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Possible answers to this question are very contingent on the
particularities of the object of study.
See comments in-line below.

On 30-Jul-11 12:35:45, Anoop Balachandran wrote:

Hi everyone,

How valid is a study which uses a sample size of 12 with a
within-subject design? Or how well can we extrapolate the
conclusions to the population studied.

Validity: Regardless of sample size, a properly executed study
is always valid -- as far as it goes. How far it goes depends,
again, on the particularities. Even then, whether it is *adequate*
for your purposes depends on how far you need it to go. It might
be that your needs are so broadly defined that a very imprecise
answer will be adequate, and a small sample may be adequate.
If your needs for precision are stringently set, and if the
variation in the population is large, then a small sample will
not be adequate, and you will need to calculate what sample
size will be adequiate. However, for that you need an estimate
of the variability in the population.

Extrapolation to population: *Very* contingent on how likely it
is that a sample of given size will capture, "representatively",
the essential features of the variation in the population.
In particular, if the population is "non-homogenous" (i.e.
consists of several groups each with distinct characteristics)
then your sample will need to be large enough to have a good
propbability of including an adequate number from each group.
Or, if you know of markers which can identify the distinct
groups, then you might consider adopting a stratified sampling
approach.

I have heard that you need at least 30 per group to get a normal
distribution. That been said,  I have read that even numbers
approaching 10 can get you close to a normal distribution.

If the distribution of values in the population being sampled
from is close to Normal, then you can validly treat your sample
as a sample from a Normal distribution, regardless of sample size.
However, varifying this condition is another question. People
often carry out some test (e.g. Kolmogorov-Smirnov) for Normality.
It will be important to choose the sample size large enough that
a deperture from Normality in the population which is large enough
to invalidate your results has high probability of being detected
(power of test).

If you have a sample from a non-Normal population, then in general
a large enough sample size means that statistics calculated in the
analysis (e.g. sample means) will have distributions which are
close to Normal. However, the sample size you would need will depend
on the degree and kind of the departure from Normality. In particular,
if the variable has (say) a positively skew distribution in the
population, then Normality for the (say) sample mean  will only
slowly be approached (skewness is particularly toxic to the Normality
of sample means from large samples).

In certain (admittedly extreme) cases, no sample size however
large can give you any improvement with respect to Normality
of (say) sample means.

For instance, imagine a machine-gun mounted on a pivot and
able to rotate in a horizontal plane. At some distance to
one side is a plaster wall. The gun is rotated rapidly about
its pivot, and continually fires a very large number of rounds.
The positions of the resulting holes in the wall are measured,
and used as the data.

Then the distribution of the positions is not Normal, and
no matter how large a sample you take, the distribution
of the sample mean will be exactly the same as that for
the original holes, so will be just as non-Normal.

For the technically minded: The circumstances of the experiment
are such that the direction in which the machine-gun is pointing
when a bullet is fired is uniformly distributed round a circle.
The distance of the point of impact from the point on the wall
nearest to the gun is the tangent of this angle (multiplied
by the distance from gun to wall). It then folllows that the
points of impact have a Cauchy distribution, which is totally
immune to the Normalising effects of large sample means!

The study I am talking about measures muscle fiber size using
a biopsy and also assess strength measurements.

You need to do a preliminary study in order to ascertain how
these variables are typically distributed. Then you will be
in a position to tackle the other issues discussed above.
With luck, you may find such data in the literature ...

In the exercise field, we sill use P-values so they haven't
reported the confidence intervals.

... or quite possibly not, if the workers in this field are
so indifferent as to the nature of their primary data.

However, there may be useful information in physiological
literature of a more serious kind!

Thanks!

Hoping it helps!
Ted.

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Date: 30-Jul-11                                       Time: 14:15:02
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