[R-sig-ME] lme4

[Peter R Law]

I have a simple question about how lme4 works. If one uses lme4 to model a given data set for several different distributional assumptions (e.g., > gamma distribution, via glmer, and Gaussian via lmer and maximum likelihood rather than REML) are the maximum likelihoods output by lme4 comparable in the sense that one could compute AIC from the MLs and compare these models?

I ran a Gaussian model and a Gamma model on the same set of responses with several predictors and for each distribution checked all predictor combinations. I was struck by the fact that all the Gamma models had lower AICs than all the Gaussian models and wondered if there was some systematic difference between the way the two distributional assumptions were modelled in lme4 that might be responsible for this fat other than the Gamma models just being better.

I would have thought the models with different distributional assumptions do have comparable likelihoods and hence AICs since the likelihood models are formulated in the same manner, Y ~ g^{-1}(Z) + \epsilon, where Y is the response, Z the systematic component and g the link function, so in each case the raw responses are not being transformed, unlike the case when the responses are log-normal, in which case lnY = Z + \epsilon, which does not take the form of a generalized linear model since the residuals are multiplicative on the expectation of Y rather than additive. I take it this is why the log-normal is not included as an option in glmer and one can't compare AIC's of the log-normal model of Y with glmer models of Y. I saw a suggestion on a forum to use

family = gaussian(link =

"log"

)

with lme4 but this model would be Y =Ln^{-1}(Z) + \epsilon = e^Z + \epsilon, i.e., one has linearized the multiplicative error to make it additive, which is not exactly

the same as requiring Y to be log-normally distributed.

Peter R Law
Research Associate
Center for African Conservation Ecology
Nelson Mandela University
South Africa

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