Layman's explanation of bootstrap confidence intervals for a regression with percentile method

Question

First of all, I am not a statistician. I only how to interpret stats and to do them with R, my understanding of the math/formulas behind them is virtually zero.

With this said, I am looking for a laymen's explanation of the way percentile bootstrap confidence intervals are calculated for mixed-effects models with the confint() function of the R package lme4 (that is, I would like to gain a basic understanding of how the math works)

Would it be correct to state that this function selects a user-defined number of subsamples from the original data, applies the regression model to them and then calculates the range within which we can be 95% sure that the true population effect of a level of a predictor falls? How does this calculation work? Would it be correct to portray it as averaging over a large number of coefficients?

Answer 1

A percentile bootstrap confidence interval starts by taking the definition of a 95% confidence interval literally: If we were to draw repeatedly random samples (with replacement) from our population and computed a coefficient in each of these samples, then 95% of these coefficients would fall within that interval. Problem is that it is usually impossible/unpractical to repeatedly sample from the population. We do have a good approximation of the population: our current sample. So the bootstrap repeatedly samples $N$ out of $N$ observations with replacement (so some observations appear twice, some three time or even more, and some won't appear at all) and computes the coefficients of interest in each of these samples. The percentile confidence interval orders these coeffcients from smallest to largest and the lower bound is chosen such that the there are only 2.5% of the coefficients smaller than that lower bound and the upper bound is chosen such that there are only 2.5% larger than that upper bound (these are the 2.5$^{\mathrm{th}}$ and 97.5$^{\mathrm{th}}$ percentiles.