# Proportion data - beta distribution v. GLM with binomial distribution and logit link

I have a fisheries dataset for which I have calculated value for each grid cell on a map. The value is the proportion of the total fishing sets in that cell for each month/year. So, I have values between 0-1, but not including 0 and 1 (the range is actually very skewed and is: 0.0005347594 to 0.1933216169). I am interested in whether the proportion of fishing sets is higher close to a specific location over time.

I have read that there are two ways to do this - either a GLM with a binomial family and logit link, or a beta regression.

I have tried both of these methods in R:

Binomial GLM:

m1 <- glm(PercentTotalSets ~ factor(SetYear) + DayLength + DistTZCF + DistNWHI,
family = binomial(link='logit'), data = Totals_CellId)


Beta:

BetaGLM <- betareg(PercentTotalSets ~ factor(SetYear) + DayLength + DistTZCF + DistNWHI,
data = Totals_CellId )


With the binomial GLM, I get very different results than I would if I ran a GLM with a gamma distribution (e.g., DistNWHI is not significant with a p-value of .9 whereas before it was significant). With the beta regression, I get very similar results to a GLM with a gamma distribution (e.g., DistNWHI is significant with similar p-value).

I think that the beta regression is the correct method, because I do not have 0s or 1s and I need to set bounds, but I am not sure if this is correct.

I'd appreciate any and all advice.

• Are these count-proportions (of n subjects, k had attribute A) or continuous proportions (this milk is 2% cream)? If count-proportions, do you have the denominators? – Glen_b Nov 10 '14 at 18:44
• They are count-proportions -- I calculated the total number of fishing sets for each month of every year. Then I divided the number of sets in each specific cell of interest (100 km x 100 km cells) for each month/year by the total number of sets in the fishery for that month/year. So, I have the total number of sets and the sum of sets in each cell for every month of 18 years. The problem is that the total number changes throughout months and years, so I thought I'd divide each cell by the total number of sets in that month/year in the whole fishery, giving me a proportion. – Meagan Nov 10 '14 at 18:55
• perhaps you can just model the raw catch? Since you are only interested in spatial comparison rather than temporal, it seems to me that the normalization is not necessary. This way you can bypass the difficulty associated with percentage data (zeros, non-negativity, skewed etc...) – qoheleth Nov 10 '14 at 22:41
• Qoheleth - I am interested in change over time, though. I am interested in both a spatial and temporal comparison: are the fishing locations closer to a specific point over the years. – Meagan Nov 11 '14 at 14:06

## 2 Answers

With count data of that form, I'd actually fit a multinomial model (at least to start with*), because several numerators are present in the denominator - each '+1' count could have gone into any of $k$ cells ('sets').

(e.g. see here)

You'll need the denominator you divided by; the model is still for the proportion, but the variability depends on the denominator you used to obtain the proportion.

* a particular concern is that you'll have dependence over both space and time (e.g. adjacent locations and adjacent times will tend to be more related than more distant locations or times - at least if there's unmodelled variation that would be accounted for by such effects)

Once you have fitted a multinomial model, you would want to assess whether you have both the variance and the correlation modelled reasonably well -- you might need mixed models (GLMM) and possibly also to account for potential remaining overdispersion in addition.

You will find a number of discussions of multinomial models here on CV.

Another possibility is to model the counts as Poisson, by allowing for offsets, factors or continuous predictors related to the variation you mentioned as the reason you scaled to proportions.

• This sounds like Poisson etc to me. If the cells vary in area, you could use area as an offset. I suspect the cells are adjacent or at least have known coordinates, so you could probably use x & y (lat & long?) coordinates as continuous predictors, & deal w/ the spatial autocorrelation. If the assumption is that there is some 'spot' where the fish are plentiful, using squared terms etc should let you locate it. – gung Nov 10 '14 at 22:52
• Offset was the word I was grasping for - especially in relation to size, thanks @gung. Sometimes the word just won't come. – Glen_b Nov 10 '14 at 23:00
• Thank you for the very helpful comments! The cells are all the same size, but the number of sets in each month of every year varies, so am I correct in understanding that I could use a Poisson GLM with total number of sets (e.g., overall effort) as an offset? I'm seen effort (usually as number of hooks) log transformed as an offset in GLMs or GAMs. Is this what you're recommending? I will also look into multinomial models now. Thank you! – Meagan Nov 11 '14 at 2:48
• And, if I model the counts with a Poisson GLM or GAM (and have total sets as an offset), do I need to create zeros for the cells that didn't have any fishing effort in that month/year or can I just do presence only? – Meagan Nov 11 '14 at 16:46
• For the second question, I'd post a new question. For the first, if it's a regular seasonal effect, a Poisson model where month is a factor might work (if it's more smoothly sinusoidal, the first few harmonics in a trigonometric seasonality formulation may be better). I'd likely try the month-as-factor approach. Is there also time-trend across years? There's a small problem with using total sets in a month as an offset, since it's actually a random variable (in fact, it's a linear function of your data). [However, if you were to do it, it should work similarly to a multinomial model.] – Glen_b Nov 11 '14 at 22:08

Based on your answer of how the proportion is calculated I believe the beta regression is most appropriate. The logistic regression for count binomial would only make sense if you have counts out of a total that is constant. Since your total changes from month to month you have a continuous proportion. Therefore beta regression is the way to go!

• I don't believe your second sentence is correct. Is there some basis for that? – Glen_b Nov 10 '14 at 21:45