From: Nathan TeBlunthuis <nathante@uw.edu>
Date: Fri, 1 Jul 2022 02:00:35 +0000 (-0700)
Subject: add mle methods from carroll
X-Git-Url: https://code.communitydata.science/ml_measurement_error_public.git/commitdiff_plain/588bdd7ed74cf8fe8fd0f15df58a6a40c26ebae5?ds=inline;hp=003733f22f42b435315803fd5f47d483c712d72d

add mle methods from carroll
---

diff --git a/simulations/01_two_covariates.R b/simulations/01_two_covariates.R
index 419403d..7b8e12e 100644
--- a/simulations/01_two_covariates.R
+++ b/simulations/01_two_covariates.R
@@ -50,8 +50,8 @@ simulate_data <- function(N, m, B0=0, Bxy=0.2, Bgy=-0.2, Bgx=0.2, y_explained_va
     }
 
     df <- df[,w_pred:=x]
-
     df <- df[sample(1:N,(1-prediction_accuracy)*N),w_pred:=(w_pred-1)**2]
+    w <- predict(glm(x ~ w_pred,data=df,family=binomial(link='logit')),type='response')
     df <- df[,':='(w=w, w_pred = w_pred)]
     return(df)
 }
@@ -61,15 +61,20 @@ parser <- add_argument(parser, "--N", default=500, help="number of observations
 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_1.feather')
+parser <- add_argument(parser, "--y_explained_variance", help='what proportion of the variance of y can be explained?', default=0.005)
+parser <- add_argument(parser, "--gx_explained_variance", help='what proportion of the variance of x can be explained by g?', default=0.15)
+parser <- add_argument(parser, "--prediction_accuracy", help='how accurate is the predictive model?', default=0.73)
+
 args <- parse_args(parser)
 
 B0 <- 0
 Bxy <- 0.2
 Bgy <- -0.2
-Bgx <- 0.5
+Bgx <- 0.4
+
+df <- simulate_data(args$N, args$m, B0, Bxy, Bgy, Bgx, seed=args$seed, y_explained_variance = args$y_explained_variance, gx_explained_variance = args$gx_explained_variance, prediction_accuracy=args$prediction_accuracy)
 
-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)
+result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy,'Bgy'=Bgy, 'Bgx'=Bgx, 'seed'=args$seed, 'y_explained_variance' = args$y_explained_variance, 'gx_explained_variance' = args$gx_explained_variance, "prediction_accuracy"=args$prediction_accuracy)
 outline <- run_simulation(df, result)
 
 outfile_lock <- lock(paste0(args$outfile, '_lock'),exclusive=TRUE)
diff --git a/simulations/02_indep_differential.R b/simulations/02_indep_differential.R
index 5a7784b..d4e0916 100644
--- a/simulations/02_indep_differential.R
+++ b/simulations/02_indep_differential.R
@@ -31,68 +31,70 @@ 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_data <- function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed, xy.explained.variance=0.01, u.explained.variance=0.1){
+simulate_data <- function(N, m, B0, Bxy, Bgy, seed, y_explained_variance=0.025, prediction_accuracy=0.73, accuracy_imbalance_difference=0.3){
     set.seed(seed)
-
-    ## the true value of x
-
-    g <- rbinom(N, 1, 0.5)
-
     # make w and y dependent
-    u <- rnorm(N,0,)
-
-    xprime <- Bgx * g + rnorm(N,0,1) 
-
-    k <- Bkx*xprime + rnorm(N,0,1.5) + 1.1*Bkx*u
-
-    x <- as.integer(logistic(scale(xprime)) > 0.5)
-
-    y <-  Bxy * x  + Bgy * g + B0 + u + rnorm(N, 0, 1)
+    g <- rbinom(N, 1, 0.5)
+    x <- rbinom(N, 1, 0.5)
 
-    df <- data.table(x=x,k=k,y=y,g=g)
+    y.var.epsilon <- (var(Bgy * g) + var(Bxy *x) + 2*cov(Bgy*g,Bxy*x)) * ((1-y_explained_variance)/y_explained_variance)
+    y.epsilon <- rnorm(N, sd = sqrt(y.var.epsilon))
+    y <- Bgy * g + Bxy * x + y.epsilon
 
-    w.model <- glm(x ~ k,df, family=binomial(link='logit')) 
+    df <- data.table(x=x,y=y,g=g)
 
-    if( m < N){
+    if(m < N){
         df <- df[sample(nrow(df), m), x.obs := x]
     } else {
         df <- df[, x.obs := x]
     }
 
-    df[, x.obs := x.obs]
+    df <- df[,w_pred:=x]
 
-    w <- predict(w.model, df) + rnorm(N, 0, 1)
-    ## y = B0 + B1x + e
+    pg <- mean(g)
+    px <- mean(x)
+    accuracy_imbalance_ratio <- (prediction_accuracy + accuracy_imbalance_difference/2) / (prediction_accuracy - accuracy_imbalance_difference/2)
 
-    df[,':='(w=w, w_pred = as.integer(w>0.5),u=u)]
-    return(df)
-}
+    # this works because of conditional probability
+    accuracy_g0 <- prediction_accuracy / (pg*(accuracy_imbalance_ratio) + (1-pg))
+    accuracy_g1 <- accuracy_imbalance_ratio * accuracy_g0
 
-schennach <- function(df){
+    dfg0 <- df[g==0]
+    ng0 <- nrow(dfg0)
+    dfg1 <- df[g==1]
+    ng1 <- nrow(dfg1)
 
-    fwx <- glm(x.obs~w, df, family=binomial(link='logit'))
-    df[,xstar_pred := predict(fwx, df)]
-    gxt <- lm(y ~ xstar_pred+g, df)
+    dfg0 <- dfg0[sample(ng0, (1-accuracy_g0)*ng0), w_pred := (w_pred-1)**2]
+    dfg1 <- dfg1[sample(ng1, (1-accuracy_g1)*ng1), w_pred := (w_pred-1)**2]
 
-}
+    df <- rbind(dfg0,dfg1)
 
+    w <- predict(glm(x ~ w_pred,data=df,family=binomial(link='logit')),type='response')
+    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=5000, help="number of observations of w")
 parser <- add_argument(parser, "--m", default=200, help="m the number of ground truth observations")
 parser <- add_argument(parser, "--seed", default=432, help='seed for the rng')
 parser <- add_argument(parser, "--outfile", help='output file', default='example_2.feather')
+parser <- add_argument(parser, "--y_explained_variance", help='what proportion of the variance of y can be explained?', default=0.01)
+parser <- add_argument(parser, "--prediction_accuracy", help='how accurate is the predictive model?', default=0.73)
+parser <- add_argument(parser, "--accuracy_imbalance_difference", help='how much more accurate is the predictive model for one class than the other?', default=0.3)
+
 args <- parse_args(parser)
 
 B0 <- 0
 Bxy <- 0.2
-Bgy <- 0
-Bkx <- 2
-Bgx <- 0
+Bgy <- -0.2
+
+df <- simulate_data(args$N, args$m, B0, Bxy, Bgy, args$seed, args$y_explained_variance, args$prediction_accuracy, args$accuracy_imbalance_difference)
+
+result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy,'Bgy'=Bgy, 'seed'=args$seed, 'y_explained_variance'=args$y_explained_variance, 'prediction_accuracy'=args$prediction_accuracy, 'accuracy_imbalance_difference'=args$accuracy_imbalance_difference)
 
+outline <- run_simulation_depvar(df=df, result)
 
-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$outfile, '_lock'),exclusive=TRUE)
 if(file.exists(args$outfile)){
diff --git a/simulations/03_depvar_differential.R b/simulations/03_depvar_differential.R
new file mode 100644
index 0000000..d52afe7
--- /dev/null
+++ b/simulations/03_depvar_differential.R
@@ -0,0 +1,113 @@
+### EXAMPLE 1: demonstrates how measurement error can lead to a type sign error in a covariate
+### What kind of data invalidates fong + tyler?
+### 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)
+setDTthreads(40)
+
+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)
+#### 
+
+## 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_data <- function(N, m, B0, Bxy, Bgy, seed, prediction_accuracy=0.73, accuracy_imbalance_difference=0.3){
+    set.seed(seed)
+    # make w and y dependent
+    g <- rbinom(N, 1, 0.5)
+    x <- rbinom(N, 1, 0.5)
+
+    ystar <- Bgy * g + Bxy * x
+    y <- rbinom(N,1,logistic(ystar))
+
+    # glm(y ~ x + g, family="binomial")
+
+    df <- data.table(x=x,y=y,ystar=ystar,g=g)
+
+    if(m < N){
+        df <- df[sample(nrow(df), m), y.obs := y]
+    } else {
+        df <- df[, y.obs := y]
+    }
+
+    df <- df[,w_pred:=y]
+
+    pg <- mean(g)
+
+    accuracy_imbalance_ratio <- (prediction_accuracy + accuracy_imbalance_difference/2) / (prediction_accuracy - accuracy_imbalance_difference/2)
+
+    # this works because of conditional probability
+    accuracy_g0 <- prediction_accuracy / (pg*(accuracy_imbalance_ratio) + (1-pg))
+    accuracy_g1 <- accuracy_imbalance_ratio * accuracy_g0
+
+    dfg0 <- df[g==0]
+    ng0 <- nrow(dfg0)
+    dfg1 <- df[g==1]
+    ng1 <- nrow(dfg1)
+
+    dfg0 <- dfg0[sample(ng0, (1-accuracy_g0)*ng0), w_pred := (w_pred-1)**2]
+    dfg1 <- dfg1[sample(ng1, (1-accuracy_g1)*ng1), w_pred := (w_pred-1)**2]
+
+    df <- rbind(dfg0,dfg1)
+
+    wmod <- glm(y.obs ~ w_pred,data=df[!is.null(y.obs)],family=binomial(link='logit'))
+    w <- predict(wmod,df,type='response')
+
+    df <- df[,':='(w=w)]
+
+    return(df)
+}
+
+parser <- arg_parser("Simulate data and fit corrected models")
+parser <- add_argument(parser, "--N", default=5000, help="number of observations of w")
+parser <- add_argument(parser, "--m", default=200, 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.feather')
+parser <- add_argument(parser, "--y_explained_variance", help='what proportion of the variance of y can be explained?', default=0.005)
+parser <- add_argument(parser, "--prediction_accuracy", help='how accurate is the predictive model?', default=0.73)
+parser <- add_argument(parser, "--accuracy_imbalance_difference", help='how much more accurate is the predictive model for one class than the other?', default=0.3)
+
+args <- parse_args(parser)
+
+B0 <- 0
+Bxy <- 0.2
+Bgy <- -0.2
+
+df <- simulate_data(args$N, args$m, B0, Bxy, Bgy, args$seed, args$prediction_accuracy, args$accuracy_imbalance_difference)
+
+result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy,'Bgy'=Bgy, 'seed'=args$seed, 'y_explained_variance'=args$y_explained_variance, 'prediction_accuracy'=args$prediction_accuracy, 'accuracy_imbalance_difference'=args$accuracy_imbalance_difference)
+
+outline <- run_simulation_depvar(df=df, result)
+
+
+outfile_lock <- lock(paste0(args$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/Makefile b/simulations/Makefile
index 97b5894..01c34fc 100644
--- a/simulations/Makefile
+++ b/simulations/Makefile
@@ -17,12 +17,12 @@ example_1.feather: example_1_jobs
 	sbatch --wait --verbose --array=3001-6001 run_simulation.sbatch 0 example_1_jobs
 
 example_2_jobs: example_2.R
-	grid_sweep.py --command "Rscript example_2.R" --arg_dict '{"N":${Ns},"m":${ms}, "seed":${seeds}, "outfile":["example_2.feather"]}' --outfile example_2_jobs
+	grid_sweep.py --command "Rscript 02_indep_differential.R" --arg_dict '{"N":${Ns},"m":${ms}, "seed":${seeds}, "outfile":["example_2.feather"]}' --outfile example_2_jobs
 
 example_2.feather: example_2_jobs
 	rm -f example_2.feather
 	sbatch --wait --verbose --array=1-3000 run_simulation.sbatch 0 example_2_jobs
-#	sbatch --wait --verbose --array=3001-6001 run_simulation.sbatch 0 example_2_jobs
+	sbatch --wait --verbose --array=3001-6001 run_simulation.sbatch 0 example_2_jobs
 
 example_2_B_jobs: example_2_B.R
 	grid_sweep.py --command "Rscript example_2_B.R" --arg_dict '{"N":${Ns},"m":${ms}, "seed":${seeds}, "outfile":["example_2_B.feather"]}' --outfile example_2_B_jobs
diff --git a/simulations/simulation_base.R b/simulations/simulation_base.R
index 345d14e..a73ed79 100644
--- a/simulations/simulation_base.R
+++ b/simulations/simulation_base.R
@@ -4,9 +4,164 @@ options(amelia.parallel="no",
         amelia.ncpus=1)
 library(Amelia)
 library(Zelig)
+library(stats4)
+
+
+## This uses the pseudolikelihood approach from Carroll page 349.
+## assumes MAR
+## assumes differential error, but that only depends on Y
+## inefficient, because pseudolikelihood
+logistic.correction.pseudo <- function(df){
+    p1.est <- mean(df[w_pred==1]$y.obs==1,na.rm=T)
+    p0.est <- mean(df[w_pred==0]$y.obs==0,na.rm=T)
+    
+    nll <- function(B0, Bxy, Bgy){
+        probs <- (1 - p0.est) + (p1.est + p0.est - 1)*plogis(B0 + Bxy * df$x + Bgy * df$g)
+
+        part1 = sum(log(probs[df$w_pred == 1]))
+        part2 = sum(log(1-probs[df$w_pred == 0]))
+        
+        return(-1*(part1 + part2))
+    }
+    
+    mlefit <- stats4::mle(minuslogl = nll, start = list(B0=0, Bxy=0.0, Bgy=0.0))
+    return(mlefit)
+
+}
+
+## This uses the likelihood approach from Carroll page 353.
+## assumes that we have a good measurement error model
+logistic.correction.liklihood <- function(df){
+    
+    ## liklihood for observed responses
+    nll <- function(B0, Bxy, Bgy, gamma0, gamma_y, gamma_g){
+        df.obs <- df[!is.na(y.obs)]
+        p.y.obs <- plogis(B0 + Bxy * df.obs$x + Bgy*df.obs$g)
+        p.y.obs[df.obs$y==0] <- 1-p.y.obs[df.obs$y==0]
+        p.s.obs <- plogis(gamma0 + gamma_y * df.obs$y + gamma_g*df.obs$g)
+        p.s.obs[df.obs$w_pred==0] <- 1 - p.s.obs[df.obs$w_pred==0]
+        
+        p.obs <- p.s.obs * p.y.obs
+
+        df.unobs <- df[is.na(y.obs)]
+
+        p.unobs.1 <- plogis(B0 + Bxy * df.unobs$x + Bgy*df.unobs$g)*plogis(gamma0 + gamma_y + gamma_g*df.unobs$g)
+        p.unobs.0 <- (1-plogis(B0 + Bxy * df.unobs$x + Bgy*df.unobs$g))*plogis(gamma0 + gamma_g*df.unobs$g)
+        p.unobs <- p.unobs.1 + p.unobs.0
+        p.unobs[df.unobs$w_pred==0] <- 1 - p.unobs[df.unobs$w_pred==0]
+
+        p <- c(p.obs, p.unobs)
+
+        return(-1*(sum(log(p))))
+    }
+
+    mlefit <- stats4::mle(minuslogl = nll, start = list(B0=1, Bxy=0,Bgy=0, gamma0=5, gamma_y=0, gamma_g=0))
+
+    return(mlefit)
+}
+
 
 logistic <- function(x) {1/(1+exp(-1*x))}
 
+run_simulation_depvar <- function(df, result){
+
+    accuracy <- df[,mean(w_pred==y)]
+    result <- append(result, list(accuracy=accuracy))
+
+    (model.true <- glm(y ~ x + g, data=df,family=binomial(link='logit')))
+    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 <- glm(y.obs~x+g,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.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 <- glm(w_pred~x+g, data=df, family=binomial(link='logit')))
+
+    naive.ci.Bxy <- confint(model.naive)['x',]
+    naive.ci.Bgy <- confint(model.naive)['g',]
+
+    result <- append(result, list(Bxy.est.naive=coef(model.naive)['x'],
+                                  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]))
+
+
+    (model.naive.cont <- lm(w~x+g, data=df))
+    naivecont.ci.Bxy <- confint(model.naive.cont)['x',]
+    naivecont.ci.Bgy <- confint(model.naive.cont)['g',]
+
+    ## my implementatoin of liklihood based correction
+    mod.caroll.lik <- logistic.correction.liklihood(df)
+    coef <- coef(mod.caroll.lik)
+    ci <- confint(mod.caroll.lik)
+
+    result <- append(result,
+                     list(Bxy.est.mle = coef['Bxy'],
+                          Bxy.ci.upper.mle = ci['Bxy','97.5 %'],
+                          Bxy.ci.lower.mle = ci['Bxy','2.5 %'],
+                          Bgy.est.mle = coef['Bgy'],
+                          Bgy.ci.upper.mle = ci['Bgy','97.5 %'],
+                          Bgy.ci.lower.mle = ci['Bgy','2.5 %']))
+                          
+
+    ## my implementatoin of liklihood based correction
+    mod.caroll.pseudo <- logistic.correction.pseudo(df)
+    coef <- coef(mod.caroll.pseudo)
+    ci <- confint(mod.caroll.pseudo)
+
+    result <- append(result,
+                     list(Bxy.est.pseudo = coef['Bxy'],
+                          Bxy.ci.upper.pseudo = ci['Bxy','97.5 %'],
+                          Bxy.ci.lower.pseudo = ci['Bxy','2.5 %'],
+                          Bgy.est.pseudo = coef['Bgy'],
+                          Bgy.ci.upper.pseudo = ci['Bgy','97.5 %'],
+                          Bgy.ci.lower.pseudo = ci['Bgy','2.5 %']))
+                          
+
+    # amelia says use normal distribution for binary variables.
+    amelia.out.k <- amelia(df, m=200, p2s=0, idvars=c('y','ystar','w_pred'))
+    mod.amelia.k <- zelig(y.obs~x+g, 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.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
+                          ))
+
+    return(result)
+
+}
+
 run_simulation <-  function(df, result){
 
     accuracy <- df[,mean(w_pred==x)]
@@ -48,19 +203,7 @@ run_simulation <-  function(df, result){
                                   Bgy.ci.lower.naive = naive.ci.Bgy[1]))
                                   
 
-    ## multiple imputation when k is observed
-    ## amelia does great at this one.
-    noms <- c()
-    if(length(unique(df$x.obs)) <=2){
-        noms <- c(noms, 'x.obs')
-    }
-
-    if(length(unique(df$g)) <=2){
-        noms <- c(noms, 'g')
-    }
-
-
-    amelia.out.k <- amelia(df, m=200, p2s=0, idvars=c('x','w_pred'),noms=noms)
+    amelia.out.k <- amelia(df, m=200, p2s=0, idvars=c('x','w_pred'))
     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))