# [R] mcmcsamp and variance ratios

Martin Henry H. Stevens hstevens at muohio.edu
Thu Jan 4 19:28:26 CET 2007

```On Jan 4, 2007, at 11:18 AM, Douglas Bates wrote:

> On 1/3/07, Martin Henry H. Stevens <hstevens at muohio.edu> wrote:
>> Hi folks,
>> I have assumed that ratios of variance components (Fst and Qst in
>> population genetics) could be estimated using the output of mcmcsamp
>> (the series on mcmc sample estimates of variance components).
>
>> What I have started to do is to use the matrix output that included
>> the log(variances), exponentiate, calculate the relevant ratio, and
>> apply either quantile or or HPDinterval to get confidence intervals.
>
>> This seems too simple but I can't think of what is wrong with it.
>
> Why bother exponentiating?  I'm not sure what ratios you want but if
> they are ratios of two of the variances that are columns of the matrix
> then you just need to take the difference of the logarithms.  I expect
> that the quantiles and HPDintervals would be better behaved, in the
> sense of being based on a distribution that is close to symmetric, on
> the scale of the logarithm of the ratio instead of the ratio itself.
>
> Quantiles calculated for the logarithm of the ratio will map to
> quantiles of the ratio.  However, if you really do feel that you must
> report an HPDinterval on the ratio then you would need to exponentiate
> the logarithm of the ratio before calculating the interval.
> Technically the HPD interval of the ratio is not the same as
> exponentiating the end points of the HPDinterval of the logarithm of
> the ratio but I doubt that the differences would be substantial.

My collaborator (the evolutionary biologist on this project) is very
skeptical of the results I have been providing. Most of the Qst ratios,

Qst = Var[population] / ( Var[population] + Var[genotype] )

have values close to 0.5 (0.45--0.55) and wider confidence intervals
(e.g. 0.2--0.8) than they have tended to see in the literature.

I suspect that this derives from our tiny sample sizes: 24 genotypes
total, distributed among 9 populations (2-3 genotypes within each
population).

Our variances (shrinkage estimates) frequently do not differ from
zero. My model building using AIC results in the removal of most of
the variance components. We only stuck the terms back in the model in
order to get SOME number for these.

The biologist (supported by a biometrician) wants to bootstrap or
jackknife the models. I will be very skeptical if all of a sudden
they get qualitatively different estimates and intervals.

Does my perspective make sense? All comments appreciated.

-Hank

Dr. Hank Stevens, Assistant Professor
338 Pearson Hall
Botany Department
Miami University
Oxford, OH 45056

Office: (513) 529-4206
Lab: (513) 529-4262
FAX: (513) 529-4243
http://www.cas.muohio.edu/~stevenmh/
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http://www.muohio.edu/botany/

"E Pluribus Unum"

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