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All examples I've found on deviance tests being performed are on generalized linear models. Is there something I can cite validating the use of deviance tests for generalized linear mixed-effects models with normal random effects, ideally outlining what the procedure looks like?

Added from the comments:

This is for submission for a peer-reviewed journal whose primary audience is MDs who are not statistically inclined. I am trying to justify my model selection method for inference. You can find my writeup of the situation here, except there is an additional term $b_2\beta_{it}$ in the model, as well as the possibility of an interation term $b_3\alpha_{it}\beta_{it}$. I used deviance testing to compare nested models which all contained the two random effects in that model, to justify which fixed effects to use, and whether or not to use the interaction term.

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  • $\begingroup$ What is your purpose for performing such a procedure ? Many things that are well defined for linear and generalised linear models do not carry across to mixed models. $\endgroup$ Oct 15 '20 at 8:58
  • $\begingroup$ @RobertLong Good question. Keep in mind this is for submission for a peer-reviewed journal whose primary audience is MDs who are not statistically inclined. I am trying to justify my model selection method for inference. You can find my writeup of the situation here, except there is an additional term $b_2\beta_{it}$. I used deviance testing to compare nested models which all contained the two random effects in that writeup, and to justify which fixed effects to use. $\endgroup$ Oct 15 '20 at 13:25
  • $\begingroup$ @RobertLong I've clarified this in the original question. Thanks. $\endgroup$ Oct 15 '20 at 13:29
  • $\begingroup$ OK I've read your other post (+1 for both btw). So, exams and students are partially crossed ? And if I've understood correctly you want to justify the inclusion of fixed effects based on some test ? Maybe I'm mising something, but why not just use a likelihood ratio test ? $\endgroup$ Oct 15 '20 at 14:05
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    $\begingroup$ That's a likelihood ratio test :) $\endgroup$ Oct 15 '20 at 14:41
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In this context the "deviance test" is another term for "likelihood ratio test" (LRT)

LRTs are well understood as a means to compare mixed effects models with different fixed effects, provided they are fitted with maximum likelihood, rather than restricted maximum likelihood.

In terms of references I don't think you need anything specifically related to mixed models, since your random structure is the same, and you are testing models with different fixed effects. So, the standard references for generalized linear models ought to suffice. For example:

McCullagh P, Nelder J (1989). Generalized Linear Models. Chapman & Hall/CRC, London.

Da Silva DN, Cordeiro GM (2009). "A Computer Program to Improve LR Tests for Generalized Linear Models." Communications in Statistics – Simulation and Computation, 38(10), 2184–2197.

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