simulations/measerr_methods.R

   1 library(formula.tools)
   2 library(matrixStats)
   3
   4 ## df: dataframe to model
   5 ## outcome_formula: formula for y | x, z
   6 ## outcome_family: family for y | x, z
   7 ## proxy_formula: formula for w | x, z, y
   8 ## proxy_family: family for w | x, z, y
   9 ## truth_formula: formula for x | z
  10 ## truth_family: family for x | z
  11
  12 ### ideal formulas for example 1
  13 # test.fit.1 <- measerr_mle(df, y ~ x + z, gaussian(), w_pred ~ x, binomial(link='logit'), x ~ z)
  14
  15 ### ideal formulas for example 2
  16 # test.fit.2 <- measerr_mle(df, y ~ x + z, gaussian(), w_pred ~ x + z + y + y:x, binomial(link='logit'), x ~ z)
  17
  18
  19 ## outcome_formula <- y ~ x + z; proxy_formula <- w_pred ~ y + x + z + x:z + x:y + z:y
  20 measerr_mle_dv <- function(df, outcome_formula, outcome_family=binomial(link='logit'), proxy_formula, proxy_family=binomial(link='logit')){
  21
  22     nll <- function(params){
  23         df.obs <- model.frame(outcome_formula, df)
  24         proxy.variable <- all.vars(proxy_formula)[1]
  25         proxy.model.matrix <- model.matrix(proxy_formula, df)
  26         response.var <- all.vars(outcome_formula)[1]
  27         y.obs <- with(df.obs,eval(parse(text=response.var)))
  28         outcome.model.matrix <- model.matrix(outcome_formula, df.obs)
  29
  30         param.idx <- 1
  31         n.outcome.model.covars <- dim(outcome.model.matrix)[2]
  32         outcome.params <- params[param.idx:n.outcome.model.covars]
  33         param.idx <- param.idx + n.outcome.model.covars
  34
  35         if((outcome_family$family == "binomial") & (outcome_family$link == 'logit')){
  36             ll.y.obs <- vector(mode='numeric', length=length(y.obs))
  37             ll.y.obs[y.obs==1] <- plogis(outcome.params %*% t(outcome.model.matrix[y.obs==1,]),log=TRUE)
  38             ll.y.obs[y.obs==0] <- plogis(outcome.params %*% t(outcome.model.matrix[y.obs==0,]),log=TRUE,lower.tail=FALSE)
  39         }
  40
  41         df.obs <- model.frame(proxy_formula,df)
  42         n.proxy.model.covars <- dim(proxy.model.matrix)[2]
  43         proxy.params <- params[param.idx:(n.proxy.model.covars+param.idx-1)]
  44
  45         param.idx <- param.idx + n.proxy.model.covars
  46         proxy.obs <- with(df.obs, eval(parse(text=proxy.variable)))
  47
  48         if( (proxy_family$family=="binomial") & (proxy_family$link=='logit')){
  49             ll.w.obs <- vector(mode='numeric',length=dim(proxy.model.matrix)[1])
  50             ll.w.obs[proxy.obs==1] <- plogis(proxy.params %*% t(proxy.model.matrix[proxy.obs==1,]),log=TRUE)
  51             ll.w.obs[proxy.obs==0] <- plogis(proxy.params %*% t(proxy.model.matrix[proxy.obs==0,]),log=TRUE, lower.tail=FALSE)
  52         }
  53
  54         ll.obs <- sum(ll.y.obs + ll.w.obs)
  55
  56         df.unobs <- df[is.na(df[[response.var]])]
  57         df.unobs.y1 <- copy(df.unobs)
  58         df.unobs.y1[[response.var]] <- 1
  59         df.unobs.y0 <- copy(df.unobs)
  60         df.unobs.y0[[response.var]] <- 0
  61
  62         ## integrate out y
  63         outcome.model.matrix.y1 <- model.matrix(outcome_formula, df.unobs.y1)
  64
  65         if((outcome_family$family == "binomial") & (outcome_family$link == 'logit')){
  66             ll.y.unobs.1 <- vector(mode='numeric', length=dim(outcome.model.matrix.y1)[1])
  67             ll.y.unobs.0 <- vector(mode='numeric', length=dim(outcome.model.matrix.y1)[1])
  68             ll.y.unobs.1 <- plogis(outcome.params %*% t(outcome.model.matrix.y1),log=TRUE)
  69             ll.y.unobs.0 <- plogis(outcome.params %*% t(outcome.model.matrix.y1),log=TRUE,lower.tail=FALSE)
  70         }
  71
  72         proxy.model.matrix.y1 <- model.matrix(proxy_formula, df.unobs.y1)
  73         proxy.model.matrix.y0 <- model.matrix(proxy_formula, df.unobs.y0)
  74         proxy.unobs <- with(df.unobs, eval(parse(text=proxy.variable)))
  75
  76         if( (proxy_family$family=="binomial") & (proxy_family$link=='logit')){
  77             ll.w.unobs.1 <- vector(mode='numeric',length=dim(proxy.model.matrix.y1)[1])
  78             ll.w.unobs.0 <- vector(mode='numeric',length=dim(proxy.model.matrix.y0)[1])
  79             ll.w.unobs.1[proxy.unobs==1] <- plogis(proxy.params %*% t(proxy.model.matrix.y1[proxy.unobs==1,]),log=TRUE)
  80             ll.w.unobs.1[proxy.unobs==0] <- plogis(proxy.params %*% t(proxy.model.matrix.y1[proxy.unobs==0,]),log=TRUE, lower.tail=FALSE)
  81
  82             ll.w.unobs.0[proxy.unobs==1] <- plogis(proxy.params %*% t(proxy.model.matrix.y0[proxy.unobs==1,]),log=TRUE)
  83             ll.w.unobs.0[proxy.unobs==0] <- plogis(proxy.params %*% t(proxy.model.matrix.y0[proxy.unobs==0,]),log=TRUE, lower.tail=FALSE)
  84         }
  85
  86         ll.unobs.1 <- ll.y.unobs.1 + ll.w.unobs.1
  87         ll.unobs.0 <- ll.y.unobs.0 + ll.w.unobs.0
  88         ll.unobs <- sum(colLogSumExps(rbind(ll.unobs.1,ll.unobs.0)))
  89         ll <- ll.unobs + ll.obs
  90         return(-ll)
  91     }
  92
  93     params <- colnames(model.matrix(outcome_formula,df))
  94     lower <- rep(-Inf, length(params))
  95     proxy.params <- colnames(model.matrix(proxy_formula, df))
  96     params <- c(params, paste0('proxy_',proxy.params))
  97     lower <- c(lower, rep(-Inf, length(proxy.params)))
  98     start <- rep(0.1,length(params))
  99     names(start) <- params
 100
 101     fit <- optim(start, fn = nll, lower=lower, method='L-BFGS-B', hessian=TRUE, control=list(maxit=1e6))
 102     return(fit)
 103 }
 104
 105 ## Experimental, and not necessary if errors are independent.
 106 measerr_irr_mle <- function(df, outcome_formula, outcome_family=gaussian(), rater_formula, proxy_formula, proxy_family=binomial(link='logit'), truth_formula, truth_family=binomial(link='logit')){
 107
 108     ### in this scenario, the ground truth also has measurement error, but we have repeated measures for it.
 109
 110     ## probability of y given observed data.
 111     df.obs <- df[!is.na(x.obs.1)]
 112     proxy.variable <- all.vars(proxy_formula)[1]
 113     df.x.obs.1 <- copy(df.obs)[,x:=1]
 114     df.x.obs.0 <- copy(df.obs)[,x:=0]
 115     y.obs <- df.obs[,y]
 116
 117     nll <- function(params){
 118         outcome.model.matrix.x.obs.0 <- model.matrix(outcome_formula, df.x.obs.0)
 119         outcome.model.matrix.x.obs.1 <- model.matrix(outcome_formula, df.x.obs.1)
 120
 121         param.idx <- 1
 122         n.outcome.model.covars <- dim(outcome.model.matrix.x.obs.0)[2]
 123         outcome.params <- params[param.idx:n.outcome.model.covars]
 124         param.idx <- param.idx + n.outcome.model.covars
 125
 126         sigma.y <- params[param.idx]
 127         param.idx <- param.idx + 1
 128
 129         ll.y.x.obs.0 <- dnorm(y.obs, outcome.params %*% t(outcome.model.matrix.x.obs.0),sd=sigma.y, log=TRUE)
 130         ll.y.x.obs.1 <- dnorm(y.obs, outcome.params %*% t(outcome.model.matrix.x.obs.1),sd=sigma.y, log=TRUE)
 131
 132         ## assume that the two coders are statistically independent conditional on x
 133         ll.x.obs.0.x0 <- vector(mode='numeric', length=nrow(df.obs))
 134         ll.x.obs.1.x0 <- vector(mode='numeric', length=nrow(df.obs))
 135         ll.x.obs.0.x1 <- vector(mode='numeric', length=nrow(df.obs))
 136         ll.x.obs.1.x1 <- vector(mode='numeric', length=nrow(df.obs))
 137
 138         rater.model.matrix.x.obs.0 <- model.matrix(rater_formula, df.x.obs.0)
 139         rater.model.matrix.x.obs.1 <- model.matrix(rater_formula, df.x.obs.1)
 140
 141         n.rater.model.covars <- dim(rater.model.matrix.x.obs.0)[2]
 142         rater.0.params <- params[param.idx:(n.rater.model.covars + param.idx - 1)]
 143         param.idx <- param.idx + n.rater.model.covars
 144
 145         rater.1.params <- params[param.idx:(n.rater.model.covars + param.idx - 1)]
 146         param.idx <- param.idx + n.rater.model.covars
 147
 148         # probability of rater 0 if x is 0 or 1
 149         ll.x.obs.0.x0[df.obs$x.obs.0==1] <- plogis(rater.0.params %*% t(rater.model.matrix.x.obs.0[df.obs$x.obs.0==1,]), log=TRUE)
 150         ll.x.obs.0.x0[df.obs$x.obs.0==0] <- plogis(rater.0.params %*% t(rater.model.matrix.x.obs.0[df.obs$x.obs.0==0,]), log=TRUE, lower.tail=FALSE)
 151         ll.x.obs.0.x1[df.obs$x.obs.0==1] <- plogis(rater.0.params %*% t(rater.model.matrix.x.obs.1[df.obs$x.obs.0==1,]), log=TRUE)
 152         ll.x.obs.0.x1[df.obs$x.obs.0==0] <- plogis(rater.0.params %*% t(rater.model.matrix.x.obs.1[df.obs$x.obs.0==0,]), log=TRUE, lower.tail=FALSE)
 153
 154         # probability of rater 1 if x is 0 or 1
 155         ll.x.obs.1.x0[df.obs$x.obs.1==1] <- plogis(rater.1.params %*% t(rater.model.matrix.x.obs.0[df.obs$x.obs.1==1,]), log=TRUE)
 156         ll.x.obs.1.x0[df.obs$x.obs.1==0] <- plogis(rater.1.params %*% t(rater.model.matrix.x.obs.0[df.obs$x.obs.1==0,]), log=TRUE, lower.tail=FALSE)
 157         ll.x.obs.1.x1[df.obs$x.obs.1==1] <- plogis(rater.1.params %*% t(rater.model.matrix.x.obs.1[df.obs$x.obs.1==1,]), log=TRUE)
 158         ll.x.obs.1.x1[df.obs$x.obs.1==0] <- plogis(rater.1.params %*% t(rater.model.matrix.x.obs.1[df.obs$x.obs.1==0,]), log=TRUE, lower.tail=FALSE)
 159
 160         proxy.model.matrix.x0 <- model.matrix(proxy_formula, df.x.obs.0)
 161         proxy.model.matrix.x1 <- model.matrix(proxy_formula, df.x.obs.1)
 162
 163         n.proxy.model.covars <- dim(proxy.model.matrix.x0)[2]
 164         proxy.params <- params[param.idx:(n.proxy.model.covars+param.idx-1)]
 165         param.idx <- param.idx + n.proxy.model.covars
 166
 167         proxy.obs <- with(df.obs, eval(parse(text=proxy.variable)))
 168
 169         if( (proxy_family$family=="binomial") & (proxy_family$link=='logit')){
 170             ll.w.obs.x0 <- vector(mode='numeric',length=dim(proxy.model.matrix.x0)[1])
 171             ll.w.obs.x1 <- vector(mode='numeric',length=dim(proxy.model.matrix.x1)[1])
 172
 173                                         # proxy_formula likelihood using logistic regression
 174             ll.w.obs.x0[proxy.obs==1] <- plogis(proxy.params %*% t(proxy.model.matrix.x0[proxy.obs==1,]),log=TRUE)
 175             ll.w.obs.x0[proxy.obs==0] <- plogis(proxy.params %*% t(proxy.model.matrix.x0[proxy.obs==0,]),log=TRUE, lower.tail=FALSE)
 176
 177             ll.w.obs.x1[proxy.obs==1] <- plogis(proxy.params %*% t(proxy.model.matrix.x1[proxy.obs==1,]),log=TRUE)
 178             ll.w.obs.x1[proxy.obs==0] <- plogis(proxy.params %*% t(proxy.model.matrix.x1[proxy.obs==0,]),log=TRUE, lower.tail=FALSE)
 179         }
 180
 181         ## assume that the probability of x is a logistic regression depending on z
 182         truth.model.matrix.obs <- model.matrix(truth_formula, df.obs)
 183         n.truth.params <- dim(truth.model.matrix.obs)[2]
 184         truth.params <- params[param.idx:(n.truth.params + param.idx - 1)]
 185
 186         ll.obs.x0 <- plogis(truth.params %*% t(truth.model.matrix.obs), log=TRUE)
 187         ll.obs.x1 <- plogis(truth.params %*% t(truth.model.matrix.obs), log=TRUE, lower.tail=FALSE)
 188
 189         ll.obs <- colLogSumExps(rbind(ll.y.x.obs.0 + ll.x.obs.0.x0 + ll.x.obs.1.x0 + ll.obs.x0 + ll.w.obs.x0,
 190                                       ll.y.x.obs.1 + ll.x.obs.0.x1 + ll.x.obs.1.x1 + ll.obs.x1 + ll.w.obs.x1))
 191
 192         ### NOW FOR THE FUN PART. Likelihood of the unobserved data.
 193         ### we have to integrate out x.obs.0, x.obs.1, and x.
 194
 195
 196         ## THE OUTCOME
 197         df.unobs <- df[is.na(x.obs)]
 198         df.x.unobs.0 <- copy(df.unobs)[,x:=0]
 199         df.x.unobs.1 <- copy(df.unobs)[,x:=1]
 200         y.unobs <- df.unobs$y
 201
 202         outcome.model.matrix.x.unobs.0 <- model.matrix(outcome_formula, df.x.unobs.0)
 203         outcome.model.matrix.x.unobs.1 <- model.matrix(outcome_formula, df.x.unobs.1)
 204
 205         ll.y.unobs.x0 <- dnorm(y.unobs, outcome.params %*% t(outcome.model.matrix.x.unobs.0), sd=sigma.y, log=TRUE)
 206         ll.y.unobs.x1 <- dnorm(y.unobs, outcome.params %*% t(outcome.model.matrix.x.unobs.1), sd=sigma.y, log=TRUE)
 207
 208
 209         ## THE UNLABELED DATA
 210
 211
 212         ## assume that the two coders are statistically independent conditional on x
 213         ll.x.unobs.0.x0 <- vector(mode='numeric', length=nrow(df.unobs))
 214         ll.x.unobs.1.x0 <- vector(mode='numeric', length=nrow(df.unobs))
 215         ll.x.unobs.0.x1 <- vector(mode='numeric', length=nrow(df.unobs))
 216         ll.x.unobs.1.x1 <- vector(mode='numeric', length=nrow(df.unobs))
 217
 218         df.x.unobs.0[,x.obs := 1]
 219         df.x.unobs.1[,x.obs := 1]
 220
 221         rater.model.matrix.x.unobs.0 <- model.matrix(rater_formula, df.x.unobs.0)
 222         rater.model.matrix.x.unobs.1 <- model.matrix(rater_formula, df.x.unobs.1)
 223
 224
 225         ## # probability of rater 0 if x is 0 or 1
 226         ## ll.x.unobs.0.x0 <- colLogSumExps(rbind(plogis(rater.0.params %*% t(rater.model.matrix.x.unobs.0), log=TRUE),
 227         ##                                      plogis(rater.0.params %*% t(rater.model.matrix.x.unobs.0), log=TRUE, lower.tail=TRUE)))
 228
 229         ## ll.x.unobs.0.x1 <- colLogSumExps(rbind(plogis(rater.0.params %*% t(rater.model.matrix.x.unobs.1), log=TRUE),
 230         ##                                        plogis(rater.0.params %*% t(rater.model.matrix.x.unobs.1), log=TRUE, lower.tail=TRUE)))
 231
 232         ## # probability of rater 1 if x is 0 or 1
 233         ## ll.x.unobs.1.x0 <- colLogSumExps(rbind(plogis(rater.1.params %*% t(rater.model.matrix.x.unobs.0), log=TRUE),
 234         ##                                      plogis(rater.1.params %*% t(rater.model.matrix.x.unobs.0), log=TRUE, lower.tail=TRUE)))
 235
 236         ## ll.x.unobs.1.x1 <- colLogSumExps(rbind(plogis(rater.1.params %*% t(rater.model.matrix.x.unobs.1), log=TRUE),
 237         ##                                      plogis(rater.1.params %*% t(rater.model.matrix.x.unobs.1), log=TRUE, lower.tail=TRUE)))
 238
 239
 240         proxy.unobs <- with(df.unobs, eval(parse(text=proxy.variable)))
 241         proxy.model.matrix.x0.unobs <- model.matrix(proxy_formula, df.x.unobs.0)
 242         proxy.model.matrix.x1.unobs <- model.matrix(proxy_formula, df.x.unobs.1)
 243
 244         if( (proxy_family$family=="binomial") & (proxy_family$link=='logit')){
 245             ll.w.unobs.x0 <- vector(mode='numeric',length=dim(proxy.model.matrix.x0)[1])
 246             ll.w.unobs.x1 <- vector(mode='numeric',length=dim(proxy.model.matrix.x1)[1])
 247
 248
 249                                         # proxy_formula likelihood using logistic regression
 250             ll.w.unobs.x0[proxy.unobs==1] <- plogis(proxy.params %*% t(proxy.model.matrix.x0.unobs[proxy.unobs==1,]),log=TRUE)
 251             ll.w.unobs.x0[proxy.unobs==0] <- plogis(proxy.params %*% t(proxy.model.matrix.x0.unobs[proxy.unobs==0,]),log=TRUE, lower.tail=FALSE)
 252
 253             ll.w.unobs.x1[proxy.unobs==1] <- plogis(proxy.params %*% t(proxy.model.matrix.x1.unobs[proxy.unobs==1,]),log=TRUE)
 254             ll.w.unobs.x1[proxy.unobs==0] <- plogis(proxy.params %*% t(proxy.model.matrix.x1.unobs[proxy.unobs==0,]),log=TRUE, lower.tail=FALSE)
 255         }
 256
 257         truth.model.matrix.unobs <- model.matrix(truth_formula, df.unobs)
 258
 259         ll.unobs.x0 <- plogis(truth.params %*% t(truth.model.matrix.unobs), log=TRUE)
 260         ll.unobs.x1 <- plogis(truth.params %*% t(truth.model.matrix.unobs), log=TRUE, lower.tail=FALSE)
 261
 262         ll.unobs <- colLogSumExps(rbind(ll.unobs.x0 + ll.w.unobs.x0 + ll.y.unobs.x0,
 263                                         ll.unobs.x1 + ll.w.unobs.x1 + ll.y.unobs.x1))
 264
 265         return(-1 *( sum(ll.obs) + sum(ll.unobs)))
 266     }
 267
 268     outcome.params <- colnames(model.matrix(outcome_formula,df))
 269     lower <- rep(-Inf, length(outcome.params))
 270
 271     if(outcome_family$family=='gaussian'){
 272         params <- c(outcome.params, 'sigma_y')
 273         lower <- c(lower, 0.00001)
 274     } else {
 275         params <- outcome.params
 276     }
 277
 278     rater.0.params <- colnames(model.matrix(rater_formula,df))
 279     params <- c(params, paste0('rater_0',rater.0.params))
 280     lower <- c(lower, rep(-Inf, length(rater.0.params)))
 281
 282     rater.1.params <- colnames(model.matrix(rater_formula,df))
 283     params <- c(params, paste0('rater_1',rater.1.params))
 284     lower <- c(lower, rep(-Inf, length(rater.1.params)))
 285
 286     proxy.params <- colnames(model.matrix(proxy_formula, df))
 287     params <- c(params, paste0('proxy_',proxy.params))
 288     lower <- c(lower, rep(-Inf, length(proxy.params)))
 289
 290     truth.params <- colnames(model.matrix(truth_formula, df))
 291     params <- c(params, paste0('truth_', truth.params))
 292     lower <- c(lower, rep(-Inf, length(truth.params)))
 293     start <- rep(0.1,length(params))
 294     names(start) <- params
 295
 296     fit <- optim(start, fn = nll, lower=lower, method='L-BFGS-B', hessian=TRUE, control=list(maxit=1e6))
 297     return(fit)
 298 }
 299
 300
 301 measerr_mle <- function(df, outcome_formula, outcome_family=gaussian(), proxy_formula, proxy_family=binomial(link='logit'), truth_formula, truth_family=binomial(link='logit')){
 302
 303     measrr_mle_nll <- function(params){
 304         df.obs <- model.frame(outcome_formula, df)
 305         proxy.variable <- all.vars(proxy_formula)[1]
 306         proxy.model.matrix <- model.matrix(proxy_formula, df)
 307         response.var <- all.vars(outcome_formula)[1]
 308         y.obs <- with(df.obs,eval(parse(text=response.var)))
 309
 310         outcome.model.matrix <- model.matrix(outcome_formula, df)
 311
 312         param.idx <- 1
 313         n.outcome.model.covars <- dim(outcome.model.matrix)[2]
 314         outcome.params <- params[param.idx:n.outcome.model.covars]
 315         param.idx <- param.idx + n.outcome.model.covars
 316
 317         ## likelihood for the fully observed data
 318         if(outcome_family$family == "gaussian"){
 319             sigma.y <- params[param.idx]
 320             param.idx <- param.idx + 1
 321
 322                                         #  outcome_formula likelihood using linear regression
 323             ll.y.obs <- dnorm(y.obs, outcome.params %*% t(outcome.model.matrix),sd=sigma.y, log=TRUE)
 324         }
 325
 326         df.obs <- model.frame(proxy_formula,df)
 327         n.proxy.model.covars <- dim(proxy.model.matrix)[2]
 328         proxy.params <- params[param.idx:(n.proxy.model.covars+param.idx-1)]
 329         param.idx <- param.idx + n.proxy.model.covars
 330         proxy.obs <- with(df.obs, eval(parse(text=proxy.variable)))
 331
 332         if( (proxy_family$family=="binomial") & (proxy_family$link=='logit')){
 333             ll.w.obs <- vector(mode='numeric',length=dim(proxy.model.matrix)[1])
 334
 335                                         # proxy_formula likelihood using logistic regression
 336             ll.w.obs[proxy.obs==1] <- plogis(proxy.params %*% t(proxy.model.matrix[proxy.obs==1,]),log=TRUE)
 337             ll.w.obs[proxy.obs==0] <- plogis(proxy.params %*% t(proxy.model.matrix[proxy.obs==0,]),log=TRUE, lower.tail=FALSE)
 338         }
 339
 340         df.obs <- model.frame(truth_formula, df)
 341         truth.variable <- all.vars(truth_formula)[1]
 342         truth.obs <- with(df.obs, eval(parse(text=truth.variable)))
 343         truth.model.matrix <- model.matrix(truth_formula,df)
 344         n.truth.model.covars <- dim(truth.model.matrix)[2]
 345
 346         truth.params <- params[param.idx:(n.truth.model.covars + param.idx - 1)]
 347
 348         if( (truth_family$family=="binomial") & (truth_family$link=='logit')){
 349             ll.x.obs <- vector(mode='numeric',length=dim(truth.model.matrix)[1])
 350
 351                                         # truth_formula likelihood using logistic regression
 352             ll.x.obs[truth.obs==1] <- plogis(truth.params %*% t(truth.model.matrix[truth.obs==1,]),log=TRUE)
 353             ll.x.obs[truth.obs==0] <- plogis(truth.params %*% t(truth.model.matrix[truth.obs==0,]),log=TRUE, lower.tail=FALSE)
 354         }
 355
 356                                         # add the three likelihoods
 357         ll.obs <- sum(ll.y.obs + ll.w.obs + ll.x.obs)
 358
 359         ## likelihood for the predicted data
 360         ## integrate out the "truth" variable.
 361
 362         if(truth_family$family=='binomial'){
 363             df.unobs <- df[is.na(eval(parse(text=truth.variable)))]
 364             df.unobs.x1 <- copy(df.unobs)
 365             df.unobs.x1[,'x'] <- 1
 366             df.unobs.x0 <- copy(df.unobs)
 367             df.unobs.x0[,'x'] <- 0
 368             outcome.unobs <- with(df.unobs, eval(parse(text=response.var)))
 369
 370             outcome.model.matrix.x0 <- model.matrix(outcome_formula, df.unobs.x0)
 371             outcome.model.matrix.x1 <- model.matrix(outcome_formula, df.unobs.x1)
 372             if(outcome_family$family=="gaussian"){
 373
 374                                         # likelihood of outcome
 375                 ll.y.x0 <- dnorm(outcome.unobs, outcome.params %*% t(outcome.model.matrix.x0), sd=sigma.y, log=TRUE)
 376                 ll.y.x1 <- dnorm(outcome.unobs, outcome.params %*% t(outcome.model.matrix.x1), sd=sigma.y, log=TRUE)
 377             }
 378
 379             if( (proxy_family$family=='binomial') & (proxy_family$link=='logit')){
 380
 381                 proxy.model.matrix.x0 <- model.matrix(proxy_formula, df.unobs.x0)
 382                 proxy.model.matrix.x1 <- model.matrix(proxy_formula, df.unobs.x1)
 383                 proxy.unobs <- df.unobs[[proxy.variable]]
 384                 ll.w.x0 <- vector(mode='numeric', length=dim(df.unobs)[1])
 385                 ll.w.x1 <- vector(mode='numeric', length=dim(df.unobs)[1])
 386
 387                                         # likelihood of proxy
 388                 ll.w.x0[proxy.unobs==1] <- plogis(proxy.params %*% t(proxy.model.matrix.x0[proxy.unobs==1,]), log=TRUE)
 389                 ll.w.x1[proxy.unobs==1] <- plogis(proxy.params %*% t(proxy.model.matrix.x1[proxy.unobs==1,]), log=TRUE)
 390
 391                 ll.w.x0[proxy.unobs==0] <- plogis(proxy.params %*% t(proxy.model.matrix.x0[proxy.unobs==0,]), log=TRUE,lower.tail=FALSE)
 392                 ll.w.x1[proxy.unobs==0] <- plogis(proxy.params %*% t(proxy.model.matrix.x1[proxy.unobs==0,]), log=TRUE,lower.tail=FALSE)
 393             }
 394
 395             if(truth_family$link=='logit'){
 396                 truth.model.matrix <- model.matrix(truth_formula, df.unobs.x0)
 397                                         # likelihood of truth
 398                 ll.x.x1 <- plogis(truth.params %*% t(truth.model.matrix), log=TRUE)
 399                 ll.x.x0 <- plogis(truth.params %*% t(truth.model.matrix), log=TRUE, lower.tail=FALSE)
 400             }
 401         }
 402
 403         ll.x0 <- ll.y.x0 + ll.w.x0 + ll.x.x0
 404         ll.x1 <- ll.y.x1 + ll.w.x1 + ll.x.x1
 405         ll.unobs <- sum(colLogSumExps(rbind(ll.x0, ll.x1)))
 406         return(-(ll.unobs + ll.obs))
 407     }
 408
 409     outcome.params <- colnames(model.matrix(outcome_formula,df))
 410     lower <- rep(-Inf, length(outcome.params))
 411
 412     if(outcome_family$family=='gaussian'){
 413         params <- c(outcome.params, 'sigma_y')
 414         lower <- c(lower, 0.00001)
 415     } else {
 416         params <- outcome.params
 417     }
 418
 419     proxy.params <- colnames(model.matrix(proxy_formula, df))
 420     params <- c(params, paste0('proxy_',proxy.params))
 421     lower <- c(lower, rep(-Inf, length(proxy.params)))
 422
 423     truth.params <- colnames(model.matrix(truth_formula, df))
 424     params <- c(params, paste0('truth_', truth.params))
 425     lower <- c(lower, rep(-Inf, length(truth.params)))
 426     start <- rep(0.1,length(params))
 427     names(start) <- params
 428
 429     fit <- optim(start, fn = measrr_mle_nll, lower=lower, method='L-BFGS-B', hessian=TRUE, control=list(maxit=1e6))
 430
 431     return(fit)
 432 }