Coverage probability calculation for LM

Question

I am trying to calculate coverage probability for a set of residual bootstrap replicates I generated on the intercept and slope of regression . Can anyone show me how to calculate coverage probability of confidence intervals? Many thanks.

Note that I manually ran the regression using Qr decomposition but you can use lm() if that's easier. I just thought doing it manually will be faster.

set.seed(42)  ## for sake of reproducibility
n <- 100
x <- rnorm(n)
e <- rnorm(n)
y <- as.numeric(50 + 25*x + e)
dd <- data.frame(id=1:n, x=x, y=y)

mo <- lm(y ~ x, data=dd)

# Manual Residual Bootstrap
resi <- residuals(mo)
fit <- fitted(mo)
ressampy <- function() fit + sample(resi, length(resi), replace=TRUE)
# Sample y values:
head(ressampy())
# Qr decomposition of X values
qrX <- qr(cbind(Intercept=1, dd[, "x", drop=FALSE]), LAPACK=TRUE)
# faster than LM
qr.coef(qrX, dd[, "y"])
# One Bootstrap replication
boot1 <- qr.coef(qrX, ressampy())
# 1000 bootstrap replications
boot <- t(replicate(1000, qr.coef(qrX, ressampy())))

EDIT Incorporating jay.sf's answer, I rewrote the code that ran the lm() method and compared the first and second approach of calculating coverage probability in the link shared by jay.sf:

library(lmtest);library(sandwich)
ci <- confint(coeftest(mo, vcov.=vcovHC(mo, type="HC3")))
ci

FUNInter <- function() {
  X <- model.matrix(mo)
  ressampy.2 <- fit + sample(resi, length(resi), replace = TRUE)
  bootmod <- lm(ressampy.2 ~ X-1)
  confint(bootmod, "X(Intercept)", level = 0.95)
}

FUNBeta <- function() {
  X <- model.matrix(mo)
  ressampy.2 <- fit + sample(resi, length(resi), replace = TRUE)
  bootmod <- lm(ressampy.2 ~ X-1)
  confint(bootmod, "Xx", level = 0.95)
}

set.seed(42)
R <- 1000
Interres <- replicate(R, FUNInter(), simplify=FALSE)
Betares <- replicate(R, FUNBeta(), simplify=FALSE)
ciinter <- t(sapply(Interres, function(x, y) x[grep(y, rownames(x)), ], "X\\(Intercept\\)"))
cibeta <- t(sapply(Betares, function(x, y) x[grep(y, rownames(x)), ], "Xx"))

#second approach of calculating CP
sum(ciinter[,"2.5 %"] <=50 & 50 <= ciinter[,"97.5 %"])/R
[1] 0.842
sum(cibeta[,"2.5 %"] <=25 & 25 <= cibeta[,"97.5 %"])/R
[1] 0.945
#first approach of calculating CP
sum(apply(ciinter, 1, function(x) {
  all(data.table::between(x, ci[1,1], ci[1,2]))
}))/R
[1] 0.076
sum(apply(cibeta, 1, function(x) {
  all(data.table::between(x, ci[2,1], ci[2,2]))
}))/R
[1] 0.405

Answer 1

According to Morris et. al 2019, Table 6, the coverage probability is defined as the probability how often real theta lies within a bootstrapped confidence interval (CI) (i.e. those of the model applied on many samples based on the actual data, or—in other words—new experiments):

Hence, we want to compute CIs based on OP's proposed i.i.d. bootstrap R times and calculate the ratio of how often theta is or is not in these CIs.

First, we estimate our model mo using the actual data.

mo <- lm(y ~ x)

To avoid unnecessary unpacking fitted values yhat, residuals u, model matrix X, and coefficients coef0 in the replications, we extract them beforehand.

yhat <- mo$fitted.values
u <- as.matrix(mo$residuals)
X <- model.matrix(mo)
theta <- c(50, 25)  ## known from data generating process of simulation

In a bootstrap function FUN we wrap all the steps we want to do in one replication. In order to apply the very fast .lm.fit, we have to calculate the white standard errors manually (identical to lmtest::coeftest(fit, vcov.=sandwich::vcovHC(fit, type="HC1"))).

FUN <- function() {
  ## resampling residuals
  y.star <- yhat + sample(u, length(u), replace=TRUE)
  ## refit model
  fit <- .lm.fit(X, y.star)
  coef <- fit$coefficients[sort.list(fit$pivot)]
  ## alternatively using QR, but `.lm.fit` is slightly faster
  # qrX <- qr(X, LAPACK=TRUE)
  # coef <- qr.coef(qrX, y.star)
  ## white standard errors
  v.cov <- chol2inv(chol(t(X) %*% X))
  meat <- t(X) %*% diag(diag(u %*% t(u))) %*% X
  ## degrees of freedom adjust (HC1)
  d <- dim(X)
  dfa <- d[1] / (d[1] - d[2])
  white.se <- sqrt(diag(v.cov %*% meat %*% v.cov)*dfa)
  ## 95% CIs
  ci <- coef + qt(1 - .025, d[1] - d[2])*white.se %*% t(c(-1, 1))
  ## coverage
  c(intercept=theta[1] >= ci[1, 1] & theta[1] <= ci[1, 2],
    x=theta[2] >= ci[2, 1] & theta[2] <= ci[2, 2])
}

Now we execute the bootstrap using replicate.

R <- 5e3
set.seed(42)
system.time(res <- t(replicate(R, FUN())))
#   user  system elapsed
#  71.19   28.25  100.28 

head(res, 3)
#      intercept    x
# [1,]      TRUE TRUE
# [2,]     FALSE TRUE
# [3,]      TRUE TRUE

The mean of TRUEs in both columns simultaneously across the rows, or in each column respectively, gives the coverage probability we are looking for.

(cp.t <- mean(rowSums(res) == ncol(res)))  ## coverage probability total  
(cp.i <- colMeans(res))  ## coverage probability individual coefs
(cp <- c(total=cp.t, cp.i))
#  total intercept         x
# 0.8954    0.9478    0.9444

## values with other R:
#   total intercept         x
# 0.90700   0.95200   0.95200  ## R ==   1k
# 0.89950   0.95000   0.94700  ## R ==   2k
# 0.89540   0.94780   0.94440  ## R ==   5k
# 0.89530   0.94570   0.94680  ## R ==  10k
# 0.89722   0.94694   0.94777  ## R == 100k

And this is how it looks like after 100k repetitions

Code for plot:

r1 <- sapply(seq(nrow(res)), \(i) mean(rowSums(res[1:i,,drop=FALSE]) == ncol(res)))
r2 <- t(sapply(seq(nrow(res)), \(i) colMeans(res[1:i,,drop=FALSE])))
r <- cbind(r1, r2)

matplot(r, type='l', col=2:4, lty=1, main='coverage probability', xlab='R', 
        ylab='cum. mean',ylim=c(.89, .955))
grid()
sapply(seq(cp), \(i) abline(h=cp[i], lty=2, col=i + 1))
legend('right', col=2:4, lty=1, legend=names(cp), bty='n')

---

Data:

set.seed(42)
n <- 1e3
x <- rnorm(n)
y <- 50 + 25*x + rnorm(n)