Re: [R] Bootstrapping Confidence Intervals for Medians

From: Tim Hesterberg <timh_at_insightful.com>
Date: Wed 31 Jan 2007 - 03:11:31 GMT


This is in followup to an earlier message about bootstrap confidence intervals for the difference in two medians. I'll touch on three topics:

* bootstrap theory for one vs two samples
* special case of median
* bootstrap small samples


ONE VS TWO SAMPLES Brian Ripley wrote (the complete posting is below):
>The 'standard' theory of bootstrap confidence intervals as implemented in
>e.g. package 'boot' is for a single-sample problem (and it would be
>pushing its justification very hard to use this for n=9). But you have
>two samples, and haven't told us how you intend to bootstrap. I guess you
>mean a stratified bootstrap, sampling with replacement independently from
>observations 1-4 and 5-9. I don't know of theory for bootstrap confidence
>intervals from that scenario: do you?

Much of the bootstrap theory was developed for a single sample, but carries over fairly easily to two samples or other stratified sampling. Davison and Hinkley do talk about stratified sampling in their book, and Canty implemented that in the 'boot' package. S+Resample also supports stratified sampling, with a front-end function 'bootstrap2' to make the two-sample case easier.

SPECIAL CASE OF MEDIANS
>Beyond this, there are considerable problems with bootstrapping medians in
>small samples as the median is a non-smooth function of the data and the
>bootstrap samples take very few values. See for example the galaxies
>dataset as discussed in MASS4. For the stratified bootstrapping I
>referred to, there are only a handful of possible values of each of the
>medians and so the bootstrap distribution is a highly non-uniform one on a
>few values. E.g.

Bootstrapping medians does present special difficulties. Particularly with a single sample and n odd, the bootstrap distribution is often a very inaccurate estimate for the sampling distribution; it takes on only a few values.

Curiously, however, bootstrap percentile intervals for a single median are not bad. They are similar to classical nonparametric intervals based on order statistics, and use either the same or an adjacent order statistic; where they differ, the classical interval is conservative (one-sided non-coverage less than 0.025, often much less), and the bootstrap interval is one order statistic narrower, with closer to the nominal coverage.

BOOTSTRAPPING SMALL VS MEDIUM SAMPLES People tend to think of bootstrapping for small samples, where they don't trust the central limit theorem, like n=9. That is misguided. They should use the bootstrap in medium size samples, where they shouldn't trust the central limit theorem, like perhaps n=20 (depending on the application) to n=1000.

For very small samples the data distribution does not reliably reflect the shape of the population, and it is better to impose parametric assumptions. In other words, with very small samples you can't accurately estimate skewness, so it may be better to use classical methods that just assume that skewness is zero.

Conversely, for medium size samples one should not just assume that skewness is zero, but instead use bootstrap or other methods that allow for skewness. Yes, n=1000 is a medium size sample -- the effect of skewness on confidence intervals and tests decreases very slowly, with size or coverage errors of O(1/sqrt(n)) for classical t tests and confidence intervals.

One beauty of the bootstrap is that it provides a nice way to estimate what size errors you make by assuming sampling distributions are normal -- by creating a bootstrap distribution and seeing how non-normal it is. It provides a nice alternative to the pre-computer rule of using normal methods if n >= 30.

Tim Hesterberg


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========================================================
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(I do talk more about these issues in the short courses.)

Brian Ripley wrote:
>On Sat, 6 Jan 2007, gilbertg@musc.edu wrote:
>
>> I apologize for this post. I am new to R (two days) and I have tried and
>> tried to calculated confidence intervals for medians. Can someone help
>> me?
>
>Later, you say you want a confidence interval for a difference in medians,
>not the same thing.
>
>For medians, see MASS4 section 5.7 for worked examples and discussion of
>the pitfalls.
>
>> Here is my data:
>>
>> institution1
>> 0.21
>> 0.16
>> 0.32
>> 0.69
>> 1.15
>> 0.9
>> 0.87
>> 0.87
>> 0.73
>>
>> The first four observations compose group 1 and observations 5 through 9
>> compose group 2. I would like to create a bootstrapped 90% confidence
>> interval on the difference of the medians (n2-n1). I have successfully
>> calculated a permutation test.
>>
>> This shouldn't be as difficult as I am making it, would someone please
>> enlighten me?
>
>
>It seems to me to be much more difficult than you have made it. We need
>to know exactly what you mean by
>
>> a bootstrapped 90% confidence interval on the difference of the medians
>
>The 'standard' theory of bootstrap confidence intervals as implemented in
>e.g. package 'boot' is for a single-sample problem (and it would be
>pushing its justification very hard to use this for n=9). But you have
>two samples, and haven't told us how you intend to bootstrap. I guess you
>mean a stratified bootstrap, sampling with replacement independently from
>observations 1-4 and 5-9. I don't know of theory for bootstrap confidence
>intervals from that scenario: do you?
>
>Beyond this, there are considerable problems with bootstrapping medians in
>small samples as the median is a non-smooth function of the data and the
>bootstrap samples take very few values. See for example the galaxies
>dataset as discussed in MASS4. For the stratified bootstrapping I
>referred to, there are only a handful of possible values of each of the
>medians and so the bootstrap distribution is a highly non-uniform one on a
>few values. E.g.
>
>x <- c(0.21, 0.16, 0.32, 0.69)
>y <- c(1.15, 0.9, 0.87, 0.87, 0.73)
>z <- numeric(10000)
>for(i in seq_len(10000))
>z[i] <- median(sample(x, replace=TRUE)) - median(sample(y, replace=TRUE))
>
> -0.99 -0.965 -0.94 -0.91 -0.885 -0.83 -0.74 -0.725 -0.715 -0.71
> 33 70 83 27 134 64 129 16 259 317
> -0.7 -0.69 -0.685 -0.66 -0.645 -0.635 -0.63 -0.605 -0.58 -0.57
> 43 370 711 1064 70 538 455 1388 424 29
> -0.55 -0.545 -0.52 -0.49 -0.475 -0.465 -0.46 -0.45 -0.445 -0.42
> 905 57 79 41 54 119 28 183 146 436
> -0.41 -0.395 -0.365 -0.305 -0.28 -0.225 -0.21 -0.18 -0.04
> 100 290 759 13 34 64 117 323 28
>
>You could use that table to give 'basic' or 'percentile' confidence
>intervals, if you have reason to believe in them.
>
>
>> Greg Gilbert, Faculty Research Associate
>> Department of Biostatistics, Bioinformatics, & Epidemiology
>> Medical University of South Carolina
>
>--
>Brian D. Ripley, ripley@stats.ox.ac.uk
>Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/
>University of Oxford, Tel: +44 1865 272861 (self)
>1 South Parks Road, +44 1865 272866 (PA)
>Oxford OX1 3TG, UK Fax: +44 1865 272595



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