On Fri, 24 Mar 2000, Bernard Higgins wrote:
>
>
> Hi Bruce
Hello Bernard.
>
> The point I was making is that when developing hypothesis tests,
> from a theoretical point of view, the sampling distribution of the
> test statistic from which critical values or p-values etc are
> obtained, is determined by the null hypothesis. We need a probability
> model to enable use to determine how likely observed patterns are.
> These probability models will often work well in practice even if we
> relax the usual assumptions. When using distribution-free tests as
> an alternative to a parametric test we may need to specify
> restrictions in order that the tests can be considered "equivalent".
Agreed.
>
> In my view the t-test is fairly robust and will work well in most
> situations where the distribution is not too skewed, and constant
> variance is reasonable. Indeed I have no problems in using it for the
> majority of problems. When comparing two independent samples using
> t-tests, lack of normality and constant variance are often not too
> serious if the samples are of similar size, always a good idea in
> planned experiments.
Agreed here too.
>
> As you say, when samples are fairly large, some say 30+ or even
> less, the sampling distribution of the mean can often be approximated
> by a normal distribution (Central Limit Theorem) and hence the use of
> an (asymptotic) Z-test is frequently used. It would not, I think, be
> strictly correct to call such a statistic t, although from a
> practical point of view there may be little difference. The formal
> definition of the single sample t-test is derived from the ratio of a
> Standard Normal random variable to a Chi-squared random variable and
> does, in theory, require independent observations from a normal
> distribution.
I think we are no longer in complete agreement here. I am not a
mathematician, but for what it's worth, here is my understanding of t-
and z-tests:
numerator = (statistic - parameter|H0)
denominator = SE(statistic)
test statistic = z if SE(statistic) is based on pop. SD
test statistic = t if SE(statistic) is based on sample SD
The most common 'statistics' in the numerator are Xbar and (Xbar1 -
Xbar2); but others are certainly possible (e.g., for large-sample
versions of rank-based tests).
An assumption of both tests is that the statistic in the numerator has a
sampling distribution that is normal. This is where the CLT comes into
play: It lays out the conditions under which the sampling distribution of
the statistic is approximately normal--and those conditions can vary
depending on what statistic you're talking about. But having a normal
sampling distribution does not mean that we can or should use a critical
z-value rather than a critical t when the population variance is unknown
(which is what I thought you were suggesting).
As you say, one can substitute critical z for critical t when n gets
larger, because the differences become negligible. But nowadays, most of
us are using computer programs that give us more or less exact p-values
anyway, so this is less of an issue than it once was.
Cheers,
Bruce
--
Bruce Weaver
[EMAIL PROTECTED]
http://www.angelfire.com/wv/bwhomedir/
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