# 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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