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

[John Maindonald]

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 result
from 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.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre 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 PM
> To: jerome.goudet at unil.ch
> Cc: Chris Brien; r-sig-mixed-models at r-project.org
> Subject: Re: [R-sig-ME] Negative estimates of variance component
> 
>  I claim that this is in the category of "consistency with older
> method-of-moments estimates" ... I know that population geneticists
> still like to think in terms of variance components, but wouldn't one
> ideally want to deal with the negative correlation built into the
> system by estimating it more or less directly (i.e. via a structured
> variance-covariance model that has nonnegative variances but could
> have negative covariances) rather than by estimating a negative
> variance 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 in
>> populations.  If individuals avoid mating with relatives, then the variance
>> component of allele frequencies among individuals within population is
>> expected to take negative values, as 2 genes taken from 2 different
>> individuals 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 less
>> correct) method-of-moments estimates that could give negative
>> estimates?
>> 
>> 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 variance
>> components. I wanted to allow for the estimate of the component to be
>> negative. My search of the documentation led me to believe that this is not
>> possible. Am I right, or did I miss something?
>> 
>> Cheers,
>> 
>> Chris Brien
>> -----
>> University of South Australia
>> ADELAIDE  5001  South Australia
>> WEB page:  <http://people.unisa.edu.au/Chris.Brien>
>> 
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>> 
>> --
>> Jérôme Goudet
>> Dept. Ecology & Evolution
>> UNIL-Sorge, CH-1015 Lausanne
>> 
>> mail:jerome.goudet at unil.ch
>> Tel:+41 (0)21 692 4242
>> Fax:+41 (0)21 692 4265
> 
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