Bootstrap
Cyrus Samii
2/26/2019
Inverse Propensity Score Weighting (IPW)
Recall that the stabilized IPW estimator for the ATE is \[ \hat{\rho}_{IPW} = \frac{\sum_{i=1}^n \frac{D_i}{\hat{e}(X_i)}Y_i}{\sum_{i=1}^n \frac{D_i}{\hat{e}(X_i)}} - \frac{\sum_{i=1}^n \frac{1-D_i}{1-\hat{e}(X_i)}Y_i}{\sum_{i=1}^n \frac{1-D_i}{1-\hat{e}(X_i)}}, \] and that this can be implementing with weighted least squares regression in which you regress \(Y\) on \(D\), using the inverse of the estimate propensity score \(\hat{e}(X_i)\) as the weights for the treated and the inverse of \(1-\hat{e}(X_i)\) as the weights for the control.
Inference for \(\hat{\rho}_{IPW}\) should account for the sampling variation in the estimated propensity scores, \(\hat{e}(\cdot)\). In the Robust Inference I lecture notes, we showed how to derived the variance when \(\hat{e}(\cdot)\) is estimated using an \(M\)-estimator. Here we will use the bootstrap instead.
Make a simulated population with confounding
We will simulate a population with confounding in \(X\). We do it in a way that makes it so that simple control specifications for \(X\) in a regression would be inadequate, and so the specification challenge is a little harder than usual:
# Make population data
set.seed(11)
N.pop <- 10000
index <- 1:N.pop
X <- .5*exp(rnorm(N.pop))
Y0 <- 50-.05*X^2+(1+.5*(max(X)-X))*rnorm(N.pop)
Y1 <- -1.5 + Y0 + 10*X + .2*X^2+ .25*X*rnorm(N.pop)
e.X <- (1+exp(1-.2*X))^(-1)
D <- rbinom(N.pop, 1, e.X)
Y <- D*Y1 + (1-D)*Y0
rho <- mean(Y1-Y0)
plot(X, Y0, col="blue", pch=19,
cex=.25, ylim=range(c(Y1,Y0)),
main="Potential outcomes")
points(X, Y1, col="red", pch=19, cex=.25)
plot(X, e.X, pch=19, main="Propensity scores")
plot(X, Y, type="n", ,
main="Realized outcomes")
points(X[D==0], Y[D==0], col="blue", pch=19, cex=.25)
points(X[D==1], Y[D==1], col="red", pch=19, cex=.25)
rho## [1] 7.124273summary(lm(Y~D))[[4]][,1]## (Intercept) D
## 49.800748 9.237881summary(lm(Y~D+X))[[4]][,1]## (Intercept) D X
## 45.379997 7.790128 5.881640summary(lm(Y~D*I(X-mean(X))))[[4]][,1]## (Intercept) D I(X - mean(X)) D:I(X - mean(X))
## 49.7601797 7.3606239 -0.5394921 11.7581545pop.data <- data.frame(index, Y, D, X)Sampling process
Next we draw a sample and estimate the effect using simple regression strategies and then IPW:
# Draw a sample:
n.samp <- 1000
samp.i <- sample(index, n.samp)
samp.data <- pop.data[samp.i,]
plot(samp.data$X, samp.data$Y, type="n")
points(samp.data$X[D==0], samp.data$Y[D==0], col="blue")
points(samp.data$X[D==1], samp.data$Y[D==1], col="red")
summary(lm(Y~D, data=samp.data))[[4]][,1]## (Intercept) D
## 49.18994 10.72004summary(lm(Y~D+X, data=samp.data))[[4]][,1]## (Intercept) D X
## 44.331840 8.891598 6.449878summary(lm(Y~D*I(X-mean(X)), data=samp.data))[[4]][,1]## (Intercept) D I(X - mean(X)) D:I(X - mean(X))
## 49.1173279 8.5252179 -0.8426236 12.3346502e.hat.X <- predict(glm(D~X, data=samp.data,
family="binomial"),
type="response")
samp.data$w <- samp.data$D*(1/e.hat.X) + (1-samp.data$D)*(1/(1-e.hat.X))
plot(samp.data$X, samp.data$Y, type="n")
points(samp.data$X[D==0], samp.data$Y[D==0],
cex=.5*samp.data$w[D==0],col="blue")
points(samp.data$X[D==1], samp.data$Y[D==1],
cex=.5*samp.data$w[D==1],col="red")
Fitting IPW model
fit.ipsw.s <- lm(Y~D, weights=samp.data$w, data=samp.data)
summary(fit.ipsw.s)[[5]][,1]## (Intercept) D
## 49.114690 8.478966Estimating the standard error
# Get bootstrap estimate
n.boot <- 500
ate.hat <- rep(NA, n.boot)
t.out <- rep(NA, n.boot)
for (i in 1:n.boot){
boot.index <- sample(samp.data$index, n.samp, replace=T)
boot.data <- samp.data[match(boot.index, samp.data$index),]
e.hat.boot <- predict(glm(D~X, data=boot.data,
family="binomial"),
type="response")
boot.data$w <- boot.data$D*(1/e.hat.boot)/sum(boot.data$D*(1/e.hat.boot)) + (1-boot.data$D)*(1/(1-e.hat.boot))/sum((1-boot.data$D)*(1/(1-e.hat.boot)))
fit.ipsw.b <- lm(Y~D, weights=boot.data$w, data=boot.data)
ate.hat[i] <- coef(fit.ipsw.b)[2]
t.out[i] <- coeftest(fit.ipsw.b , vcov.= vcovHC(fit.ipsw.b, "HC2"))[2,3]
cat(paste(i, " ")); flush.console() # progress indicator
}## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500hist(ate.hat, breaks=50)
abline(v=coef(fit.ipsw.s)[2], col="blue")
abline(v=rho, col="red")
aCI.out <- coeftest(fit.ipsw.s , vcov.= vcovHC(fit.ipsw.s, "HC2"))
# Naive analytical asymptotic CI ignoring pscore estimation
aaCI <- c(aCI.out[2,1]-qt(.975, fit.ipsw.s[[9]])*aCI.out[2,2], aCI.out[2,1]+qt(.975, fit.ipsw.s[[9]])*aCI.out[2,2])
# Bootstrap-b CI
bbCI <- quantile(ate.hat, c(0.025, .975))
# Bootstrap-t CI
btCI <- aCI.out[2,2]*quantile(t.out, c(0.025, .975))
coef(fit.ipsw.s)[2]## D
## 8.478966aaCI## [1] 6.520593 10.437340bbCI## 2.5% 97.5%
## 6.614625 10.355653btCI## 2.5% 97.5%
## 6.580765 10.348629Comparing to the true sampling distribution
n.iter <- 500
ate.hat.s <- rep(NA, n.iter)
t.out.s <- rep(NA, n.iter)
for(j in 1:n.iter){
samp.i <- sample(index, n.samp)
samp.data <- pop.data[samp.i,]
e.hat.X <- predict(glm(D~X, data=samp.data,
family="binomial"),
type="response")
samp.data$w <- samp.data$D*(1/e.hat.X) + (1-samp.data$D)*(1/(1-e.hat.X))
fit.ipsw <- lm(Y~D, weights=samp.data$w, data=samp.data)
ate.hat.s[j] <- coef(fit.ipsw)[2]
t.out.s[j] <- coeftest(fit.ipsw , vcov.= vcovHC(fit.ipsw, "HC2"))[2,3]
cat(paste(j, " ")); flush.console() # progress indicator
}## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500# True coef mean
mean(ate.hat.s)## [1] 7.405926# Estimate
coef(fit.ipsw.s)[2]## D
## 8.478966# True sampling sd
sd(ate.hat.s)## [1] 0.7989762# Estimates
# Naive analytical
aCI.out[2,2]## [1] 0.9979767# bootstrap-b
sd(ate.hat)## [1] 0.9788662# True t stat dist
plot(density(t.out.s-mean(t.out.s)),
main="Sampling distribution",
xlim=c(-4,6))
# Estimates:
# Analytical
points( seq(-4,4,.01),
dt(seq(-4,4,.01),
df=fit.ipsw.s[[9]]),
type="l",
col="red")
# Bootstrap-t
points(density(t.out-mean(t.out)), type="l",col="blue")
legend(2,.3, legend=c("True","Analytical","Boot-t"),
col=c("black","red","blue"),
lty=c("solid","solid","solid"),
cex=.75)
Bootstrap failure
We will show the failure of the bootstrap for estimates of the maximum, using \(\max\{X_i\}\) for units \(i\) in the sample. This is a non-smooth statistic:
First, we can view our target, the population maximum, relative to what we get in our sample:
par(mfrow=c(1,1))
pophist <- hist(pop.data$X, breaks=100, freq=TRUE, main="Hist. of X")
hist(samp.data$X, add=TRUE, breaks=pophist$breaks, col="black")
max(pop.data$X)## [1] 20.21729max(samp.data$X)## [1] 12.70117Now look at the sampling distribution of \(\max\{X_i\}\):
max.s <- rep(NA, n.iter)
for(j in 1:n.iter){
samp.i <- sample(index, n.samp)
samp.data <- pop.data[samp.i,]
max.s[j] <- max(samp.data$X)
}
hist(max.s, freq=F, breaks=pophist$breaks)
Now look at how the bootstrap attempts to approximate this:
max.b <- rep(NA, n.iter)
for (i in 1:n.boot){
boot.index <- sample(samp.data$index, n.samp, replace=T)
boot.data <- samp.data[match(boot.index, samp.data$index),]
max.b[i] <- max(boot.data$X)
}
hist(max.b, freq=F, breaks=pophist$breaks)
sd(max.s)## [1] 3.907458sd(max.b)## [1] 0.6836264