Bootstrap intervals: percentile and BCa

In brief

The bootstrap repeatedly draws new samples from the existing cases (with replacement), recalculates the statistic each time and takes an interval from the distribution of these values. Under every result Statable names the type of interval, the number of draws (B) and the seed of the random numbers, so the calculation can be repeated.

Percentile or BCa?

• Percentile interval (default for the mediation): the middle 95% of the draws. Hayes has recommended it since PROCESS 3 because the bias-corrected interval raises the error rate in mediation (Hayes & Scharkow 2013, Psychological Science 24(10), 1918-1927; Hayes 2022).
• BCa (bias-corrected and accelerated, Efron 1987): shifts the limits by a bias correction z0 and an acceleration a. The acceleration comes from leaving out one case at a time (jackknife). BCa is regarded as more accurate for skewed distributions. It can be chosen under "More options".

Chance and the seed

Statable always starts the random numbers with the same seed (20261008). The same data therefore give the same interval. Another seed moves the limits a little; the lower BCa limit varies more than the upper one. This is not a mistake but the random error of the method, and it gets smaller with more draws. The random numbers of Statable are not those of R: even with the same seed the single draws differ, and the intervals agree within the random error.

What Statable reports

• Edge: if a limit lies at the edge of the draws, BCa stands on shaky ground there. Statable says so and advises more draws or the percentile interval.
• Not computable: if all draws lie on one side of the estimate, BCa cannot be calculated. Statable then shows the percentile interval and says so; a missing number never appears in the table.
• Many equal draws: if more than 5% of the draws give exactly the estimate (typical for the median and proportions), BCa is uncertain; the percentile interval is then the better choice.
• Invalid draws (for example without variation) do not count; their number is given in the note.
• Small samples: with fewer than 30 people bootstrap intervals are usually too narrow (Hesterberg 2015, The American Statistician 69(4)). They should be read with care.

For specialists

Quantiles and BCa follow the R package boot (Canty & Ripley): interpolation on the normal scale between the order statistics at rank (B + 1) · α (boot:::norm.inter), z0 from the share of draws strictly below the estimate, a from the jackknife (empinf, type = "jack"). Checked: on fixed distributions from R the calculation matches boot.ci to 1e-9; the draws of Statable agree statistically with a reference with 400,000 draws. The bootstrap in the learning lab has its own random stream and is not connected to this base. Sources: Efron & Tibshirani (1993), An Introduction to the Bootstrap; Davison & Hinkley (1997), Bootstrap Methods and their Application, section 5.3.