How to calculate credibility interval limits

Hi everyone.
My name is Choi. I'm working on detecting drug adverse reaction signals using
health insurance data. I have a problem in getting credibility interval limits due to
my poor math and statistics below is what I quoted from a paper.
'Ωcan be viewed as the logarithm of the posterior mean of an unknown rate of
incidence μ under the natural assumption that n111 is Po(μ·E111)-distributed with log2μ=Ω and
a gamma prior distribution (or random effects model in a likelihood-based analysis) for μ: G(α,α),
with expected value 1. The choice of prior is made mainly for mathematical convenience, since
due to conjugacy the posterior distribution for μ will also be gamma (but with parameters n111+α
and E111+α, expected value (n111+α)/(E111+α) and variance (n111+α)/(E111+α)^2).
With the Bayesian approach, exact credibility interval limits for μ can be found numerically as
solutions to the following equation, for appropriate posterior quantiles μq :

**(equation 20) **

Specifically, the logarithm of the solutions to (20) for q=0.025 and 0.975, respectively, provides
the upper and lower limits of a two-sided 95 per cent credibility interval for Ω: Ω025 and Ω975.'
I used calculation tools to figure out uq when q=0.025. But they made errors and failed
to get the value. I worder how I can solve this problem.. seeing the table or calculus
tools or whatever.


TS Contributor
So I just see you are trying to calculate the quantile of the posterior (gamma) distribution. What is your error? Can you provide more information?
Thanks for the reply. I wanted to the practical tool to get the value.
Anyway I've got the answer. Below is from the author of the paper.
For details of how to calculate the credibility intervals from the observed and expected values I would recommend you to read the newly published article: Norén GN, Hopstadius J, Bate A. Shrinkage observed-to-expected ratios for robust and transparent large-scale pattern discovery. Statistical Methods in Medical Research. June 24, 2011

The key in the calculations is that the ratio (O+1/2)/(E+1/2) is Gamma G(O+1/2, E+1/2) distributed making the lower limit of the 95% credibility interval to be calculated from the quantiles of the gamma distribution. As an example, in R (open source statistical software) IC and IC025 may be calculated from the variables observed and expected:

omega <- log2( (observed+0.5)/(expected+0.5) )

omega_025 <- log2(qgamma(p=0.025 , shape=(observed +0.5), rate=(expected +0.5)))

Some implementations (e.g. the quantile function in SAS) uses the parameter scale = 1/rate.

In the referred paper there is an approximation possible to use if you don’t have access to any statistical packages.