Comparing mixed models

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

Comparing mixed models

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