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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.