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I've been reading a lot these last few days about how to get a p-value from bootstrap for regression models (not by permutation). For each coefficient of the model, the null hypothesis is that the coefficient equals 0 and H1 is that it is different to 0 (bilateral test).

The most noticeable similar subjects are the following two questions on stackexchange, but the answers confuse me a lot:

From what I've read, I noticed several approaches, but I can't figure out which is the valid one:

  1. Should my bootstrap function return the test statistic calculated for each sample, or the estimate?

  2. Should I calculate the proportion of the test statistic/estimate above 0 or above the point estimate of the base model?

  3. Should I multiply the result by 2 because the test is bilateral or use absolute values?

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Suppose we want to test the null hypothesis that a regression coefficient = 0 using bootstrap, and say we decide 0.05 to be the level of significance. Now, we can generate the sampling distribution for each coefficient using bootstrap. It is easy to check if 0 falls within 95% confidence interval, thus we can easily decide whether we can reject the null or not.

To get a p-value, we need to check what is the quantile value of 0 in the sampling distribution. (I am using a quantile based approach there are other methods to do it which can be found here Fox on Regression) After you get the quantile,Q, the p-value is 2*Q or 2*(1-Q) depending on whether Q > 0.5 or less than 0.5.

As an illustration of the approach, consider this

library(faraway)

Build linear model

mdl <- lm(divorce ~ ., data = divusa)

Bootstrap

bootTest <- sapply(1:1e4,function(x){
  rows <- sample(nrow(divusa),nrow(divusa),replace = T)
  mdl <- lm(divorce ~ ., data = divusa[rows,])
  return(mdl$coefficients)
})

Here is the model

summary(mdl)

Call:
lm(formula = divorce ~ ., data = divusa)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.9087 -0.9212 -0.0935  0.7447  3.4689 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 380.14761   99.20371   3.832 0.000274 ***
year         -0.20312    0.05333  -3.809 0.000297 ***
unemployed   -0.04933    0.05378  -0.917 0.362171    
femlab        0.80793    0.11487   7.033 1.09e-09 ***
marriage      0.14977    0.02382   6.287 2.42e-08 ***
birth        -0.11695    0.01470  -7.957 2.19e-11 ***
military     -0.04276    0.01372  -3.117 0.002652 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.513 on 70 degrees of freedom
Multiple R-squared:  0.9344,    Adjusted R-squared:  0.9288 
F-statistic: 166.2 on 6 and 70 DF,  p-value: < 2.2e-16

Notice the p-values

Subset of regression coefficient generated using bootstrap

bootTest[,1:5]
                    [,1]          [,2]         [,3]         [,4]         [,5]
(Intercept) 335.70970574 372.525260160 569.85830341 338.70069977 344.69261238
year         -0.18107568  -0.201798080  -0.30579380  -0.18125215  -0.18328105
unemployed   -0.02916575   0.006828023   0.01197723  -0.05610887  -0.11463230
femlab        0.79078784   0.842924808   1.02607863   0.77527548   0.76472406
marriage      0.17372382   0.199571033   0.18782967   0.15289119   0.15693996
birth        -0.11613752  -0.118507758  -0.11998122  -0.11666450  -0.13344442
military     -0.04051730  -0.056277118  -0.04062756  -0.05167556  -0.07251748

Generate p-values with bootstrap

pvals <- sapply(1:nrow(bootTest),function(x) {
  distribution <- ecdf(bootTest[x,])
  qt0 <- distribution(0)
  if(qt0 < 0.5){
        return(2*qt0)
      } else {
        return(2*(1-qt0))
      }
})

Comparing p-values from t-test and bootstrap

T test

summary(mdl)$coefficients[,4]
 (Intercept)         year   unemployed       femlab     marriage        birth     military 
2.744830e-04 2.966776e-04 3.621708e-01 1.085196e-09 2.419284e-08 2.191964e-11 2.652003e-03 

Bootstrap

> pvals
[1] 0.0008 0.0008 0.2196 0.0000 0.0000 0.0000 0.0188

The highly significant p-values with coefficients < 1e-8 are all 0 with bootstrap with 1e4 iterations. Furthermore, the ranking of p-values is comparable as well.

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