Power analysis for matched poisson variables I have two matched groups consisting of 22 observations each for a Poisson distributed variable. I am interested in evaluating the power to detect a 20% decrease in the outcome for the second group. I have no idea of the correlation between matched groups and hence will have to assume independence. I am planning on evaluating the power using a simulation study and have the following ideas in mind. 
1) Define rate0 and rate1 = rate0-(0.20*rate0)
2) Simulate X1~Poisson(22,rate0) and X2~Poisson(22,rate)
3) Compute pairwise differences X1-X2 and conduct a paired Wilcoxon test or a paired ttest.
4) Conduct simulation nsim times and compute the times pvalue < 0.05
I would appreciate some feedback/suggestions regarding step 3. Would anyone have any other ideas? 
 A: The power analysis by simulation is ok, I think what you’re really asking for is a way to compare matched Poisson variables, other than Wilcoxon test or paired t-test.


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*A brute force approach would be: use as test statistic $\sum_i X_{1i} - X_{2i}$ ; assume H0 (same rate in two groups), estimate the common rate $\lambda$ using pooled data, and simulate N (N = big) times two groups of 22 variables $\sim \mathcal P(\lambda)$ to get an empirical distribution of you test statistic.

*If your rates are big enough, you could also use the fact that if $X\sim\mathcal P (\lambda)$, then the distribution of $2 \sqrt X$ is approximated by a normal $\mathcal N(2\sqrt\lambda,1)$. This leads to normal based test (the distribution of $X_{1i} - X_{2i}$ is known under $H_0$, independently of $\lambda$), and can lead to a nice paper-pen computation to the power. The figure below illustrate the (in)accuracy of this approximation for various $\lambda$ (in black, the cdf of $2 \sqrt X$, in red, the normal approximation).

