RE: Correlations greater than 1
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RE: Correlations greater than 1

Hi Kim,
I think that correlations reported by ASREML would be rounding errors. 
You will get slightly different answers if you truncate or round the
(co)variance estimates.  For some traits, values near one are expected
but other cases it may illustrate a bigger issue.  All values should be
reported as the actual value - although it is very much like how
negative estimates of variance were truncated at zero when reported. 
This allows the readers to make their own mind about the true value
weighted by have you describe the results.

The issue of parameter space and land of no constraint have varying
context.  I am presuming your query relates to the computational
aspects.  By standard definition, variances are bounded by zero and plus
infinity, covariances are bounded by minus and plus infinity but
correlations are defined between minus one and plus one.  Further, there
are additional constraints that depend on model imposed such as genetic
models as the heritability is defined between zero and one.  For
example, a sire model forces a particular constraint on residual
variance.  Therefore, constraints are required so that estimates are
within their associated parameter spaces.

The vast majority of methods (and the associated computational
algorithms) to estimate variance components do not account for these
constraints.  For example,  ANOVA permits negative estimates of variance
components. An additional aspect is the type of algorithm used.  Use of
second-derivative methods are not guaranteed to converge and may
converge to a local maximum. Most maximum likelihood methods usually
only refer only to the variance components such that these remain in the
parameter space. But does not guarantee that the correlations are within
the parameter space.  I believe that most programs attempt to
re-estimate the (co)variance components if the correlations are out of
the parameter space.  But generally these end up with estimates on the
boundary space hence correlations around +/-1.  It is questionable if
these are real values, at a local maximum or due to a lack of sufficient
information. Hence, it is user beware and user be knowledgeable of the
data, model and method.

I do not know how ASREML implements the !GP option.  However, I don't
understand the concern about biased estimates because maximum likelihood
is biased in the first place!  

Hope this helps,
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