# Poisson for percentage data if values are low?

I have percentage data for diet per area (example here.....)

I have no data on the individuals contributing to this diet, only for the population as a whole for each area.

I want to assess whether the size of the area significantly affects the percentage of grains in the diet (so a GLM with area as the explanatory variable, and % grains as the response, ignoring all other diet items for now..)

I just want to check which distribution these data would fall under...I fitted the model using a Poisson distribution as, although %, all the values are fairly low (47 is the highest for the whole dataset). It was overdispersed, so I then tried a negative binomial which seemed to correct the problem.

Does this sound like the correct approach?

• Poisson (& negative binomial) is a distribution for counts, ie, 0, 1, 2, ... That isn't what you have here, so I'm not sure if I understand your question. Where do these %s come from? Are they eg number of people eating grains out of a total population in an area, or the % of calories that come from grains in those people's diets, or something else? Commented Feb 25, 2015 at 19:53
• I agree with @gung. The distributions you are thinking of are unsuitable for percents. What you have is closer to binomial despite being continuous; or to beta. The variance must go to 0 as the mean goes to 0 (as the mean can only be 0 if all values are 0) and also as the mean goes to 100. So, qualitatively the variance must be highest for intermediate percents. Many analysts use here either beta regression or GLMs with binomial family and logit link. There is detail about getting standard errors about right and whether your implementation of GLM indulges this. Commented Feb 25, 2015 at 20:02
• Thanks for the comments so far! Gung - essentially the data are %calories attributed to grains out of the total diet (so in area 1, the population gained 5% of their calories from grains over the study period).
– Lau9
Commented Feb 26, 2015 at 11:16
• Thanks Nick. I'm struggling a bit! Wouldn't your argument for variance going to 0 when the mean goes to 0 hold for poisson as well? Assuming you cant have negative values? Is it just the other end of the scale that is important (i.e. the fact that it is bound at 100?). I had used the logic that, as none of the results for grains were above 50%, whatever is happening at the other (high) end of the scale wouldn't have an effect and therefore poisson (actually neg binomial) was valid. Is that a load of rubbish??? I may well be drawing stupid assumptions based on my very basic understanding...!
– Lau9
Commented Feb 26, 2015 at 11:25
• There is no "just" about the other end of a scale: bounded data can't behave like unbounded data. The variance-mean relationship is only one facet; you want a model whose predictions are guaranteed to respect the range of the response. It's true that for small percents behaviour may not be that different but that is no argument for using one distribution family when there is a more appropriate one to hand. Commented Feb 27, 2015 at 9:36

## 1 Answer

These percentages look like continuous proportions, like the percentage of fat in milk, and as such shouldn't normally be analyzed with count-data models like the Poisson.

[In addition, the Poisson is unbounded on the right. I simply wouldn't use the Poisson on this problem at all.]

A more typical model for such proportions would be the beta. It's bounded in (0,1) and is often used for such continuous proportions. Search here (and elsewhere) on beta regression, for example.

Many packages offer beta regression in some form, or something that would be suitable for continuous proportions. You might looks at a transformation (either to linearize the mean and weight for the variance function or to stablize the variance and use a nonlinear model), but I'd see if you could do beta regression or something else suited for modelling continuous proportions first.

In some cases*, you might get away with a quasi-binomial model for the proportion since it has the same mean-variance relationship as the beta; the logit link (or one of the other typical links for the binomial) may also make sense for your problem. You may have to do some small amount of fiddling to get it to work (the $n$ will essentially be arbitrary, but you have a compensating scale factor in the dispersion parameter).

*(I'd at least want proportions not too close to 0 or 1 and probably largish sample sizes as well)

• Thanks for this, and to those who commented above. I went with binomial family with logit link in the end - I tried beta regression as well which gave a similar result, so I suppose that is reassuring!
– Lau9
Commented Apr 1, 2015 at 12:56