Say I am comparing two integers, like the number of bacterial colonies that grew in the absence or presence of chemical X. If I get 2 colonies with and 5 colonies without, that's likely to be by chance. But if I get 200 with and 500 without, that's more likely to be the effect of the chemical. Is there a statistical tests that calculates the p for such results?
That's count data. You can use a count model (Poisson, negative binomial) to test the difference. The tests of hypotheses about coefficients in count models are usually approximate likelihood-ratio tests based on the Chi²-distribution (see Wilk's theorem). So the answer is actually "Chi²-tests", but this answer may be quite misleading when the background is not understood (as it is likely mistaken as Pearson's Chi²-test, a goodness-of-fit test). Look for "analysing count data".
Poisson or negbinolmial should only be used if the number of colonies is very large (practically no upper boundary). If it is small you should rather use logistic (aka binomial) regression or betabinomial regression.
Generally, it is preferred to use negbinomial or betabinomial, because Poisson and binomial have a fixed mean-variance relation. Frequently, error variance is larger than allowed by these two models (overdispersion). Negbinomial or betabinomial allow for extra variance. See chapter 7.4 of my book draft: https://www.researchgate.net/project/Book-New-statistics-for-the-design-researcher
Matrin, maybe I got you wrong, but I think there is a conceptional mistake in your answer. You wrote that logistic regression should be used when the number of colonies is small. As I understand, logistic regression requires to know the fixed upper bound of colonies. Like for instance, you have n colonies, and you determine how many of those n colonies have some property (are infected, are transposed, are "blue", whatever). You actually determine a proportion. So your data is: k out of n colonies are counted as "success", and n-k are counted as "failure". This would call for a binomial model.* However, as I understood Andrzej, he counted colonies that happen to be on a substrate, where (idealized) one can have any unlimited number of colonies. The existence of a colony is an "event" within a spacial "interval" (the culture plate). That calls for some kind of Poisson model (if the event rate is constant, it's an actual Poisson model; if the rate is not constant, it may be modelled by a quasi-Poisson or a negative-binomial model [linear or quadratic variance/mean-relation]).
I also think that betabinomial is not for such non-binomial data.* Could it be that you meant beta-negativebinomial? This further relaxes the assumption the negative binomial model (i.e. the variance/mean relation may not even be quadratic).
Please correct me when I am wrong.
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* the binomial model assumes a conctant "success probability" (p). The beta-binomial model relaxes this by specifying that p can be beta-distributed. The variance of the binomial model is n*p/(1-p), the variance of the beta-binomial model can be greater than that.
Jochen Wilhelm, there is nothing wrong with what either of us said. I just didn't know how many colonies we are speaking, as I am not familiar with the research domain. Regard the two *if* at the beginning ;)
From a theoretical point of view, there must always is an upper limit, as nothing can be truly infinite. Still, when abundance is high and probability of event is moderate or low, Poisson (or negbimomial) are good approximations, especially, when one does not know the upper limit.
But, maybe there is an upper limit, for example when colonies compete for food or space, such that there is max number of colonies on the dish. Dunno!
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