From: Nathan TeBlunthuis <nathante@uw.edu>
Date: Tue, 28 Jun 2022 05:48:20 +0000 (-0700)
Subject: make first simulation with precise accuracies and explained variances
X-Git-Url: https://code.communitydata.science/ml_measurement_error_public.git/commitdiff_plain/f8f58301e0285118f7b669a96ed9367a9914ba02?ds=inline;hp=--cc

make first simulation with precise accuracies and explained variances
---

f8f58301e0285118f7b669a96ed9367a9914ba02
diff --git a/simulations/01_two_covariates.R b/simulations/01_two_covariates.R
new file mode 100644
index 0000000..c52a3dc
--- /dev/null
+++ b/simulations/01_two_covariates.R
@@ -0,0 +1,85 @@
+### EXAMPLE 2_b: demonstrates how measurement error can lead to a type sign error in a covariate
+### This is the same as example 2, only instead of x->k we have k->x.
+### Even when you have a good predictor, if it's biased against a covariate you can get the wrong sign.
+### Even when you include the proxy variable in the regression.
+### But with some ground truth and multiple imputation, you can fix it.
+
+library(argparser)
+library(mecor)
+library(ggplot2)
+library(data.table)
+library(filelock)
+library(arrow)
+library(Amelia)
+library(Zelig)
+library(predictionError)
+options(amelia.parallel="no",
+        amelia.ncpus=1)
+
+source("simulation_base.R")
+
+## SETUP:
+### we want to estimate x -> y; x is MAR
+### we have x -> k; k -> w; x -> w is used to predict x via the model w.
+### A realistic scenario is that we have an NLP model predicting something like "racial harassment" in social media comments
+### The labels x are binary, but the model provides a continuous predictor
+
+### simulation:
+#### how much power do we get from the model in the first place? (sweeping N and m)
+#### 
+
+simulate_data <- function(N, m, B0=0, Bxy=0.2, Bgy=-0.2, Bgx=0.2, y_explained_variance=0.025, gx_explained_variance=0.15, prediction_accuracy=0.73, seed=1){
+    set.seed(seed)
+    g <- rbinom(N, 1, 0.5)
+
+    x.var.epsilon <- var(Bgx *g) * ((1-gx_explained_variance)/gx_explained_variance)
+    x.epsilon <- rnorm(N,sd=sqrt(x.var.epsilon))
+    xprime <- Bgx * g + x.epsilon
+    x <- as.integer(logistic(scale(xprime)) > 0.5)
+
+    y.var.epsilon <- (var(Bgy * g) + var(Bxy *x) + 2*cov(Bxy*x,Bgy*g)) * ((1-y_explained_variance)/y_explained_variance)
+    y.epsilon <- rnorm(N, sd = sqrt(y.var.epsilon))
+    y <- Bgy * g + Bxy * x + y.epsilon
+
+    df <- data.table(x=x,xprime=xprime,y=y,g=g)
+
+    if(m < N){
+        df <- df[sample(nrow(df), m), x.obs := x]
+    } else {
+        df <- df[, x.obs := x]
+    }
+
+    df <- df[,w_pred:=x]
+
+    df <- df[sample(1:N,(1-prediction_accuracy)*N),w_pred:=(w_pred-1)**2]
+    df <- df[,':='(w=w, w_pred = w_pred)]
+    return(df)
+}
+
+parser <- arg_parser("Simulate data and fit corrected models")
+parser <- add_argument(parser, "--N", default=500, help="number of observations of w")
+parser <- add_argument(parser, "--m", default=100, help="m the number of ground truth observations")
+parser <- add_argument(parser, "--seed", default=4321, help='seed for the rng')
+parser <- add_argument(parser, "--outfile", help='output file', default='example_2_B.feather')
+args <- parse_args(parser)
+
+B0 <- 0
+Bxy <- 0.2
+Bgy <- -0.2
+Bgx <- 0.5
+
+df <- simulate_data(args$N, args$m, B0, Bxy, Bgy, Bgx, seed=args$seed, y_explained_variance = 0.025, gx_explained_variance = 0.15)
+result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy,'Bgy'=Bgy, 'Bgx'=Bgx, 'seed'=args$seed)
+outline <- run_simulation(df, result)
+
+outfile_lock <- lock(paste0(args$outfile, '_lock'),exclusive=TRUE)
+if(file.exists(args$outfile)){
+    logdata <- read_feather(args$outfile)
+    logdata <- rbind(logdata,as.data.table(outline))
+} else {
+    logdata <- as.data.table(outline)
+}
+
+print(outline)
+write_feather(logdata, args$outfile)
+unlock(outfile_lock)
diff --git a/simulations/example_3.R b/simulations/02_indep_differential.R
similarity index 70%
rename from simulations/example_3.R
rename to simulations/02_indep_differential.R
index 389dba1..5a7784b 100644
--- a/simulations/example_3.R
+++ b/simulations/02_indep_differential.R
@@ -1,6 +1,5 @@
-### EXAMPLE 2_b: demonstrates how measurement error can lead to a type sign error in a covariate
+### EXAMPLE 1: demonstrates how measurement error can lead to a type sign error in a covariate
 ### What kind of data invalidates fong + tyler?
-### This is the same as example 2, only instead of x->k we have k->x.
 ### Even when you have a good predictor, if it's biased against a covariate you can get the wrong sign.
 ### Even when you include the proxy variable in the regression.
 ### But with some ground truth and multiple imputation, you can fix it.
@@ -32,7 +31,7 @@ source("simulation_base.R")
 
 ## one way to do it is by adding correlation to x.obs and y that isn't in w.
 ## in other words, the model is missing an important feature of x.obs that's related to y.
-simulate_latent_cocause <- function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed){
+simulate_data <- function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed, xy.explained.variance=0.01, u.explained.variance=0.1){
     set.seed(seed)
 
     ## the true value of x
@@ -40,7 +39,7 @@ simulate_latent_cocause <- function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed){
     g <- rbinom(N, 1, 0.5)
 
     # make w and y dependent
-    u <- rnorm(N,0,Bxy)
+    u <- rnorm(N,0,)
 
     xprime <- Bgx * g + rnorm(N,0,1) 
 
@@ -48,7 +47,7 @@ simulate_latent_cocause <- function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed){
 
     x <- as.integer(logistic(scale(xprime)) > 0.5)
 
-    y <-  Bxy * x  + Bgy * g + rnorm(N, 0, 1) + B0 + u
+    y <-  Bxy * x  + Bgy * g + B0 + u + rnorm(N, 0, 1)
 
     df <- data.table(x=x,k=k,y=y,g=g)
 
@@ -69,42 +68,6 @@ simulate_latent_cocause <- function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed){
     return(df)
 }
 
-## simulate_latent_cocause_2 <- function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed){
-##     set.seed(seed)
-
-##     ## the true value of x
-
-##     g <- rbinom(N, 1, 0.5)
-
-##     # make w and y dependent
-##     u <- rnorm(N,0,5)
-
-##     xprime <- Bgx * g + rnorm(N,0,1)
-
-##     k <- Bkx*xprime + rnorm(N,0,3)
-
-##     x <- as.integer(logistic(scale(xprime+0.3)) > 0.5)
-
-##     y <-  Bxy * x  + Bgy * g + rnorm(N, 0, 1) + B0 + u
-
-##     df <- data.table(x=x,k=k,y=y,g=g)
-
-##     w.model <- glm(x ~ k, df, family=binomial(link='logit')) 
-
-##     if( m < N){
-##         df <- df[sample(nrow(df), m), x.obs := x]
-##     } else {
-##         df <- df[, x.obs := x]
-##     }
-
-##     w <- predict(w.model,data.frame(k=k)) + u 
-##     ## y = B0 + B1x + e
-
-##     df[,':='(w=w, w_pred = as.integer(w>0.5),u=u)]
-##     return(df)
-## }
-
-
 schennach <- function(df){
 
     fwx <- glm(x.obs~w, df, family=binomial(link='logit'))
@@ -128,7 +91,7 @@ Bkx <- 2
 Bgx <- 0
 
 
-outline <- run_simulation(simulate_latent_cocause(args$N, args$m, B0, Bxy, Bgy, Bkx, Bgx, args$seed)
+outline <- run_simulation(simulate_data(args$N, args$m, B0, Bxy, Bgy, Bkx, Bgx, args$seed)
                          ,list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy,'Bgy'=Bgy, 'Bkx'=Bkx, 'Bgx'=Bgx, 'seed'=args$seed))
 
 outfile_lock <- lock(paste0(args$outfile, '_lock'),exclusive=TRUE)
diff --git a/simulations/example_2_B.R b/simulations/example_2_B.R
deleted file mode 100644
index 91623a1..0000000
--- a/simulations/example_2_B.R
+++ /dev/null
@@ -1,276 +0,0 @@
-### EXAMPLE 2_b: demonstrates how measurement error can lead to a type sign error in a covariate
-### This is the same as example 2, only instead of x->k we have k->x.
-### Even when you have a good predictor, if it's biased against a covariate you can get the wrong sign.
-### Even when you include the proxy variable in the regression.
-### But with some ground truth and multiple imputation, you can fix it.
-
-library(argparser)
-library(mecor)
-library(ggplot2)
-library(data.table)
-library(filelock)
-library(arrow)
-library(Amelia)
-library(Zelig)
-library(predictionError)
-options(amelia.parallel="no",
-        amelia.ncpus=1)
-
-source("simulation_base.R")
-
-## SETUP:
-### we want to estimate x -> y; x is MAR
-### we have x -> k; k -> w; x -> w is used to predict x via the model w.
-### A realistic scenario is that we have an NLP model predicting something like "racial harassment" in social media comments
-### The labels x are binary, but the model provides a continuous predictor
-
-### simulation:
-#### how much power do we get from the model in the first place? (sweeping N and m)
-#### 
-logistic <- function(x) {1/(1+exp(-1*x))}
-
-simulate_latent_cocause <- function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed){
-    set.seed(seed)
-
-    ## the true value of x
-
-    g <- rbinom(N, 1, 0.5)
-    xprime <- Bgx * g + rnorm(N,0,1)
-
-    k <- Bkx*xprime + rnorm(N,0,3)
-
-    x <- as.integer(logistic(scale(xprime)) > 0.5)
-
-    y <-  Bxy * x  + Bgy * g + rnorm(N, 0, 2) + B0
-    df <- data.table(x=x,k=k,y=y,g=g)
-
-    if( m < N){
-        df <- df[sample(nrow(df), m), x.obs := x]
-    } else {
-        df <- df[, x.obs := x]
-    }
-
-    w.model <- glm(x ~ k,df, family=binomial(link='logit'))
-    w <- predict(w.model,data.frame(k=k)) + rnorm(N,0,1)
-    ## y = B0 + B1x + e
-
-    df[,':='(w=w, w_pred = as.integer(w>0.5))]
-    return(df)
-}
-
-schennach <- function(df){
-
-    fwx <- glm(x.obs~w, df, family=binomial(link='logit'))
-    df[,xstar_pred := predict(fwx, df)]
-    gxt <- lm(y ~ xstar_pred+g, df)
-
-}
-
-parser <- arg_parser("Simulate data and fit corrected models")
-parser <- add_argument(parser, "--N", default=100, help="number of observations of w")
-parser <- add_argument(parser, "--m", default=20, help="m the number of ground truth observations")
-parser <- add_argument(parser, "--seed", default=4321, help='seed for the rng')
-parser <- add_argument(parser, "--outfile", help='output file', default='example_2_B.feather')
-args <- parse_args(parser)
-
-rows <- list()
-
-B0 <- 0
-Bxy <- 0.2
-Bgy <- -0.2
-Bkx <- 1
-Bgx <- 3
-
-outline <- run_simulation(simulate_latent_cocause(args$N, args$m, B0, Bxy, Bgy, Bkx, Bgx, args$seed)
-                         ,list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy,'Bgy'=Bgy, 'Bkx'=Bkx, 'Bgx'=Bgx, 'seed'=args$seed))
-
-outfile_lock <- lock(paste0(args$outfile, '_lock'),exclusive=TRUE)
-if(file.exists(args$outfile)){
-    logdata <- read_feather(args$outfile)
-    logdata <- rbind(logdata,as.data.table(outline))
-} else {
-    logdata <- as.data.table(outline)
-}
-
-print(outline)
-write_feather(logdata, args$outfile)
-unlock(outfile_lock)
-
-## Ns <- c(1e6, 5e4, 1000)
-## ms <- c(100, 250, 500, 1000)
-## seeds <- 1:500
-
-## rowssets <- list()
-## library(doParallel)
-## options(mc.cores = parallel::detectCores())
-## cl <- makeCluster(20)
-## registerDoParallel(cl)
-
-## ## library(future)
-
-## ## plan(multiprocess,workers=40,gc=TRUE)
-
-## for(N in Ns){
-##     print(N)
-##     for(m in ms){
-##         if(N>m){
-##             new.rows <- foreach(iter=seeds, .combine=rbind, .packages = c('mecor','Amelia','Zelig','predictionError','data.table'),
-##                                 .export = c("run_simulation","simulate_latent_cocause","logistic","N","m","B0","Bxy","Bgy","Bkx","Bgx")) %dopar%
-                
-##                 {run_simulation(simulate_latent_cocause(N, m, B0, Bxy, Bgy, Bkx, Bgx, iter)
-##                                ,list('N'=N,'m'=m,'B0'=B0,'Bxy'=Bxy,'Bgy'=Bgy, 'Bkx'=Bkx, 'Bgx'=Bgx, 'seed'=iter))}
-##             rowsets <- append(rowssets, list(data.table(new.rows)))
-##        }
-
-##     }
-##                 ## rows <- append(rows, list(future({run_simulation(simulate_latent_cocause(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed)
-## }
-                ##                                                 ,list(N=N,m=m,B0=B0,Bxy=Bxy,Bgy=Bgy, Bkx=Bkx, Bgx=Bgx, seed=seed))w},
-                ##                                  packages=c('mecor','Amelia','Zelig','predictionError'),
-                ##                                  seed=TRUE)))
-
-
-## df <- rbindlist(rowsets)
-
-## write_feather(df,"example_2B_simulation.feather")
-
-## run_simulation <-  function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed){
-##     result <- list()
-##     df <- simulate_latent_cocause(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed)
-
-##     result <- append(result, list(N=N,
-##                                   m=m,
-##                                   B0=B0,
-##                                   Bxy=Bxy,
-##                                   Bgy=Bgy,
-##                                   Bkx=Bkx,
-##                                   seed=seed))
-
-##     (accuracy <- df[,.(mean(w_pred==x))])
-##     result <- append(result, list(accuracy=accuracy))
-
-##     (model.true <- lm(y ~ x + g, data=df))
-##     true.ci.Bxy <- confint(model.true)['x',]
-##     true.ci.Bgy <- confint(model.true)['g',]
-
-##     result <- append(result, list(Bxy.est.true=coef(model.true)['x'],
-##                                   Bgy.est.true=coef(model.true)['g'],
-##                                   Bxy.ci.upper.true = true.ci.Bxy[2],
-##                                   Bxy.ci.lower.true = true.ci.Bxy[1],
-##                                   Bgy.ci.upper.true = true.ci.Bgy[2],
-##                                   Bgy.ci.lower.true = true.ci.Bgy[1]))
-                                  
-##     (model.feasible <- lm(y~x.obs+g,data=df))
-
-##     feasible.ci.Bxy <- confint(model.feasible)['x.obs',]
-##     result <- append(result, list(Bxy.est.feasible=coef(model.feasible)['x.obs'],
-##                                   Bxy.ci.upper.feasible = feasible.ci.Bxy[2],
-##                                   Bxy.ci.lower.feasible = feasible.ci.Bxy[1]))
-
-##     feasible.ci.Bgy <- confint(model.feasible)['g',]
-##     result <- append(result, list(Bgy.est.feasible=coef(model.feasible)['g'],
-##                                   Bgy.ci.upper.feasible = feasible.ci.Bgy[2],
-##                                   Bgy.ci.lower.feasible = feasible.ci.Bgy[1]))
-
-##     (model.naive <- lm(y~w+g, data=df))
-    
-##     naive.ci.Bxy <- confint(model.naive)['w',]
-##     naive.ci.Bgy <- confint(model.naive)['g',]
-
-##     result <- append(result, list(Bxy.est.naive=coef(model.naive)['w'],
-##                                   Bgy.est.naive=coef(model.naive)['g'],
-##                                   Bxy.ci.upper.naive = naive.ci.Bxy[2],
-##                                   Bxy.ci.lower.naive = naive.ci.Bxy[1],
-##                                   Bgy.ci.upper.naive = naive.ci.Bgy[2],
-##                                   Bgy.ci.lower.naive = naive.ci.Bgy[1]))
-                                  
-
-##     ## multiple imputation when k is observed
-##     ## amelia does great at this one.
-##     amelia.out.k <- amelia(df, m=200, p2s=0, idvars=c('x','w_pred'),noms=c("x.obs","g"),lgstc=c('w'))
-##     mod.amelia.k <- zelig(y~x.obs+g, model='ls', data=amelia.out.k$imputations, cite=FALSE)
-##     (coefse <- combine_coef_se(mod.amelia.k, messages=FALSE))
-
-##     est.x.mi <- coefse['x.obs','Estimate']
-##     est.x.se <- coefse['x.obs','Std.Error']
-##     result <- append(result,
-##                      list(Bxy.est.amelia.full = est.x.mi,
-##                           Bxy.ci.upper.amelia.full = est.x.mi + 1.96 * est.x.se,
-##                           Bxy.ci.lower.amelia.full = est.x.mi - 1.96 * est.x.se
-##                           ))
-
-##     est.g.mi <- coefse['g','Estimate']
-##     est.g.se <- coefse['g','Std.Error']
-
-##     result <- append(result,
-##                      list(Bgy.est.amelia.full = est.g.mi,
-##                           Bgy.ci.upper.amelia.full = est.g.mi + 1.96 * est.g.se,
-##                           Bgy.ci.lower.amelia.full = est.g.mi - 1.96 * est.g.se
-##                           ))
-
-##     ## What if we can't observe k -- most realistic scenario. We can't include all the ML features in a model.
-##     amelia.out.nok <- amelia(df, m=200, p2s=0, idvars=c("x","k"), noms=c("x.obs",'g'),lgstc = c("w"))
-##     mod.amelia.nok <- zelig(y~x.obs+g, model='ls', data=amelia.out.nok$imputations, cite=FALSE)
-##     (coefse <- combine_coef_se(mod.amelia.nok, messages=FALSE))
-
-##     est.x.mi <- coefse['x.obs','Estimate']
-##     est.x.se <- coefse['x.obs','Std.Error']
-##     result <- append(result,
-##                      list(Bxy.est.amelia.nok = est.x.mi,
-##                           Bxy.ci.upper.amelia.nok = est.x.mi + 1.96 * est.x.se,
-##                           Bxy.ci.lower.amelia.nok = est.x.mi - 1.96 * est.x.se
-##                           ))
-
-##     est.g.mi <- coefse['g','Estimate']
-##     est.g.se <- coefse['g','Std.Error']
-
-##     result <- append(result,
-##                      list(Bgy.est.amelia.nok = est.g.mi,
-##                           Bgy.ci.upper.amelia.nok = est.g.mi + 1.96 * est.g.se,
-##                           Bgy.ci.lower.amelia.nok = est.g.mi - 1.96 * est.g.se
-##                           ))
-
-##     p <- v <- train <- rep(0,N)
-##     M <- m
-##     p[(M+1):(N)] <- 1
-##     v[1:(M)] <- 1
-##     df <- df[order(x.obs)]
-##     y <- df[,y]
-##     x <- df[,x.obs]
-##     g <- df[,g]
-##     w <- df[,w]
-##     # gmm gets pretty close
-##     (gmm.res <- predicted_covariates(y, x, g, w, v, train, p, max_iter=100, verbose=FALSE))
-
-##     result <- append(result,
-##                      list(Bxy.est.gmm = gmm.res$beta[1,1],
-##                           Bxy.ci.upper.gmm = gmm.res$confint[1,2],
-##                           Bxy.ci.lower.gmm = gmm.res$confint[1,1]))
-
-##     result <- append(result,
-##                      list(Bgy.est.gmm = gmm.res$beta[2,1],
-##                           Bgy.ci.upper.gmm = gmm.res$confint[2,2],
-##                           Bgy.ci.lower.gmm = gmm.res$confint[2,1]))
-
-
-##     mod.calibrated.mle <- mecor(y ~ MeasError(w, reference = x.obs) + g, df, B=400, method='efficient')
-##     (mod.calibrated.mle)
-##     (mecor.ci <- summary(mod.calibrated.mle)$c$ci['x.obs',])
-##     result <- append(result, list(
-##                                  Bxy.est.mecor = mecor.ci['Estimate'],
-##                                  Bxy.upper.mecor = mecor.ci['UCI'],
-##                                  Bxy.lower.mecor = mecor.ci['LCI'])
-##                      )
-
-##     (mecor.ci <- summary(mod.calibrated.mle)$c$ci['g',])
-
-##     result <- append(result, list(
-##                                  Bxy.est.mecor = mecor.ci['Estimate'],
-##                                  Bxy.upper.mecor = mecor.ci['UCI'],
-##                                  Bxy.lower.mecor = mecor.ci['LCI'])
-##                      )
-
-
-##     return(result)
-## }
-
diff --git a/simulations/simulation_base.R b/simulations/simulation_base.R
index 26916ab..345d14e 100644
--- a/simulations/simulation_base.R
+++ b/simulations/simulation_base.R
@@ -82,26 +82,26 @@ run_simulation <-  function(df, result){
                           ))
 
     ## What if we can't observe k -- most realistic scenario. We can't include all the ML features in a model.
-    amelia.out.nok <- amelia(df, m=200, p2s=0, idvars=c("x","k","w_pred"), noms=noms)
-    mod.amelia.nok <- zelig(y~x.obs+g, model='ls', data=amelia.out.nok$imputations, cite=FALSE)
-    (coefse <- combine_coef_se(mod.amelia.nok, messages=FALSE))
-
-    est.x.mi <- coefse['x.obs','Estimate']
-    est.x.se <- coefse['x.obs','Std.Error']
-    result <- append(result,
-                     list(Bxy.est.amelia.nok = est.x.mi,
-                          Bxy.ci.upper.amelia.nok = est.x.mi + 1.96 * est.x.se,
-                          Bxy.ci.lower.amelia.nok = est.x.mi - 1.96 * est.x.se
-                          ))
-
-    est.g.mi <- coefse['g','Estimate']
-    est.g.se <- coefse['g','Std.Error']
-
-    result <- append(result,
-                     list(Bgy.est.amelia.nok = est.g.mi,
-                          Bgy.ci.upper.amelia.nok = est.g.mi + 1.96 * est.g.se,
-                          Bgy.ci.lower.amelia.nok = est.g.mi - 1.96 * est.g.se
-                          ))
+    ## amelia.out.nok <- amelia(df, m=200, p2s=0, idvars=c("x","w_pred"), noms=noms)
+    ## mod.amelia.nok <- zelig(y~x.obs+g, model='ls', data=amelia.out.nok$imputations, cite=FALSE)
+    ## (coefse <- combine_coef_se(mod.amelia.nok, messages=FALSE))
+
+    ## est.x.mi <- coefse['x.obs','Estimate']
+    ## est.x.se <- coefse['x.obs','Std.Error']
+    ## result <- append(result,
+    ##                  list(Bxy.est.amelia.nok = est.x.mi,
+    ##                       Bxy.ci.upper.amelia.nok = est.x.mi + 1.96 * est.x.se,
+    ##                       Bxy.ci.lower.amelia.nok = est.x.mi - 1.96 * est.x.se
+    ##                       ))
+
+    ## est.g.mi <- coefse['g','Estimate']
+    ## est.g.se <- coefse['g','Std.Error']
+
+    ## result <- append(result,
+    ##                  list(Bgy.est.amelia.nok = est.g.mi,
+    ##                       Bgy.ci.upper.amelia.nok = est.g.mi + 1.96 * est.g.se,
+    ##                       Bgy.ci.lower.amelia.nok = est.g.mi - 1.96 * est.g.se
+    ##                       ))
 
     N <- nrow(df)
     m <- nrow(df[!is.na(x.obs)])
@@ -148,8 +148,8 @@ run_simulation <-  function(df, result){
                      )
 
 ##    clean up memory
-    rm(list=c("df","y","x","g","w","v","train","p","amelia.out.k","amelia.out.nok", "mod.calibrated.mle","gmm.res","mod.amelia.k","mod.amelia.nok", "model.true","model.naive","model.feasible"))
+##    rm(list=c("df","y","x","g","w","v","train","p","amelia.out.k","amelia.out.nok", "mod.calibrated.mle","gmm.res","mod.amelia.k","mod.amelia.nok", "model.true","model.naive","model.feasible"))
     
-    gc()
+##    gc()
     return(result)
 }