# Modeling count data to poisson distribution when expected is zero

I am working on a project where we saw an event occur in 5 of 39 patients. We don't expect this event to occur at all (though we know they occur very rarely, we don't expect to see any in this instance), and my advisor wants me to model to a Poisson distribution (with a p-value) to show that this is more than we would expect by random chance.

He did the math, which I don't have anymore, but I'm not convinced that the values he used were appropriate for Poisson. I didn't think you could do this if you expect 0 events.

I'm hoping someone can explain to me: 1) If possible, how to do this math appropriately 2) If there is another statistical test to show that this event is occurring more than we'd expect by random chance.

• I don't follow: if you adopt a model in which the event cannot occur, and it occurs, then haven't you demonstrated the model is incorrect? Could you clarify what you mean by "expect 0 events" and provide enough information to formulate a null hypothesis for some kind of test? – whuber Feb 16 at 21:35
• Well, if you didn't expect this at all, then you don't need any distributional assumptions to extract a p value. Your model is "we don't expect this at all", and your test statistic (the number of events you observed) is more extreme than anything your model would have expected, so $p=0.00$. (This is one of the rare cases where you can actually write "$p=0.00$", instead of only "$p<0.0001$".) Which is just a fancy reformulation of what @whuber commented. – Stephan Kolassa Feb 17 at 6:15
• It's a very rare genetic event that occurs normally once every few thousand or more years...so not entirely zero. But pretty close. And we're saying it's happening de novo in cancer (which no one believes is true but we found evidence that it is in fact true). I said pretty much what @whuber did, but my advisor really wanted to model to a poisson dist and get a p-value, and I was at a loss of how to do that. ::shrug:: Does what I just added change any of what either of you commented? If it doesn't make sense, I can try to add more detail in the Q – Gaius Augustus Feb 17 at 8:03
• I think @Stephen may have pointed the way to satisfying your advisor. Take a Poisson rate parameter suggested by the last thousand years of experience and compute the chances. For the longer term (such as after you get your degree), you might want to make a point of ignoring any advice your advisor gives you about statistical analyses :-). – whuber Feb 17 at 16:06