Bootstrap intervals
The bootstrap resamples observations or other sampling units with replacement, recomputes the statistic, and uses the resulting distribution to estimate uncertainty. The resampling unit must match the design: clustered, paired, or time-dependent data cannot be treated as independent rows without justification.
| Interval | Construction | Main caution |
|---|---|---|
| Percentile | Uses bootstrap-distribution quantiles | Can inherit bias and skew limitations. |
| Basic | Reflects bootstrap errors around the observed estimate | Transformation behavior differs from percentile. |
| BCa | Adjusts for estimated bias and acceleration | More refined, but still depends on a valid resampling scheme. |
Efron’s BCa method was designed to improve coverage through bias and acceleration adjustments.2
Credible intervals
A Bayesian interval comes from the posterior distribution. “There is 95% posterior probability that the parameter lies in this interval” is valid only under the specified likelihood, prior, model, and data. It is not a model-free certainty.
Confidence interval interpretation covers the repeated-sampling meaning people often reverse. Confidence versus prediction intervals distinguishes uncertainty in a mean response from uncertainty for a new observation.
What to report
Name the interval type, confidence/credible level, resampling unit or prior, number of resamples or posterior computation, and any transformation. Do not label every nonparametric-looking interval “bootstrap.”
Same endpoints, different statements
Suppose two analyses both print an interval of [1.2, 3.8]. If one is a 95% BCa bootstrap interval, read it as an uncertainty procedure whose repeated-sampling performance depends on the sampling design and resampling scheme. If the other is a 95% Bayesian credible interval, read it as containing 95% posterior probability under the stated likelihood and prior.
Matching endpoints do not make the interpretations interchangeable. First identify the procedure, then state the corresponding uncertainty claim. The numerical range comes after that identification step.
A defensible reporting sentence
“The 95% BCa bootstrap interval was [1.2, 3.8], based on resampling the independent study units. This is a repeated-sampling uncertainty interval, not a 95% posterior-probability statement.”