Will Hopkins wrote:
>
> Responses to various folks. And to everyone touchy about one-tailed
> tests, let me make it quite clear that I am only promoting them as a
> way of making a sensible statement about probability. A two-tailed p
> value has no real meaning, because no real effects are ever null. A
> one-tailed p value, for a normally distributed statistic, does have a
> real meaning, as I pointed out. But precision of
> estimation--confidence limits--is paramount. Hypothesis testing is
> passe.
>
...............................
The only use for a test
> statistic is to help you work out a confidence interval. Don't ever
> report them in your papers.
>
This is arguably the case for research matters when estimating/testing a
mean - a confidence interval and a test are two ways of approaching the
same thing. Even there, the hypothesis testing approach is a useful way
of thinking. It is exactly the scientific method writ small. I also
happen to think that all tests should be one tailed, but almost
certainly not for the same reasons as Will's.
In 'practical statistics' such as quality control, one is only
interested if the sample mean is sufficiently close to what it should be
that one can proceed as if it does equal what it should - that is,
accept the null model and proceed - or not. If it is not, the 'true
value' (meaningless phrase!) is of no interest, so obtaining a
confidence interval is a waste of time. It could be done, but offers
nothing.
Hypothesis testing is essentially a method of selecting between models.
Should I use the model with mu = 0, or a model with mu not= 0? If the
latter, what value of mu should I use?
A more illuminating example is simple linear regression. Should I use
the model with beta = 0 (that is, the 'constant mean' model, Y = mu +
epsilon) or the model with beta not= 0 (that is, the varying mean model,
Y = alpha + beta*X)? This is clearly a choice between two different
models. Again one can resolve it by using a test statistic or by
calculating a confidence interval, but in both cases you are doing the
same thing - deciding between the two models.
The questionable thing about hypothesis testing is the fact that the
null model is privileged over the alternative. But this is resolved as
follows: if a test statistic is not significant (or equivalently, if the
confidence interval includes zero) then it does not matter which model
you choose. But you do have to choose, at least tentatively. (In a
quality control application you have to decide really; in research, you
choose tentatively.) All this means is that you make your decision on
some other basis than the statistics. For the regression example, we
would decide on the basis of simplicity. In a court case we decide on
the basis of fairness. In the case of research we decide on the basis of
accepted theory.
Hypothesis testing is certainly not passe!
Regards,
Alan
--
Alan McLean ([EMAIL PROTECTED])
Department of Econometrics and Business Statistics
Monash University, Caulfield Campus, Melbourne
Tel: +61 03 9903 2102 Fax: +61 03 9903 2007
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