parser <- add_argument(parser, "--y_explained_variance", help='what proportion of the variance of y can be explained?', default=0.05)
# parser <- add_argument(parser, "--zx_explained_variance", help='what proportion of the variance of x can be explained by z?', default=0.3)
parser <- add_argument(parser, "--prediction_accuracy", help='how accurate is the predictive model?', default=0.73)
-parser <- add_argument(parser, "--Bzx", help='coefficient of z on x?', default=1)
-args <- parse_args(parser)
+parser <- add_argument(parser, "--outcome_formula", help='formula for the outcome variable', default="y~x+z")
+parser <- add_argument(parser, "--proxy_formula", help='formula for the proxy variable', default="w_pred~x")
+
+parser <- add_argument(parser, "--truth_formula", help='formula for the true variable', default="x~z")
+parser <- add_argument(parser, "--Bzx", help='Effect of z on x', default=0.3)
+parser <- add_argument(parser, "--Bzy", help='Effect of z on y', default=-0.3)
+parser <- add_argument(parser, "--Bxy", help='Effect of z on y', default=0.3)
+args <- parse_args(parser)
B0 <- 0
-Bxy <- 0.3
-Bzy <- -0.3
+Bxy <- args$Bxy
+Bzy <- args$Bzy
Bzx <- args$Bzx
if (args$m < args$N){
df <- simulate_data(args$N, args$m, B0, Bxy, Bzy, Bzx, seed=args$seed + 500, y_explained_variance = args$y_explained_variance, prediction_accuracy=args$prediction_accuracy)
- result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy,'Bzy'=Bzy, 'Bzx'=Bzx, 'seed'=args$seed, 'y_explained_variance' = args$y_explained_variance, 'zx_explained_variance' = args$zx_explained_variance, "prediction_accuracy"=args$prediction_accuracy, "error"="")
+ result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy, Bzx=Bzx, 'Bzy'=Bzy, 'seed'=args$seed, 'y_explained_variance'=args$y_explained_variance, 'prediction_accuracy'=args$prediction_accuracy, 'accuracy_imbalance_difference'=args$accuracy_imbalance_difference, 'outcome_formula'=args$outcome_formula, 'proxy_formula'=args$proxy_formula,truth_formula=args$truth_formula, error='')
- outline <- run_simulation(df, result)
+ outline <- run_simulation(df, result, outcome_formula=as.formula(args$outcome_formula), proxy_formula=as.formula(args$proxy_formula), truth_formula=as.formula(args$truth_formula))
outfile_lock <- lock(paste0(args$outfile, '_lock'),exclusive=TRUE)
if(file.exists(args$outfile)){
## 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, Bzx, Bzy, seed, y_explained_variance=0.025, prediction_accuracy=0.73, y_bias=-0.8){
+simulate_data <- function(N, m, B0, Bxy, Bzx, Bzy, seed, y_explained_variance=0.025, prediction_accuracy=0.73, y_bias=-0.8,accuracy_imbalance_difference=0.3){
set.seed(seed)
# make w and y dependent
- z <- rbinom(N, 1, 0.5)
- x <- rbinom(N, 1, Bzx * z + 0.5)
+ z <- rbinom(N, 1, plogis(qlogis(0.5)))
+ x <- rbinom(N, 1, plogis(Bzx * z + qlogis(0.5)))
y.var.epsilon <- (var(Bzy * z) + var(Bxy *x) + 2*cov(Bzy*z,Bxy*x)) * ((1-y_explained_variance)/y_explained_variance)
y.epsilon <- rnorm(N, sd = sqrt(y.var.epsilon))
}
## probablity of an error is correlated with y
- p.correct <- plogis(y_bias*scale(y) + qlogis(prediction_accuracy))
+ ## pz <- mean(z)
+ ## accuracy_imbalance_ratio <- (prediction_accuracy + accuracy_imbalance_difference/2) / (prediction_accuracy - accuracy_imbalance_difference/2)
- acc.x0 <- p.correct[df[,x==0]]
- acc.x1 <- p.correct[df[,x==1]]
+ ## # this works because of conditional probability
+ ## accuracy_z0 <- prediction_accuracy / (pz*(accuracy_imbalance_ratio) + (1-pz))
+ ## accuracy_z1 <- accuracy_imbalance_ratio * accuracy_z0
- df[x==0,w:=rlogis(.N,qlogis(1-acc.x0))]
- df[x==1,w:=rlogis(.N,qlogis(acc.x1))]
+ ## z0x0 <- df[(z==0) & (x==0)]$x
+ ## z0x1 <- df[(z==0) & (x==1)]$x
+ ## z1x0 <- df[(z==1) & (x==0)]$x
+ ## z1x1 <- df[(z==1) & (x==1)]$x
- df[,w_pred := as.integer(w>0.5)]
+ ## yz0x0 <- df[(z==0) & (x==0)]$y
+ ## yz0x1 <- df[(z==0) & (x==1)]$y
+ ## yz1x0 <- df[(z==1) & (x==0)]$y
+ ## yz1x1 <- df[(z==1) & (x==1)]$y
+
+ ## nz0x0 <- nrow(df[(z==0) & (x==0)])
+ ## nz0x1 <- nrow(df[(z==0) & (x==1)])
+ ## nz1x0 <- nrow(df[(z==1) & (x==0)])
+ ## nz1x1 <- nrow(df[(z==1) & (x==1)])
+
+ ## yz1 <- df[z==1]$y
+ ## yz1 <- df[z==1]$y
+
+ ## # tranform yz0.1 into a logistic distribution with mean accuracy_z0
+ ## acc.z0x0 <- plogis(0.5*scale(yz0x0) + qlogis(accuracy_z0))
+ ## acc.z0x1 <- plogis(0.5*scale(yz0x1) + qlogis(accuracy_z0))
+ ## acc.z1x0 <- plogis(1.5*scale(yz1x0) + qlogis(accuracy_z1))
+ ## acc.z1x1 <- plogis(1.5*scale(yz1x1) + qlogis(accuracy_z1))
+
+ ## w0z0x0 <- (1-z0x0)**2 + (-1)**(1-z0x0) * acc.z0x0
+ ## w0z0x1 <- (1-z0x1)**2 + (-1)**(1-z0x1) * acc.z0x1
+ ## w0z1x0 <- (1-z1x0)**2 + (-1)**(1-z1x0) * acc.z1x0
+ ## w0z1x1 <- (1-z1x1)**2 + (-1)**(1-z1x1) * acc.z1x1
+
+ ## ##perrorz0 <- w0z0*(pyz0)
+ ## ##perrorz1 <- w0z1*(pyz1)
+
+ ## w0z0x0.noisy.odds <- rlogis(nz0x0,qlogis(w0z0x0))
+ ## w0z0x1.noisy.odds <- rlogis(nz0x1,qlogis(w0z0x1))
+ ## w0z1x0.noisy.odds <- rlogis(nz1x0,qlogis(w0z1x0))
+ ## w0z1x1.noisy.odds <- rlogis(nz1x1,qlogis(w0z1x1))
+
+ ## df[(z==0)&(x==0),w:=plogis(w0z0x0.noisy.odds)]
+ ## df[(z==0)&(x==1),w:=plogis(w0z0x1.noisy.odds)]
+ ## df[(z==1)&(x==0),w:=plogis(w0z1x0.noisy.odds)]
+ ## df[(z==1)&(x==1),w:=plogis(w0z1x1.noisy.odds)]
+
+ ## df[,w_pred:=as.integer(w > 0.5)]
+ ## print(mean(df[z==0]$x == df[z==0]$w_pred))
+ ## print(mean(df[z==1]$x == df[z==1]$w_pred))
+ ## print(mean(df$w_pred == df$x))
+
+ odds.x1 <- qlogis(prediction_accuracy) + y_bias*qlogis(pnorm(scale(df[x==1]$y)))
+ odds.x0 <- qlogis(prediction_accuracy,lower.tail=F) + y_bias*qlogis(pnorm(scale(df[x==0]$y)))
+
+ ## acc.x0 <- p.correct[df[,x==0]]
+ ## acc.x1 <- p.correct[df[,x==1]]
+
+ df[x==0,w:=plogis(rlogis(.N,odds.x0))]
+ df[x==1,w:=plogis(rlogis(.N,odds.x1))]
+
+ df[,w_pred := as.integer(w > 0.5)]
print(mean(df[z==0]$x == df[z==0]$w_pred))
print(mean(df[z==1]$x == df[z==1]$w_pred))
print(mean(df$w_pred == df$x))
print(mean(df[y>=0]$w_pred == df[y>=0]$x))
print(mean(df[y<=0]$w_pred == df[y<=0]$x))
-
return(df)
}
parser <- arg_parser("Simulate data and fit corrected models")
parser <- add_argument(parser, "--N", default=1000, help="number of observations of w")
-parser <- add_argument(parser, "--m", default=500, help="m the number of ground truth observations")
+aparser <- add_argument(parser, "--m", default=500, help="m the number of ground truth observations")
parser <- add_argument(parser, "--seed", default=51, 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, "--y_explained_variance", help='what proportion of the variance of y can be explained?', default=0.1)
+parser <- add_argument(parser, "--prediction_accuracy", help='how accurate is the predictive model?', default=0.8)
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)
parser <- add_argument(parser, "--Bzx", help='Effect of z on x', default=0.3)
parser <- add_argument(parser, "--Bzy", help='Effect of z on y', default=-0.3)
parser <- add_argument(parser, "--Bxy", help='Effect of z on y', default=0.3)
-parser <- add_argument(parser, "--proxy_formula", help='formula for the proxy variable', default="w_pred~x*y")
+parser <- add_argument(parser, "--outcome_formula", help='formula for the outcome variable', default="y~x+z")
+parser <- add_argument(parser, "--proxy_formula", help='formula for the proxy variable', default="w_pred~y*z*x")
parser <- add_argument(parser, "--y_bias", help='coefficient of y on the probability a classification is correct', default=-0.75)
+parser <- add_argument(parser, "--truth_formula", help='formula for the true variable', default="x~z")
args <- parse_args(parser)
df <- simulate_data(args$N, args$m, B0, Bxy, Bzx, Bzy, args$seed, args$y_explained_variance, args$prediction_accuracy, y_bias=args$y_bias)
- ## df.pc <- df[,.(x,y,z,w_pred)]
+ ## df.pc <- df[,.(x,y,z,w_pred,w)]
## # df.pc <- df.pc[,err:=x-w_pred]
## pc.df <- pc(suffStat=list(C=cor(df.pc),n=nrow(df.pc)),indepTest=gaussCItest,labels=names(df.pc),alpha=0.05)
## plot(pc.df)
- result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy, Bzx=args$Bzx, 'Bzy'=Bzy, 'seed'=args$seed, 'y_explained_variance'=args$y_explained_variance, 'prediction_accuracy'=args$prediction_accuracy, 'accuracy_imbalance_difference'=args$accuracy_imbalance_difference, 'y_bias'=args$y_bias,error='')
+ result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy, Bzx=args$Bzx, 'Bzy'=Bzy, 'seed'=args$seed, 'y_explained_variance'=args$y_explained_variance, 'prediction_accuracy'=args$prediction_accuracy, 'accuracy_imbalance_difference'=args$accuracy_imbalance_difference, 'y_bias'=args$y_bias,'outcome_formula'=args$outcome_formula, 'proxy_formula'=args$proxy_formula,truth_formula=args$truth_formula, error='')
- outline <- run_simulation(df, result, outcome_formula=y~x+z, proxy_formula=as.formula(args$proxy_formula), truth_formula=x~z)
+ outline <- run_simulation(df, result, outcome_formula=as.formula(args$outcome_formula), proxy_formula=as.formula(args$proxy_formula), truth_formula=as.formula(args$truth_formula))
- outfile_lock <- lock(paste0(args$outfile, '_lock'),exclusive=TRUE)
+
+ 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), fill=TRUE)
--- /dev/null
+### 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, Bzy, seed, prediction_accuracy=0.73){
+ set.seed(seed)
+
+ # make w and y dependent
+ z <- rbinom(N, 1, 0.5)
+ x <- rbinom(N, 1, 0.5)
+
+ ystar <- Bzy * z + Bxy * x + B0
+ y <- rbinom(N,1,plogis(ystar))
+
+ # glm(y ~ x + z, family="binomial")
+
+ df <- data.table(x=x,y=y,ystar=ystar,z=z)
+
+ if(m < N){
+ df <- df[sample(nrow(df), m), y.obs := y]
+ } else {
+ df <- df[, y.obs := y]
+ }
+
+ odds.y1 <- qlogis(prediction_accuracy)
+ odds.y0 <- qlogis(prediction_accuracy,lower.tail=F)
+
+ df[y==0,w:=plogis(rlogis(.N,odds.y0))]
+ df[y==1,w:=plogis(rlogis(.N,odds.y1))]
+
+ df[,w_pred := as.integer(w > 0.5)]
+
+ print(mean(df[x==0]$y == df[x==0]$w_pred))
+ print(mean(df[x==1]$y == df[x==1]$w_pred))
+ print(mean(df$w_pred == df$y))
+ return(df)
+}
+
+parser <- arg_parser("Simulate data and fit corrected models")
+parser <- add_argument(parser, "--N", default=1000, help="number of observations of w")
+parser <- add_argument(parser, "--m", default=500, help="m the number of ground truth observations")
+parser <- add_argument(parser, "--seed", default=17, help='seed for the rng')
+parser <- add_argument(parser, "--outfile", help='output file', default='example_2.feather')
+parser <- add_argument(parser, "--y_explained_variance", help='what proportion of the variance of y can be explained?', default=0.005)
+parser <- add_argument(parser, "--prediction_accuracy", help='how accurate is the predictive model?', default=0.72)
+## parser <- add_argument(parser, "--x_bias_y1", help='how is the classifier biased when y = 1?', default=-0.75)
+## parser <- add_argument(parser, "--x_bias_y0", help='how is the classifier biased when y = 0 ?', default=0.75)
+parser <- add_argument(parser, "--Bxy", help='coefficient of x on y', default=0.3)
+parser <- add_argument(parser, "--Bzy", help='coeffficient of z on y', default=-0.3)
+parser <- add_argument(parser, "--outcome_formula", help='formula for the outcome variable', default="y~x+z")
+parser <- add_argument(parser, "--proxy_formula", help='formula for the proxy variable', default="w_pred~y")
+
+args <- parse_args(parser)
+
+B0 <- 0
+Bxy <- args$Bxy
+Bzy <- args$Bzy
+
+
+if(args$m < args$N){
+ df <- simulate_data(args$N, args$m, B0, Bxy, Bzy, args$seed, args$prediction_accuracy)
+
+# result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy,'Bzy'=Bzy, 'seed'=args$seed, 'y_explained_variance'=args$y_explained_variance, 'prediction_accuracy'=args$prediction_accuracy, 'x_bias_y0'=args$x_bias_y0,'x_bias_y1'=args$x_bias_y1,'outcome_formula' = args$outcome_formula, 'proxy_formula' = args$proxy_formula)
+ result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy,'Bzy'=Bzy, 'seed'=args$seed, 'y_explained_variance'=args$y_explained_variance, 'prediction_accuracy'=args$prediction_accuracy, 'outcome_formula' = args$outcome_formula, 'proxy_formula' = args$proxy_formula)
+
+ outline <- run_simulation_depvar(df, result, outcome_formula = as.formula(args$outcome_formula), proxy_formula = as.formula(args$proxy_formula))
+
+ outfile_lock <- lock(paste0(args$outfile, '_lock'),exclusive=TRUE)
+
+ if(file.exists(args$outfile)){
+ logdata <- read_feather(args$outfile)
+ logdata <- rbind(logdata,as.data.table(outline),fill=TRUE)
+ } else {
+ logdata <- as.data.table(outline)
+ }
+
+ print(outline)
+ write_feather(logdata, args$outfile)
+ unlock(outfile_lock)
+}
## one way to do it is by adding correlation to x.obs and y that isn't in w.
## in other words, the model is missing an important feature of x.obs that's related to y.
-simulate_data <- function(N, m, B0, Bxy, Bzy, seed, prediction_accuracy=0.73, accuracy_imbalance_difference=0.3){
+simulate_data <- function(N, m, B0, Bxy, Bzy, seed, prediction_accuracy=0.73, x_bias=-0.75){
set.seed(seed)
+
# make w and y dependent
z <- rbinom(N, 1, 0.5)
x <- rbinom(N, 1, 0.5)
- ystar <- Bzy * z + Bxy * x
+ ystar <- Bzy * z + Bxy * x + B0
y <- rbinom(N,1,plogis(ystar))
# glm(y ~ x + z, family="binomial")
} else {
df <- df[, y.obs := y]
}
-
- df <- df[,w_pred:=y]
-
- pz <- mean(z)
-
- accuracy_imbalance_ratio <- (prediction_accuracy + accuracy_imbalance_difference/2) / (prediction_accuracy - accuracy_imbalance_difference/2)
-
- # this works because of conditional probability
- accuracy_z0 <- prediction_accuracy / (pz*(accuracy_imbalance_ratio) + (1-pz))
- accuracy_z1 <- accuracy_imbalance_ratio * accuracy_z0
-
-
- yz0 <- df[z==0]$y
- yz1 <- df[z==1]$y
- nz1 <- nrow(df[z==1])
- nz0 <- nrow(df[z==0])
-
- acc_z0 <- plogis(0.7*scale(yz0) + qlogis(accuracy_z0))
- acc_z1 <- plogis(1.3*scale(yz1) + qlogis(accuracy_z1))
-
- w0z0 <- (1-yz0)**2 + (-1)**(1-yz0) * acc_z0
- w0z1 <- (1-yz1)**2 + (-1)**(1-yz1) * acc_z1
- w0z0.noisy.odds <- rlogis(nz0,qlogis(w0z0))
- w0z1.noisy.odds <- rlogis(nz1,qlogis(w0z1))
- df[z==0,w:=plogis(w0z0.noisy.odds)]
- df[z==1,w:=plogis(w0z1.noisy.odds)]
+ odds.y1 <- qlogis(prediction_accuracy) + x_bias*df[y==1]$x
+ odds.y0 <- qlogis(prediction_accuracy,lower.tail=F) + x_bias*df[y==0]$x
- df[,w_pred:=as.integer(w > 0.5)]
+ df[y==0,w:=plogis(rlogis(.N,odds.y0))]
+ df[y==1,w:=plogis(rlogis(.N,odds.y1))]
- print(mean(df[y==0]$y == df[y==0]$w_pred))
- print(mean(df[y==1]$y == df[y==1]$w_pred))
- print(mean(df$w_pred == df$y))
+ df[,w_pred := as.integer(w > 0.5)]
+ print(mean(df[x==0]$y == df[x==0]$w_pred))
+ print(mean(df[x==1]$y == df[x==1]$w_pred))
+ print(mean(df$w_pred == df$y))
return(df)
}
parser <- add_argument(parser, "--seed", default=17, help='seed for the rng')
parser <- add_argument(parser, "--outfile", help='output file', default='example_2.feather')
parser <- add_argument(parser, "--y_explained_variance", help='what proportion of the variance of y can be explained?', default=0.005)
-parser <- add_argument(parser, "--prediction_accuracy", help='how accurate is the predictive model?', default=0.73)
-parser <- add_argument(parser, "--accuracy_imbalance_difference", help='how much more accurate is the predictive model for one class than the other?', default=0.3)
+parser <- add_argument(parser, "--prediction_accuracy", help='how accurate is the predictive model?', default=0.8)
+## parser <- add_argument(parser, "--x_bias_y1", help='how is the classifier biased when y = 1?', default=-0.75)
+## parser <- add_argument(parser, "--x_bias_y0", help='how is the classifier biased when y = 0 ?', default=0.75)
+parser <- add_argument(parser, "--x_bias", help='how is the classifier biased?', default=0.75)
+parser <- add_argument(parser, "--Bxy", help='coefficient of x on y', default=0.3)
+parser <- add_argument(parser, "--Bzy", help='coeffficient of z on y', default=-0.3)
+parser <- add_argument(parser, "--outcome_formula", help='formula for the outcome variable', default="y~x+z")
+parser <- add_argument(parser, "--proxy_formula", help='formula for the proxy variable', default="w_pred~y*x")
args <- parse_args(parser)
B0 <- 0
-Bxy <- 0.7
-Bzy <- -0.7
+Bxy <- args$Bxy
+Bzy <- args$Bzy
+
if(args$m < args$N){
- df <- simulate_data(args$N, args$m, B0, Bxy, Bzy, args$seed, args$prediction_accuracy, args$accuracy_imbalance_difference)
+ df <- simulate_data(args$N, args$m, B0, Bxy, Bzy, args$seed, args$prediction_accuracy, args$x_bias_y0, args$x_bias_y1)
- result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy,'Bzy'=Bzy, 'seed'=args$seed, 'y_explained_variance'=args$y_explained_variance, 'prediction_accuracy'=args$prediction_accuracy, 'accuracy_imbalance_difference'=args$accuracy_imbalance_difference)
+# result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy,'Bzy'=Bzy, 'seed'=args$seed, 'y_explained_variance'=args$y_explained_variance, 'prediction_accuracy'=args$prediction_accuracy, 'x_bias_y0'=args$x_bias_y0,'x_bias_y1'=args$x_bias_y1,'outcome_formula' = args$outcome_formula, 'proxy_formula' = args$proxy_formula)
+ result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy,'Bzy'=Bzy, 'seed'=args$seed, 'y_explained_variance'=args$y_explained_variance, 'prediction_accuracy'=args$prediction_accuracy, 'x_bias'=args$x_bias,'x_bias'=args$x_bias,'outcome_formula' = args$outcome_formula, 'proxy_formula' = args$proxy_formula)
- outline <- run_simulation_depvar(df, result, outcome_formula = y ~ x + z, proxy_formula = w_pred ~ y*x + y*z + z*x)
+ outline <- run_simulation_depvar(df, result, outcome_formula = as.formula(args$outcome_formula), proxy_formula = as.formula(args$proxy_formula))
outfile_lock <- lock(paste0(args$outfile, '_lock'),exclusive=TRUE)
--- /dev/null
+### EXAMPLE 1: demonstrates how measurement error can lead to a type sign error in a covariate
+### What kind of data invalidates fong + tyler?
+### Even when you have a good predictor, if it's biased against a covariate you can get the wrong sign.
+### Even when you include the proxy variable in the regression.
+### But with some ground truth and multiple imputation, you can fix it.
+
+library(argparser)
+library(mecor)
+library(ggplot2)
+library(data.table)
+library(filelock)
+library(arrow)
+library(Amelia)
+library(Zelig)
+library(predictionError)
+options(amelia.parallel="no",
+ amelia.ncpus=1)
+setDTthreads(40)
+
+source("simulation_base.R")
+
+## SETUP:
+### we want to estimate x -> y; x is MAR
+### we have x -> k; k -> w; x -> w is used to predict x via the model w.
+### A realistic scenario is that we have an NLP model predicting something like "racial harassment" in social media comments
+### The labels x are binary, but the model provides a continuous predictor
+
+### simulation:
+#### how much power do we get from the model in the first place? (sweeping N and m)
+####
+
+## one way to do it is by adding correlation to x.obs and y that isn't in w.
+## in other words, the model is missing an important feature of x.obs that's related to y.
+simulate_data <- function(N, m, B0, Bxy, Bzy, seed, prediction_accuracy=0.73, x_bias=-0.75){
+ set.seed(seed)
+
+ # make w and y dependent
+ z <- rbinom(N, 1, 0.5)
+ x <- rbinom(N, 1, 0.5)
+
+ ystar <- Bzy * z + Bxy * x + B0
+ y <- rbinom(N,1,plogis(ystar))
+
+ # glm(y ~ x + z, family="binomial")
+
+ df <- data.table(x=x,y=y,ystar=ystar,z=z)
+
+ if(m < N){
+ df <- df[sample(nrow(df), m), y.obs := y]
+ } else {
+ df <- df[, y.obs := y]
+ }
+
+ odds.y1 <- qlogis(prediction_accuracy) + x_bias*df[y==1]$x
+ odds.y0 <- qlogis(prediction_accuracy,lower.tail=F) + x_bias*df[y==0]$x
+
+ df[y==0,w:=plogis(rlogis(.N,odds.y0))]
+ df[y==1,w:=plogis(rlogis(.N,odds.y1))]
+
+ df[,w_pred := as.integer(w > 0.5)]
+
+ print(mean(df[x==0]$y == df[x==0]$w_pred))
+ print(mean(df[x==1]$y == df[x==1]$w_pred))
+ print(mean(df$w_pred == df$y))
+ return(df)
+}
+
+parser <- arg_parser("Simulate data and fit corrected models")
+parser <- add_argument(parser, "--N", default=1000, help="number of observations of w")
+parser <- add_argument(parser, "--m", default=500, help="m the number of ground truth observations")
+parser <- add_argument(parser, "--seed", default=17, help='seed for the rng')
+parser <- add_argument(parser, "--outfile", help='output file', default='example_2.feather')
+parser <- add_argument(parser, "--y_explained_variance", help='what proportion of the variance of y can be explained?', default=0.005)
+parser <- add_argument(parser, "--prediction_accuracy", help='how accurate is the predictive model?', default=0.8)
+## parser <- add_argument(parser, "--x_bias_y1", help='how is the classifier biased when y = 1?', default=-0.75)
+## parser <- add_argument(parser, "--x_bias_y0", help='how is the classifier biased when y = 0 ?', default=0.75)
+parser <- add_argument(parser, "--x_bias", help='how is the classifier biased?', default=0.75)
+parser <- add_argument(parser, "--Bxy", help='coefficient of x on y', default=0.3)
+parser <- add_argument(parser, "--Bzy", help='coeffficient of z on y', default=-0.3)
+parser <- add_argument(parser, "--outcome_formula", help='formula for the outcome variable', default="y~x+z")
+parser <- add_argument(parser, "--proxy_formula", help='formula for the proxy variable', default="w_pred~y+x")
+
+args <- parse_args(parser)
+
+B0 <- 0
+Bxy <- args$Bxy
+Bzy <- args$Bzy
+
+
+if(args$m < args$N){
+ df <- simulate_data(args$N, args$m, B0, Bxy, Bzy, args$seed, args$prediction_accuracy, args$x_bias)
+
+# result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy,'Bzy'=Bzy, 'seed'=args$seed, 'y_explained_variance'=args$y_explained_variance, 'prediction_accuracy'=args$prediction_accuracy, 'x_bias_y0'=args$x_bias_y0,'x_bias_y1'=args$x_bias_y1,'outcome_formula' = args$outcome_formula, 'proxy_formula' = args$proxy_formula)
+ result <- list('N'=args$N,'m'=args$m,'B0'=B0,'Bxy'=Bxy,'Bzy'=Bzy, 'seed'=args$seed, 'y_explained_variance'=args$y_explained_variance, 'prediction_accuracy'=args$prediction_accuracy, 'x_bias'=args$x_bias,'outcome_formula' = args$outcome_formula, 'proxy_formula' = args$proxy_formula)
+
+ outline <- run_simulation_depvar(df, result, outcome_formula = as.formula(args$outcome_formula), proxy_formula = as.formula(args$proxy_formula))
+
+ outfile_lock <- lock(paste0(args$outfile, '_lock'),exclusive=TRUE)
+
+ if(file.exists(args$outfile)){
+ logdata <- read_feather(args$outfile)
+ logdata <- rbind(logdata,as.data.table(outline),fill=TRUE)
+ } else {
+ logdata <- as.data.table(outline)
+ }
+
+ print(outline)
+ write_feather(logdata, args$outfile)
+ unlock(outfile_lock)
+}
SHELL=bash
-Ns=[1000,3600,14400]
-ms=[75,150,300]
-seeds=[$(shell seq -s, 1 250)]
+Ns=[1000, 2000, 4000, 8000]
+ms=[100, 200, 400, 800]
+seeds=[$(shell seq -s, 1 100)]
explained_variances=[0.1]
all:remembr.RDS
# sbatch --wait --verbose --array=3001-6001 run_simulation.sbatch 0 example_1_jobs
example_2_jobs: 02_indep_differential.R simulation_base.R
- grid_sweep.py --command "Rscript 02_indep_differential.R" --arg_dict '{"N":${Ns},"m":${ms}, "seed":${seeds}, "outfile":["example_2.feather"],"y_explained_variance":${explained_variances}, "accuracy_imbalance_difference":[0.3], "Bzy":[0.3]}' --outfile example_2_jobs
+ grid_sweep.py --command "Rscript 02_indep_differential.R" --arg_dict '{"N":${Ns},"m":${ms}, "seed":${seeds}, "outfile":["example_2.feather"],"y_explained_variance":${explained_variances}, "Bzy":[-0.3],"Bxy":[0.3],"Bzx":[0.3], "outcome_formula":["y~x+z"], "proxy_formula":["w_pred~y*z*x"], "truth_formula":["x~z"]}' --outfile example_2_jobs
example_2.feather: example_2_jobs
rm -f example_2.feather
# rm -f example_2_B.feather
# sbatch --wait --verbose --array=1-3000 run_simulation.sbatch 0 example_2_B_jobs
-example_3_jobs: 03_depvar_differential.R simulation_base.R
- grid_sweep.py --command "Rscript 03_depvar_differential.R" --arg_dict '{"N":${Ns},"m":${ms}, "seed":${seeds}, "outfile":["example_3.feather"], "y_explained_variance":${explained_variances}}' --outfile example_3_jobs
+example_3_jobs: 03_depvar.R simulation_base.R
+ grid_sweep.py --command "Rscript 03_depvar.R" --arg_dict '{"N":${Ns},"m":${ms}, "seed":${seeds}, "outfile":["example_3.feather"], "y_explained_variance":${explained_variances}}' --outfile example_3_jobs
example_3.feather: example_3_jobs
rm -f example_3.feather
sbatch --wait --verbose --array=1-$(shell cat example_3_jobs | wc -l) run_simulation.sbatch 0 example_3_jobs
+example_4_jobs: 04_depvar_differential.R simulation_base.R
+ grid_sweep.py --command "Rscript 04_depvar_differential.R" --arg_dict '{"N":${Ns},"m":${ms}, "seed":${seeds}, "outfile":["example_4.feather"], "y_explained_variance":${explained_variances}}' --outfile example_4_jobs
-remembr.RDS:example_1.feather example_2.feather example_3.feather plot_example.R plot_dv_example.R
+example_4.feather: example_4_jobs
+ rm -f example_4.feather
+ sbatch --wait --verbose --array=1-$(shell cat example_4_jobs | wc -l) run_simulation.sbatch 0 example_4_jobs
+
+remembr.RDS:example_1.feather example_2.feather example_3.feather example_4.feather plot_example.R plot_dv_example.R
rm -f remembr.RDS
${srun} Rscript plot_example.R --infile example_1.feather --name "plot.df.example.1"
${srun} Rscript plot_example.R --infile example_2.feather --name "plot.df.example.2"
${srun} Rscript plot_dv_example.R --infile example_3.feather --name "plot.df.example.3"
+ ${srun} Rscript plot_dv_example.R --infile example_4.feather --name "plot.df.example.4"
clean:
rm *.feather
df.unobs.y1 <- copy(df.unobs)
df.unobs.y1[[response.var]] <- 1
df.unobs.y0 <- copy(df.unobs)
- df.unobs.y0[[response.var]] <- 1
+ df.unobs.y0[[response.var]] <- 0
## integrate out y
outcome.model.matrix.y1 <- model.matrix(outcome_formula, df.unobs.y1)
if(outcome_family$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)
}
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)
}
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
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$family=="gaussian"){
+
+ # likelihood of outcome
ll.y.x0 <- dnorm(outcome.unobs, outcome.params %*% t(outcome.model.matrix.x0), sd=sigma.y, log=TRUE)
ll.y.x1 <- dnorm(outcome.unobs, outcome.params %*% t(outcome.model.matrix.x1), sd=sigma.y, log=TRUE)
}
ll.w.x0 <- vector(mode='numeric', length=dim(df.unobs)[1])
ll.w.x1 <- vector(mode='numeric', length=dim(df.unobs)[1])
+ # likelihood of proxy
ll.w.x0[proxy.unobs==1] <- plogis(proxy.params %*% t(proxy.model.matrix.x0[proxy.unobs==1,]), log=TRUE)
ll.w.x1[proxy.unobs==1] <- plogis(proxy.params %*% t(proxy.model.matrix.x1[proxy.unobs==1,]), log=TRUE)
if(truth_family$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)
}
}
paste0('B',coefname,'y.ci.lower.',suffix),
paste0('B',coefname,'y.ci.upper.',suffix),
'y_explained_variance',
- 'Bzy',
- 'accuracy_imbalance_difference'
+ 'Bzy'
),
with=FALSE]
variable=coefname,
method=suffix
),
- by=c("N","m",'Bzy','accuracy_imbalance_difference','y_explained_variance')
+ by=c("N","m",'Bzy','y_explained_variance')
]
return(part.plot)
plot.df <- read_feather(args$infile)
+print(unique(plot.df$N))
# df <- df[apply(df,1,function(x) !any(is.na(x)))]
## Zhang got this model from Hausman 1998
### I think this is actually eqivalent to the pseudo.mle method
zhang.mle.iv <- function(df){
- nll <- function(B0=0, Bxy=0, Bzy=0, sigma_y=0.1, ppv=0.9, npv=0.9){
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
-
- ll.w1x1.obs <- with(df.obs[(w_pred==1)], dbinom(x.obs,size=1,prob=ppv,log=T))
- ll.w0x0.obs <- with(df.obs[(w_pred==0)], dbinom(1-x.obs,size=1,prob=npv,log=T))
## 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)
- ll <- ll + sum(ll.w1x1.obs) + sum(ll.w0x0.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 == 0
lls.x.0 <- colLogSumExps(rbind(log(1-npv) + ll.x.1, log(npv) + ll.x.0))
- lls <- colLogSumExps(rbind(lls.x.1, lls.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,ppv=0.00001, npv=0.00001),
- upper=list(sigma_y=Inf, B0=Inf, Bxy=Inf, Bzy=Inf, ppv=0.99999,npv=0.99999),method='L-BFGS-B')
+ 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 Carolla
-zhang.mle.dv <- function(df){
+## 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)]
+## 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))
+## ## 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))
+## ## 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)
+## }
- ## 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))
+zhang.mle.dv <- function(df){
+ df.obs <- df[!is.na(y.obs)]
+ df.unobs <- df[is.na(y.obs)]
- lls <- colLogSumExps(rbind(lls.y.1, lls.y.0))
- ll <- ll + sum(lls)
- return(-ll)
+ 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=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))
+ 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){
naivecont.ci.Bxy <- confint(model.naive.cont)['x',]
naivecont.ci.Bzy <- confint(model.naive.cont)['z',]
- ## my implementatoin of liklihood based correction
+ ## my implementation of liklihood based correction
temp.df <- copy(df)
temp.df[,y:=y.obs]
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({
## outcome_formula, proxy_formula, and truth_formula are passed to measerr_mle
-run_simulation <- function(df, result, outcome_formula=y~x+z, proxy_formula=w_pred~x, truth_formula=x~z){
+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))
tryCatch({
- amelia.out.k <- amelia(df, m=200, p2s=0, idvars=c('x','w_pred'))
+ 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))
projects / ml_measurement_error_public.git / commitdiff