I have a density metric (# of territories/km) which I would like to model in a LMM or GLMM. Using the lme4 and glmmamdb packages in R I am successfully able to model it as such with a Gaussian family distribution. I have Year (as a factor) and proportion stream impacted (from 0-1, e.g. 0.35 for 35%) as fixed effects, and Stream (as factor, the "sites" of study) as a random factor. I am essentially seeing whether Year or percent impact affects density controlled for Stream variability.
However, any examples I've found online of modeling density as a response variable use *count* of territories instead (# territories) to model density, with area (stream length) monitored (which varied by year) as an offset. This would make the data Poisson for count data, and I ran into a slew of errors in getting these models to work.
So my question in essence is from a philosophical/statistical standpoint which is technically the "proper" way to model a density variable (Gaussian, poisson, or something else)? If I already accounted for stream length distances in my density metric (# terrs/km) then I don't see the need to make the model more complicated (e.g. use poisson), but I am curious how others would approach this problem.
*Part of the issue too is that density is slightly "non-normal" according to a Shapiro test and histogram, but with a ggplot quantiles look mostly fine with some minimal skirting at both tails. Depending on how non-normal is considered "non-normal," I wouldn't be able to use "Gaussian" unless I transform the variable, which I don't want to do either. My understanding is that I would have to seek a GLMM alternative. The density value ranges from 0.64-2.66 and mean 1.33 (so
There are multiple ways of looking at your data (as always). If you want tot keep your data already corrected for stream area, then the Poisson method is broken (the Poisson distribution is only meant for non-negative integers). If you are willing to take a step back and know stream reach area, then it makes sense to work at the level of territories and model them as counts (as the Poisson distribution assumes), with stream area as a covariate.
However, if you already have a measure of density (as it sounds like you do), then I think you are on the right track with treating the response (density) as 'gaussian'. Although, as you mention, you may need to treat it as log-normal (just a log transform of the response). This wouldn't be surprising, and it is easy to interpret responses on the log scale (just google it). So, my advice is forget generalized linear mixed models, I think you can just use a linear mixed model.
It should be possible to treat you data in a LMM framework, even without log-transforming your dependent variable (the normality is often considered to be the less important assumption of a linear model). However, in my opinion, you should perform a precise model validation in order to verify if the other assumptions of a linear model are met, and to decide accordingly whether to adopt this technique or switch to something different (GLMM). For instance, if you will find strong patterns in the residuals, or negative fitted values (the prediction of a negative density doesn't make sense in biological terms), probably you will have to try with a poisson GLMM, or something different in the case of over-dispersion (quasi-poisson, negative binomial, zero-inflation).
You can also find a lot of tips, generalities on the different distribution and suggestion on model validation in this book:
Zuur A.F., Ieno E.N., Walker N.J., Savaliev A.A. & Smith G.M., 2009 - Mixed effect models and extensions in ecology with R. Ed. Springer, Berlin, 574 p.
It is more statistically correct to model counts using a Poisson model.
Part of the reason is that you want to estimate the effect of your various independent factors on bird abundance. But combining area and count into one measure of density confuses the effect. Separating area/length from abundance (count) ensures your model estimates effects on abundance (estimated by lambda in the Poisson model), which is what you really want to know about.
I can't tell from your question whether your density was calculated as count per km of stream OR count per some unit area (e.g., 1 km2). If the former, you can correct the sampling unit to the same by using "log(# km of stream)" as an offset. If the latter, you don't need an offset, because the sampling unit is the same.
A relevant conversation occurred here: https://www.researchgate.net/post/Is_it_possible_to_use_quasi-poisson_or_negative_binominal_regression_with_continuous_dependent_variable
And another very useful explanation of offsets here: http://stats.stackexchange.com/questions/66791/where-does-the-offset-go-in-poisson-negative-binomial-regression
Hi Everyone, thank you for your inputs! It sounds like while using the density metric wouldn't be wrong in terms of being able to using a simpler and more powerful LMM, I decided to go the route of a GLMM with poisson family. Using a "count" response variable, I did not have any issues with overdispersion and was more "normal" than my density metric (although some non-normality). This way too as Nicole mentioned I wouldn't potentially confound my estimate. My offset is differing lengths of stream monitored (for example, 4.0 km in year 2009 for stream "A" but 4.2km in next and following years). So originally if stream A had 4 territories in 2009, my density metric was 1.0 terr/km. Now my response variable is just 4 (terrs). As such in lme4, my model is set up as glmer(Count~StreamImpact+Year+(1|Stream)+offset(log(StreamLength)), data=density.d, family="poisson"). So far, density seems to be significantly different by year, but proportion of stream impacted does not seem to explain density. The two fixed effects are not correlated and the model seems to perform better with both still included.
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