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Calculate Bias-Corrected and Accelerated (BCa) bootstrap confidence interval

Source: R/bca_ci.R
bca_ci.Rd

This function calculates a Bias-Corrected and Accelerated (BCa) confidence interval from a bootstrap sample. It is used by calculate_bootstrap_ci().

Usage

bca_ci(t0, t, a, conf = 0.95, h = function(t) t, hinv = function(t) t)

Arguments

t0

Original statistic.

t

Numeric vector of bootstrap replicates.

a

Acceleration constant. See also calculate_acceleration().

conf

A numeric value specifying the confidence level of the interval. Default is 0.95 (95 % confidence level).

h

A function defining a transformation. The intervals are calculated on the scale of h(t) and the inverse function hinv applied to the resulting intervals. It must be a function of one variable only. The default is the identity function.

hinv

A function, like h, which returns the inverse of h. It is used to transform the intervals calculated on the scale of h(t) back to the original scale. The default is the identity function. If h is supplied but hinv is not, then the intervals returned will be on the transformed scale.

Value

A matrix with four columns:

  • conf: confidence level

  • rk_lower: rank of lower endpoint (interpolated)

  • rk_upper: rank of upper endpoint (interpolated)

  • ll: lower confidence limit

  • ul: lower confidence limit

Details

Adjusts for bias and acceleration. Bias refers to the systematic difference between the observed statistic from the original dataset and the center of the bootstrap distribution of the statistic. The bias correction term is calculated as follows:

$$\hat{z}_0 = \Phi^{-1}\left(\frac{\#(\hat{\theta}^*_b < \hat{\theta})}{B}\right)$$

where \(\#\) is the counting operator, counting the number of times \(\hat{\theta}^*_b\) is smaller than \(\hat{\theta}\), and \(\Phi^{-1}\) the inverse cumulative density function of the standard normal distribution.\(B\) is the number of bootstrap samples.

Acceleration quantifies how sensitive the variability of the statistic is to changes in the data. See calculate_acceleration() on how this is calculated.

  • \(a=0\): The statistic's variability does not depend on the data (e.g., symmetric distribution)

  • \(a>0\): Small changes in the data have a large effect on the statistic's variability (e.g., positive skew)

  • \(a<0\): Small changes in the data have a smaller effect on the statistic's variability (e.g., negative skew).

The bias and acceleration estimates are then used to calculate adjusted percentiles.

\(\alpha_1 = \Phi\left( \hat{z}_0 + \frac{\hat{z}_0 + z_{\alpha/2}}{1 - \hat{a}(\hat{z}_0 + z_{\alpha/2})} \right)\), \(\alpha_2 = \Phi\left( \hat{z}_0 + \frac{\hat{z}_0 + z_{1 - \alpha/2}}{1 - \hat{a}(\hat{z}_0 + z_{1 - \alpha/2})} \right)\)

So, we get

$$CI_{bca} = \left[ \hat{\theta}^*_{(\alpha_1)}, \hat{\theta}^*_{(\alpha_2)} \right]$$

Note

This function is adapted from the internal function bca.ci() in the boot package (Canty & Ripley, 1999).

References

Canty, A., & Ripley, B. (1999). boot: Bootstrap Functions (Originally by Angelo Canty for S) [Computer software]. https://CRAN.R-project.org/package=boot

Davison, A. C., & Hinkley, D. V. (1997). Bootstrap Methods and their Application (1st ed.). Cambridge University Press. doi:10.1017/CBO9780511802843

See also

Other interval_calculation: basic_ci(), norm_ci(), perc_ci()

Examples

set.seed(123)
boot_reps <- rnorm(1000)
t0 <- mean(boot_reps)

# Example acceleration value (normally estimated via jackknife)
a <- 0.01

# BCa bootstrap CI
bca_ci(t0, boot_reps, a, conf = 0.95)
#>      conf rk_lower rk_upper        ll       ul
#> [1,] 0.95    27.62   978.46 -1.930133 2.112914