# Confidence interval calculation from bootstrap samples.

### Description

`bootCI`

calculates five different confidence intervals from bootstrap samples: see details: `bootCIlogit`

calculates corrections on the logit scale and back-transforms.

### Usage

1 2 | ```
bootCI(t0, bt, conf = 0.95)
bootCIlogit(t0, bt, conf = 0.95)
``` |

### Arguments

`t0` |
the statistic estimated from the original sample |

`bt` |
a vector of bootstrap statistics. |

`conf` |
a (single!) confidence interval to estimate. |

### Details

Let t = true value of the statistic,

t0 = estimate of t based on the original sample,

bt = bootstrap estimates.

If bootstrap sampling introduces no bias, E[mean(bt)] = t0, otherwise BS bias = mean(bt) - t0.

Assuming that the original sampling causes the same bias as the bootstrap sampling, we write: mean(bt) - t0 = t0 - t, and hence calculate a bias-corrected estimate, t1 = 2 x t0 - mean(bt).

The percentiles CI, “perc”, gives quantiles of the bootstrap values, interpolated if necessary. However, in general, the bootstrap estimates are biased, so “perc” should be corrected.

“basic” is a bias-corrected version of “perc”, analogous to t1: 2 x t0 - `perc`

.

“norm” gives tail cutoffs for a normal distribution with mean = t1 and sd = sd(bt).

These three are equivalent to the confidence intervals returned by `boot::boot.ci`

. “basic” and “norm” are appropriate if you are using the bias-corrected estimator, t1. If you use the uncorrected estimator, t0, you should use “basic0” or “norm0”:

“basic0” is `perc`

- mean(bt) + t0.

“norm0” gives tail cutoffs as before, but with mean = t0 instead of t1.

The "logit" versions perform the corrections on the logit scale and then back transform. This would be appropriate for probabilities or proportions.

### Value

A named matrix with 2 columns for lower and upper limits and a row for each type of estimate. Values will be NA if the bootstrap sample is too small (after removing NAs) for estimation: 40 is the minimum for a 95% confidence interval, 200 for 99% (though for stable estimates you need at least 999 bootstrap estimates, preferably 10,000).

### Author(s)

Mike Meredith

### See Also

`boot.ci`

in package `boot`

.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
# Generate some sample data:
set.seed(123)
dat <- runif(20)
t0 <- sd(dat)
# and 999 nonparametric bootstrap estimates:
bootmat <- matrix(sample(dat, 20*999, replace=TRUE), 20, 999)
bt <- apply(bootmat, 2, sd)
mean(bt)
mean(bt) - t0 # bootstrap bias
bootCI(t0, bt)
bootCIlogit(t0, bt)
bootCI(t0, bt, conf=0.05)
# Which of these 5% CIs include the point estimate, t0?
bootCI(t0, bt, conf=0.05) < t0 # Should be "TRUE FALSE"
``` |

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