[ADMB Users] NLMM Model Selection

[Chris Gast]

This isn't precisely an ADMB topic, but it seems as though ADMB users might
be knowledgeable in this regard.

I've searched the archives and haven't found a lot of discussion regarding
model selection in nonlinear mixed models. For a given dataset, I have a
series of models which differ in combinations of structure, number of
effects considered random, and assumed distribution of random effect
components, and would like some (preferably likelihood-based) method to rank
them. Burnham and Anderson (Model Selection and Multimodel Inference, 2002,
page 310) describe a method based on shrinkage estimators where the penalty
term is computed somewhere between 1 and the number of random components,
but this appears to require both a single random effect and a fit of the
model where each random component is considered a parameter; neither of
these is feasible with my models (or, I suspect, many others). I can't
simply use LRTs to decide between a mixed model and its fixed counterpart,
because the value of interest for the sigma parameter lies on the boundary
of its space, 0.

I have found some instances where the problem is basically ignored (Hall,
D.B. and Clutter, M. 2004. Multivariate multilevel nonlinear mixed effects
models for timer yield predictions. Biometrics, 60:16-24). To quote: "...the
first-order approximate log likelihood is treated as the true log
likelihood, and standard errors for parameter estimates, likelihood ratio
tests for nested models, and model selection criteria such as AIC and BIC
are formed in the usual way. Although the formal justification of this
“approximately asymptotic” approach to inference is an open problem, it is
commonly used in practice, and we adopt it for our purposes in this
article."

One simple method would be to choose the model that best reconstructs the
original data as measured by the chi-squared test statistic sum((O-E)^2/E),
but again, it would be nice to have something likelihood-based such that the
framework is a cohesive, and the principle of parsimony is in effect.

One additional question: these models also may include covariates.  Holding
all other model features of a mixed-model constant, LRTs should be justified
for model selection of covariates only, as they result from a mathematical
restriction of some beta=0, correct? I see plenty of information about the
LASSO for covariate selection in NLMMs, but haven't yet found the time to
learn this technique.

I would appreciate any and all thoughts or opinions.

Thanks,

Chris Gast
University of Washington
Quantitative Ecology and Resource Management
cmgast at gmail.com
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