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I have two vectors (a and b), which are each a sample of n = 10000 from the poisson distribution with lambda = 10.

x <- data.frame("a" = rpois(10000,1000), "b" = rpois(10000,1000))

I conduct 10000 two-sample poisson tests between each element of a and the corresponding element of b, where the null hypothesis is that the two numbers are from the same poisson distribution (the rate ratio is 1). 

x$p.value <- NA
for (i in 1:nrow(x)) {
  temp <- poisson.test(c(x[i, "a"], x[i, "b"]), T = c(1,1), alternative = "two.sided")
  x$p.value[i] <- temp$p.value }

As the null hypothesis is obviously true for this simulated data, I would expect the 10000 resulting p-values to be evenly distributed between 0 and 1. However, this is not the case. Instead the histogram shows a significant rightward skew towards 1.

hist(x$p.value, breaks = seq(0,1, by = 0.01))

enter image description here

Why is this the case?

GenomeGnome
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    You are not reading the histogram correctly. By definition, histograms represent probability by means of *area,* not height. Look closely at the right hand side and notice that the high bars are balanced by neighboring low bars, producing a very uniform density across the board. This phenomenon (of bar heights bouncing around) is *aliasing* in the binning process, as discussed at https://stats.stackexchange.com/questions/401692/what-is-this-phenomenon-called. Use a uniform probability plot instead. – whuber May 16 '19 at 12:28
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    Thanks for the clarification and relevant links, this has solved the problem for me! – GenomeGnome May 17 '19 at 09:38

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