--- /dev/null
+### EXAMPLE 2: demonstrates how measurement error can lead to a type sign error in a covariate
+### 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)
+
+options(amelia.parallel="multicore",
+ amelia.ncpus=40)
+
+## SETUP:
+### we want to estimate g -> y and x -> y; g is observed, x is MAR
+### we have k -> x; g -> x; g->k; k is used to predict x via the model w.
+### we have k -> w; x -> w; w is observed.
+### for illustration, g is binary (e.g., gender==male).
+### A realistic scenario is that we have an NLP model predicting something like "racial harassment" in social media comments
+### Whether a comment is "racial harassment" depends on context, like the kind of person (i.e.,) the race of the person making the comment
+### e.g., a Black person saying "n-word" is less likely to be racial harassement than if a white person does it.
+### Say we have a language model that predicts "racial harassment," but it doesn't know the race of the writer.
+### 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.
+### 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.
+
+### simulation:
+#### how much power do we get from the model in the first place? (sweeping N and m)
+####
+logistic <- function(x) {1/(1+exp(-1*x))}
+
+simulate_latent_cocause <- function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed){
+ set.seed(seed)
+
+ ## the true value of x
+
+ g <- rbinom(N, 1, 0.5)
+ k <- rnorm(N, 0, 1)
+ xprime <- Bkx*k + Bgx * g
+ x <- rbinom(N, 1, logistic(xprime - mean(xprime)))
+ w.model <- glm(x ~ k,family='binomial')
+ w <- as.integer(predict(w.model,data.frame(k=k),type='response') > 0.5)
+ ## y = B0 + B1x + e
+ y <- Bxy * x + Bgy * g + rnorm(N, 0, 1) + B0
+ df <- data.table(x=x,k=k,y=y,w=w,g=g)
+ if( m < N){
+ df <- df[sample(nrow(df), m), x.obs := x]
+ } else {
+ df <- df[, x.obs := x]
+ }
+
+ return(df)
+}
+
+
+run_simulation <- function(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed){
+ result <- list()
+ df <- simulate_latent_cocause(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed)
+
+ result <- append(result, list(N=N,
+ m=m,
+ B0=B0,
+ Bxy=Bxy,
+ Bgy=Bgy,
+ Bkx=Bkx,
+ seed=seed))
+
+ accuracy <- df[,.(mean(w==x))]$V1
+ result <- append(result, list(accuracy=accuracy))
+
+ model.true <- lm(y ~ x + g, data=df)
+ true.ci.Bxy <- confint(model.true)['x',]
+ true.ci.Bgy <- confint(model.true)['g',]
+
+ result <- append(result, list(Bxy.est.true=coef(model.true)['x'],
+ Bgy.est.true=coef(model.true)['g'],
+ Bxy.ci.upper.true = true.ci.Bxy[2],
+ Bxy.ci.lower.true = true.ci.Bxy[1],
+ Bgy.ci.upper.true = true.ci.Bgy[2],
+ Bgy.ci.lower.true = true.ci.Bgy[1]))
+
+
+ model.naive <- lm(y~w+g, data=df)
+
+ naive.ci.Bxy <- confint(model.naive)['w',]
+ naive.ci.Bgy <- confint(model.naive)['g',]
+
+ result <- append(result, list(Bxy.est.naive=coef(model.naive)['w'],
+ Bgy.est.naive=coef(model.naive)['g'],
+ Bxy.ci.upper.naive = naive.ci.Bxy[2],
+ Bxy.ci.lower.naive = naive.ci.Bxy[1],
+ Bgy.ci.upper.naive = naive.ci.Bgy[2],
+ Bgy.ci.lower.naive = naive.ci.Bgy[1]))
+
+
+ ## multiple imputation when k is observed
+ amelia.out.k <- amelia(df, m=200, p2s=0, idvars=c('x'),noms=c("x.obs","w","g"))
+ mod.amelia.k <- zelig(y~x.obs+g, model='ls', data=amelia.out.k$imputations, cite=FALSE)
+ coefse <- combine_coef_se(mod.amelia.k, messages=FALSE)
+
+ 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.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
+ ))
+
+ ## 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","k"), noms=c("x.obs","w",'g'))
+ 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
+ ))
+
+ return(result)
+}
+
+Ns <- c(100, 200, 300, 400, 500, 1000, 2500, 5000, 7500)
+ms <- c(30, 50, 100, 200, 300, 500)
+B0 <- 0
+Bxy <- 1
+Bgy <- 0.3
+Bkx <- 3
+Bgx <- -4
+seeds <- 1:100
+
+rows <- list()
+
+for(N in Ns){
+ print(N)
+ for(m in ms){
+ if(N>m){
+ for(seed in seeds){
+ rows <- append(rows, list(run_simulation(N, m, B0, Bxy, Bgy, Bkx, Bgx, seed)))
+ }
+ }
+ }
+}
+
+result <- rbindlist(rows)
+write_feather(result, "example_2_simulation.feather")
projects / ml_measurement_error_public.git / blobdiff