Here I introduce a R function made by myself to calculate several confidence intervals of mean/median including exact CI, basic bootstrap CI, percentile bootstrap CI & studentized bootstrap CI.
R code
Arguments
There are five arguments in this R function bootCI():
x— a numeric vector specifying $x_i$alpha = 0.05— a numeric specifying $\alpha$ valuealternative = c("t", "l", "g")— a single character string specifying two-sided, less or greater tailB = 1999— a integer specifying number of bootstrap samplesquantileAlgorithm = 7— a integer specifying the argumenttypepassed toquantile()
Examples
We can define a numeric vector $x = [5\, 2\, 3\, 6\, 8\, 19\, 1.5]$ including $N=7$ items as num:
> num <- c(5, 2, 3, 6, 8, 19, 1.5)
After loading the above function, we can call the function to calculate CIs:
> bootCI(num)
Summary of x
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.500 2.500 5.000 6.357 7.000 19.000
CIs of mu
2.5% 97.5%
$CI.exact 0.7778728 11.936413
$CI.basic 1.7125000 9.714286
$CI.percentile 3.0000000 11.001786
$CI.studentized 2.6785623 17.669412
The above results show that
\begin{aligned} \text{exact CI} & = ( \hat{\mu} + t_{\frac{\alpha}{2}} \cdot \hat{se}_{\hat{\mu}} \;,\; \hat{\mu} + t_{1-\frac{\alpha}{2}} \cdot \hat{se}_{\hat{\mu}} ) \\ \text{basic CI} & = ( 2\hat{\mu} - \hat{\mu}_{1-\frac{\alpha}{2}}^{*} \;,\; 2\hat{\mu} - \hat{\mu}_{\frac{\alpha}{2}}^{*} ) \\ \text{percentile CI} & = ( \hat{\mu}_{\frac{\alpha}{2}}^* \;,\; \hat{\mu}_{1-\frac{\alpha}{2}}^* ) \\ \text{studentized CI} & = ( \hat{\mu} + t^*_{\frac{\alpha}{2}} \cdot \hat{se}_{\hat{\mu}} \;,\; \hat{\mu} + t^*_{1-\frac{\alpha}{2}} \cdot \hat{se}_{\hat{\mu}} ) \\ \end{aligned}where $\hat{\mu}$ is mean of $x$, $t_{\frac{\alpha}{2}}$ is lower $\alpha/2$ critical value for the $t$ distribution given $\text{df} = N-1$, $\hat{se}_{\hat{\mu}}$ is the standard error of the mean of $x$, $\hat{\mu}_{\frac{\alpha}{2}}^*$ is $\alpha/2$ percentile of the mean of bootstrapped $x$, and $t^*_{\frac{\alpha}{2}} = \frac{\hat{\mu^*}-\hat{\mu}}{\hat{se}^*_{\hat{\mu^*}}}$ is t-value based on bootstrapped $x$. See
- J. Carpenter and J. Bithell (2000). Bootstrap confidence intervals: when, which, what? A practical guide for medical statisticians. Statist. Med. 19:1141-1164
- Wikipedia contributors, 'Bootstrapping (statistics)', Wikipedia, The Free Encyclopedia, 17 December 2013, 21:28 UTC, <http://en.wikipedia.org/w/index.php?title=Bootstrapping_(statistics)&oldid=586549893> [accessed 31 December 2013]
for more details.
We can assign a different $\alpha$, direction or number of bootstrap samples:
> bootCI(num, alpha=0.01)
Summary of x
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.500 2.500 5.000 6.357 7.000 19.000
CIs of mu
0.5% 99.5%
$CI.exact -2.0962642 14.81055
$CI.basic -0.1428571 10.28571
$CI.percentile 2.4285714 12.85714
$CI.studentized 1.1394952 25.22030
> bootCI(num, alternative="g")
Summary of x
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.500 2.500 5.000 6.357 7.000 19.000
CIs of mu
5% 100%
$CI.exact 1.926445 Inf
$CI.basic 2.714286 Inf
$CI.percentile 3.285714 Inf
$CI.studentized 3.332740 Inf
> bootCI(num, B=99)
Summary of x
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.500 2.500 5.000 6.357 7.000 19.000
CIs of mu
2.5% 97.5%
$CI.exact 0.7778728 11.936413
$CI.basic 1.6053571 9.330357
$CI.percentile 3.3839286 11.108929
$CI.studentized 2.5532298 15.532786
The returned list may be helpful for someone:
> myCI <- bootCI(num)
> myCI$CI.percentile
2.5% 97.5%
3.00000 10.85714
> str(myCI)
List of 8
$ x : num [1:7] 5 2 3 6 8 19 1.5
$ alpha : num 0.05
$ alternative : chr "t"
$ B : num 1999
$ CI.exact : Named num [1:2] 0.778 11.936
..- attr(*, "names")= chr [1:2] "2.5%" "97.5%"
$ CI.basic : Named num [1:2] 1.86 9.71
..- attr(*, "names")= chr [1:2] "2.5%" "97.5%"
$ CI.percentile : Named num [1:2] 3 10.9
..- attr(*, "names")= chr [1:2] "2.5%" "97.5%"
$ CI.studentized: Named num [1:2] 2.65 17.73
..- attr(*, "names")= chr [1:2] "2.5%" "97.5%"
- attr(*, "class")= chr "bootCI"
