What is a statistical significance test for two Poisson distributions? Say that I have two Poisson distributions. These were modelling after count data.
How would I determine statistical significance between these two distributions? That is, how would I determine whether these two poisson distributions are statistically different? 
Could I apply any non-parametric test (because they don't assume anything about the distribution of the data)? A simple Google search doesn't seem to provide direct answers.
 A: Note that Poisson distributions are entirely determined by their parameter, so a test of equality of their mean parameter is a test for whether the distributions are the same.
Some possible tests:


*

*If you have two samples which you treat as iid Poisson each with its own parameter, which you want to test for equality of that parameter; in that case you can simply combine all the observations in each group into a single Poisson count. 
a.  You could condition on the total count and do a test of proportions (a binomial test in exact form, or via normal approximation, or equivalently a chi-squared test). For example, this binomial test is what you get if you do poisson.test on two samples in R.
b. You could do a likelihood ratio test. 
(There are a number of other possibilities under this option.)

*If you don't necessarily want to treat them as Poisson except as a rough approximation (but do treat them as iid), you would keep all the individual values. 
a. You could then do a permutation test of the means. 
b. You could do a Wilcoxon-Mann-Whitney or even a goodness of fit test (e.g. a Kolmogorov-Smirnov test) but you will have to deal with the discreteness of the distributions. 
c. If you expect that the means won't be very small, you could perform (say) a t-test (under the null the samples should have equal variance, so it's not important whether you do the equal-variance version).

*If instead of being identically distributed, they are of known but different exposures, you could combine into single counts as in option 1, but also combine the exposures into a single exposure for each. You could then follow the approaches in 1.

*If they have unknown exposure but the exposures of pairs of observations will be the same, you effectively have pairing. You could perform a paired permutation test -- permuting the group labels within each pair (which corresponds to putting + and - signs on each absolute pair difference of counts).  You could also do a sign test, or since (under the null) the differences would be symmetric you could consider a signed rank test (again properly accounting for ties).
