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Re: st: RE: one-tailed tests


From   Roger Newson <r.newson@imperial.ac.uk>
To   "statalist@hsphsun2.harvard.edu" <statalist@hsphsun2.harvard.edu>
Subject   Re: st: RE: one-tailed tests
Date   Thu, 8 Jul 2010 17:39:49 +0100

My point was that tests are a means to the end of defining confidence regions. These confidence regions may be confidence intervals for a scalar parameter (eg a difference between 2 probabilities of voting Democrat), or k-dimensional confidence regions for a k-vector parameter (eg the Bonferroni boxes recommended for Bea Potter's parameter (b1,b2)), or even power-set-valued confidence regions for a set-valued parameter (eg "the set of null hypotheses that are true", estimated using multiple-test procedures).

The confidence region is defined (at least approximately) as the set of parameter values that pass the test. Most of these applications work best when the tests are 2-tailed. In the case of confidence intervals, the test must usually be 2-tailed for the confidence interval to have both an upper limit and a lower limit. In the case of multiple-test procedures, it helps if the underlying tests are 2-tailed, because, that way, the P-values are non-negatively correlated (even if the test statistics may be negatively correlated), and this makes the multiple-test procedure less conservative. Occasional exceptions arise when the distribution of the test statistic, under some hypotheses, really is one-tailed. This typically leads to confidence intervals bounded only on one side (eg confidence intervals for odds ratios extending from a lower limit to infinity), or to a requirement for conservative multiple-test procedures (eg the Holm procedure).

I hope this helps.

Best wishes

Roger


Roger B Newson BSc MSc DPhil
Lecturer in Medical Statistics
Respiratory Epidemiology and Public Health Group
National Heart and Lung Institute
Imperial College London
Royal Brompton Campus
Room 33, Emmanuel Kaye Building
1B Manresa Road
London SW3 6LR
UNITED KINGDOM
Tel: +44 (0)20 7352 8121 ext 3381
Fax: +44 (0)20 7351 8322
Email: r.newson@imperial.ac.uk
Web page: http://www.imperial.ac.uk/nhli/r.newson/
Departmental Web page:
http://www1.imperial.ac.uk/medicine/about/divisions/nhli/respiration/popgenetics/reph/

Opinions expressed are those of the author, not of the institution.

On 08/07/2010 15:01, Nick Cox wrote:
I don't know why anyone would want to do any test if you can get a
confidence interval. After all, most of the best science should be
quantitative, and a confidence interval is a report on the measurement
scale of interest: it summarizes information, it indicates uncertainty
and it can be compared with your qualitative ideas on sign or direction
quite easily. If you think that something should be positive, or
increase, or whatever, you can see whether the c.i. lies the correct
side of zero, or whatever. In contrast, a yes-no test decision or even a
P-value is a more cryptic and problematic reduction of the data.

Actually, I don't know why anyone would get a confidence interval if you
can put most or all of the data on a graph. After all, a good graph
shows the main features of the data and detailed departures from those
features, and it can give you ideas on what next to do.

I have to estimate the extent to which Eric or Roger was talking tongue
in cheek, and you will have to do the same about this.

Nick
n.j.cox@durham.ac.uk

Eric Uslaner

Bea Potter asked how to do one-tailed tests and Roger Newson responded:
"I don't know why anybody would want to do a 1-tailed test, except if
the
distribution of the test statistic, under the null hypothesis, really IS
one-tailed."

I don't understand why anyone would do a two-tailed test.  If you have a
theory, that theory is directional: The more A, the more B.  A
two-tailed test is completely inappropriate for this.  E.g., in studies
of voting behavior in political science, we usually posit that party
identification is the dominant force behind voter choice.  So if you are
a strong Democrat, you will vote Democratic.  A two-tailed test would
imply that a strong Republican would be equally likely as a strong
Democrat to vote Democratic.  This is plain silly.  If you don't have a
theoretical focus, you don't have a research design worth submitting
anywhere.

So I don't understand why anyone does two-tailed tests and why Stata
(together with other statistical programs) report them as defaults.


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