bca: Confidence Interval - Bias-Corrected and Accelerated
In jeksterslabds/jeksterslabRboot: Jekster's Lab Linear Bootstrap Utilities

Description Usage Arguments Details Value Author(s) References See Also

Calculates bias-corrected and accelerated confidence intervals.

bca(
  thetahatstar,
  thetahat,
  data,
  fitFUN,
  ...,
  alpha = c(0.001, 0.01, 0.05),
  wald = FALSE,
  null = 0,
  dist = "z",
  df,
  eval = FALSE,
  theta = 0,
  par = TRUE,
  ncores = NULL,
  mc = TRUE,
  lb = FALSE,
  cl_eval = FALSE,
  cl_export = FALSE,
  cl_expr,
  cl_vars
)

`thetahatstar`	Numeric vector. The bootstrap sampling distribution ≤ft( \boldsymbol{\hat{θ}^{}} \right), that is, the sampling distribution of `thetahat` estimated for each `b` bootstrap sample. \hat{θ}_{≤ft( 1 \right)}, \hat{θ}_{≤ft( 2 \right)}, \hat{θ}_{≤ft( 3 \right)}, \hat{θ}_{≤ft( b \right)}, …, \hat{θ}_{≤ft( B \right)}* .
`thetahat`	Numeric. Parameter estimate ≤ft( \hat{θ} \right) from the original sample data.
`data`	Vector, matrix, or data frame. Sample data. Ignored if `thetahatstarjack` is provided.
`fitFUN`	Function. Fit function to use on `data`. The first argument should correspond to `data`. Other arguments can be passed to `fitFUN` using `...`. `fitFUN` should return a single value. Ignored if `thetahatstarjack` is provided.
`...`	Arguments to pass to `fitFUN`.
`alpha`	Numeric vector. Significance level ≤ft( α \right) . By default, `alpha` is set to conventional significance levels `alpha = c(0.001, 0.01, 0.05)`.
`wald`	Logical. If `TRUE`, calculates the square root of the Wald test statistic and p-value. The estimated bootstrap standard error is used. The arguments `null`, `dist`, and `df` are used to calculate the square root of the Wald test statistic and p-value and are NOT used in constructing the bootstrap confidence interval. If `FALSE`, returns `statistic = NA` and `p = NA` If `FALSE`, the arguments `null`, `dist`, and `df` are ignored.
`null`	Numeric. Hypothesized value of `theta` ≤ft( θ_{0} \right). Set to zero by default.
`dist`	Character string. `dist = "z"` for the standard normal distribution. `dist = "t"` for the t distribution.
`df`	Numeric. Degrees of freedom (df) if `dist = "t"`. Ignored if `dist = "z"`.
`eval`	Logical. Evaluate confidence intervals using `zero_hit()`, `theta_hit()`, `len()`, and `shape()`.
`theta`	Numeric. Population parameter ≤ft( θ \right) .
`par`	Logical. If `TRUE`, use multiple cores. If `FALSE`, use `lapply()`.
`ncores`	Integer. Number of cores to use if `par = TRUE`. If unspecified, defaults to `detectCores() - 1`.
`mc`	Logical. If `TRUE`, use `parallel::mclapply()`. If `FALSE`, use `parallel::parLapply()` or `parallel::parLapplyLB()`. Ignored if `par = FALSE`.
`lb`	Logical. If `TRUE` use `parallel::parLapplyLB()`. If `FALSE`, use `parallel::parLapply()`. Ignored if `par = FALSE` and `mc = TRUE`.
`cl_eval`	Logical. Execute `parallel::clusterEvalQ()` using `cl_expr`. Ignored if `mc = TRUE`.
`cl_export`	Logical. Execute `parallel::clusterExport()` using `cl_vars`. Ignored if `mc = TRUE`.
`cl_expr`	Expression. Expression passed to `parallel::clusterEvalQ()` Ignored if `mc = TRUE`.
`cl_vars`	Character vector. Names of objects to pass to `parallel::clusterExport()` Ignored if `mc = TRUE`.

The estimated bootstrap standard error is given by

\widehat{\mathrm{se}}_{\mathrm{B}} ≤ft( \hat{θ} \right) = √{ \frac{1}{B - 1} ∑_{b = 1}^{B} ≤ft[ \hat{θ}^{*} ≤ft( b \right) - \hat{θ}^{*} ≤ft( \cdot \right) \right]^2 }

where

\hat{θ}^{*} ≤ft( \cdot \right) = \frac{1}{B} ∑_{b = 1}^{B} \hat{θ}^{*} ≤ft( b \right) .

In addition to the bias-correction \hat{z}_{0} discussed in bc(), the acceleration \hat{a}, which refers to the rate of change of the standard error of \hat{θ} with respect to the true parameter value θ, is factored in.

The acceleration \hat{a} is given by

\hat{a} = \frac{ ∑_{i = 1}^{n} ≤ft[ \hat{θ}_{ ≤ft( \cdot \right) } - \hat{θ}_{ ≤ft( i \right) } \right]^{3} } { 6 ≤ft\{ ∑_{i = 1}^{n} ≤ft[ \hat{θ}_{ ≤ft( \cdot \right) } - \hat{θ}_{ ≤ft( i \right)} \right]^{2} \right\}^{3/2} }

where

\hat{θ}_{ ≤ft( i \right) } = ≤ft\{ \hat{θ}_{≤ft( 1 \right)}, \hat{θ}_{≤ft( 2 \right)}, \hat{θ}_{≤ft( 3 \right)}, …, \hat{θ}_{≤ft( n \right)} \right\}

is the jackknife sampling distribution and

\hat{θ}_{ ≤ft( \cdot \right) } = \frac{1}{n} ∑_{i = 1}^{n} \hat{θ}_{ ≤ft( i \right) }

is the jackknife mean. See jack() and jack_hat() .

Using \hat{z}_{0} and \hat{a} we can obtain the adjusted z-scores as follows

z_{ \mathrm{BCa}_{\mathrm{lo}} } = \hat{z}_{0} + \frac{ \hat{z}_{0} + z_{ ≤ft( \frac{ α } { 2 } \right) } } { 1 - \hat{a} ≤ft[ \hat{z}_{0} + z_{ ≤ft( \frac{ α } { 2 } \right) } \right] } ,

z_{ \mathrm{BCa}_{\mathrm{up}} } = \hat{z}_{0} + \frac{ \hat{z}_{0} + z_{ ≤ft[ 1 - \frac{ α } { 2 } \right] } } { 1 - \hat{a} ≤ft[ \hat{z}_{0} + z_{ ≤ft( 1 - \frac{ α } { 2 } \right) } \right] } .

The adjusted z-scores are used to determine the adjusted percentile ranks to form the confidence interval.

The bias-corrected and accelerated confidence interval is given by

≤ft[ \hat{θ}_{\mathrm{lo}}, \hat{θ}_{\mathrm{up}} \right] = ≤ft[ \hat{θ}^{*}_{ ≤ft( z_{ \mathrm{BCa}_{ \mathrm{lo} } } \right) }, \hat{θ}^{*}_{ ≤ft( z_{ \mathrm{BCa}_{ \mathrm{up} } } \right) } \right] .

For more details and examples see the following vignettes:

Notes: Introduction to Nonparametric Bootstrapping

Notes: Introduction to Parametric Bootstrapping

Returns a vector with the following elements:

statistic: Square root of Wald test statistic. NA if wald = FALSE.
p: p-value. NA if wald = FALSE.
se: Estimated bootstrap standard error ≤ft( \widehat{\mathrm{se}}_{\mathrm{B}} ≤ft( \hat{θ} \right) \right).
ci_: Estimated bias-corrected and accelerated confidence limits corresponding to alpha from the bootstrap sampling distribution thetahatstar ≤ft( \boldsymbol{\hat{θ}^{*}} \right).