small sample size confidence interval by bootstrap

[HelponR]

Hi, All:

I only have 4 samples. I wish to get a confidence interval around the mean.
Is it reasonable? If not, is there a way to compute a confidence interval
for such small sample size's mean?

Many thanks,

U

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[Prof Brian Ripley]

On Fri, 31 Mar 2006, Urania Sun wrote:

> I only have 4 samples. I wish to get a confidence interval around the mean.
> Is it reasonable? If not, is there a way to compute a confidence interval
> for such small sample size's mean?

(BTW, the CI is for the population mean, not the sample mean.  I'll also
assume that you are prepared to assume that you have a single random
sample of size 4 from a location family.)

For a confidence interval, you need to make some assumptions about the
distribution.  If you assume normality, you can use t.test, but the
estimate of the standard deviation (on just 3 df) will be very variable
and this will be reflected in the length of the CI.

Your subject line mentions the bootstrap.  You could use one of several
different types of bootstrap CI but they also make assumptions, weaker
assumptions that lead to even more variability.  For a sample of size 4
there are (at most) 36 distinct means of bootstrap resamples, so none of
the methods I know of will work adequately (and most not at all).

As an example to ponder, the Cauchy distribution does not even have a
mean, but from small samples you will have no idea that is very
long-tailed.  And getting a CI for a location parameter is often better
done from a robust estimator of location than from the sample mean.
Alternatively, your true distribution might be a discrete distribution on
5 points, and you have no idea at all about the 5th value.

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Brian D. Ripley,                  [hidden email]
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  [phone] (self)
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[J Dougherty]

On Friday 31 March 2006 18:21, Urania Sun wrote:
> Hi, All:
>
> I only have 4 samples. I wish to get a confidence interval around the mean.
> Is it reasonable? If not, is there a way to compute a confidence interval
> for such small sample size's mean?
>
> Many thanks,
>
With a sample that small, it is far safer to simply consider them as four
examples and leave it at that.  In a population where there is little
variation (say an archaeological projectile point type with a nech width that
varies between 3 and 5 mm), the examples are likely to be close to typical,
and the difference isn't really llikely to be important anyway.  However, in
a population with considerable variation (for example height in humans) you
can see that trying to make any generalizations from 4 examples is going to
be more likely to be misleading than anything else.  

If your sample of four is your entire population, you have all the information
possible through simple measurements.  But, if the population were 100 the
number of possible samples of size 4 is, as Gnumeric assures me, about 4 x
10^306, which, to put it scientifically, is a whole bunch.  It'is better not
to generalize from small samples.

JD

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