X-Git-Url: https://code.communitydata.science/ml_measurement_error_public.git/blobdiff_plain/46e2d1fe4876a9ed906b723f9e5f74fcc949e339..d8bc08f18f8c2128369ee959196e0e6080a11689:/simulations/measerr_methods.R

diff --git a/simulations/measerr_methods.R b/simulations/measerr_methods.R
index ab87d71..63f8bc1 100644
--- a/simulations/measerr_methods.R
+++ b/simulations/measerr_methods.R
@@ -1,6 +1,7 @@
 library(formula.tools)
 library(matrixStats)
-
+library(optimx)
+library(bbmle)
 ## df: dataframe to model
 ## outcome_formula: formula for y | x, z
 ## outcome_family: family for y | x, z
@@ -17,7 +18,7 @@ library(matrixStats)
 
 
 ## outcome_formula <- y ~ x + z; proxy_formula <- w_pred ~ y + x + z + x:z + x:y + z:y 
-measerr_mle_dv <- function(df, outcome_formula, outcome_family=binomial(link='logit'), proxy_formula, proxy_family=binomial(link='logit')){
+measerr_mle_dv <- function(df, outcome_formula, outcome_family=binomial(link='logit'), proxy_formula, proxy_family=binomial(link='logit'),method='optim'){
 
     nll <- function(params){
         df.obs <- model.frame(outcome_formula, df)
@@ -57,7 +58,7 @@ measerr_mle_dv <- function(df, outcome_formula, outcome_family=binomial(link='lo
         df.unobs.y1 <- copy(df.unobs)
         df.unobs.y1[[response.var]] <- 1
         df.unobs.y0 <- copy(df.unobs)
-        df.unobs.y0[[response.var]] <- 1
+        df.unobs.y0[[response.var]] <- 0
         
         ## integrate out y
         outcome.model.matrix.y1 <- model.matrix(outcome_formula, df.unobs.y1)
@@ -98,23 +99,33 @@ measerr_mle_dv <- function(df, outcome_formula, outcome_family=binomial(link='lo
     start <- rep(0.1,length(params))
     names(start) <- params
     
-    fit <- optim(start, fn = nll, lower=lower, method='L-BFGS-B', hessian=TRUE, control=list(maxit=1e6))
+    if(method=='optim'){
+        fit <- optim(start, fn = nll, lower=lower, method='L-BFGS-B', hessian=TRUE, control=list(maxit=1e6))
+    } else {
+        quoted.names <- gsub("[\\(\\)]",'',names(start))
+        print(quoted.names)
+        text <- paste("function(", paste0(quoted.names,'=',start,collapse=','),"){params<-c(",paste0(quoted.names,collapse=','),");return(nll(params))}")
+
+        measerr_mle_nll <- eval(parse(text=text))
+        names(start) <- quoted.names
+        names(lower) <- quoted.names
+        fit <- mle2(minuslogl=measerr_mle_nll, start=start, lower=lower, parnames=params,control=list(maxit=1e6),method='L-BFGS-B')
+    }
     return(fit)
 }
 
-measerr_mle <- function(df, outcome_formula, outcome_family=gaussian(), proxy_formula, proxy_family=binomial(link='logit'), truth_formula, truth_family=binomial(link='logit')){
 
-    measrr_mle_nll <- function(params){
-        df.obs <- model.frame(outcome_formula, df)
-        
-        proxy.variable <- all.vars(proxy_formula)[1]
-        proxy.model.matrix <- model.matrix(proxy_formula, df)
+measerr_mle <- function(df, outcome_formula, outcome_family=gaussian(), proxy_formula, proxy_family=binomial(link='logit'), truth_formula, truth_family=binomial(link='logit'),method='optim'){
 
-        response.var <- all.vars(outcome_formula)[1]
-        y.obs <- with(df.obs,eval(parse(text=response.var)))
-        
-        outcome.model.matrix <- model.matrix(outcome_formula, df)
+    df.obs <- model.frame(outcome_formula, df)
+    response.var <- all.vars(outcome_formula)[1]
+    proxy.variable <- all.vars(proxy_formula)[1]
+    truth.variable <- all.vars(truth_formula)[1]
+    outcome.model.matrix <- model.matrix(outcome_formula, df)
+    proxy.model.matrix <- model.matrix(proxy_formula, df)
+    y.obs <- with(df.obs,eval(parse(text=response.var)))
 
+    measerr_mle_nll <- function(params){
         param.idx <- 1
         n.outcome.model.covars <- dim(outcome.model.matrix)[2]
         outcome.params <- params[param.idx:n.outcome.model.covars]
@@ -124,6 +135,7 @@ measerr_mle <- function(df, outcome_formula, outcome_family=gaussian(), proxy_fo
         if(outcome_family$family == "gaussian"){
             sigma.y <- params[param.idx]
             param.idx <- param.idx + 1
+                                        #  outcome_formula likelihood using linear regression
             ll.y.obs <- dnorm(y.obs, outcome.params %*% t(outcome.model.matrix),sd=sigma.y, log=TRUE)
         }
         
@@ -135,12 +147,14 @@ measerr_mle <- function(df, outcome_formula, outcome_family=gaussian(), proxy_fo
 
         if( (proxy_family$family=="binomial") & (proxy_family$link=='logit')){
             ll.w.obs <- vector(mode='numeric',length=dim(proxy.model.matrix)[1])
+
+                                        # proxy_formula likelihood using logistic regression
             ll.w.obs[proxy.obs==1] <- plogis(proxy.params %*% t(proxy.model.matrix[proxy.obs==1,]),log=TRUE)
             ll.w.obs[proxy.obs==0] <- plogis(proxy.params %*% t(proxy.model.matrix[proxy.obs==0,]),log=TRUE, lower.tail=FALSE)
         }
 
         df.obs <- model.frame(truth_formula, df)
-        truth.variable <- all.vars(truth_formula)[1]
+
         truth.obs <- with(df.obs, eval(parse(text=truth.variable)))
         truth.model.matrix <- model.matrix(truth_formula,df)
         n.truth.model.covars <- dim(truth.model.matrix)[2]
@@ -149,10 +163,13 @@ measerr_mle <- function(df, outcome_formula, outcome_family=gaussian(), proxy_fo
 
         if( (truth_family$family=="binomial") & (truth_family$link=='logit')){
             ll.x.obs <- vector(mode='numeric',length=dim(truth.model.matrix)[1])
+
+                                        # truth_formula likelihood using logistic regression
             ll.x.obs[truth.obs==1] <- plogis(truth.params %*% t(truth.model.matrix[truth.obs==1,]),log=TRUE)
             ll.x.obs[truth.obs==0] <- plogis(truth.params %*% t(truth.model.matrix[truth.obs==0,]),log=TRUE, lower.tail=FALSE)
         }
         
+                                        # add the three likelihoods
         ll.obs <- sum(ll.y.obs + ll.w.obs + ll.x.obs)
 
         ## likelihood for the predicted data
@@ -169,8 +186,10 @@ measerr_mle <- function(df, outcome_formula, outcome_family=gaussian(), proxy_fo
             outcome.model.matrix.x0 <- model.matrix(outcome_formula, df.unobs.x0)
             outcome.model.matrix.x1 <- model.matrix(outcome_formula, df.unobs.x1)
             if(outcome_family$family=="gaussian"){
-            ll.y.x0 <- dnorm(outcome.unobs, outcome.params %*% t(outcome.model.matrix.x0), sd=sigma.y, log=TRUE)
-            ll.y.x1 <- dnorm(outcome.unobs, outcome.params %*% t(outcome.model.matrix.x1), sd=sigma.y, log=TRUE)
+
+                                        # likelihood of outcome
+                ll.y.x0 <- dnorm(outcome.unobs, outcome.params %*% t(outcome.model.matrix.x0), sd=sigma.y, log=TRUE)
+                ll.y.x1 <- dnorm(outcome.unobs, outcome.params %*% t(outcome.model.matrix.x1), sd=sigma.y, log=TRUE)
             }
 
             if( (proxy_family$family=='binomial') & (proxy_family$link=='logit')){
@@ -181,6 +200,7 @@ measerr_mle <- function(df, outcome_formula, outcome_family=gaussian(), proxy_fo
                 ll.w.x0 <- vector(mode='numeric', length=dim(df.unobs)[1])
                 ll.w.x1 <- vector(mode='numeric', length=dim(df.unobs)[1])
 
+                                        # likelihood of proxy
                 ll.w.x0[proxy.unobs==1] <- plogis(proxy.params %*% t(proxy.model.matrix.x0[proxy.unobs==1,]), log=TRUE)
                 ll.w.x1[proxy.unobs==1] <- plogis(proxy.params %*% t(proxy.model.matrix.x1[proxy.unobs==1,]), log=TRUE)
 
@@ -190,8 +210,9 @@ measerr_mle <- function(df, outcome_formula, outcome_family=gaussian(), proxy_fo
 
             if(truth_family$link=='logit'){
                 truth.model.matrix <- model.matrix(truth_formula, df.unobs.x0)
-                ll.x.x0 <- plogis(truth.params %*% t(truth.model.matrix), log=TRUE)
-                ll.x.x1 <- plogis(truth.params %*% t(truth.model.matrix), log=TRUE, lower.tail=FALSE)
+                                        # likelihood of truth
+                ll.x.x1 <- plogis(truth.params %*% t(truth.model.matrix), log=TRUE)
+                ll.x.x0 <- plogis(truth.params %*% t(truth.model.matrix), log=TRUE, lower.tail=FALSE)
             }
         }
 
@@ -220,8 +241,356 @@ measerr_mle <- function(df, outcome_formula, outcome_family=gaussian(), proxy_fo
     lower <- c(lower, rep(-Inf, length(truth.params)))
     start <- rep(0.1,length(params))
     names(start) <- params
+
+    if(method=='optim'){
+        fit <- optim(start, fn = measerr_mle_nll, lower=lower, method='L-BFGS-B', hessian=TRUE, control=list(maxit=1e6))
+    } else { # method='mle2'
+                
+        quoted.names <- gsub("[\\(\\)]",'',names(start))
+
+        text <- paste("function(", paste0(quoted.names,'=',start,collapse=','),"){params<-c(",paste0(quoted.names,collapse=','),");return(measerr_mle_nll(params))}")
+
+        measerr_mle_nll_mle <- eval(parse(text=text))
+        names(start) <- quoted.names
+        names(lower) <- quoted.names
+        fit <- mle2(minuslogl=measerr_mle_nll_mle, start=start, lower=lower, parnames=params,control=list(maxit=1e6),method='L-BFGS-B')
+    }
+
+    return(fit)
+}
+
+## Experimental, but probably works. 
+measerr_irr_mle <- function(df, outcome_formula, outcome_family=gaussian(), coder_formulas=list(x.obs.0~x, x.obs.1~x), proxy_formula, proxy_family=binomial(link='logit'), truth_formula, truth_family=binomial(link='logit'),method='optim'){
+
+    ### in this scenario, the ground truth also has measurement error, but we have repeated measures for it. 
+    # this time we never get to observe the true X
+    outcome.model.matrix <- model.matrix(outcome_formula, df)
+    proxy.model.matrix <- model.matrix(proxy_formula, df)
+    response.var <- all.vars(outcome_formula)[1]
+    proxy.var <- all.vars(proxy_formula)[1]
+    param.var <- all.vars(truth_formula)[1]
+    truth.var<- all.vars(truth_formula)[1]
+    y <- with(df,eval(parse(text=response.var)))
+
+    nll <- function(params){
+        param.idx <- 1
+        n.outcome.model.covars <- dim(outcome.model.matrix)[2]
+        outcome.params <- params[param.idx:n.outcome.model.covars]
+        param.idx <- param.idx + n.outcome.model.covars
+
+
+        if(outcome_family$family == "gaussian"){
+            sigma.y <- params[param.idx]
+            param.idx <- param.idx + 1
+        }
+
+
+        n.proxy.model.covars <- dim(proxy.model.matrix)[2]
+        proxy.params <- params[param.idx:(n.proxy.model.covars+param.idx-1)]
+        param.idx <- param.idx + n.proxy.model.covars
+        
+        df.temp <- copy(df)
+
+        if((truth_family$family == "binomial")
+           & (truth_family$link=='logit')){
+            integrate.grid <- expand.grid(replicate(1 + length(coder_formulas), c(0,1), simplify=FALSE))
+            ll.parts <- matrix(nrow=nrow(df),ncol=nrow(integrate.grid))
+            for(i in 1:nrow(integrate.grid)){
+                # setup the dataframe for this row
+                row <- integrate.grid[i,]
+
+                df.temp[[param.var]] <- row[[1]]
+                ci <- 2
+                for(coder_formula in coder_formulas){
+                    coder.var <- all.vars(coder_formula)[1]
+                    df.temp[[coder.var]] <- row[[ci]]
+                    ci <- ci + 1 
+                }
+                
+                outcome.model.matrix <- model.matrix(outcome_formula, df.temp)                
+                if(outcome_family$family == "gaussian"){
+                    ll.y <- dnorm(y, outcome.params %*% t(outcome.model.matrix), sd=sigma.y, log=TRUE)
+                }
+
+                if(proxy_family$family=="binomial" & (proxy_family$link=='logit')){
+                    proxy.model.matrix <- model.matrix(proxy_formula, df.temp)
+                    ll.w <- vector(mode='numeric', length=dim(proxy.model.matrix)[1])
+                    proxyvar <- with(df.temp,eval(parse(text=proxy.var)))
+                    ll.w[proxyvar==1] <- plogis(proxy.params %*% t(proxy.model.matrix[proxyvar==1,]),log=TRUE)
+                    ll.w[proxyvar==0] <- plogis(proxy.params %*% t(proxy.model.matrix[proxyvar==0,]),log=TRUE,lower.tail=FALSE)
+                }
+
+                ## probability of the coded variables
+                coder.lls <- matrix(nrow=nrow(df.temp),ncol=length(coder_formulas))
+                ci <- 1
+                for(coder_formula in coder_formulas){
+                    coder.model.matrix <- model.matrix(coder_formula, df.temp)
+                    n.coder.model.covars <- dim(coder.model.matrix)[2]
+                    coder.params <- params[param.idx:(n.coder.model.covars + param.idx - 1)]
+                    param.idx <- param.idx + n.coder.model.covars
+                    coder.var <- all.vars(coder_formula)[1]
+                    x.obs <- with(df.temp, eval(parse(text=coder.var)))
+                    true.codervar <- df[[all.vars(coder_formula)[1]]]
+
+                    ll.coder <- vector(mode='numeric', length=dim(coder.model.matrix)[1])
+                    ll.coder[x.obs==1] <- plogis(coder.params %*% t(coder.model.matrix[x.obs==1,]),log=TRUE)
+                    ll.coder[x.obs==0] <- plogis(coder.params %*% t(coder.model.matrix[x.obs==0,]),log=TRUE,lower.tail=FALSE)
+
+                    # don't count when we know the observed value, unless we're accounting for observed value
+                    ll.coder[(!is.na(true.codervar)) & (true.codervar != x.obs)] <- NA
+                    coder.lls[,ci] <- ll.coder
+                    ci <- ci + 1
+                }
+                
+                truth.model.matrix <- model.matrix(truth_formula, df.temp)
+                n.truth.model.covars <- dim(truth.model.matrix)[2]
+                truth.params <- params[param.idx:(n.truth.model.covars + param.idx - 1)]
+
+                for(coder_formula in coder_formulas){
+                    coder.model.matrix <- model.matrix(coder_formula, df.temp)
+                    n.coder.model.covars <- dim(coder.model.matrix)[2]
+                    param.idx <- param.idx - n.coder.model.covars
+                }
+
+                x <- with(df.temp, eval(parse(text=truth.var)))
+                ll.truth <- vector(mode='numeric', length=dim(truth.model.matrix)[1])
+                ll.truth[x==1] <- plogis(truth.params %*% t(truth.model.matrix[x==1,]), log=TRUE)
+                ll.truth[x==0] <- plogis(truth.params %*% t(truth.model.matrix[x==0,]), log=TRUE, lower.tail=FALSE)
+
+                true.truthvar <- df[[all.vars(truth_formula)[1]]]
+                
+                if(!is.null(true.truthvar)){
+                                        # ll.truth[(!is.na(true.truthvar)) & (true.truthvar != truthvar)] <- -Inf
+                    # ll.truth[(!is.na(true.truthvar)) & (true.truthvar == truthvar)] <- 0
+                }
+                ll.parts[,i] <- ll.y + ll.w + apply(coder.lls,1,sum) + ll.truth
+                
+            }
+
+            lls <- rowLogSumExps(ll.parts,na.rm=TRUE)
+
+        ## likelihood of observed data 
+            target <- -1 * sum(lls)
+            return(target)
+        }
+    }
+        
+    outcome.params <- colnames(model.matrix(outcome_formula,df))
+    lower <- rep(-Inf, length(outcome.params))
+
+    if(outcome_family$family=='gaussian'){
+        params <- c(outcome.params, 'sigma_y')
+        lower <- c(lower, 0.00001)
+    } else {
+        params <- outcome.params
+    }
+
+    proxy.params <- colnames(model.matrix(proxy_formula, df))
+    params <- c(params, paste0('proxy_',proxy.params))
+    positive.params <- paste0('proxy_',truth.var)
+    lower <- c(lower, rep(-Inf, length(proxy.params)))
+    names(lower) <- params
+    lower[positive.params] <- 0.01
+    ci <- 0
+    
+    for(coder_formula in coder_formulas){
+        coder.params <- colnames(model.matrix(coder_formula,df))
+        params <- c(params, paste0('coder_',ci,coder.params))
+        positive.params <- paste0('coder_', ci, truth.var)
+        ci <- ci + 1
+        lower <- c(lower, rep(-Inf, length(coder.params)))
+        names(lower) <- params
+        lower[positive.params] <- 0.01
+    }
+
+    truth.params <- colnames(model.matrix(truth_formula, df))
+    params <- c(params, paste0('truth_', truth.params))
+    lower <- c(lower, rep(-Inf, length(truth.params)))
+    start <- rep(0.1,length(params))
+    names(start) <- params
+    names(lower) <- params
+    
+    if(method=='optim'){
+        print(start)
+        fit <- optim(start, fn = nll, lower=lower, method='L-BFGS-B', hessian=TRUE, control=list(maxit=1e6))
+    } else {
+                
+        quoted.names <- gsub("[\\(\\)]",'',names(start))
+        print(quoted.names)
+        text <- paste("function(", paste0(quoted.names,'=',start,collapse=','),"){params<-c(",paste0(quoted.names,collapse=','),");return(nll(params))}")
+
+        measerr_mle_nll <- eval(parse(text=text))
+        names(start) <- quoted.names
+        names(lower) <- quoted.names
+        fit <- mle2(minuslogl=measerr_mle_nll, start=start, lower=lower, method='L-BFGS-B',control=list(maxit=1e6))
+    }
+
+    return(fit)
+}
+
+## Experimental, and does not work.
+measerr_irr_mle_dv <- function(df, outcome_formula, outcome_family=binomial(link='logit'), coder_formulas=list(y.obs.0~y+w_pred+y.obs.1,y.obs.1~y+w_pred+y.obs.0), proxy_formula=w_pred~y, proxy_family=binomial(link='logit'),method='optim'){
+    integrate.grid <- expand.grid(replicate(1 + length(coder_formulas), c(0,1), simplify=FALSE))
+    print(integrate.grid)
+
+
+    outcome.model.matrix <- model.matrix(outcome_formula, df)
+    n.outcome.model.covars <- dim(outcome.model.matrix)[2]
+
+
+    ### in this scenario, the ground truth also has measurement error, but we have repeated measures for it. 
+    # this time we never get to observe the true X
+    nll <- function(params){
+        param.idx <- 1
+        outcome.params <- params[param.idx:n.outcome.model.covars]
+        param.idx <- param.idx + n.outcome.model.covars
+        proxy.model.matrix <- model.matrix(proxy_formula, df)
+        n.proxy.model.covars <- dim(proxy.model.matrix)[2]
+        response.var <- all.vars(outcome_formula)[1]
+
+        if(outcome_family$family == "gaussian"){
+            sigma.y <- params[param.idx]
+            param.idx <- param.idx + 1
+        }
+
+        proxy.params <- params[param.idx:(n.proxy.model.covars+param.idx-1)]
+        param.idx <- param.idx + n.proxy.model.covars
+
+        df.temp <- copy(df)
+
+        if((outcome_family$family == "binomial")
+           & (outcome_family$link=='logit')){
+            ll.parts <- matrix(nrow=nrow(df),ncol=nrow(integrate.grid))
+            for(i in 1:nrow(integrate.grid)){
+                # setup the dataframe for this row
+                row <- integrate.grid[i,]
+
+                df.temp[[response.var]] <- row[[1]]
+                ci <- 2
+                for(coder_formula in coder_formulas){
+                    codervar <- all.vars(coder_formula)[1]
+                    df.temp[[codervar]] <- row[[ci]]
+                    ci <- ci + 1 
+                }
+                
+                outcome.model.matrix <- model.matrix(outcome_formula, df.temp)                
+                if(outcome_family$family == "gaussian"){
+                    ll.y <- dnorm(df.temp[[response.var]], outcome.params %*% t(outcome.model.matrix), sd=sigma.y, log=T)
+                }
+
+                if(outcome_family$family == "binomial" & (outcome_family$link=='logit')){
+                    ll.y <- vector(mode='numeric',length=nrow(df.temp))
+                    ll.y[df.temp[[response.var]]==1] <- plogis( outcome.params %*% t(outcome.model.matrix), log=TRUE)
+                    ll.y[df.temp[[response.var]]==0] <- plogis( outcome.params %*% t(outcome.model.matrix), log=TRUE,lower.tail=FALSE)
+                }
+
+                if(proxy_family$family=="binomial" & (proxy_family$link=='logit')){
+                    proxy.model.matrix <- model.matrix(proxy_formula, df.temp)
+                    ll.w <- vector(mode='numeric', length=dim(proxy.model.matrix)[1])
+                    proxyvar <- with(df.temp,eval(parse(text=all.vars(proxy_formula)[1])))
+                    ll.w[proxyvar==1] <- plogis(proxy.params %*% t(proxy.model.matrix[proxyvar==1,]),log=TRUE)
+                    ll.w[proxyvar==0] <- plogis(proxy.params %*% t(proxy.model.matrix[proxyvar==0,]),log=TRUE,lower.tail=FALSE)
+                }
+
+                ## probability of the coded variables
+                coder.lls <- matrix(nrow=nrow(df.temp),ncol=length(coder_formulas))
+                ci <- 1
+                for(coder_formula in coder_formulas){
+                    coder.model.matrix <- model.matrix(coder_formula, df.temp)
+                    n.coder.model.covars <- dim(coder.model.matrix)[2]
+                    coder.params <- params[param.idx:(n.coder.model.covars + param.idx - 1)]
+                    param.idx <- param.idx + n.coder.model.covars
+                    codervar <- with(df.temp, eval(parse(text=all.vars(coder_formula)[1])))
+                    true.codervar <- df[[all.vars(coder_formula)[1]]]
+
+                    ll.coder <- vector(mode='numeric', length=dim(coder.model.matrix)[1])
+                    ll.coder[codervar==1] <- plogis(coder.params %*% t(coder.model.matrix[codervar==1,]),log=TRUE)
+                    ll.coder[codervar==0] <- plogis(coder.params %*% t(coder.model.matrix[codervar==0,]),log=TRUE,lower.tail=FALSE)
+
+                    # don't count when we know the observed value, unless we're accounting for observed value
+                    ll.coder[(!is.na(true.codervar)) & (true.codervar != codervar)] <- NA
+                    coder.lls[,ci] <- ll.coder
+                    ci <- ci + 1
+                }
+
+                for(coder_formula in coder_formulas){
+                    coder.model.matrix <- model.matrix(coder_formula, df.temp)
+                    n.coder.model.covars <- dim(coder.model.matrix)[2]
+                    param.idx <- param.idx - n.coder.model.covars
+                }
+
+                ll.parts[,i] <- ll.y + ll.w + apply(coder.lls,1,function(x) sum(x)) 
+                
+            }
+
+            lls <- rowLogSumExps(ll.parts,na.rm=TRUE)
+
+            ## likelihood of observed data 
+            target <- -1 * sum(lls)
+            print(target)
+            print(params)
+            return(target)
+        }
+    }
+        
+    outcome.params <- colnames(model.matrix(outcome_formula,df))
+    response.var <- all.vars(outcome_formula)[1]
+    lower <- rep(-Inf, length(outcome.params))
+
+    if(outcome_family$family=='gaussian'){
+        params <- c(outcome.params, 'sigma_y')
+        lower <- c(lower, 0.00001)
+    } else {
+        params <- outcome.params
+    }
+
+    ## constrain the model of the coder and proxy vars
+    ## this is to ensure identifiability
+    ## it is a safe assumption because the coders aren't hostile (wrong more often than right)
+    ## so we can assume that y ~Bw, B is positive
+    proxy.params <- colnames(model.matrix(proxy_formula, df))
+    positive.params <- paste0('proxy_',response.var)
+    params <- c(params, paste0('proxy_',proxy.params))
+    lower <- c(lower, rep(-Inf, length(proxy.params)))
+    names(lower) <- params
+    lower[positive.params] <- 0.001
+
+    ci <- 0
+    for(coder_formula in coder_formulas){
+        coder.params <- colnames(model.matrix(coder_formula,df))
+        latent.coder.params <- coder.params %in% response.var
+        params <- c(params, paste0('coder_',ci,coder.params))
+        positive.params <- paste0('coder_',ci,response.var)
+        ci <- ci + 1
+        lower <- c(lower, rep(-Inf, length(coder.params)))
+        names(lower) <-params
+        lower[positive.params] <- 0.001
+    }
+
+    ## init by using the "loco model"
+    temp.df <- copy(df)
+    temp.df <- temp.df[y.obs.1 == y.obs.0, y:=y.obs.1]
+    loco.model <- glm(outcome_formula, temp.df, family=outcome_family)
     
-    fit <- optim(start, fn = measrr_mle_nll, lower=lower, method='L-BFGS-B', hessian=TRUE, control=list(maxit=1e6))
+    start <- rep(1,length(params))
+    names(start) <- params
+    start[names(coef(loco.model))] <- coef(loco.model)
+    names(lower) <- params
+    if(method=='optim'){
+        print(lower)
+        fit <- optim(start, fn = nll, lower=lower, method='L-BFGS-B', hessian=TRUE,control=list(maxit=1e6))
+    } else {
+                
+        quoted.names <- gsub("[\\(\\)]",'',names(start))
+        print(quoted.names)
+        text <- paste("function(", paste0(quoted.names,'=',start,collapse=','),"){params<-c(",paste0(quoted.names,collapse=','),");return(nll(params))}")
+
+        measerr_mle_nll <- eval(parse(text=text))
+        names(start) <- quoted.names
+        names(lower) <- quoted.names
+        fit <- mle2(minuslogl=measerr_mle_nll, start=start, lower=lower, parnames=params,control=list(maxit=1e6),method='L-BFGS-B')
+    }
 
     return(fit)
 }
+