Testing for differences between populations of individuals with binomial data So I am trying to compare between populuations and determine whether there is a difference in count based data. 
My data look like this:
Individual    Population    Total_Reads    Positive_Reads
indiv1        A             14             5
indiv2        B             12             8
indiv3        C             15             8
indiv4        A             8              4
etc. with ~7 populations and ~6 individuals per population

and I would prefer to compare them in such a way that (a) I can identify what groups are significantly different from one another and (b) I retain the count-based nature of the data. 
Does anybody have any suggestions or ideas? A friend of mine suggested a multivariate, binomial model which will produce an effect size for each population, but I'm not sure that will address the question. 
 A: I would suggest to look on Mixed Effects Logistic Regression. I believe it's a common way to approach this question. The idea is that the probability of having a positive read :
- Is influenced by the population (fixed effects)
- Get noised by random effects. The most obvious one is at the scale of the individual : Some people would have a tendency to get more positive read than others, even if they are in the same population. 
Your goal is to assess the significativity of your fixed effects. 
In r you would end up with something like 
m <- glmer(cbind(Positive_Read,Total_Reads-Positive_Reads) ~ Population + (1 | Individual),family = binomial,data=mydata)

The first term ($Population$) is the fixed effects, the second term ($(1 | Individual)$) is a random intercept grouped by individual which explains the disparity between individuals by a random effect.

Disclaimer : I am currently a noob in Mixed Effects Logistic Regression. However I feel it's a legitimate way to address the problem. Maybe the specification of the random effects is incomplete, I picked the simplest form but I do not know if more sophisticated random effects could be specified here.
