# [R-sig-ME] marginal R square calculation in zero truncated poisson model

Paul Johnson paul.johnson at glasgow.ac.uk
Sat Apr 1 20:08:10 CEST 2017

```Hi Stephanie,

Starting with the caveat that I might have misunderstood your model…

It initially seemed strange to me to use a count distribution such as Poisson to model a continuous outcome such as a time interval. A zero-truncated Poisson is typically used when zeros exist but can’t be observed (e.g. number of eggs laid by an insect on leaves, where clutches of zero eggs can’t be observed). There might be no zeroes in your time interval responses, but that doesn’t mean that zeroes exist but are unobservable. However, I can see how a ZT Poisson might fit well, if the interval is measured in years and a 1 year interval is common while a zero-year interval never happens, and usually if the model fits well that’s enough to justify using it. In the case of a ZT Poisson however the estimates will refer to the untruncated model, i.e. the mean interval predicted by the model will be lower than the actual mean. If you’re only interested in the effect estimates, again that might not matter, but you might also want to consider other strictly positive distributions (e.g. gamma).

Re r-squared for ZT Poisson...

I don’t think anyone has worked out R-squared for a truncated Poisson GLMM, although I’m aware that Shinichi Nakagawa and Holger Schielzeth are working on extending their R-squared statistics to more distributions. It might be possible to calculate an R-squared that refers to the untruncated model that includes the missing zeroes. The problem is that the estimates will refer to the untruncated data (in you case including the “unobserved” zero intervals) but when it comes to calculating the fixed effects variance you only have access to the truncated data. I haven’t had time to think this through, so will just leave it here in the hope that someone else picks it up.

Some more general arm waving about R-squared for GLMMs... I’m a little sceptical anyway about how useful R-squared is for GL(M)Ms that don’t have an identity link, because of the difficulty of using variance components on the link scale (the log scale in your case) to explain variation in a response that is of course observed on a different scale (the count scale here). Nakagawa and Schielzeth devised some clever ways to get around this but we’re still left with the problem that a statistic which is meant to summarise explained variance in an intuitive way refers to an unintuitive scale. R-squared might be useful for explaining the amount of variance explained by different models fitted to the same data, but I find it hard to see how a Poisson model with a log link, a binomial model with a logit link and a Gaussian model with an identity link, fitted to different data sets, that all have marginal R-squareds of 30%, can all be said to have the same explanatory power. I guess your opinion on this will depend on whether you think marginal R-squared should have such a context-free interpretation, but I think this is how many people interpret it.

Best wishes,
Paul

> On 31 Mar 2017, at 13:41, Stephanie Kalberer <stephanie.kalberer at uni-bielefeld.de> wrote:
>
> Dear Professor Bolker,
> I am looking into life history data of sea lions at the moment and would like to test what influences the length of the inter-birth interval. My response variable shows a clear poisson distribution but as I don't have any zeros in my inter-birth interval, I use a zero truncated model. The full model:
> glmmadmb(IBI..years.~ sex.first.offspring + SST.Jan.May. + IBI.of.previous.pup + birthyear_mother.num + birthyear_pup.num + first.offspring.of.interval.died.within.1st.year + (1|AnimalID),
> family="truncpoiss", data=IBI_successive_pups_sexmat_OF_sex)
> I couldn't find any information though how to calculate the marginal R square in that case, could you point me towards a document or solution for it?
> Thanks a lot and best,
> Stephanie Kalberer
> --
> ___________________________
>
> Stephanie Kalberer
> PhD Candidate
> Galapagos Sea Lion Project
> Department of Animal Behaviour
> Bielefeld University
> Germany
>
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>
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