## Alternative to Poisson distribution

Forum for discussion of general questions related to study design and/or analysis of existing data - software neutral.

### Alternative to Poisson distribution

I've been using spatial explicit capture recapture (secr) to estimate density. I'm using those density estimates in mixed models. Visually the data seem to fit a Poisson distribution. The catch is that the density estimates from secr are not whole numbers.

I could round the density estimates up to nearest whole number. I feel like I'm loosing some information by doing this, particularly for rare species that have densities <1 per hectare.

Are there any methods for dealing with Poisson distributed data that are not whole numbers? This seems like something that must come up fairly often with density data.

Thanks for any ideas.
Bryan Hamilton

Posts: 111
Joined: Thu Jun 15, 2006 11:36 am
Location: Great Basin National Park

### Re: Alternative to Poisson distribution

I know others will have more sophisticated responses, so stay tuned. Also, I imagine folks will mention pitfalls of doing statistics on statistics, probably for good reasons.

However, one work-around I've used for the same situation is multiplying all abundance estimates by 100 then rounding to whole numbers, which will be the response variable in the GLMM. Then you maintain more precision. You need to divide estimates from the GLMM by 100. You can also run the same model with abundance x 100 and with just abundance and see how different estimates are.

Also, you could look at model fit with a linear mixed model with abundance log-transformed, assuming you don't have many zeros.
Although, keep in mind: Do not log-transform count data. 2010. Methods in Ecology and Evolution.

Joe

Posts: 67
Joined: Mon Oct 13, 2014 3:53 pm

### Re: Alternative to Poisson distribution

Thanks for the reply. Much appreciated.

I wondered about the 'stats on stats' approach. In my case, the inferences are similar whether I use density or raw count data. An entirely separate issue that I'd like to raise in the future is about the value of using some of these models when they are closely correlated with count data. In my case it really doesn't make much difference. What does make a difference though is the distribution I chose for the mixed model.

That's a good idea with the transformation. And I wasn't familiar with "Do not log-transform count data." paper. Thanks for that.
Bryan Hamilton

Posts: 111
Joined: Thu Jun 15, 2006 11:36 am
Location: Great Basin National Park

### Re: Alternative to Poisson distribution

The log transformation paper is really interesting. If I remember right, alot of diversity analyses imediately square root or log transform the data.

Multiplying by 100 before rounding just blows up the mixed models. All the p values are 1 and the z values are zero. I'll have to spend some time figuring our what is happening.
Bryan Hamilton

Posts: 111
Joined: Thu Jun 15, 2006 11:36 am
Location: Great Basin National Park

### Re: Alternative to Poisson distribution

Does the poisson GLMM run/converge when you use rounded abundance without multiplying by 100 first?

Yea, the log transformation issue is interesting. I recently did an analysis of abundance data where an LMM with log transformed response fit the data much better than a Poisson GLMM. My counts were fairly large, which maybe explains it. As the authors say in that paper, "We found that the transformations performed poorly, except when the dispersion was small and the mean counts were large."

Posts: 67
Joined: Mon Oct 13, 2014 3:53 pm

### Re: Alternative to Poisson distribution

The models run fine when I just round up. It doesn't make sense that multiplying by 100 would matter but glmmadmb can be pretty finicky. I think I might need to 'center' the larger density estimates. When I use year as random effect without centering, the models fall apart too.

Any thoughts on log transforming cover data? I have an analyses that works great with the log transformed data. I feel like robust datasets (like your count data) can handle some pretty extreme transformations. Its the sparse or strangely data where the transformations make a difference.
Bryan Hamilton

Posts: 111
Joined: Thu Jun 15, 2006 11:36 am
Location: Great Basin National Park

### Re: Alternative to Poisson distribution

You could try running the models with the glmer function in the lme4 package in R and see if the results make more sense.

Log transforming percent cover data, which are predictors? If the range of percents is fairly large then I usually format them as decimals so a one unit change in the predictor is equivalent to a 10% change in cover. 1% increments are just so small. Obviously proportions are bounded, so if you log transform with say base 2, then for each doubling of the predictor, the response changes by the coefficient estimate for that predictor, but once you get above 50% cover, doubling no longer makes sense. I haven't done much with standardizing / centering predictors, but it sounds like it generally improves convergence if that is the issue.

Posts: 67
Joined: Mon Oct 13, 2014 3:53 pm

### Re: Alternative to Poisson distribution

Just to quickly chime in:

The estimates of density from the package 'secr' are not Poisson random variables: these (the number of activity centers and their locations) are marginalized out of the likelihood, and the output parameter D-hat is more like the average of a Poisson spatial point process...so more like a lambda than a y1, y2, etc. If using a Bayesian approach, you retain the latent activity center variables, but the posterior distribution is not likely to be an integer either.

At any rate, I think most people would argue that it is more robust to model variation in density within the SCR or SECR model itself (in the same way that you are better off modeling variation in survival directly within a CJS model rather than running regressions on the parameter estimates afterwards). The issue (as I vaguely understand it) is that running stats on stats fails to properly account for variance: each d-hat has a standard error that is related to detection parameters with a standard error, and running an unweighted regression on d-hat post-hoc is not really accounting for the variance in the estimate itself. In other words, treating a parameter with some error value like a perfectly observed data point. Which I guess you could get around if you used some sort of weighted glmm, if this sort of thing exists. But an additional reason to consider variation in the state parameter within a scr or c-r model is that it may influence estimation within the nested detection model as well.

It is not terribly difficult within the 'secr' package (Murray has written a vignette on doing so), and you are fully capable of considering group, session, or spatial covariates within the linear predictor (log-link, identity link, spline).
JDJC

Posts: 10
Joined: Fri Sep 28, 2012 4:05 pm

### Re: Alternative to Poisson distribution

Great points.

Link. 1999. Modeling patterns in collections of parameters. Journal of Wildlife Management.
- Nice comparison of different modeling approaches, including a 2-stage 'stats on stats' approach and the associated pitfalls (and benefits).

Joe