Re: please help

From: S. F. Thomas (sf.thomas@verizon.net)
Date: Thu Jun 14 2001 - 05:09:30 MET DST

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    Kelly wrote:
    >
    > I have the gage repeatability & reproducibility(gage R&R) analysis
    > done on two instruments, what hyphoses test can I use to test that the
    > repeatability variance(expected sigma values of repeatability) of the
    > two instruments are significantly different form each other or to say
    > one has a lower variance than the other.
    > Any insight will be greatly appreciated.
    > Thanks in advance for your help.

    One approach is to form the likelihood function in each case and to
    eliminate the nuisance parameters (the means) by marginalization.
    Although it is well known that marginalization by maximization will
    give misleading answers for both the location and precision of your
    estimate of the variances, I have shown how another method based on
    marginalization by the rule of product-sum can avoid the problems
    known to exist with respect to the former. (See _Fuzziness and
    Probability_ (ACG Press, 1995)). This method also avoids the
    assumptions of the Bayesian approach -- effectively a method of
    marginalization by integration -- which have been considered and
    rejected, and with good reason in my opinion, by those of the
    classical school. The product-sum method may be relatively easily
    implemented within an extensible stat package such as R, and I would
    be happy to apply my implementation of it to your problem if you
    would send me the two datasets. Essentially, once the nuisance
    parameters (the one or more means) are eliminated, what is left in
    each case is the (marginal) likelihood function of the variance, and
    one could effectively compare directly the plots of the two variance
    marginal likelihoods, and also, if need be, the likelihood function
    of the difference, to see how different this is from zero. This is
    not a classicist's answer, but tests of hypothesis and all that can
    be obviated if the likelihood function can be directly manipulated in
    the way I describe. This has been the whole point of the Bayesian
    method, except of course for the inadequate justification provided
    not only for its insistent subjectiveness, but also for treating
    model parameters as though they were random variables in their own
    right. Hope this is helpful.

    Regards,
    S. F. Thomas

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