Date: Fri, 26 Nov 2010 18:37:21 -0500 "Viel, Kevin" "SAS(r) Discussion" "Viel, Kevin" Exact Poisson CI text/plain; charset="us-ascii"

On page 73 of "Analyzing Categorical Data" by Jeffrey Simonoff, the exact CI limits of a rate, assuming a Poisson distribution, are given as:

LB = 0, if x = 0 [Chi-sq(1-alpha)/2, 2x)]/(2n), otherwise

UB = [Chi-sq(alpha/2, 2x+2)]/(2n)

Given Pr(Y=y) = lambda**y * exp( -lambda) / y!

I come up with:

Pr(Y=y_LB|lambda_LB) = Sigma(n=y_LB,infinity) lambda_LB**n * exp( -lambda_LB) / n! Pr(Y=y_LB|lambda_LB) = 1 - Sigma(n=0,y_LB - 1) lambda_LB**n * exp( -lambda_LB) / n!

If we take the 95% CI, we have

0.025 = 1 - Sigma(n=0,y_LB - 1) lambda_LB**n * exp( -lambda_LB) / n! Sigma(n=0,y_LB - 1) lambda_LB**n * exp( -lambda_LB) / n! = 0.975

By Brute Force and Awkwardness, we just have to iterate through values of lambda_LB until we find one that approaches 0.975 to our satisfication (I chose an absolute difference of 0.0005).

I came up with LB = 1.5485. Another source suggests that the above book may have erred by swapping the p levels. Using that formula I arrive at:

cinv( alpha / 2 , 6 ) = 1.54

Note that 1.5485/40 = 0.039 ~ 0.038 that the books lists as its lower bound for the hurricane example (6 in 40 years).

Two questions: 1) Can anyone verify or correct the p level of the chi-squared formula? 2) Can anyone suggest a reference to guide the "analytical derivation" of these formulas? I suspect one starts by using the definition of lower bound that I used above.

I was surprised that this is not laid out anywhere on the web. You'd think by now, some stats instructor would have asked for a step-by-step derivation as an homework assignment that he or she could then post to the internet. It actually matters if one is going to pursue sample size calculations, for instance :)

Thanks,

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