# Can I use a binomial model with logit link function when dealing with continuous proportions?

I have continuous proportion data (i.e. ranges from 0 to 1). Or it could be percentage too if multiplied by 100. This is the proportion/percentage of overlap in home range between animals, therefore I do not have denominators or a case of #successes/#trials.

Can I analyze this using a binomial glm with logit link function? I cannot seem to get a clear answer on this on the web.

One method is to arcsine transform the percentage value and use it in a linear model but it seems there is much argument against this.

Also the data are extremely skewed to the right (almost looks like a classic Poisson distribution) with many zeros and some 1's too, which means I can't use a beta regression either.

What is the best method to analyse this? Any advice would help. I am an R user.

I would consider a fractional logit model, which can be implemented as a GLM with binomial family, a logit link, and heteroskedasticity-robust standard errors. This can accommodate 0s and 1s if those occur through the same process as the intermediate proportions.

However, that sounds like that is not the case, so you might consider a zero-one inflated beta regression. This is essentially a maximum likelihood estimator with a

1. a logistic regression model for whether or not the proportion equals 0,
2. a logistic regression model for whether or not the proportion equals 1,
3. a beta model for the proportions between 0 and 1.

Another option is to transform your outcome, nudging your 0s and 1s toward the middle and to use beta regression. Smithson and Verkuilen (2006) propose using

$$p’ = \frac{p \cdot (N - 1) + 0.5}{N}$$

• Thank you. I am not sure what you mean with 0s and 1s that "occur through the same process as the intermediate proportions." The proportions (which can also be seen as percentages) is the overlap in the home-range of two animals. Which means a 0 = no overlap and 1 - 100% overlap. It is generated the same as the intermediate proportions...I think...if I understood you correctly. How do I specify heteroskedasticity-robust standard errors? If you have any good suggested reading that would be much appreciated. – MiMi May 18 '16 at 18:30
• @Mimi I am not a biologist, so hopefully this makes sense. If 0s and 1s represent very low or very high proportions that “by accident” resulted in a proportion of 0 or 1, then the fractional logit is OK. If 0s and 1s represent distinct processes, then they have to be modeled differently. If you see mass points at zero and one, the latter case is more reasonable. – Dimitriy V. Masterov May 18 '16 at 18:45
• @Mimi Another example is fraction of disposable income spent on vacations. You might imagine that number of kids makes zeros more likely (negative effects), but conditional on going on vacation, more kids makes the fraction increase (positive effect). So the process generating the zero is different than the process generating the non-zeros. – Dimitriy V. Masterov May 18 '16 at 18:45
• @Mimi I would read the references in here. – Dimitriy V. Masterov May 18 '16 at 18:49
• Thank you very much Dimitriy. I appreciate the effort. I will first have to go think about my problem a bit further: i.e. to use fractional response regression or ZOIB's. – MiMi May 18 '16 at 19:36