There are numerous issues involved in Steve’s original message. Some of this is philosophical and some is technical. If you are modeling R-side (co-)variation with a GLMM, you may be performing a strictly quasi-likelihood analysis, whether you realize it or not. For instance, if the conditional distribution (i.e., conditional on the G-side random effects) does not have a free scale parameter (binomial and Poisson, for instance), then any R-side modeling is incorporating a multiplicative scale parameter that would not be there for these distributions. As stated on page 128 of Stroup (2012), “No actual probability distribution exists with this structure, but in many cases it adequately models the distribution of the count [or proportion] data, and the quasi-likelihood is perfectly well defined for ... estimation and inference purposes.” So, this is one way of getting at the usually intractable marginal distribution. If you don’t have random (G-side) effects, you are getting GEE analysis with this situation, which is used very successfully for GLMs without other random effects. We “see” marginal distributions, that is, observations are from marginal distributions. But, it can be argued that observations are generated from conditional distributions, that is, a conditional model comes closer to capturing the data-generating mechanism. This is certainly a take-home message in Stroup’s book (although I am sure that I am greatly oversimplifying a much bigger topic—sorry). This theme is found throughout his book. In the marginal-vs.-conditional debate, it is often overlooked that the two kinds of models are targeting different parameters; Stroup makes a compelling argument that the typical investigator is more interested in the targeted parameter from the conditional model (such as the conditional binomial probability). I basically agree with this, but I am sure this can be debated. The more I learn about GLMMs, the more I am leaning to the conditional-model approach to analysis. However, there can be important uses for marginal models, so I am not going to get into any major on-line debates about this. However, in terms of repeated measures, I have a difficult time conceptualizing what an autoregressive (or other structure) means for the multiplicative scale parameter (say, with overdispersion for a “binomial” distribution). I can conceptualize this with a random effect in a conditional model. For exponential-family distributions with a free-scale parameter (e.g., gamma, negative binomial, and other two-parameter conditional distributions), R-side analysis (with RANDOM _RESIDUAL_ / …) makes sense as a true likelihood analysis (not quasi-likelihood). But one must be careful in fitting a model. This is technical issue with the analysis. For instance, a RANDOM _RESIDUAL_; statement here would create another multiplicative scale parameter, so that the overall scaling would be the product of two constants; there would be no unique estimates for the two scale terms (a form of overparameterization). However, statements like RANDOM _RESIDUAL_ / group=TRT; would be useful to indicate that there is separate scale parameter for each treatment (etc.). When you get into repeated measures analysis for the gamma and negative binomial, things can get very messy. If you specify, for instance, an AR(1) structure for R-side analysis, you are defining a working correlation matrix. As stated by Stroup (page 435), “it is not clear how the working correlation parameters co-exist with the scale parameters intrinsic to the [conditional] distribution… The area in need of further development is clearly the two-parameter [non-normal] exponential family.” My view is that a lot is unknown about R-side analysis for two-parameter non-normal distributions—good research opportunities for statisticians.
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