# How do I compare two event rates?

Say I have two event streams, modeled as Poisson processes. These streams omit events at rates $\lambda_1$ and $\lambda_2$ respectively, so the count of observed events in $t$ time is $\lambda_1t$ and $\lambda_2t$. For some desired confidence, how long do I need to observe in order to be confident that either $\lambda_1 > \lambda_2$ or vice versa?

Intuition suggests that the answer will be some distribution that's a function of $\lambda_1$, $\lambda_2$, $t$, and my desired confidence. I did some searching and just found tests for inequality of $\lambda_1$ and $\lambda_2$, not a specific parameterized distribution.

• You need to decide on a parametrization appropriate for what you're modelling. Often the parameter of interest would be a function of either $\lambda_1 - \lambda_2$ or $\frac{\lambda_1}{\lambda_2}$, with $\lambda_1+\lambda_2$ being a nuisance parameter. I've added the power-analysis tag. – Scortchi Jun 1 '18 at 11:13
• Thanks. I'm not sure which parameterization is appropriate. I'm just trying to detect whether $\lambda_1$ and $\lambda_2$ are different. Ideally, I could detect whether any of $\lambda_1, \lambda_2, ..., \lambda_n$ are greater than the others. – tom Jun 1 '18 at 14:31
• You'll need to wait longer to detect a difference of one event per hour than to detect a difference of a hundred events per hour. Typically you'd set the Type I error of your test - the probability of rejecting the null hypothesis of no difference when it is in fact true - to 1%, say, & the power - the probability of rejecting the null hypothesis when a given alternative is in fact true - to 90%, say, for an effect size of your choosing; & then solve for the required observation time. So you do need to pick a particular parametrization first. – Scortchi Jun 1 '18 at 15:10
• You'll also - not that this need be given in your question - have to make a guess, typically a pessimistic one, at the overall event rate (or one of the individual rates). Clearly if you expect to wait a day or so to observe any events at all across both streams, the study will take longer than if you expect many over the course of an hour. – Scortchi Jun 1 '18 at 15:30
• @Scortchi That was my thought as well...pick a Type 1 error and Type 2 error tolerance, and solve for t. I was hoping someone would say "Oh, that's just a XYZ distribution parameterized as follows", but it looks like I may have to do the math myself. – tom Jun 1 '18 at 20:30