[R-sig-ME] FW: Negative estimates of variance component

[chris.brien at iinet.net.au]

Hi Jerome,

The example that John gave is a nice one. There are others in references that
discuss negative estimates. Two older references are:

Nelder, J. A. (1954). "The interpretation of negative components of variance."
Biometrika 41(3/4): 544-548.
Nelder, J. A. (1977). "A reformulation of linear models (with discussion)."
Journal of the Royal Statistical Society, Series A 140(1): 48-77.

There is also a lot of discussion of negative estimates in Searle, S. R., G.
Casella, and C.E. McCulloch. (1992). Variance components. New York, And Wiley.

and some in Littell, R. C., G. A. Milliken, et al. (2006). SAS for Mixed Models.
Cary, N.C., SAS Press.

As for implementation in lme4, that is outside what I know. However, the usual
way used in other packages is to allow negative estimates of negative components
as a surrogate for components of excess covariance, but, as John has pointed out,
with the overall covariance matrix remaining positive definite.

Cheers,

Chris

PS Apologies for multiple messages caused by mail client problems

>From:
>Jerome Goudet [mailto:jerome.goudet at unil.ch] 
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>Sent: Monday, 2 August 2010 12:44 PM
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>To: John Maindonald
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>Cc: Chris Brien; Ben Bolker; r-sig-mixed-models at r-project.org
>
>Subject: Re: [R-sig-ME] Negative estimates of variance component
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>I'd be delighted to learn more about this, and how it could
>be implemented in lme4. Any lead that I could follow?  Cheers, Jerome
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>John Maindonald wrote: 
>
>The negative estimates are really then parameters in the variance-covariance
matrix.  (There is no escaping from the requirement that this matrix, however
parameterised, should be positive definite.)  If a parameter is negative to an
extent that excludes statistical error, it cannot then be interpreted as a
variances, but provides an indication that the variance-covariance structure has
been wrongly formulated.  This can be useful diagnostic information. In results
from a block design, it might for example resultfrom choosing plots in a manner
that increases rather than decreases heterogeneity between plots.  For example,
blocks may be chosen so that plots are at increasing distances from a river
bank.  I have from time to time encountered scientists who though that a choice
of this type (maximising heterogeneity) was the right thing to do. John
Maindonald             email: john.maindonald at anu.edu.auphone : +61 2
(6125)3473    fax  : +61 2(6125)5549Centre for Mathematics & Its Applications,
Room 1194,John Dedman Mathematical Sciences Building (Building 27)Australian
National University, Canberra ACT 0200.http://www.maths.anu.edu.au/~johnm On
31/07/2010, at 7:38 AM, Chris Brien wrote:   
>
>Dear all, Doug, thanks for the information that negative estimates are not
possible with lme4 and nlme. As for the questions from others about why would you
want to do it, there are a number of reasons. Of course, variance components must
be positive. However, one reason, given by Littel et al in  SAS for Mixed Models,
to allow negative estimates is that it controls Type I error better and in
certain circumstance may give better power. This is essentially related to the
issue of pooling non-significant error terms. In SAS MIXED there is an option to
allow negative estimates.  Another reason is that variance components might be
interpreted as components of excess variation or even excess covariance. Then
negative values indicate less covariance between than within groups, as Jerome
explained. However, as suggested in the "Heywood case" scenario, reasons for
negative estimates need to be investigated to make sure that it really is
negative covariance. As for fitting using structured covariance matrices as
suggested by Ben, this is a nice way to go about it but requires that the
software has this capability. Lme4 does not have it and nlme is not good at
fitting crossed factors - a gotcha' situation. A third reason is that in order to
approximate a randomization analysis, one needs to allow for negative estimates.
I reiterate that here I am thinking of variance components as surrogates for
components of excess covariance. So, I do not agree that it is in the category of
"consistency with older method-of-moments estimates". The issue is more that
there is no coherent way to specify the types of models of which I am thinking in
current software packages. Cheers,  Chris  -----Original Message-----From: Ben
Bolker [mailto:bbolker at gmail.com] Sent: Friday, 30 July 2010 7:02 PMTo:
jerome.goudet at unil.chCc: Chris Brien; r-sig-mixed-models at r-project.orgSubject:
Re: [R-sig-ME] Negative estimates of variance component  I claim that this is in
the category of "consistency with oldermethod-of-moments estimates" ... I know
that population geneticistsstill like to think in terms of variance components,
but wouldn't oneideally want to deal with the negative correlation built into
thesystem by estimating it more or less directly (i.e. via a
structuredvariance-covariance model that has nonnegative variances but couldhave
negative covariances) rather than by estimating a negativevariance component? 
Ben Bolker (not a population geneticist so possibly missing the point) On Fri,
Jul 30, 2010 at 12:39 PM, Jerome Goudet <jerome.goudet at unil.ch> wrote:    
>
>Hi all, Here is an example: genes are nested in individuals, themselves nested
inpopulations.  If individuals avoid mating with relatives, then the
variancecomponent of allele frequencies among individuals within population
isexpected to take negative values, as 2 genes taken from 2 differentindividuals
are more similar than two genes taken from the same individual.See Weir &
Cockerham (1984) Evolution for instance.    Ben Bolker wrote:  Pardon my asking,
but why? For consistency with older (arguably lesscorrect) method-of-moments
estimates that could give negativeestimates? On Fri, Jul 30, 2010 at 1:32 AM,
Chris Brien <Chris.Brien at unisa.edu.au>wrote:  Dear mixed modellers, I have a data
set that gives me an estimate of 0 for one of the variancecomponents. I wanted to
allow for the estimate of the component to benegative. My search of the
documentation led me to believe that this is notpossible. Am I right, or did I
miss something? Cheers, Chris Brien-----University of South AustraliaADELAIDE 
5001  South AustraliaWEB page: 
<http://people.unisa.edu.au/Chris.Brien> _______________________________________________R-sig-mixed-models at r-project.org
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mailing listhttps://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models   --Jérôme
GoudetDept. Ecology & EvolutionUNIL-Sorge, CH-1015
Lausanne mail:jerome.goudet at unil.chTel:+41 (0)21 692 4242Fax:+41 (0)21 692 4265      
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>-- Jérôme GoudetDept. Ecology & EvolutionUNIL-Sorge, CH-1015
Lausanne mail:jerome.goudet at unil.chTel:+41 (0)21 692 4242Fax:+41 (0)21 692 4265
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