.211 0.002 0.031 -7.87 0.002
200 10 0.3 0.299 -0.431 0.011 0.304 1.408 0.012 0.292 -2.51 0.011
200 30 0.0333 0.034 2.874 0.001 0.034 2.688 0.001 0.033 -0.44 0.001
200 30 0.3 0.295 -1.759 0.009 0.298 -0.819 0.011 0.290 -3.42 0.009

Relative Efficiency by Sample Sizes

c <- sdat %>%
  group_by(Estimator, N2, N1) %>%
  summarise(mu = weighted.mean(muRE, wi),
            mw = weighted.mean(mwRE, wi),
            uw = weighted.mean(uwRE, wi))

c1 <- c[ c$Estimator == 'MLR', c( 'N2','N1', 'mu', 'mw', 'uw')]
colnames(c1) <- c('N2','N1', c('MLR/ULSMV', 'MLR/WLSMV', 'ULSMV/WLSMV'))

kable(c1, format='html', digits=3, row.names = F) %>%
  kable_styling(full_width = T)
N2 N1 MLR/ULSMV MLR/WLSMV ULSMV/WLSMV
30 5 1.579 2.01 1.48
30 10 1.054 1.48 1.41
30 30 1.021 1.31 1.27
50 5 1.247 1.60 1.28
50 10 1.040 1.31 1.24
50 30 0.985 1.12 1.14
100 5 1.100 1.22 1.11
100 10 1.004 1.13 1.12
100 30 0.974 1.05 1.07
200 5 0.997 1.06 1.06
200 10 0.980 1.03 1.05
200 30 0.975 1.01 1.04

sessionInfo()
R version 3.6.1 (2019-07-05)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 18362)

Matrix products: default

locale:
[1] LC_COLLATE=English_United States.1252 
[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] xtable_1.8-4          kableExtra_1.1.0      MplusAutomation_0.7-3
 [4] data.table_1.12.6     patchwork_1.0.0       forcats_0.4.0        
 [7] stringr_1.4.0         dplyr_0.8.3           purrr_0.3.3          
[10] readr_1.3.1           tidyr_1.0.0           tibble_2.1.3         
[13] ggplot2_3.2.1         tidyverse_1.3.0      

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.3        lubridate_1.7.4   lattice_0.20-38   assertthat_0.2.1 
 [5] zeallot_0.1.0     rprojroot_1.3-2   digest_0.6.23     R6_2.4.1         
 [9] cellranger_1.1.0  plyr_1.8.4        backports_1.1.5   reprex_0.3.0     
[13] evaluate_0.14     coda_0.19-3       highr_0.8         httr_1.4.1       
[17] pillar_1.4.2      rlang_0.4.2       lazyeval_0.2.2    readxl_1.3.1     
[21] rstudioapi_0.10   texreg_1.36.23    rmarkdown_1.18    gsubfn_0.7       
[25] labeling_0.3      proto_1.0.0       webshot_0.5.2     pander_0.6.3     
[29] munsell_0.5.0     broom_0.5.2       compiler_3.6.1    httpuv_1.5.2     
[33] modelr_0.1.5      xfun_0.11         pkgconfig_2.0.3   htmltools_0.4.0  
[37] tidyselect_0.2.5  workflowr_1.5.0   viridisLite_0.3.0 crayon_1.3.4     
[41] dbplyr_1.4.2      withr_2.1.2       later_1.0.0       grid_3.6.1       
[45] nlme_3.1-140      jsonlite_1.6      gtable_0.3.0      lifecycle_0.1.0  
[49] DBI_1.0.0         git2r_0.26.1      magrittr_1.5      scales_1.1.0     
[53] cli_1.1.0         stringi_1.4.3     reshape2_1.4.3    farver_2.0.1     
[57] fs_1.3.1          promises_1.1.0    xml2_1.2.2        generics_0.0.2   
[61] vctrs_0.2.0       boot_1.3-22       tools_3.6.1       glue_1.3.1       
[65] hms_0.5.2         parallel_3.6.1    yaml_2.2.0        colorspace_1.4-1 
[69] rvest_0.3.5       knitr_1.26        haven_2.2.0