> That is: today, with more experience, I think
> I *can* imagine an odd data circumstance (one with
> small Rs) where it would be useful estimate the Rs from
> cov(XY)/ sqrt[var(X)var(Y)]
> where each var() and covar() was based on its own
> most-complete estimate from the data.
This is actually a very common problem when using covariance component
estimates to estimate correlations (or anything else derived from covariance
matrices). This occurs naturally when trying to partition the sources of
variance and covariance using, say, an ANOVA approach. A typical application
is the estimation of genetic variance and covariance among traits in
quantitative genetics. Here, the sampling distributions of these things
definitely lead to problems in analysis. One can try to use a ML approach to
get around this, although in practice this usually means setting negative
variance components to zero, which undoubtedly really messes up the
eigenstructure of the matrix.
See:
Hill WG, Thompson R (1978) Probabilities of non-positive definite
between-group or genetic covariance matrices. Biometrics 34, 429-439.
Hayes JF, Hill WG (1981) Modification of estimates of parameters in the
construction of genetic selection indices ('bending'). Biometrics 37,
483-493.
Kirkpatrick M, Lofsvold D, Bulmer M (1990) Analysis of the inheritance,
selection, and evolution of growth trajectories. Genetics 124, 979-993.
Phillips PC, Arnold SJ (1999) Hierarchical comparison of genetic
variance-covariance matrices. I. using the Flury hierarchy. Evolution 53,
1506-1515.
--
Patrick Phillips
Center for Ecology and Evolutionary Biology
University of Oregon
.
.
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