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I'm looking for the mathematical notation equivalent of:

glmer(ret15i~AARC_Ret+(1|METRO)+ret02i)

I have a model; it is neither a HLM nor a GLM, but both. It is referred to (variously) as a hierarchical generalized linear model, a generalized linear mixed model, a multi-level non-linear mixed effect model, etc. Generalized Linear Models require a link function. The hierarchical nature of my data means I'll have both fixed and random effects. I know what distribution I'm using: Poisson. As formulated in R, this suggests my random effects are limited to random intercept.

From: STATA Multilevel Mixed Effects Reference Manual Release 13.

User1865345
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Mox
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  • A similar question, with no answer: https://stats.stackexchange.com/questions/74318/mathematical-equation-for-a-generalized-linear-mixed-model-with-interactions – Mox Sep 17 '18 at 21:32
  • Good explanation of Random vs. fixed effects and their notation: https://stats.stackexchange.com/questions/21760/what-is-a-difference-between-random-effects-fixed-effects-and-marginal-model/241950#241950 – Mox Sep 17 '18 at 21:33
  • Related wiki: https://en.wikipedia.org/wiki/Hierarchical_generalized_linear_model – Mox Sep 17 '18 at 21:35
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    The formula you have there won't work (there's an extra parenthesis, and +| isn't allowed). Should it be glmer(ret15i ~ AARC_Ret + (1|METRO) + ret02i, family=poisson) ? – Ben Bolker Sep 17 '18 at 22:12
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    Do you have multiple measurements of ret15i for each METRO and if yes, how many? Are these counts collected over time or space? If over time/space, is the time/area span underlying each count the same across multiple counts for a given level of METRO? How many levels does METRO have? Are AARC_Ret and ret02i continuous/discrete? Are the values of AARC_Ret and ret02i changing over time for a given level of METRO? In my view, one would need to know all of this before suggesting a mathematical notation for your specific model. – Isabella Ghement Sep 17 '18 at 23:50
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    Ret15i is Retail employment in 2015; ret02i is retail employment in 2002. I have multiple measurements within each metro, each representing a single transit station. Both are counts. AARC is continuous/discrete. METRO is a factor (grouping) variable. – Mox Sep 19 '18 at 23:08

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GLMM notation:

This is the best thing I've found so far (on the internet). enter image description here

"Generalized linear mixed models are typically constructed by incorporating random effects into the linear predictor of a conditionally independent exponential family model. I now formalize that notion with the following definition of a generalized linear mixed model: In this definition we see the usual ingredients of a generalized linear model. First, the distribution of Yi from an exponential family (in this case the distribution is assumed to hold conditional on the random effects u). Second, a link function, g(-) is applied to the conditional mean of Yi given u to obtain the conditional linear predictor. Finally, the linear predictor is assumed to consist of two components, the fixed effects portion, described by x'13 and the random effects portion, z'u, for which a distribution is assigned to u"

McCulloch, C. E., & Neuhaus, J. M. (2014). Generalized linear mixed models. Wiley StatsRef: Statistics Reference Online.

Mox
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    It would help if you could indicate whether you would still like a general formulation for Poisson mixed effects models or a specific formulation for your own model. In general, people answer their own question only if nobody else has done so and/or they figured out the answer all by themselves after posting the initial question. I think answering your own question may inadvertently suggest that you are no longer seeking an answer from member of this forum. – Isabella Ghement Sep 17 '18 at 23:52
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    I was hoping someone would provide a better answer, and I could upvote that. So yes, I am still seeking such a formulation. Or perhaps derivation is the proper word? – Mox Sep 19 '18 at 23:10