I agree with Brian that bootstrapping or likelihood profiles are a good way
to determine confidence intervals on your parameters. But I disagree with
his suggestion that ability to estimate parameters is a proper component of
model selection. It's important to realize that the width of confidence
regions is an indication of the information content of the data relative to
those parameters. The fact that two parameter values have equal or very
similar likelihoods means that the data are agnostic as to their relative
explanatory value. To put it another way, if you are asking a question
that hinges on the value of a parameter and the confidence interval for
that parameter is so wide that your question goes unanswered, this is
because the data do not contain an answer to your question. Therefore, the
only way to get such an answer is to get better data. One is, of course,
always free to make additional or different assumptions (such as a
different model as Brian suggests or imposing a prior), but do not make the
mistake of confusing assumptions with information. This is what you would
be doing if you were, for example, to reject a model with a high likelihood
but poorly identified parameters in favor of one with a low likelihood but
precise estimates.
--
Aaron A. King, Ph.D.
Ecology & Evolutionary Biology
Mathematics
Center for the Study of Complex Systems
University of Michigan
GPG Public Key: 0x15780975
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