ANOVA-like test for bounded variables (percentages)

Question

I'm interested in determining whether two or more groups of data share the same mean, and it seems like the ANOVA framework is a good way to approach this. However, ANOVA assumes residuals are normally distributed, while each of my data is a number between 0 and 100 (a percentage). Because normal distributions have support on all the real line, my data cannot adhere to the normality assumption.

My question: Is there an ANOVA-like test which is suited to predicting percentages (i.e., a variables that is bounded)?

Answer 1

Is the percentage the most raw data you have, or did you compute the percentage from some sort of binomial count data? If the latter, then you should submit the raw 1s and 0s to a logistic regression. In R, check out glm:

glm(
    response ~ group
    , family = binomial
)

If each individual of each group contributes multiple 1s an 0s, then use lmer from the lme4 package to model the individuals as random effects:

lmer(
    response ~ (1|individual) + group
    , family = binomial
)

If all you have are the percentages, then maybe consider either bootstrapping confidence interval on the group means and differences, or employ a permutation test for differences.

Answer 2

Percentage values may have normal distributions, e.g. cholesterol levels across humans is approximately normally distributed, and it remains normal even if expressed as percentage of the maximal cholesterol level seen. In such cases you need not worry about the fact that your data cover only a narrow interval.

However, there may be a very different mechanism which leads to your percentage. Often percentages summarize many binary measurements on the same subject (e.g. proportion of malignant cells in the tissue sample from the subject). In such cases it's best to use a model that takes the appropriate distribution into consideration (e.g. binomial distribution). To get more specific advice on such a model, tell more about your percentage values!

Other generic methods include transforming the percentage variable, or using a non-parametric test (e.g. Kruskal-Wallis).

Answer 3

As suggested by Mike, for binomial data (e.g. counts over a total - or binary data), logistic regression has greater interpretability and higher power than ANOVAs of transformed data. However, for non-binomial data you can try the logit transformation log (p/(1-p)) is proposed as an alternative trasnformation. The arcsine classic transformation for proportion data seems o became an historical antiquity. See the paper by Warton and Hui,