S. Shapiro wrote:
>
> I have a set of six numbers, as follows:
>
> 6.77597
> 7.04532
> 7.17026
> 7.13235
> 7.56820
> 6.97272
>
> which represent results from six different measurements of
> the same thing in six different trials, one measurement
> per trial. (As a consequence of measurement the samples
> are destroyed, so it is not possible to measure the same
> sample six different times. Therefore, I had to set up
> six separate, independent experiments and measure my
> parameter of interest once in each experiment.)
>
> The question I seek to answer is: are the 6 values
> obtained in the measuring process reproducible within
> statistically meaningful boundaries? I suppose another
> way of asking the same question is: is the null hypothesis
> Ho satisfied with respect to this series of measured
> values?
Firstly: do you *have* a reasonable null hypothesis H0 prior to your
observations? I would suggest that for single-sample scenarios the default
is that there is not. Ask yourself - is there any value which had any
special plausibiity before the data were collected? (EG: if these were pH
values of supposedly pure samples of water, H0: pH = 7.0 might be plausible;
but if they were samples of industrial waste, you would probably not have
any one pH value with a high enough prior plausibility to justify hypothesis
testing.) If not, you should be thinking exclusively in terms of interval
estimation, not hypothesis testing; _you_have_no_hypothesis_to_test_. A
basic rule of thumb:
NEVER test a null hypothesis derived from your data.
> Using MINITAB 11.21 (the only statistics programme
> available to me) I saw that the population distribution
> for these six values is _sort of_ symmetric though not
> quite normal. This observation, plus the fact that the
> sample size is so small (n = 6) suggested that I might
> obtain the answer I seek using Wilcoxon's Signed Rank
> Test.
With a sample size of 6 you cannot draw any strong conclusions about
population distribution. Try the following MINITAB simulation:
MTB> random 6 c1;
SUBC>normal 0 1. #this line and the semicolon above may actually be omitted
MTB> boxp c1
MTB> hist c1
MTB> dotp c1
repeating it 20 times. You will see that the plot often does not look
normal.
Now try the same thing substituting
SUBC> uniform 0 1.
and you will see that you often get a distribution that *might* be normal;
SUBC> exponential 1.
from a very skewed population sometimes gives a rather symmetric sample with
n=6; and
SUBC> cauchy 0 1.
drawn from a distribution so bad that the central limit theorem does not
apply to it, will sometimes produce plausibly normal-looking plots (about
one time in three,in my experience.)
Question to ask: is there any reason outside the data to assume
approximate normality, or approximate symmetry? You have assumed the second
and not the first based on EDA (exploratory data analysis); I would claim
that a sample this small *cannot* support either assumption, though in
fairly extreme circumstances it may make one or both appear very *un*likely.
In the absence of either assumption, you can only use a sign
test/interval; you can get a 97% confidence interval (tail areas 1/64) of
the form (Ymin,Ymax) - here, (6.77,7.56) - and that is about the best you
can do. [With this few data I do not trust MINITAB's 95% nonlinear
interpolation interval as far as I could spit a rat. There is not enough
information to justify any interpolation model.] Note that this is an
estimate for the median, not the mean; and you have no reason to suppose
that they coincide.
Now, as to the rest of your posting: I am not quite sure what you mean
by "reproducibility" in this context, but that is not what a confidence
interval represents. A _prediction_interval_ would give the range in which
you would expect 95% of future values to fall. With this few data, you
cannot create a nonparametric prediction interval. I think the answer that
you must bear back to the Direktor is something like:
We do not have enough data to reach any honest conclusions about future
measurements, and we have only a rather vague interval estimate of the
median. More data are needed.
If you had 20 to 30 data, the following would happen:
(1) the criteria for "approximate normality" for a t confidence interval
would be fairly loose. For instance, an exponential or uniform distribution
would be just fine.
(CAUTION: with a moderate sample size and skewed distribution, the real
question is whether the mean is the best measure of location, or whether the
median would be more relevant. This cannot be answered _in_vacuo_; it
depends what you want the estimate for. For symmetric non-normal
distributions, OTOH, the mean and median coincide.)
(WARNING: prediction intervals cannot rely on the CLT to "make it all
normal", as they use the original distribution more or less directly.)
(2) you would have enough data to make an informed decision as to
whether the criteria for a CLT-based CI were satisfied.
(3) you would have enough data to make an informed decision as to
whether the stricter criteria for a normal-model PI were satisfied.
If data are very hard to get, a statistician, working hands-on, could
advise you as to how few you can get away with.
If there just are no more data to be had, you may have to accept that
the gods have not granted you an answer. This is not an unknown outcome.
-Robert Dawson
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