|Date: ||Fri, 26 Nov 2010 18:37:21 -0500|
|Reply-To: ||"Viel, Kevin" <kviel@SJHA.ORG>|
|Sender: ||"SAS(r) Discussion" <SAS-L@LISTSERV.UGA.EDU>|
|From: ||"Viel, Kevin" <kviel@SJHA.ORG>|
|Subject: ||Exact Poisson CI|
|Content-Type: ||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).
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 :)
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