# Differences between poisson.test and E-test when testing Poisson parameters

Suppose we have two independent Poisson-distributed variables $X_1$ and $X_2$. We want to test whether the Poisson parameters are equal, i.e. whether $\lambda_1=\lambda_2$.

Now we have 4 distinct statistical exact tests to choose:

1. E-test (see Krishnamoorthy and Thomson, A more powerful test for comparing two Poisson means, Journal of Statistical Planning and Inference 119 (2004) 23–35; see also Checking if two Poisson samples have the same mean on Cross Validated)
2. poisson.exact(tsmethod="central")
3. poisson.exact(tsmethod="minlike")
4. poisson.exact(tsmethod="balker") (from exactci R package)

Now, given all those tests are labeled as "exact", one would expect all yield the same p-values. Contrary to this, the quoted paper clearly illustrates that methods 2-4 give different significance. Furthermore, I personally implemented the E-test and found that this test gives yet another, distinct result. Why is that?

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I am not familiar with all the methods so I am reluctant to make this comment and answer. It is really just a guess on my part. It is possible to have different tests for the same null hypothesis that use different test statistics. It is the distribution of the test statistic under the null hypothesis that determines the p-value for the specific test. Saying that a test is exact level alpha just means that it is constructed using the exact distirbution of the test statistic under the null hypothesis rather than an asymptotic approximation. –  Michael Chernick Jul 26 '12 at 22:47
So it shouldn't be surprising that different tests on the same data set give different p-value. Now if the p-value is less than 0.05 for one test and greater for another one will reject and the other will not (at the 5% level). Some tests are more powerful than others (this could be universal or under specific conditions). So this can happen and is not rare. –  Michael Chernick Jul 26 '12 at 22:51
@MichaelChernick Are you sure about the meaning of exactness ? I think an exact confidence interval at the nominal $\alpha$ level is a confidence interval whose effective confidence level is at least $\alpha$ (but it is true that such tests are usually (always?) based on the exact distribution of the test statistic and not asymptotic approximation) –  Stéphane Laurent Jul 27 '12 at 7:27
Adam, where do you see an illustration of "significant" difference between the methods ? The paper gives an example for which the three methods yield close confidence intervals. –  Stéphane Laurent Jul 27 '12 at 7:29
@StéphaneLaurent My point about exact tests is that an 0.05 level test means that the exact significance level is less thanor equal to 0.05. For absolutely continous distributions an exact test will have exact level 0.05 for all sample sizes. But for test statistics that have discrete distributions for most sample sizes the exact level will be less than 0.05 and yet we will call the test exact. Examples would be an exact binomial test test using the Clopper-Pearson method or Fisher's exact test for contingency tables. –  Michael Chernick Jul 27 '12 at 8:41
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