188 lines
7.1 KiB
R
188 lines
7.1 KiB
R
### EXAMPLE 2: demonstrates how measurement error can lead to a type sign error in a covariate
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### Even when you have a good predictor, if it's biased against a covariate you can get the wrong sign.
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### Even when you include the proxy variable in the regression.
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### But with some ground truth and multiple imputation, you can fix it.
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library(argparser)
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library(mecor)
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library(ggplot2)
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library(data.table)
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library(filelock)
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library(arrow)
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library(Amelia)
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library(Zelig)
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library(predictionError)
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options(amelia.parallel="multicore",
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amelia.ncpus=40)
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## SETUP:
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### we want to estimate g -> y and x -> y; g is observed, x is MAR
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### we have k -> x; g -> x; g->k; k is used to predict x via the model w.
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### we have k -> w; x -> w; w is observed.
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### for illustration, g is binary (e.g., gender==male).
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### A realistic scenario is that we have an NLP model predicting something like "racial harassment" in social media comments
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### Whether a comment is "racial harassment" depends on context, like the kind of person (i.e.,) the race of the person making the comment
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### e.g., a Black person saying "n-word" is less likely to be racial harassement than if a white person does it.
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### Say we have a language model that predicts "racial harassment," but it doesn't know the race of the writer.
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### Our content analyzers can see signals of the writer's race (e.g., a profile or avatar). So our "ground truth" takes this into accont.
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### Our goal is to predict an outcome (say that someone gets banned from the platform) as a function of whether they made a racial harassing comment and of their race.
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### simulation:
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#### how much power do we get from the model in the first place? (sweeping N and m)
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####
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logistic <- function(x) {1/(1+exp(-1*x))}
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simulate_latent_cocause <- function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed){
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set.seed(seed)
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## the true value of x
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g <- rbinom(N, 1, 0.5)
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k <- rnorm(N, 0, 1)
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x <- Bkx*k + Bgx * g + rnorm(N,0,1)
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w.model <- lm(x ~ k)
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w <- predict(w.model,data.frame(k=k)) + rnorm(N,0,1)
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## y = B0 + B1x + e
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y <- Bxy * x + Bgy * g + rnorm(N, 0, 1) + B0
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df <- data.table(x=x,k=k,y=y,w=w,g=g)
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if( m < N){
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df <- df[sample(nrow(df), m), x.obs := x]
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} else {
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df <- df[, x.obs := x]
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}
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return(df)
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}
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run_simulation <- function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed){
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result <- list()
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df <- simulate_latent_cocause(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed)
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result <- append(result, list(N=N,
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m=m,
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B0=B0,
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Bxy=Bxy,
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Bgy=Bgy,
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Bkx=Bkx,
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seed=seed))
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correlation <- cor(df$w,df$x)
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result <- append(result, list(correlation=correlation))
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model.true <- lm(y ~ x + g, data=df)
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true.ci.Bxy <- confint(model.true)['x',]
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true.ci.Bgy <- confint(model.true)['g',]
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result <- append(result, list(Bxy.est.true=coef(model.true)['x'],
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Bgy.est.true=coef(model.true)['g'],
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Bxy.ci.upper.true = true.ci.Bxy[2],
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Bxy.ci.lower.true = true.ci.Bxy[1],
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Bgy.ci.upper.true = true.ci.Bgy[2],
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Bgy.ci.lower.true = true.ci.Bgy[1]))
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model.naive <- lm(y~w+g, data=df)
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naive.ci.Bxy <- confint(model.naive)['w',]
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naive.ci.Bgy <- confint(model.naive)['g',]
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result <- append(result, list(Bxy.est.naive=coef(model.naive)['w'],
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Bgy.est.naive=coef(model.naive)['g'],
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Bxy.ci.upper.naive = naive.ci.Bxy[2],
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Bxy.ci.lower.naive = naive.ci.Bxy[1],
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Bgy.ci.upper.naive = naive.ci.Bgy[2],
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Bgy.ci.lower.naive = naive.ci.Bgy[1]))
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## multiple imputation when k is observed
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amelia.out.k <- amelia(df, m=200, p2s=0, idvars=c('x'))
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mod.amelia.k <- zelig(y~x.obs+g+k, model='ls', data=amelia.out.k$imputations, cite=FALSE)
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coefse <- combine_coef_se(mod.amelia.k, messages=FALSE)
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est.x.mi <- coefse['x.obs','Estimate']
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est.x.se <- coefse['x.obs','Std.Error']
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result <- append(result,
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list(Bxy.est.amelia.full = est.x.mi,
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Bxy.ci.upper.amelia.full = est.x.mi + 1.96 * est.x.se,
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Bxy.ci.lower.amelia.full = est.x.mi - 1.96 * est.x.se
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))
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est.g.mi <- coefse['g','Estimate']
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est.g.se <- coefse['g','Std.Error']
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result <- append(result,
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list(Bgy.est.amelia.full = est.g.mi,
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Bgy.ci.upper.amelia.full = est.g.mi + 1.96 * est.g.se,
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Bgy.ci.lower.amelia.full = est.g.mi - 1.96 * est.g.se
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))
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## What if we can't observe k -- most realistic scenario. We can't include all the ML features in a model.
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amelia.out.nok <- amelia(df, m=200, p2s=0, idvars=c("x","k"))
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mod.amelia.nok <- zelig(y~x.obs+g, model='ls', data=amelia.out.nok$imputations, cite=FALSE)
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coefse <- combine_coef_se(mod.amelia.nok, messages=FALSE)
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est.x.mi <- coefse['x.obs','Estimate']
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est.x.se <- coefse['x.obs','Std.Error']
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result <- append(result,
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list(Bxy.est.amelia.nok = est.x.mi,
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Bxy.ci.upper.amelia.nok = est.x.mi + 1.96 * est.x.se,
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Bxy.ci.lower.amelia.nok = est.x.mi - 1.96 * est.x.se
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))
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est.g.mi <- coefse['g','Estimate']
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est.g.se <- coefse['g','Std.Error']
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result <- append(result,
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list(Bgy.est.amelia.nok = est.g.mi,
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Bgy.ci.upper.amelia.nok = est.g.mi + 1.96 * est.g.se,
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Bgy.ci.lower.amelia.nok = est.g.mi - 1.96 * est.g.se
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))
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p <- v <- train <- rep(0,N)
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M <- m
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p[(M+1):(N)] <- 1
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train[0:(M/2)] <- 1
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v[(M/2+1):(M)] <- 1
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df <- df[order(x.obs)]
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y <- df[,y]
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x <- df[,x.obs]
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g <- df[,g]
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w <- df[,w]
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gmm.res <- predicted_covariates(y, x, g, w, v, train, p, max_iter=100, verbose=FALSE)
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result <- append(result,
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list(Bxy.est.gmm = gmm.res$beta[1,1],
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Bxy.ci.upper.gmm = gmm.res$confint[1,2],
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Bxy.ci.lower.gmm = gmm.res$confint[1,1],
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Bgy.est.gmm = gmm.res$beta[2,1],
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Bgy.ci.upper.gmm = gmm.res$confint[2,2],
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Bgy.ci.lower.gmm = gmm.res$confint[2,1]))
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return(result)
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}
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Ns <- c(100, 200, 300, 400, 500, 1000, 2500, 5000, 7500)
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ms <- c(30, 50, 100, 200, 300, 500)
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B0 <- 0
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Bxy <- 1
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Bgy <- 0.3
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Bkx <- 3
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Bgx <- -4
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seeds <- 1:100
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rows <- list()
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for(N in Ns){
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print(N)
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for(m in ms){
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if(N>m){
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for(seed in seeds){
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rows <- append(rows, list(run_simulation(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed)))
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}
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}
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}
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}
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result <- rbindlist(rows)
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write_feather(result, "example_2_simulation_continuous.feather")
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