library(predictionError) library(mecor) options(amelia.parallel="no", amelia.ncpus=1) library(Amelia) library(Zelig) library(bbmle) library(matrixStats) # for numerically stable logsumexps source("measerr_methods.R") ## for my more generic function. ## This uses the pseudolikelihood approach from Carroll page 349. ## assumes MAR ## assumes differential error, but that only depends on Y ## inefficient, because pseudolikelihood ## This uses the pseudo-likelihood approach from Carroll page 346. my.pseudo.mle <- function(df){ p1.est <- mean(df[w_pred==1]$y.obs==1,na.rm=T) p0.est <- mean(df[w_pred==0]$y.obs==0,na.rm=T) nll <- function(B0, Bxy, Bzy){ pw <- vector(mode='numeric',length=nrow(df)) dfw1 <- df[w_pred==1] dfw0 <- df[w_pred==0] pw[df$w_pred==1] <- plogis(B0 + Bxy * dfw1$x + Bzy * dfw1$z, log=T) pw[df$w_pred==0] <- plogis(B0 + Bxy * dfw0$x + Bzy * dfw0$z, lower.tail=FALSE, log=T) probs <- colLogSumExps(rbind(log(1 - p0.est), log(p1.est + p0.est - 1) + pw)) return(-1*sum(probs)) } mlefit <- mle2(minuslogl = nll, start = list(B0=0.0, Bxy=0.0, Bzy=0.0), control=list(maxit=1e6),method='L-BFGS-B') return(mlefit) } ## model from Zhang's arxiv paper, with predictions for y ## Zhang got this model from Hausman 1998 ### I think this is actually eqivalent to the pseudo.mle method zhang.mle.iv <- function(df){ df.obs <- df[!is.na(x.obs)] df.unobs <- df[is.na(x.obs)] tn <- df.obs[(w_pred == 0) & (x.obs == w_pred),.N] pn <- df.obs[(w_pred==0), .N] npv <- tn / pn tp <- df.obs[(w_pred==1) & (x.obs == w_pred),.N] pp <- df.obs[(w_pred==1),.N] ppv <- tp / pp nll <- function(B0=0, Bxy=0, Bzy=0, sigma_y=0.1){ ## fpr = 1 - TNR ### Problem: accounting for uncertainty in ppv / npv ## fnr = 1 - TPR ll.y.obs <- with(df.obs, dnorm(y, B0 + Bxy * x + Bzy * z, sd=sigma_y,log=T)) ll <- sum(ll.y.obs) # unobserved case; integrate out x ll.x.1 <- with(df.unobs, dnorm(y, B0 + Bxy + Bzy * z, sd = sigma_y, log=T)) ll.x.0 <- with(df.unobs, dnorm(y, B0 + Bzy * z, sd = sigma_y,log=T)) ## case x == 1 lls.x.1 <- colLogSumExps(rbind(log(ppv) + ll.x.1, log(1-ppv) + ll.x.0)) ## case x == 0 lls.x.0 <- colLogSumExps(rbind(log(1-npv) + ll.x.1, log(npv) + ll.x.0)) lls <- colLogSumExps(rbind(df.unobs$w_pred * lls.x.1, (1-df.unobs$w_pred) * lls.x.0)) ll <- ll + sum(lls) return(-ll) } mlefit <- mle2(minuslogl = nll, control=list(maxit=1e6), lower=list(sigma_y=0.0001, B0=-Inf, Bxy=-Inf, Bzy=-Inf), upper=list(sigma_y=Inf, B0=Inf, Bxy=Inf, Bzy=Inf),method='L-BFGS-B') return(mlefit) } ## this is equivalent to the pseudo-liklihood model from Caroll ## zhang.mle.dv <- function(df){ ## nll <- function(B0=0, Bxy=0, Bzy=0, ppv=0.9, npv=0.9){ ## df.obs <- df[!is.na(y.obs)] ## ## fpr = 1 - TNR ## ll.w0y0 <- with(df.obs[y.obs==0],dbinom(1-w_pred,1,npv,log=TRUE)) ## ll.w1y1 <- with(df.obs[y.obs==1],dbinom(w_pred,1,ppv,log=TRUE)) ## # observed case ## ll.y.obs <- vector(mode='numeric', length=nrow(df.obs)) ## ll.y.obs[df.obs$y.obs==1] <- with(df.obs[y.obs==1], plogis(B0 + Bxy * x + Bzy * z,log=T)) ## ll.y.obs[df.obs$y.obs==0] <- with(df.obs[y.obs==0], plogis(B0 + Bxy * x + Bzy * z,log=T,lower.tail=FALSE)) ## ll <- sum(ll.y.obs) + sum(ll.w0y0) + sum(ll.w1y1) ## # unobserved case; integrate out y ## ## case y = 1 ## ll.y.1 <- vector(mode='numeric', length=nrow(df)) ## pi.y.1 <- with(df,plogis(B0 + Bxy * x + Bzy*z, log=T)) ## ## 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) ## lls.y.1 <- colLogSumExps(rbind(log(ppv) + pi.y.1, log(1-ppv) + pi.y.1)) ## ## case y = 0 ## ll.y.0 <- vector(mode='numeric', length=nrow(df)) ## pi.y.0 <- with(df,plogis(B0 + Bxy * x + Bzy*z, log=T,lower.tail=FALSE)) ## ## 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) ## lls.y.0 <- colLogSumExps(rbind(log(npv) + pi.y.0, log(1-npv) + pi.y.0)) ## lls <- colLogSumExps(rbind(lls.y.1, lls.y.0)) ## ll <- ll + sum(lls) ## return(-ll) ## } ## 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), ## upper=list(B0=Inf, Bxy=Inf, Bzy=Inf,ppv=0.999,npv=0.999)) ## return(mlefit) ## } zhang.mle.dv <- function(df){ df.obs <- df[!is.na(y.obs)] df.unobs <- df[is.na(y.obs)] fp <- df.obs[(w_pred==1) & (y.obs != w_pred),.N] p <- df.obs[(w_pred==1),.N] fpr <- fp / p fn <- df.obs[(w_pred==0) & (y.obs != w_pred), .N] n <- df.obs[(w_pred==0),.N] fnr <- fn / n nll <- function(B0=0, Bxy=0, Bzy=0){ ## observed case ll.y.obs <- vector(mode='numeric', length=nrow(df.obs)) ll.y.obs[df.obs$y.obs==1] <- with(df.obs[y.obs==1], plogis(B0 + Bxy * x + Bzy * z,log=T)) ll.y.obs[df.obs$y.obs==0] <- with(df.obs[y.obs==0], plogis(B0 + Bxy * x + Bzy * z,log=T,lower.tail=FALSE)) ll <- sum(ll.y.obs) pi.y.1 <- with(df,plogis(B0 + Bxy * x + Bzy*z, log=T)) pi.y.0 <- with(df,plogis(B0 + Bxy * x + Bzy*z, log=T,lower.tail=FALSE)) lls <- with(df.unobs, colLogSumExps(rbind(w_pred * colLogSumExps(rbind(log(fpr), log(1 - fnr - fpr)+pi.y.1)), (1-w_pred) * colLogSumExps(rbind(log(1-fpr), log(1 - fnr - fpr)+pi.y.0))))) ll <- ll + sum(lls) return(-ll) } mlefit <- mle2(minuslogl = nll, control=list(maxit=1e6),method='L-BFGS-B',lower=c(B0=-Inf, Bxy=-Inf, Bzy=-Inf), upper=c(B0=Inf, Bxy=Inf, Bzy=Inf)) return(mlefit) } ## This uses the likelihood approach from Carroll page 353. ## assumes that we have a good measurement error model my.mle <- function(df){ ## liklihood for observed responses nll <- function(B0, Bxy, Bzy, gamma0, gamma_y, gamma_z, gamma_yz){ df.obs <- df[!is.na(y.obs)] yobs0 <- df.obs$y==0 yobs1 <- df.obs$y==1 p.y.obs <- vector(mode='numeric', length=nrow(df.obs)) p.y.obs[yobs1] <- plogis(B0 + Bxy * df.obs[yobs1]$x + Bzy*df.obs[yobs1]$z,log=T) p.y.obs[yobs0] <- plogis(B0 + Bxy * df.obs[yobs0]$x + Bzy*df.obs[yobs0]$z,lower.tail=FALSE,log=T) wobs0 <- df.obs$w_pred==0 wobs1 <- df.obs$w_pred==1 p.w.obs <- vector(mode='numeric', length=nrow(df.obs)) 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) 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) p.obs <- p.w.obs + p.y.obs df.unobs <- df[is.na(y.obs)] p.unobs.0 <- vector(mode='numeric',length=nrow(df.unobs)) p.unobs.1 <- vector(mode='numeric',length=nrow(df.unobs)) wunobs.0 <- df.unobs$w_pred == 0 wunobs.1 <- df.unobs$w_pred == 1 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) 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) 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]$z, log=T) p.unobs.1[wunobs.0] <- plogis(B0 + Bxy * df.unobs[wunobs.0]$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) p.unobs <- colLogSumExps(rbind(p.unobs.1, p.unobs.0)) p <- c(p.obs, p.unobs) return(-1*(sum(p))) } 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') return(mlefit) } run_simulation_depvar <- function(df, result, outcome_formula=y~x+z, proxy_formula=w_pred~y){ accuracy <- df[,mean(w_pred==y)] result <- append(result, list(accuracy=accuracy)) (model.true <- glm(y ~ x + z, data=df,family=binomial(link='logit'))) true.ci.Bxy <- confint(model.true)['x',] true.ci.Bzy <- confint(model.true)['z',] result <- append(result, list(Bxy.est.true=coef(model.true)['x'], Bzy.est.true=coef(model.true)['z'], Bxy.ci.upper.true = true.ci.Bxy[2], Bxy.ci.lower.true = true.ci.Bxy[1], Bzy.ci.upper.true = true.ci.Bzy[2], Bzy.ci.lower.true = true.ci.Bzy[1])) (model.feasible <- glm(y.obs~x+z,data=df,family=binomial(link='logit'))) feasible.ci.Bxy <- confint(model.feasible)['x',] result <- append(result, list(Bxy.est.feasible=coef(model.feasible)['x'], Bxy.ci.upper.feasible = feasible.ci.Bxy[2], Bxy.ci.lower.feasible = feasible.ci.Bxy[1])) feasible.ci.Bzy <- confint(model.feasible)['z',] result <- append(result, list(Bzy.est.feasible=coef(model.feasible)['z'], Bzy.ci.upper.feasible = feasible.ci.Bzy[2], Bzy.ci.lower.feasible = feasible.ci.Bzy[1])) (model.naive <- glm(w_pred~x+z, data=df, family=binomial(link='logit'))) naive.ci.Bxy <- confint(model.naive)['x',] naive.ci.Bzy <- confint(model.naive)['z',] result <- append(result, list(Bxy.est.naive=coef(model.naive)['x'], Bzy.est.naive=coef(model.naive)['z'], Bxy.ci.upper.naive = naive.ci.Bxy[2], Bxy.ci.lower.naive = naive.ci.Bxy[1], Bzy.ci.upper.naive = naive.ci.Bzy[2], Bzy.ci.lower.naive = naive.ci.Bzy[1])) (model.naive.cont <- lm(w~x+z, data=df)) naivecont.ci.Bxy <- confint(model.naive.cont)['x',] naivecont.ci.Bzy <- confint(model.naive.cont)['z',] ## my implementation of liklihood based correction temp.df <- copy(df) temp.df[,y:=y.obs] mod.caroll.lik <- measerr_mle_dv(temp.df, outcome_formula=outcome_formula, proxy_formula=proxy_formula) fisher.info <- solve(mod.caroll.lik$hessian) coef <- mod.caroll.lik$par ci.upper <- coef + sqrt(diag(fisher.info)) * 1.96 ci.lower <- coef - sqrt(diag(fisher.info)) * 1.96 result <- append(result, list(Bxy.est.mle = coef['x'], Bxy.ci.upper.mle = ci.upper['x'], Bxy.ci.lower.mle = ci.lower['x'], Bzy.est.mle = coef['z'], Bzy.ci.upper.mle = ci.upper['z'], Bzy.ci.lower.mle = ci.lower['z'])) ## my implementatoin of liklihood based correction mod.zhang <- zhang.mle.dv(df) coef <- coef(mod.zhang) ci <- confint(mod.zhang,method='quad') result <- append(result, list(Bxy.est.zhang = coef['Bxy'], Bxy.ci.upper.zhang = ci['Bxy','97.5 %'], Bxy.ci.lower.zhang = ci['Bxy','2.5 %'], Bzy.est.zhang = coef['Bzy'], Bzy.ci.upper.zhang = ci['Bzy','97.5 %'], Bzy.ci.lower.zhang = ci['Bzy','2.5 %'])) # amelia says use normal distribution for binary variables. tryCatch({ amelia.out.k <- amelia(df, m=200, p2s=0, idvars=c('y','ystar','w')) mod.amelia.k <- zelig(y.obs~x+z, model='ls', data=amelia.out.k$imputations, cite=FALSE) (coefse <- combine_coef_se(mod.amelia.k, messages=FALSE)) est.x.mi <- coefse['x','Estimate'] est.x.se <- coefse['x','Std.Error'] result <- append(result, list(Bxy.est.amelia.full = est.x.mi, Bxy.ci.upper.amelia.full = est.x.mi + 1.96 * est.x.se, Bxy.ci.lower.amelia.full = est.x.mi - 1.96 * est.x.se )) est.z.mi <- coefse['z','Estimate'] est.z.se <- coefse['z','Std.Error'] result <- append(result, list(Bzy.est.amelia.full = est.z.mi, Bzy.ci.upper.amelia.full = est.z.mi + 1.96 * est.z.se, Bzy.ci.lower.amelia.full = est.z.mi - 1.96 * est.z.se )) }, error = function(e){ message("An error occurred:\n",e) result$error <- paste0(result$error,'\n', e) }) return(result) } ## outcome_formula, proxy_formula, and truth_formula are passed to measerr_mle run_simulation <- function(df, result, outcome_formula=y~x+z, proxy_formula=NULL, truth_formula=NULL){ accuracy <- df[,mean(w_pred==x)] result <- append(result, list(accuracy=accuracy)) (model.true <- lm(y ~ x + z, data=df)) true.ci.Bxy <- confint(model.true)['x',] true.ci.Bzy <- confint(model.true)['z',] result <- append(result, list(Bxy.est.true=coef(model.true)['x'], Bzy.est.true=coef(model.true)['z'], Bxy.ci.upper.true = true.ci.Bxy[2], Bxy.ci.lower.true = true.ci.Bxy[1], Bzy.ci.upper.true = true.ci.Bzy[2], Bzy.ci.lower.true = true.ci.Bzy[1])) (model.feasible <- lm(y~x.obs+z,data=df)) feasible.ci.Bxy <- confint(model.feasible)['x.obs',] result <- append(result, list(Bxy.est.feasible=coef(model.feasible)['x.obs'], Bxy.ci.upper.feasible = feasible.ci.Bxy[2], Bxy.ci.lower.feasible = feasible.ci.Bxy[1])) feasible.ci.Bzy <- confint(model.feasible)['z',] result <- append(result, list(Bzy.est.feasible=coef(model.feasible)['z'], Bzy.ci.upper.feasible = feasible.ci.Bzy[2], Bzy.ci.lower.feasible = feasible.ci.Bzy[1])) (model.naive <- lm(y~w_pred+z, data=df)) naive.ci.Bxy <- confint(model.naive)['w_pred',] naive.ci.Bzy <- confint(model.naive)['z',] result <- append(result, list(Bxy.est.naive=coef(model.naive)['w_pred'], Bzy.est.naive=coef(model.naive)['z'], Bxy.ci.upper.naive = naive.ci.Bxy[2], Bxy.ci.lower.naive = naive.ci.Bxy[1], Bzy.ci.upper.naive = naive.ci.Bzy[2], Bzy.ci.lower.naive = naive.ci.Bzy[1])) tryCatch({ amelia.out.k <- amelia(df, m=200, p2s=0, idvars=c('x','w')) mod.amelia.k <- zelig(y~x.obs+z, model='ls', data=amelia.out.k$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.z.mi <- coefse['z','Estimate'] est.z.se <- coefse['z','Std.Error'] result <- append(result, list(Bzy.est.amelia.full = est.z.mi, Bzy.ci.upper.amelia.full = est.z.mi + 1.96 * est.z.se, Bzy.ci.lower.amelia.full = est.z.mi - 1.96 * est.z.se )) }, error = function(e){ message("An error occurred:\n",e) result$error <-paste0(result$error,'\n', e) } ) tryCatch({ temp.df <- copy(df) temp.df <- temp.df[,x:=x.obs] mod.caroll.lik <- measerr_mle(temp.df, outcome_formula=outcome_formula, proxy_formula=proxy_formula, truth_formula=truth_formula) fisher.info <- solve(mod.caroll.lik$hessian) coef <- mod.caroll.lik$par ci.upper <- coef + sqrt(diag(fisher.info)) * 1.96 ci.lower <- coef - sqrt(diag(fisher.info)) * 1.96 result <- append(result, list(Bxy.est.mle = coef['x'], Bxy.ci.upper.mle = ci.upper['x'], Bxy.ci.lower.mle = ci.lower['x'], Bzy.est.mle = coef['z'], Bzy.ci.upper.mle = ci.upper['z'], Bzy.ci.lower.mle = ci.lower['z'])) }, error = function(e){ message("An error occurred:\n",e) result$error <- paste0(result$error,'\n', e) }) tryCatch({ mod.zhang.lik <- zhang.mle.iv(df) coef <- coef(mod.zhang.lik) ci <- confint(mod.zhang.lik,method='quad') result <- append(result, list(Bxy.est.zhang = coef['Bxy'], Bxy.ci.upper.zhang = ci['Bxy','97.5 %'], Bxy.ci.lower.zhang = ci['Bxy','2.5 %'], Bzy.est.zhang = coef['Bzy'], Bzy.ci.upper.zhang = ci['Bzy','97.5 %'], Bzy.ci.lower.zhang = ci['Bzy','2.5 %'])) }, error = function(e){ message("An error occurred:\n",e) result$error <- paste0(result$error,'\n', e) }) ## 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","w_pred"), noms=noms) ## mod.amelia.nok <- zelig(y~x.obs+g, model='ls', data=amelia.out.nok$imputations, cite=FALSE) ## (coefse <- combine_coef_se(mod.amelia.nok, messages=FALSE)) ## est.x.mi <- coefse['x.obs','Estimate'] ## est.x.se <- coefse['x.obs','Std.Error'] ## result <- append(result, ## list(Bxy.est.amelia.nok = est.x.mi, ## Bxy.ci.upper.amelia.nok = est.x.mi + 1.96 * est.x.se, ## Bxy.ci.lower.amelia.nok = est.x.mi - 1.96 * est.x.se ## )) ## est.g.mi <- coefse['g','Estimate'] ## est.g.se <- coefse['g','Std.Error'] ## result <- append(result, ## list(Bgy.est.amelia.nok = est.g.mi, ## Bgy.ci.upper.amelia.nok = est.g.mi + 1.96 * est.g.se, ## Bgy.ci.lower.amelia.nok = est.g.mi - 1.96 * est.g.se ## )) N <- nrow(df) m <- nrow(df[!is.na(x.obs)]) p <- v <- train <- rep(0,N) M <- m p[(M+1):(N)] <- 1 v[1:(M)] <- 1 df <- df[order(x.obs)] y <- df[,y] x <- df[,x.obs] z <- df[,z] w <- df[,w_pred] # gmm gets pretty close (gmm.res <- predicted_covariates(y, x, z, w, v, train, p, max_iter=100, verbose=TRUE)) result <- append(result, list(Bxy.est.gmm = gmm.res$beta[1,1], Bxy.ci.upper.gmm = gmm.res$confint[1,2], Bxy.ci.lower.gmm = gmm.res$confint[1,1], gmm.ER_pval = gmm.res$ER_pval )) result <- append(result, list(Bzy.est.gmm = gmm.res$beta[2,1], Bzy.ci.upper.gmm = gmm.res$confint[2,2], Bzy.ci.lower.gmm = gmm.res$confint[2,1])) tryCatch({ mod.calibrated.mle <- mecor(y ~ MeasError(w_pred, reference = x.obs) + z, 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.ci.upper.mecor = mecor.ci['UCI'], Bxy.ci.lower.mecor = mecor.ci['LCI']) ) (mecor.ci <- summary(mod.calibrated.mle)$c$ci['z',]) result <- append(result, list( Bzy.est.mecor = mecor.ci['Estimate'], Bzy.ci.upper.mecor = mecor.ci['UCI'], Bzy.ci.lower.mecor = mecor.ci['LCI']) ) }, error = function(e){ message("An error occurred:\n",e) result$error <- paste0(result$error, '\n', e) } ) ## 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")) ## gc() return(result) }