# A MOVER Confidence interval for the ratio of two Poisson rates

If X and Y be two independent Poisson rates with $\hat{\lambda}_1$ and $\hat{\lambda}_2$ how do we obtain this confidence interval for the ratio of $\hat{\lambda}_2/\hat{\lambda}_1$? Where $(l_i,u_i)$ are seperate confidence itervals for $\hat{\lambda}_1$ and $\hat{\lambda}_2$ Then the lower and upper bound is: $L=\hat{\lambda}_2 / \hat{\lambda}_1-\sqrt{(\hat{\lambda}_2 / \hat{\lambda}_1 )^2-l_1 (2\hat{\lambda}_2-l_1 )[u_2 (2\hat{\lambda}_1-u_2 )] )}/(u_2 (2\hat{\lambda}_1-u_2 ) )$ $U=\hat{\lambda}_2 / \hat{\lambda}_1+\sqrt{(\hat{\lambda}_2 / \hat{\lambda}_1 )^2-u_1 (2\hat{\lambda}_2-u_1 )[l_2 (2\hat{\lambda}_1-l_2 )] )}/(l_2 (2\hat{\lambda}_1-l_2 ))$

We have obtained a confidence interval for the difference between two independent Poisson rates. Then by using MOVER method we obtain these bounds: $L'=λ ̂_2-λ ̂_1-√((λ ̂_2-l_2 )^2+(u_1-λ ̂_1 )^2 )$ $U'=λ ̂_2-λ ̂_1+√((u_2-λ ̂_2 )^2+(λ ̂_1-l_1 )^2 )$

Now we want to obtain a confidence interval for the ratio of them. So we want to get $Pr(L≤λ_2/λ_1 ≤U)=1-α$ or equality $Pr(λ_2-Uλ_1≤0≤λ_2-Lλ_1 )=1-α$ Now we can apply $L'$ and $U'$ for $λ_2-Lλ_1$ and $λ_2-Uλ_1$ and then obtain the confidence intervals which is above. But how?

• you need to either clear up the expression or clarify your notation here for this question to be answerable. – AdamO May 15 '17 at 17:16
• You seem to be familiar with Fieller's theorem. Have you tried actually plugging in the parameters? You know the mean and standard deviations, and that the two are not correlated, so there will be no covariance term: en.wikipedia.org/wiki/Fieller%27s_theorem – Alex R. May 15 '17 at 17:20
• I have tried to solve this with finding square roots but i have faced an issue – Raziye May 15 '17 at 17:35
• Exactly how were these rates estimated? It's possible the underlying estimates were based on their logarithms using Poisson regression, which would make it relatively easy to estimate the ratio and construct a CI for it. – whuber May 15 '17 at 17:57
• I have read the article of this issue... But when I read the references of this article I faced a different formula... In the references it was exactly which is I obtain.. I obtain the same indexes in the parentheses. What shoul I do? – Raziye May 17 '17 at 15:55

Your formula does not seem to match the actual Fieller's Theorem, which according to Wiki, has a correction due to the correlation of the two variables.

If your goal is to construct an approximate CI for a ratio of poisson RVs using the upper and lower bounds of the CIs for two Poisson means, this is impossible: you must assume those variables are independent or you must know their covariance.

If they are independent, there is no reason to appeal to Fiellers theorem, the CI is a direct result of the $\delta$-method. In either case, your only task is to identify the SE from the CI which I describe below.

The first task, for poisson RVs, is to identify the scale on which a normal approximation is made. On the linear scale, the SE of lambda is $\sqrt{\hat{\lambda}/n}$, on a log scale the SE of the log lambda is $\sqrt{2 \hat{\lambda}/n}$ which results in a better coverage 95% non-symmetric CI when re-exponentiated.

In either case, it is only algebra to transform the upper and lower bounds of a CI into the SE for the estimate (either lambda or log-lambda). Then you can use the general form of Fieller's theorem to obtain the SE and CI for the ratio or set the covariance to 0 for the special case. Or you can derive the SE yourself with $\delta$ method.

• Thank you.. But i could not solve it yet – Raziye May 17 '17 at 15:40

In a Bayesian framework, you can obtain posterior samples for $\lambda_1$ and $\lambda_2$, and divide them to get posterior samples for their fraction. Then get a Bayesian credibility interval from those samples.

You can use pymc3 in Python or stan in R to obtain these samples, and Poisson models are easy and straightforward to sample for.

• This is a computational solution, but not an analytical one which was the nature of the question. – AdamO May 15 '17 at 17:19
• You're right, I misread the 'this' confidence interval part. – Gijs May 15 '17 at 17:22
• Thank you but i need the proof of this problem – Raziye May 15 '17 at 19:07
• Yes, I understand, this isn't helpful in that direction. – Gijs May 15 '17 at 19:27