MedCalc  # A technical note on the computation of BCa bootstrap confidence intervals

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

In their paper on bootstrap confidence intervals, DiCiccio and Efron (1996) estimate the bias-correction factor $\hat{z}_0$ for the BCa bootstrap confidence interval by

$$\hat{z}_0 = Φ^{-1} \Biggl( \frac { \# \left \{ \hat{\theta}\!\,^{ *} (b) < \hat{\theta} \right \} } { B } \Biggr)$$

Note that when half of the $\hat{\theta}\!\,^{ *} (b)$ values are less than or equal to $\hat{\theta}$ then $\hat{z}_0$ 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 $\hat{\theta}$ 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 $\hat{z}_0$ equal to 0.

MedCalc therefore uses the following correction in the formula for $\hat{z}_0$:

$$\hat{z}_0 = Φ^{-1} \Biggl( \dfrac { \# \left \{ \hat{\theta}\!\,^{ *} (b) < \hat{\theta} \right \} + \dfrac { \# \left \{ \hat{\theta}\!\,^{ *} (b) = \hat{\theta} \right \} } {2} } { B } \Biggr)$$

With this correction we obtain $\hat{z}_0$ = 0 when the same proportion of $\hat{\theta}\!\,^{ *} (b)$ is lower and higher than $\hat{\theta}$, 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.