simulations/simulation_base.R

   1 library(predictionError)
   2 library(mecor)
   3 options(amelia.parallel="no",
   4         amelia.ncpus=1)
   5 library(Amelia)
   6 library(Zelig)
   7 library(bbmle)
   8 library(matrixStats) # for numerically stable logsumexps
   9
  10 source("measerr_methods.R") ## for my more generic function.
  11
  12 ## This uses the pseudolikelihood approach from Carroll page 349.
  13 ## assumes MAR
  14 ## assumes differential error, but that only depends on Y
  15 ## inefficient, because pseudolikelihood
  16
  17 ## This uses the pseudo-likelihood approach from Carroll page 346.
  18 my.pseudo.mle <- function(df){
  19     p1.est <- mean(df[w_pred==1]$y.obs==1,na.rm=T)
  20     p0.est <- mean(df[w_pred==0]$y.obs==0,na.rm=T)
  21
  22     nll <- function(B0, Bxy, Bzy){
  23
  24         pw <- vector(mode='numeric',length=nrow(df))
  25         dfw1 <- df[w_pred==1]
  26         dfw0 <- df[w_pred==0]
  27         pw[df$w_pred==1] <- plogis(B0 + Bxy * dfw1$x + Bzy * dfw1$z, log=T)
  28         pw[df$w_pred==0] <- plogis(B0 + Bxy * dfw0$x + Bzy * dfw0$z, lower.tail=FALSE, log=T)
  29
  30         probs <- colLogSumExps(rbind(log(1 - p0.est), log(p1.est + p0.est - 1) + pw))
  31         return(-1*sum(probs))
  32     }
  33
  34     mlefit <- mle2(minuslogl = nll, start = list(B0=0.0, Bxy=0.0, Bzy=0.0), control=list(maxit=1e6),method='L-BFGS-B')
  35     return(mlefit)
  36
  37 }
  38
  39
  40 ## model from Zhang's arxiv paper, with predictions for y
  41 ## Zhang got this model from Hausman 1998
  42 ### I think this is actually eqivalent to the pseudo.mle method
  43 zhang.mle.iv <- function(df){
  44     nll <- function(B0=0, Bxy=0, Bzy=0, sigma_y=0.1, ppv=0.9, npv=0.9){
  45     df.obs <- df[!is.na(x.obs)]
  46     df.unobs <- df[is.na(x.obs)]
  47
  48     ## fpr = 1 - TNR
  49     ### Problem: accounting for uncertainty in ppv / npv
  50
  51     ll.w1x1.obs <- with(df.obs[(w_pred==1)], dbinom(x.obs,size=1,prob=ppv,log=T))
  52     ll.w0x0.obs <- with(df.obs[(w_pred==0)], dbinom(1-x.obs,size=1,prob=npv,log=T))
  53
  54     ## fnr = 1 - TPR
  55     ll.y.obs <- with(df.obs, dnorm(y, B0 + Bxy * x + Bzy * z, sd=sigma_y,log=T))
  56     ll <- sum(ll.y.obs)
  57     ll <- ll + sum(ll.w1x1.obs) + sum(ll.w0x0.obs)
  58
  59     # unobserved case; integrate out x
  60     ll.x.1 <- with(df.unobs, dnorm(y, B0 + Bxy + Bzy * z, sd = sigma_y, log=T))
  61     ll.x.0 <- with(df.unobs, dnorm(y, B0 + Bzy * z, sd = sigma_y,log=T))
  62
  63     ## case x == 1
  64     lls.x.1 <- colLogSumExps(rbind(log(ppv) + ll.x.1, log(1-ppv) + ll.x.0))
  65
  66     ## case x == 0
  67     lls.x.0 <- colLogSumExps(rbind(log(1-npv) + ll.x.1, log(npv) + ll.x.0))
  68
  69     lls <- colLogSumExps(rbind(lls.x.1, lls.x.0))
  70     ll <- ll + sum(lls)
  71     return(-ll)
  72     }
  73     mlefit <- mle2(minuslogl = nll, control=list(maxit=1e6), lower=list(sigma_y=0.0001, B0=-Inf, Bxy=-Inf, Bzy=-Inf,ppv=0.00001, npv=0.00001),
  74                    upper=list(sigma_y=Inf, B0=Inf, Bxy=Inf, Bzy=Inf, ppv=0.99999,npv=0.99999),method='L-BFGS-B')
  75     return(mlefit)
  76 }
  77
  78 ## this is equivalent to the pseudo-liklihood model from Carolla
  79 zhang.mle.dv <- function(df){
  80
  81     nll <- function(B0=0, Bxy=0, Bzy=0, ppv=0.9, npv=0.9){
  82     df.obs <- df[!is.na(y.obs)]
  83
  84     ## fpr = 1 - TNR
  85     ll.w0y0 <- with(df.obs[y.obs==0],dbinom(1-w_pred,1,npv,log=TRUE))
  86     ll.w1y1 <- with(df.obs[y.obs==1],dbinom(w_pred,1,ppv,log=TRUE))
  87
  88     # observed case
  89     ll.y.obs <- vector(mode='numeric', length=nrow(df.obs))
  90     ll.y.obs[df.obs$y.obs==1] <- with(df.obs[y.obs==1], plogis(B0 + Bxy * x + Bzy * z,log=T))
  91     ll.y.obs[df.obs$y.obs==0] <- with(df.obs[y.obs==0], plogis(B0 + Bxy * x + Bzy * z,log=T,lower.tail=FALSE))
  92
  93     ll <- sum(ll.y.obs) + sum(ll.w0y0) + sum(ll.w1y1)
  94
  95     # unobserved case; integrate out y
  96     ## case y = 1
  97     ll.y.1 <- vector(mode='numeric', length=nrow(df))
  98     pi.y.1 <- with(df,plogis(B0 + Bxy * x + Bzy*z, log=T))
  99     ## P(w=1| y=1)P(y=1) + P(w=0|y=1)P(y=1) = P(w=1,y=1) + P(w=0,y=1)
 100     lls.y.1 <- colLogSumExps(rbind(log(ppv) + pi.y.1, log(1-ppv) + pi.y.1))
 101
 102     ## case y = 0
 103     ll.y.0 <- vector(mode='numeric', length=nrow(df))
 104     pi.y.0 <- with(df,plogis(B0 + Bxy * x + Bzy*z, log=T,lower.tail=FALSE))
 105
 106     ## P(w=1 | y=0)P(y=0) + P(w=0|y=0)P(y=0) = P(w=1,y=0) + P(w=0,y=0)
 107     lls.y.0 <- colLogSumExps(rbind(log(npv) + pi.y.0, log(1-npv) + pi.y.0))
 108
 109     lls <- colLogSumExps(rbind(lls.y.1, lls.y.0))
 110     ll <- ll + sum(lls)
 111     return(-ll)
 112     }
 113     mlefit <- mle2(minuslogl = nll, control=list(maxit=1e6),method='L-BFGS-B',lower=list(B0=-Inf, Bxy=-Inf, Bzy=-Inf, ppv=0.001,npv=0.001),
 114                    upper=list(B0=Inf, Bxy=Inf, Bzy=Inf,ppv=0.999,npv=0.999))
 115     return(mlefit)
 116 }
 117
 118 ## This uses the likelihood approach from Carroll page 353.
 119 ## assumes that we have a good measurement error model
 120 my.mle <- function(df){
 121
 122     ## liklihood for observed responses
 123     nll <- function(B0, Bxy, Bzy, gamma0, gamma_y, gamma_z, gamma_yz){
 124         df.obs <- df[!is.na(y.obs)]
 125         yobs0 <- df.obs$y==0
 126         yobs1 <- df.obs$y==1
 127         p.y.obs <- vector(mode='numeric', length=nrow(df.obs))
 128
 129         p.y.obs[yobs1] <- plogis(B0 + Bxy * df.obs[yobs1]$x + Bzy*df.obs[yobs1]$z,log=T)
 130         p.y.obs[yobs0] <- plogis(B0 + Bxy * df.obs[yobs0]$x + Bzy*df.obs[yobs0]$z,lower.tail=FALSE,log=T)
 131
 132         wobs0 <- df.obs$w_pred==0
 133         wobs1 <- df.obs$w_pred==1
 134         p.w.obs <- vector(mode='numeric', length=nrow(df.obs))
 135
 136         p.w.obs[wobs1] <- plogis(gamma0 + gamma_y * df.obs[wobs1]$y + gamma_z*df.obs[wobs1]$z + df.obs[wobs1]$z*df.obs[wobs1]$y* gamma_yz, log=T)
 137         p.w.obs[wobs0] <- plogis(gamma0 + gamma_y * df.obs[wobs0]$y + gamma_z*df.obs[wobs0]$z + df.obs[wobs0]$z*df.obs[wobs0]$y* gamma_yz, lower.tail=FALSE, log=T)
 138
 139         p.obs <- p.w.obs + p.y.obs
 140
 141         df.unobs <- df[is.na(y.obs)]
 142
 143         p.unobs.0 <- vector(mode='numeric',length=nrow(df.unobs))
 144         p.unobs.1 <- vector(mode='numeric',length=nrow(df.unobs))
 145
 146         wunobs.0 <- df.unobs$w_pred == 0
 147         wunobs.1 <- df.unobs$w_pred == 1
 148
 149         p.unobs.0[wunobs.1] <- plogis(B0 + Bxy * df.unobs[wunobs.1]$x + Bzy*df.unobs[wunobs.1]$z, log=T) + plogis(gamma0 + gamma_y + gamma_z*df.unobs[wunobs.1]$z + df.unobs[wunobs.1]$z*gamma_yz, log=T)
 150
 151         p.unobs.0[wunobs.0] <- plogis(B0 + Bxy * df.unobs[wunobs.0]$x + Bzy*df.unobs[wunobs.0]$z, log=T) + plogis(gamma0 + gamma_y + gamma_z*df.unobs[wunobs.0]$z + df.unobs[wunobs.0]$z*gamma_yz, lower.tail=FALSE, log=T)
 152
 153         p.unobs.1[wunobs.1] <- plogis(B0 + Bxy * df.unobs[wunobs.1]$x + Bzy*df.unobs[wunobs.1]$z, log=T, lower.tail=FALSE) + plogis(gamma0 + gamma_z*df.unobs[wunobs.1]$z, log=T)
 154
 155         p.unobs.1[wunobs.0] <- plogis(B0 + Bxy * df.unobs[wunobs.0]$x + Bzy*df.unobs[wunobs.0]$z, log=T, lower.tail=FALSE) + plogis(gamma0 + gamma_z*df.unobs[wunobs.0]$z, lower.tail=FALSE, log=T)
 156
 157         p.unobs <- colLogSumExps(rbind(p.unobs.1, p.unobs.0))
 158
 159         p <- c(p.obs, p.unobs)
 160
 161         return(-1*(sum(p)))
 162     }
 163
 164     mlefit <- mle2(minuslogl = nll, start = list(B0=0, Bxy=0,Bzy=0, gamma0=0, gamma_y=0, gamma_z=0, gamma_yz=0), control=list(maxit=1e6),method='L-BFGS-B')
 165
 166     return(mlefit)
 167 }
 168
 169 run_simulation_depvar <- function(df, result, outcome_formula=y~x+z, proxy_formula=w_pred~y){
 170
 171     accuracy <- df[,mean(w_pred==y)]
 172     result <- append(result, list(accuracy=accuracy))
 173
 174     (model.true <- glm(y ~ x + z, data=df,family=binomial(link='logit')))
 175     true.ci.Bxy <- confint(model.true)['x',]
 176     true.ci.Bzy <- confint(model.true)['z',]
 177
 178     result <- append(result, list(Bxy.est.true=coef(model.true)['x'],
 179                                   Bzy.est.true=coef(model.true)['z'],
 180                                   Bxy.ci.upper.true = true.ci.Bxy[2],
 181                                   Bxy.ci.lower.true = true.ci.Bxy[1],
 182                                   Bzy.ci.upper.true = true.ci.Bzy[2],
 183                                   Bzy.ci.lower.true = true.ci.Bzy[1]))
 184
 185     (model.feasible <- glm(y.obs~x+z,data=df,family=binomial(link='logit')))
 186
 187     feasible.ci.Bxy <- confint(model.feasible)['x',]
 188     result <- append(result, list(Bxy.est.feasible=coef(model.feasible)['x'],
 189                                   Bxy.ci.upper.feasible = feasible.ci.Bxy[2],
 190                                   Bxy.ci.lower.feasible = feasible.ci.Bxy[1]))
 191
 192     feasible.ci.Bzy <- confint(model.feasible)['z',]
 193     result <- append(result, list(Bzy.est.feasible=coef(model.feasible)['z'],
 194                                   Bzy.ci.upper.feasible = feasible.ci.Bzy[2],
 195                                   Bzy.ci.lower.feasible = feasible.ci.Bzy[1]))
 196
 197     (model.naive <- glm(w_pred~x+z, data=df, family=binomial(link='logit')))
 198
 199     naive.ci.Bxy <- confint(model.naive)['x',]
 200     naive.ci.Bzy <- confint(model.naive)['z',]
 201
 202     result <- append(result, list(Bxy.est.naive=coef(model.naive)['x'],
 203                                   Bzy.est.naive=coef(model.naive)['z'],
 204                                   Bxy.ci.upper.naive = naive.ci.Bxy[2],
 205                                   Bxy.ci.lower.naive = naive.ci.Bxy[1],
 206                                   Bzy.ci.upper.naive = naive.ci.Bzy[2],
 207                                   Bzy.ci.lower.naive = naive.ci.Bzy[1]))
 208
 209
 210     (model.naive.cont <- lm(w~x+z, data=df))
 211     naivecont.ci.Bxy <- confint(model.naive.cont)['x',]
 212     naivecont.ci.Bzy <- confint(model.naive.cont)['z',]
 213
 214     ## my implementatoin of liklihood based correction
 215
 216     temp.df <- copy(df)
 217     temp.df[,y:=y.obs]
 218     mod.caroll.lik <- measerr_mle_dv(temp.df, outcome_formula=outcome_formula, proxy_formula=proxy_formula)
 219     fisher.info <- solve(mod.caroll.lik$hessian)
 220     coef <- mod.caroll.lik$par
 221     ci.upper <- coef + sqrt(diag(fisher.info)) * 1.96
 222     ci.lower <- coef - sqrt(diag(fisher.info)) * 1.96
 223     result <- append(result,
 224                      list(Bxy.est.mle = coef['x'],
 225                           Bxy.ci.upper.mle = ci.upper['x'],
 226                           Bxy.ci.lower.mle = ci.lower['x'],
 227                           Bzy.est.mle = coef['z'],
 228                           Bzy.ci.upper.mle = ci.upper['z'],
 229                           Bzy.ci.lower.mle = ci.lower['z']))
 230
 231
 232     ## my implementatoin of liklihood based correction
 233     mod.zhang <- zhang.mle.dv(df)
 234     coef <- coef(mod.zhang)
 235     ci <- confint(mod.zhang,method='quad')
 236
 237     result <- append(result,
 238                      list(Bxy.est.zhang = coef['Bxy'],
 239                           Bxy.ci.upper.zhang = ci['Bxy','97.5 %'],
 240                           Bxy.ci.lower.zhang = ci['Bxy','2.5 %'],
 241                           Bzy.est.zhang = coef['Bzy'],
 242                           Bzy.ci.upper.zhang = ci['Bzy','97.5 %'],
 243                           Bzy.ci.lower.zhang = ci['Bzy','2.5 %']))
 244
 245
 246     # amelia says use normal distribution for binary variables.
 247     tryCatch({
 248         amelia.out.k <- amelia(df, m=200, p2s=0, idvars=c('y','ystar','w'))
 249         mod.amelia.k <- zelig(y.obs~x+z, model='ls', data=amelia.out.k$imputations, cite=FALSE)
 250         (coefse <- combine_coef_se(mod.amelia.k, messages=FALSE))
 251         est.x.mi <- coefse['x','Estimate']
 252         est.x.se <- coefse['x','Std.Error']
 253         result <- append(result,
 254                          list(Bxy.est.amelia.full = est.x.mi,
 255                               Bxy.ci.upper.amelia.full = est.x.mi + 1.96 * est.x.se,
 256                               Bxy.ci.lower.amelia.full = est.x.mi - 1.96 * est.x.se
 257                               ))
 258
 259         est.z.mi <- coefse['z','Estimate']
 260         est.z.se <- coefse['z','Std.Error']
 261
 262         result <- append(result,
 263                          list(Bzy.est.amelia.full = est.z.mi,
 264                               Bzy.ci.upper.amelia.full = est.z.mi + 1.96 * est.z.se,
 265                               Bzy.ci.lower.amelia.full = est.z.mi - 1.96 * est.z.se
 266                               ))
 267
 268     },
 269     error = function(e){
 270         message("An error occurred:\n",e)
 271         result$error <- paste0(result$error,'\n', e)
 272     })
 273
 274
 275     return(result)
 276
 277 }
 278
 279
 280 ## outcome_formula, proxy_formula, and truth_formula are passed to measerr_mle
 281 run_simulation <-  function(df, result, outcome_formula=y~x+z, proxy_formula=w_pred~x, truth_formula=x~z){
 282
 283     accuracy <- df[,mean(w_pred==x)]
 284     result <- append(result, list(accuracy=accuracy))
 285
 286     (model.true <- lm(y ~ x + z, data=df))
 287     true.ci.Bxy <- confint(model.true)['x',]
 288     true.ci.Bzy <- confint(model.true)['z',]
 289
 290     result <- append(result, list(Bxy.est.true=coef(model.true)['x'],
 291                                   Bzy.est.true=coef(model.true)['z'],
 292                                   Bxy.ci.upper.true = true.ci.Bxy[2],
 293                                   Bxy.ci.lower.true = true.ci.Bxy[1],
 294                                   Bzy.ci.upper.true = true.ci.Bzy[2],
 295                                   Bzy.ci.lower.true = true.ci.Bzy[1]))
 296
 297     (model.feasible <- lm(y~x.obs+z,data=df))
 298
 299     feasible.ci.Bxy <- confint(model.feasible)['x.obs',]
 300     result <- append(result, list(Bxy.est.feasible=coef(model.feasible)['x.obs'],
 301                                   Bxy.ci.upper.feasible = feasible.ci.Bxy[2],
 302                                   Bxy.ci.lower.feasible = feasible.ci.Bxy[1]))
 303
 304     feasible.ci.Bzy <- confint(model.feasible)['z',]
 305     result <- append(result, list(Bzy.est.feasible=coef(model.feasible)['z'],
 306                                   Bzy.ci.upper.feasible = feasible.ci.Bzy[2],
 307                                   Bzy.ci.lower.feasible = feasible.ci.Bzy[1]))
 308
 309     (model.naive <- lm(y~w_pred+z, data=df))
 310
 311     naive.ci.Bxy <- confint(model.naive)['w_pred',]
 312     naive.ci.Bzy <- confint(model.naive)['z',]
 313
 314     result <- append(result, list(Bxy.est.naive=coef(model.naive)['w_pred'],
 315                                   Bzy.est.naive=coef(model.naive)['z'],
 316                                   Bxy.ci.upper.naive = naive.ci.Bxy[2],
 317                                   Bxy.ci.lower.naive = naive.ci.Bxy[1],
 318                                   Bzy.ci.upper.naive = naive.ci.Bzy[2],
 319                                   Bzy.ci.lower.naive = naive.ci.Bzy[1]))
 320
 321
 322     tryCatch({
 323     amelia.out.k <- amelia(df, m=200, p2s=0, idvars=c('x','w_pred'))
 324     mod.amelia.k <- zelig(y~x.obs+z, model='ls', data=amelia.out.k$imputations, cite=FALSE)
 325     (coefse <- combine_coef_se(mod.amelia.k, messages=FALSE))
 326
 327     est.x.mi <- coefse['x.obs','Estimate']
 328     est.x.se <- coefse['x.obs','Std.Error']
 329     result <- append(result,
 330                      list(Bxy.est.amelia.full = est.x.mi,
 331                           Bxy.ci.upper.amelia.full = est.x.mi + 1.96 * est.x.se,
 332                           Bxy.ci.lower.amelia.full = est.x.mi - 1.96 * est.x.se
 333                           ))
 334
 335     est.z.mi <- coefse['z','Estimate']
 336     est.z.se <- coefse['z','Std.Error']
 337
 338     result <- append(result,
 339                      list(Bzy.est.amelia.full = est.z.mi,
 340                           Bzy.ci.upper.amelia.full = est.z.mi + 1.96 * est.z.se,
 341                           Bzy.ci.lower.amelia.full = est.z.mi - 1.96 * est.z.se
 342                           ))
 343
 344     },
 345     error = function(e){
 346         message("An error occurred:\n",e)
 347         result$error <-paste0(result$error,'\n', e)
 348     }
 349     )
 350
 351     tryCatch({
 352         temp.df <- copy(df)
 353         temp.df <- temp.df[,x:=x.obs]
 354         mod.caroll.lik <- measerr_mle(temp.df, outcome_formula=outcome_formula, proxy_formula=proxy_formula, truth_formula=truth_formula)
 355         fisher.info <- solve(mod.caroll.lik$hessian)
 356         coef <- mod.caroll.lik$par
 357         ci.upper <- coef + sqrt(diag(fisher.info)) * 1.96
 358         ci.lower <- coef - sqrt(diag(fisher.info)) * 1.96
 359
 360
 361         result <- append(result,
 362                          list(Bxy.est.mle = coef['x'],
 363                               Bxy.ci.upper.mle = ci.upper['x'],
 364                               Bxy.ci.lower.mle = ci.lower['x'],
 365                               Bzy.est.mle = coef['z'],
 366                               Bzy.ci.upper.mle = ci.upper['z'],
 367                               Bzy.ci.lower.mle = ci.lower['z']))
 368     },
 369
 370     error = function(e){
 371         message("An error occurred:\n",e)
 372         result$error <- paste0(result$error,'\n', e)
 373     })
 374
 375     tryCatch({
 376
 377         mod.zhang.lik <- zhang.mle.iv(df)
 378         coef <- coef(mod.zhang.lik)
 379         ci <- confint(mod.zhang.lik,method='quad')
 380         result <- append(result,
 381                          list(Bxy.est.zhang = coef['Bxy'],
 382                               Bxy.ci.upper.zhang = ci['Bxy','97.5 %'],
 383                               Bxy.ci.lower.zhang = ci['Bxy','2.5 %'],
 384                               Bzy.est.zhang = coef['Bzy'],
 385                               Bzy.ci.upper.zhang = ci['Bzy','97.5 %'],
 386                               Bzy.ci.lower.zhang = ci['Bzy','2.5 %']))
 387     },
 388
 389     error = function(e){
 390         message("An error occurred:\n",e)
 391         result$error <- paste0(result$error,'\n', e)
 392     })
 393
 394     ## What if we can't observe k -- most realistic scenario. We can't include all the ML features in a model.
 395     ## amelia.out.nok <- amelia(df, m=200, p2s=0, idvars=c("x","w_pred"), noms=noms)
 396     ## mod.amelia.nok <- zelig(y~x.obs+g, model='ls', data=amelia.out.nok$imputations, cite=FALSE)
 397     ## (coefse <- combine_coef_se(mod.amelia.nok, messages=FALSE))
 398
 399     ## est.x.mi <- coefse['x.obs','Estimate']
 400     ## est.x.se <- coefse['x.obs','Std.Error']
 401     ## result <- append(result,
 402     ##                  list(Bxy.est.amelia.nok = est.x.mi,
 403     ##                       Bxy.ci.upper.amelia.nok = est.x.mi + 1.96 * est.x.se,
 404     ##                       Bxy.ci.lower.amelia.nok = est.x.mi - 1.96 * est.x.se
 405     ##                       ))
 406
 407     ## est.g.mi <- coefse['g','Estimate']
 408     ## est.g.se <- coefse['g','Std.Error']
 409
 410     ## result <- append(result,
 411     ##                  list(Bgy.est.amelia.nok = est.g.mi,
 412     ##                       Bgy.ci.upper.amelia.nok = est.g.mi + 1.96 * est.g.se,
 413     ##                       Bgy.ci.lower.amelia.nok = est.g.mi - 1.96 * est.g.se
 414     ##                       ))
 415
 416     N <- nrow(df)
 417     m <- nrow(df[!is.na(x.obs)])
 418     p <- v <- train <- rep(0,N)
 419     M <- m
 420     p[(M+1):(N)] <- 1
 421     v[1:(M)] <- 1
 422     df <- df[order(x.obs)]
 423     y <- df[,y]
 424     x <- df[,x.obs]
 425     z <- df[,z]
 426     w <- df[,w_pred]
 427     # gmm gets pretty close
 428     (gmm.res <- predicted_covariates(y, x, z, w, v, train, p, max_iter=100, verbose=TRUE))
 429
 430     result <- append(result,
 431                      list(Bxy.est.gmm = gmm.res$beta[1,1],
 432                           Bxy.ci.upper.gmm = gmm.res$confint[1,2],
 433                           Bxy.ci.lower.gmm = gmm.res$confint[1,1],
 434                           gmm.ER_pval = gmm.res$ER_pval
 435                           ))
 436
 437     result <- append(result,
 438                      list(Bzy.est.gmm = gmm.res$beta[2,1],
 439                           Bzy.ci.upper.gmm = gmm.res$confint[2,2],
 440                           Bzy.ci.lower.gmm = gmm.res$confint[2,1]))
 441
 442
 443     tryCatch({
 444     mod.calibrated.mle <- mecor(y ~ MeasError(w_pred, reference = x.obs) + z, df, B=400, method='efficient')
 445     (mod.calibrated.mle)
 446     (mecor.ci <- summary(mod.calibrated.mle)$c$ci['x.obs',])
 447     result <- append(result, list(
 448                                  Bxy.est.mecor = mecor.ci['Estimate'],
 449                                  Bxy.ci.upper.mecor = mecor.ci['UCI'],
 450                                  Bxy.ci.lower.mecor = mecor.ci['LCI'])
 451                      )
 452
 453     (mecor.ci <- summary(mod.calibrated.mle)$c$ci['z',])
 454
 455     result <- append(result, list(
 456                                  Bzy.est.mecor = mecor.ci['Estimate'],
 457                                  Bzy.ci.upper.mecor = mecor.ci['UCI'],
 458                                  Bzy.ci.lower.mecor = mecor.ci['LCI'])
 459                      )
 460     },
 461     error = function(e){
 462         message("An error occurred:\n",e)
 463         result$error <- paste0(result$error, '\n', e)
 464     }
 465     )
 466 ##    clean up memory
 467 ##    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"))
 468
 469 ##    gc()
 470     return(result)
 471 }