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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


On 20130228, Gaj Vidmar <gaj.vidmar@guest.arnes.si> 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 sortofcontrolchart 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., 1alpha) 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
> (1aplha) = (n0.5)/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 kth 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. 325346. Here d is the number of degrees of freedom for the alternative, such as testing a pdimensional null in a p+ddimensinal 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 pvalue 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, coauthorship > 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, 143170. > 2. Berry S, Viele K. Adjusting the alphalevel for sample size. > Carnegie Mellon University Department of Statistics Technical Report 1995; > 635. > http://www.stat.cmu.edu/tr/tr635/tr635.ps > 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): 7587. > http://www.jdsonline.com/file_download/159/JDS380.pdf
 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 hrubin@stat.purdue.edu Phone: (765)4946054 FAX: (765)4940558



