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I want to determine if there is significant variation in the effect of a treatment amongst my replicates. The treatment effect is measured as a proportion.

Let me describe:

I study heart defects (in mice) and I found that babies from older mothers have a higher incidence of heart defects compared to babies from young mothers. Simply put, each mother is given a treatment (age), and we compare the incidence of heart defects in her babies before and after treatment. I simply compared the population-wide percent of babies with heart defects from when the mothers are young to the same mothers when they get old.

Now I want to know if there is quantifiable variation in this age effect between mothers. Simply put, I want to know if some mothers have a higher treatment (age) effect than other mothers in the population. The goal is to implicate genetic variation as the source of the variation in age effect.

I am having problems because of the proportion scale measurement. I looked at Cochran's Q Test, but it did not seem to fit my problem. I think I want a statistic that tests for homogeneity of differences between paired proportions that also accounts for unequal sample sizes (some mothers have more babies than others). I don't think that test exists, but any suggestions would be greatly appreciated.

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I figured out the solution to this problem, but can't figure out how to delete the question. So I will tell what I did.

I used a generalized linear mixed model (glmer in R package lme4) to model the fixed effect of age (the treatment) on risk of heart defects in the offspring. The effect of mother was modeled as a random effect. I tested whether there was variation in the magnitude of the age effect depending on mother by comparing a standard glm with only the fixed effect to the full glmm using the anova method. The model fit was improved by including the random effect of mother, suggesting that there is significant variation in the per-mother age effect. (When doing the anova test, make sure to list the glmer model before the glm model or it will not work.)

I hope this helps anyone else that has a similar issue.

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