[R-sig-ME] Comparing mixed models

Paul Debes paul.debes at utu.fi
Wed May 11 10:04:59 CEST 2016

ASReml-R does allow for negative variances, but you have to explicitly  
specify it via the component constraints. I also think this may be  
advisable to do for testing what is going on, especially when an important  
design term variance converged to zero. The variance may either simply be  
very small, which may just ask for a response / covariate rescaling or  
changing the threshold when the software considers a component to be zero,  
or be really negative. Otherwise, for 'boundary' variance terms ASReml-R  
appears to estimate the random effects (you can still extract them from  
the model) but it does not estimate the variance among them.

My guess is that designs described by Nelder occur more often than thought  
because I still see mention of 'pooling variance' of design terms (or  
'stepwise reducing models for non-significant terms'), so it remains  
unknown what was really going on with these removed design terms. I worked  
with different fish populations, kept due to space limitations in the same  
tanks; tanks were the experimental treatment units (split plot design of  
fish type within treatment tank). Now the fish populations had very  
different growth for families across treatments (wild vs. aquaculture -  
what a surprise), leading to a negative variance among tank effects, like  
what Nelder described. I think this block design in the stream you  
describe may have exhibited a similar pattern (I think I already read  
about it in an older post).
Back then, I really struggled how to deal with this practically, without  
running into controversies (I'm a biologist - impossible to be further  
away from being a statistician), until Geert Molenbeek helped me with  
bringing up (covered, if I remember correctly, also by some of his  
publications) that it may be easiest to interpret a negative variance if  
specified as correlation at the residual level. I did this and was able to  
include tank effects that did not converge to zero (as I accounted for the  
negative correlation elsewhere). Thus, I could happily report the negative  
variance as negative correlation, include tank effects, and report F-test  
results with the correct denominator degrees of freedom, though the model  
was more complicated than I wished for.
However, for more complicated experimental designs where a negative  
variance occurs at a level that cannot be moved to the residuals (or be  
specified directly as a covariance/correlation between other random effect  
groups, which may also have been a solution for my problem back then), one  
may have to deal with a negative variance component and risk being fried  
by reviewers.

On Wed, 11 May 2016 09:49:41 +0300, John Maindonald  
<john.maindonald at anu.edu.au> wrote:

> I have argued for allowing negative random effect estimates to be
> output, as was and I expect still is the case for Genstat mixed model
> fits.  What does asreml-R do? The negative value is needed so that
> the variance-covariance matrix, which does have to be positive definite
> (or at least semi-definite) is correctly estimated.
> The negative value, if more negative than can be ascribed to chance, is
> a useful warning device.  Someone at Rothamsted told me about getting
> data where blocks had been chosen in which treatment plots moved
> successively further away from the stream.  The additional systematic
> within block variance thereby induced called for a negative between
> blocks random effect so that the variance-covariance matrix would come
> out ‘right’.  Maybe Nelder’s paper mentions this specific type of effect?
> John Maindonald             email: john.maindonald at anu.edu.au
>> On 11/05/2016, at 17:39, Paul Debes <paul.debes at utu.fi> wrote:
>> Dear Jean-Philippe,
>> There are some papers that deal with the special case that the variance  
>> of an experimental design random term becomes negative due to a  
>> negative intraclass correlation. In old ANOVA models this could be  
>> detected as negative variance (this term will earn head shaking...),  
>> whereas in mixed models, where the design term is modeled at the random  
>> level, this is often not detectable because the design term variance  
>> may just be fixed at zero / converge to zero (if restrained to be  
>> positive). As a consequence, it happens that people tend to remove  
>> design terms from their models (because a zero variance random term  
>> clearly does not improve the model) and make inferences about, let's  
>> say treatments, based on observational rather than experimental units  
>> (that would only be represented by including the experimental design  
>> term) and this can lead to unrepeatable and overconfident inferences.
>> This problem cannot always be simply accounted for by leaving the  
>> random design term with a zero variance in the model. For example  
>> asreml-R does not account for zero-variance terms in F-tests (the  
>> denominator degrees of freedom inflate to observational level numbers),  
>> not sure what happens in lme4 / nlme models.
>> Here are some references about this very special topic that only covers  
>> the issue of zero-variance design terms that may in fact be negative,  
>> and how the experimental design can be accounted for at the residual  
>> level (with the associated consequences on prediction ability) in  
>> alternative to having zero-variance random terms:
>> Nelder, J. A. 1954. The interpretation of negative components of  
>> variance. Biometrika 41:544-548.
>> Wang, C. S., B. S. Yandell, and J. J. Rutledge. 1992. The dilemma of  
>> negative analysis of variance estimators of intraclass correlation.  
>> Theoretical and Applied Genetics 85:79-88.
>> Pryseley, A., C. Tchonlafi, G. Verbeke, and G. Molenberghs. 2011.  
>> Estimating negative variance components from Gaussian and non-Gaussian  
>> data: A mixed models approach. Computational Statistics & Data Analysis  
>> 55:1071-1085.
>> I hope that is not too special case for your question, but I think it  
>> is a very important case for making inferences that account for an  
>> experimental design, i.e., when a non-significant random term should be  
>> left in the model.
>> Best,
>> Paul
>> On Wed, 11 May 2016 05:52:24 +0300, Jean-Philippe Laurenceau  
>> <jlaurenceau at psych.udel.edu> wrote:
>>> Dear Ben et al.--I agree with the general practice of trying to  
>>> estimate and retain as many random effects as possible (without  
>>> estimation issues) in a mixed model. However, I was wondering whether  
>>> anyone had some references recommending or arguing for this approach.  
>>> I am aware of a paper on this topic with some simulation work by Barr  
>>> et al. (2013; Journal of Memory and Language), but I would be  
>>> interested in whether there are others. Thanks, J-P
>>> Jean-Philippe Laurenceau, Ph.D.
>>> Department of Psychological & Brain Sciences
>>> University of Delaware
>>> -----Original Message-----
>>> From: R-sig-mixed-models  
>>> [mailto:r-sig-mixed-models-bounces at r-project.org] On Behalf Of Ben  
>>> Bolker
>>> Sent: Saturday, May 7, 2016 11:35 AM
>>> To: Carlos Barboza <carlosambarboza at gmail.com>
>>> Cc: r-sig-mixed-models at r-project.org
>>> Subject: Re: [R-sig-ME] Comparing mixed models
>>>  My only other comment would be that my standard approach would be to  
>>> retain all random effects in the model unless they are causing  
>>> difficulty in model fitting -- this depends on your goal  
>>> (confirmation/testing, prediction, exploration)
>>> On Sat, May 7, 2016 at 11:26 AM, Carlos Barboza  
>>> <carlosambarboza at gmail.com>
>>> wrote:
>>>> Dear Dr. Ben Bolker
>>>> My name is Carlos Barboza and I am a Marine Biologist from the Rio de
>>>> Janeiro University, Brazil. First it's a pleasure to again have the
>>>> opportunity to send you a message.The reason for it is a simple doubt:
>>>> Can I compare AIC from:
>>>> 1. glmmADMB: Density ~ 1 + 1|Site
>>>> 2. glmmADMB: Density ~ Sector + 1|Site + Cage
>>>> Note that they have different random and fixed structures. I know that
>>>> this is not the best choice to model selection but, I think that the
>>>> AIC values can be compared.
>>>> thank you very much for your attention
>>>>  is Cage a random effect?  Are you intentionally leaving out the
>>>> intercept in the second case (it will be included anyway unless you
>>>> use -1)?  In any case, I don't see any obvious reason you can't
>>>> compare AIC values; see
>>>> https://rawgit.com/bbolker/mixedmodels-misc/master/glmmFAQ.html#can-i-
>>>> use-aic-for-mixed-models-how-do-i-count-the-number-of-degrees-of-freed
>>>> om-for-a-random-effect
>>>>  Follow-ups to r-sig-mixed-models at r-project.org, please ...
>>>> sorry, yes, cage was included only to examplify a different random
>>>> structure in the second case...it should be coded (1|Site) + (1|Cage)
>>>> yes, I know that the intercept will be included in the second model
>>>> it's an example of comparing AIC values from mixed models with
>>>> different fixed and random structures:
>>>> 1. Density ~ 1 + 1|Site
>>>> 2. Density ~ Sector + 1|Site + 1|Cage
>>>> comparing AIC...I beleive that both values can be compared
>>>> again, thank you very much for your very fast message
>>> 	[[alternative HTML version deleted]]
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>> --
>> Paul V. Debes
>> DFG Research Fellow
>> Division of Genetics and Physiology
>> Department of Biology
>> University of Turku
>> PharmaCity, 7th floor
>> Itainen Pitkakatu 4
>> 20014 Finland
>> Email: paul.debes at utu.fi
>> _______________________________________________
>> R-sig-mixed-models at r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models

Paul V. Debes
DFG Research Fellow

Division of Genetics and Physiology
Department of Biology
University of Turku
PharmaCity, 7th floor
Itainen Pitkakatu 4
20014 Finland

Email: paul.debes at utu.fi

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