How to obtain p-values of coefficients from bootstrap regression?

Question

From Robert Kabacoff's Quick-R I have

# Bootstrap 95% CI for regression coefficients 
library(boot)
# function to obtain regression weights 
bs <- function(formula, data, indices) {
  d <- data[indices,] # allows boot to select sample 
  fit <- lm(formula, data=d)
  return(coef(fit)) 
} 
# bootstrapping with 1000 replications 
results <- boot(data=mtcars, statistic=bs, 
     R=1000, formula=mpg~wt+disp)

# view results
results
plot(results, index=1) # intercept 
plot(results, index=2) # wt 
plot(results, index=3) # disp 

# get 95% confidence intervals 
boot.ci(results, type="bca", index=1) # intercept 
boot.ci(results, type="bca", index=2) # wt 
boot.ci(results, type="bca", index=3) # disp

How can I obtain the p-values $H_0:\, b_j=0$ of the bootstrap regression coefficients?

Answer 1

Just another variant that is somewhat simplistic but I think deliver the message without explicitly using the library boot that may confuse some people with the syntax it uses.

We have a linear model: $y = X \beta + \epsilon$, $\quad \epsilon \sim N(0,\sigma^2)$

The following is a parametric bootstrap for that linear model, that means that we do not resample our original data but actually we generate new data from our fitted model. Additionally we assume that the bootstrapped distribution of the regression coefficient $\beta$ is symmetric and that is translation invariant. (Very roughly speaking that we can move the axis of it with affecting its properties) The idea behind is that the fluctuations in the $\beta$ 's are due to $\epsilon$ and therefore with enough samples they should provide a good approximation of the true distribution of $\beta$ 's. As before we test again $H_0 : 0 = \beta_j$ and we defined our p-values as "the probability, given a null hypothesis for the probability distribution of the data, that the outcome would be as extreme as, or more extreme than, the observed outcome" (where the observed outcomes in this case are the $\beta$ 's we got for our original model). So here goes:

# Sample Size
N           <- 2^12;
# Linear Model to Boostrap          
Model2Boot  <- lm( mpg ~ wt + disp, mtcars)
# Values of the model coefficients
Betas       <- coefficients(Model2Boot)
# Number of coefficents to test against
M           <- length(Betas)
# Matrix of M columns to hold Bootstraping results
BtStrpRes   <- matrix( rep(0,M*N), ncol=M)

for (i in 1:N) {
# Simulate data N times from the model we assume be true
# and save the resulting coefficient in the i-th row of BtStrpRes
BtStrpRes[i,] <-coefficients(lm(unlist(simulate(Model2Boot)) ~wt + disp, mtcars))
}

#Get the p-values for coefficient
P_val1 <-mean( abs(BtStrpRes[,1] - mean(BtStrpRes[,1]) )> abs( Betas[1]))
P_val2 <-mean( abs(BtStrpRes[,2] - mean(BtStrpRes[,2]) )> abs( Betas[2]))
P_val3 <-mean( abs(BtStrpRes[,3] - mean(BtStrpRes[,3]) )> abs( Betas[3]))

#and some parametric bootstrap confidence intervals (2.5%, 97.5%) 
ConfInt1 <- quantile(BtStrpRes[,1], c(.025, 0.975))
ConfInt2 <- quantile(BtStrpRes[,2], c(.025, 0.975))
ConfInt3 <- quantile(BtStrpRes[,3], c(.025, 0.975))

As mentioned the whole idea is that you have the bootstrapped distribution of $\beta$ 's approximates their true one. (Clearly this code is optimized for speed but for readability. :) )

Answer 2

The community and @BrianDiggs may correct me if I am wrong, but I believe you can get a p-value for your problem as follows. A p-value for a two sided test is defined as

$$2*\text{min}[P(X \le x|H_0),P(X \ge x|H_0)]$$

So if you order the bootstrapped coefficients by size and then determine the proportions larger and smaller zero, the minimum proportion times two should give you a p-value.

I normally use the following function in such a situation:

twosidep<-function(data){
  p1<-sum(data>0)/length(data)
  p2<-sum(data<0)/length(data)
  p<-min(p1,p2)*2
  return(p)
}

Answer 3

The bootstrap can be used to compute $p$-values, but it would need a substantial change to your code. As I am not familiar with R I can only give you a reference in which you can look up what you would need to do: chapter 4 of (Davison and Hinkley 1997).

Davison, A.C. and Hinkley, D.V. 1997. Bootstrap methods and their application. Cambridge: Cambridge University Press.