Mann Whitney test, unequal sample sizes, different=shaped continuous proportions plus many zeros

I was hoping someone could help, I am comparing proportional data in to two different groups. One group has a sample size of 22 and the other 530. The data are not normally distributed and have different shaped distributions. Is a Mann Whitney test OK to use for this kind of data. I have tried GLM but my residuals are very strange.

The response values are proportions of overlap in home ranges. Values from 0 (no overlap) to 1 (full overlap). I tried to fit a binomial GLM. Response was overlap, predictor was same animal (0) diff animal (1). Standardized residuals plotted for normality resulted in a s-shaped curve.

They have quite a few zero values.

• What kind of proportional data (is this continuous proportions?) What GLM did you fit? What was 'very strange', exactly? – Glen_b Jan 22 '15 at 11:50
• Related posts: Post 1; Post 2. Also somewhat relevant. – Glen_b Jan 22 '15 at 11:58
• This is proportions of overlap in home ranges. Values from 0 (no overlap) to 1 (full overlap). I tried to fit a binomial GLM. Response was overlap, predictor was same animal (0) diff animal (1). Standardized residuals plotted for normality resulted in a s-shaped curve. – user3237130 Jan 22 '15 at 12:14
• Thank you Glen_b for the posts. They are helpful. Do you know if it matters that my data are proportions and that they have quite a few zero values in there? – user3237130 Jan 22 '15 at 14:31
• I had thought a binomial GLM would be suitable as my predictor values are either 0 or 1? Could you recommend a zero-inflated model to try please? – user3237130 Jan 22 '15 at 16:22

The difference in sample size is not an issue.

However, it sounds like your situation may present several possible problems for the Mann-Whitney (some of which might be overcome - I'll come back to this).

It also sounds like you a GLM may have problems and you may need a zero-inflated model - since it's the conditional distribution of the response that you need to worry about.

So three suggestions to consider:

1. Zero inflated beta: Perhaps a zero-inflated beta might work okay. In R there are several packages that can fit one, like gamlss or zoib.

2. Tweedie GLM: Another possibility is a Tweedie GLM (which can be continuous with a spike at zero for the $p$ parameter a little above 1). For that, see Peter Dunn's tweedie package in R; this might be suitable if most values of the response are nowhere near the upper bound (i.e. if the data tends to be well below 1). If that's satisfied you might also zero-inflated gamma or zero inflated lognormal models.

3. Mann-Whitney, permutation test conditional on pattern of ranks: To return to Mann-Whitney again, the large number of zeros will lead to heavy ties, but the small/large n's would suggest that neither exact methods nor asymptotic methods would necessarily work well. In that case, you could use simulation to obtain p-values with the Mann-Whitney. I don't know what the power is like in this situation though (indeed, aside heavy ties at zero I don't know what your data look like).