# Can ANOVA suffice for comparing mean of percent/proportion data?

I've been handed off some data to analyze and I'm looking for something simple and straightforward. The 1 response variable/dependent variable is average percent mortality from 7 treatment groups (with 1 independent variable). These data can be analyzed using the the proportions (example: 30 of 50 trees died). I'm worried about meeting the assumptions of ANOVA and have considered non-parametric approach using kruskal-wallace. Any suggestions?

• 1) Do you have one variable that is a binary alive/dead and another variable that gives the groups to which those alive/dead values correspond? // 2) Is there a time component to this? Should there be?
– Dave
Commented Jul 19, 2023 at 0:53

Depends how simple and straightforward you want.

• The classic way to analyze proportion data in an ANOVA is to arcsine-square-root transform them, which does some combination (I forget exactly) of normalizing and stabilizing the variance.
• If you have the numbers of trials per observation (in your example, "30 of 50 died", 50 is the number of trials), then a binomial GLM is the more modern way to do the problem, and is also super-easy with any stats package (Warton and Hui [2011] feel strongly that this is the right method: "For binomial data, logistic regression has greater interpretability and higher power than analyses of transformed data")
• a non-parametric analysis such as Kruskal-Wallis would also be fine.
• it's possible that an ANOVA would be OK if your sample sizes are large and your proportions aren't too close to 0 or 1

(I assume you have more than one sample per treatment group ...)

If you have a large number of all-zero or all-one responses in your data, it's likely that the data are overdispersed, i.e. that your data are not consistent with some of the assumptions (independent trials, equal probabilities) of the binomial distribution. In this case the easiest thing you can do is to use a quasilikelihood model, which allows the variance to be scaled relative to what would be expected from a simple (e.g. binomial) model. In R, this would be as simple as saying family = quasibinomial rather than family = binomial. A slightly more complex alternative is a beta-binomial model, which you could fit (e.g.) in the glmmTMB package with family = betabinomial. (It's also possible that you have zero-inflated responses, but then we're starting to get away from "simple and straightforward" ...)

Warton, David I., and Francis K. C. Hui. “The Arcsine Is Asinine: The Analysis of Proportions in Ecology.” Ecology 92, no. 1 (January 2011): 3–10. https://doi.org/10.1890/10-0340.1.

• Is it binomial if you have the proportion and not the integers whose ratio have the proportion?
– Dave
Commented Jul 19, 2023 at 0:54
• If you have only the proportions things get a little harder. The analog of the binomial GLM is Beta regression (and this gets further complicated if you have exact values of 0 and 1 in your data). But the wording of the OP's question suggests that they know the number of trials ("example: 30 of 50 trees died"). Commented Jul 19, 2023 at 0:58
• But I do agree that I jumped to some possibly incorrect conclusions about what the OP's data looks like ... Commented Jul 19, 2023 at 1:00
• @BenBolker and Dave, thanks for your kind responses. I do have the original data/number of trials. I looked into binomial GLM- that makes me feel more confident that I'm eliminating options with some reasoning. The data has a large number of 0's (no trees dies) and 1's (all trees died). I tried Kruskal Wallisfollowing with Wilcoxon comparisons, but there are too many ties. Here's what I'm thinking for the glm: glm(formula= dead trees~herbicide treatment+site+days after treatment+diameter at breast height, family=binomial (link=logit). I haven't used GLM in a few years, so I'm a little rusty
– E10
Commented Jul 20, 2023 at 12:35