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Topic: Adjustment of alpha to sample size (Prof Rubin, Prof Koopman et al., could you help this amateur again?)
Replies: 2   Last Post: Mar 4, 2013 5:31 PM

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Herman Rubin

Posts: 399
Registered: 2/4/10
Re: Adjustment of alpha to sample size (Prof Rubin, Prof Koopman et al., could you help this amateur again?)
Posted: Mar 1, 2013 2:20 PM
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On 2013-02-28, Gaj Vidmar <> wrote:
> Some time ago I asked a question here about adjustment of alpha to sample
> size.

> I'm using it (broadly speaking, to roughly clarify) in the field of SPC/SQC
> ("eclectic", "pragmatic", i.e., anything goes that allegedly works :)
> My sort-of-control-chart invloves prediction interval from regression
> (through origin, if anyone remembers or wants to search for the original
> post),
> where I adjust the confidence level (i.e., 1-alpha) so that there is always
> one half of a point to be expected outside the limits
> (maybe bizzare or silly, maybe not - let's leave that aside). So my formula
> is

> (1-aplha) = (n-0.5)/n

This was proposed in the 19th century to test for outliers.
The argument was not in any way Bayesian.

> At first, I had called this approach (half seriously) idiot's FDR, but
> further reading
> convinced me that albeit not entirely unrelated (as a concept),
> FDR is always and only about multiple test (to put it simply).
> Fortunately, I've found three Bayesian references that
> (at least my blockheadness guesses so)
> argue for this type of thinking (listed below; the closest to something I
> can at least
> partly understand is #3). Should be enough, especially because Bayesian is
> even
> more "in" than FDR (although I'm even more clueless about it, but the
> referees
> don't know that, and I'm skilled at conning scientific journal readers :o)

> However, in the newsgroup, Prof Herman Rubin had written that "The level
> should
> decrease with increasing sample size. In low dimensional problems,
> with the cost of incorrect acceptance going as the k-th power of the error,
> the rate at which the level decreases should be about 1/n^((d+k/2)."

> So it would be wonderful if one of you wizzards came up with some "simple
> algebra" (as it's so often said in the literature when the opposite is true
> for mere mortals) that relates Prof Rubin's formula to mine!

> I can guess that "the level" refers to alpha and that my problem is
> low dimensional (1D). "Incorrect acceptance" most probably means
> "incorrect acceptance of the null hypothesis".
> And let's say I take k to be 2 (by analogy with the notorious Taguchi,
> or because quadratic loss sounds familiar to many people in many fields).
> So far so good; but what does d stand for??

I suggest you read my paper with Sethuraman, "Bayes risk efficiency",
in _Sankhya_ 1965, pp. 325-346. Here d is the number of degrees
of freedom for the alternative, such as testing a p-dimensional
null in a p+d-dimensinal parameter space, and k is the exponent
in the Type II loss. One is not restricted to the Bayes procedure,
but a prior Bayes risk minimization for a given test procedure is
considered. The rate can have factors whose logarithm is o(ln(n)),
and convergence is not great.

A simple example is testing that the mean of a normal distribution
with identity covariance matrix in 2 dimensions is 0, with a
constant Type II loss and constant prior density for the alternative;
do not worry about the infinite integral. The probability of
rejecting if the mean square of the observations exceeds 2K is
exp(-nK), and the integrated probability of a Type II error is
a multiple of K. So the prior Bayes risk is Aexp(-nK) + BK,
and this is minimized if exp(-nK) = B/nA. So in this case the
p-value should be B/nA. Other cases do not come out in nice
closed form.

> Fortunately (or un-, from a broader perspective) my paper (currently under
> review)
> should get accepted even without such ellegant justification. And all I can
> do to
> return the favour is an acknowledgement . (Needless to say, co-authorship
> is not something that the two Profs I mention in the title -- and other
> newsgroup
> "heavyweights", especially the retired ones -- want or need, anyway).

> So, as usual, thanks in advance for any help.

> Gaj Vidmar

> References:
> 1. Seidenfeld T, Schervish MJ, Kadane JB. Decisions without ordering.
> In "Acting and Reflecting", Sieg W (ed). Kluwer: Dordrecht, 1990, 143-170.
> 2. Berry S, Viele K. Adjusting the alpha-level for sample size.
> Carnegie Mellon University Department of Statistics Technical Report 1995;
> 635.
> 3. Berry S, Viele K. A note on hypothesis testing with random sample sizes
> and its
> relationship to Bayes factors. Journal of Data Science 2008; 6(1): 75-87.

This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University Phone: (765)494-6054 FAX: (765)494-0558

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