# A technical note on the computation of BC_{a} bootstrap confidence intervals

MedCalc uses a correction to the usual formula for the bias-correction factor used in the estimation of BC_{a} bootstrap confidence intervals.

In their paper on bootstrap confidence intervals, DiCiccio and Efron (1996) estimate the bias-correction factor for the BC_{a} bootstrap confidence interval by

Note that when half of the values are less than or equal to then should equal zero (Efron & Tibshirai, 1994).

In the presence of ties however this formula underestimates the proportion of bootstrap replications that are less than or equal to and therefore overestimates bias.

When for example, the original estimate of a parameter of interest is 5 and 9 bootstrap replications are 1, 2, 3, 4, 5, 6, 7, 8, and 9, then there is no bias, but the proportion of replications that are below 5 is 4/9 according to the formula given above, whereas it should be 4.5/9 = 0.5 to obtain a equal to 0.

MedCalc therefore uses the following correction in the formula for :

With this correction we obtain = 0 when the same proportion of is lower and higher than , also in the presence of ties.

## References

- DiCiccio TJ, Efron B (1996) Bootstrap Confidence Intervals. Statistical Science 11:189-228.
- Efron B, Tibshirai RJ (1994) An introduction to the bootstrap. Chapman & Hall/CRC: Boca Raton; p 186.