Hi Sofia,

Thank you for your response. The exact method using Poisson distribution to compute RIR was not illustrated in the article and I can't find specific pages in the citation number 22 that illustrate the formulae. Last week, I emailed the first author but my email was undelivered because the author's email address is no longer valid. 

It will be nice to find a software that can compute confidence interval for 'ratio of two ratios". Anyone knows? Thanks!

It seems to be an interesting book which is freely available. See the link. I guess that  your requested formula is on chapter 19 of this book. Take a look at page 137 too.

https://archive.org/details/StatisticalMethodsInMedicalResearch

Hi Kam,

The statistic looks similar to Relative Risk.

Please have a look at the link below -->

http://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_confidence_intervals/BS704_Confidence_Intervals2.html

Bootstrapping.

Thank you for your responses.

@Sofia_Cividini

https://www.researchgate.net/profile/Sofia_Cividini

@Mehmet_Iyisoy

https://www.researchgate.net/profile/Mehmet_Iyisoy

@Paer_Nilsson2

https://www.researchgate.net/profile/Paer_Nilsson2

@Rohit_Ojha

https://www.researchgate.net/profile/Rohit_Ojha

@Boon_How_Chew2

https://www.researchgate.net/profile/Boon_How_Chew2

@Kam_Cheong_Wong

https://www.researchgate.net/profile/Kam_Cheong_Wong

I am sorry for my over-simplified question in the initial post. Actually, my question is how to compute 95%CI for an index that contains 4 ratios as follows:

An index for “Relative Illness Ratio of female to male”

= RIR for female / RIR for male

=  {(Ci/Σ Ci) / (Ni/Σ Ni)}female / {(Ci/Σ Ci) / (Ni/Σ Ni)male}

Where:

Relative Illness Ratio (RIR)= (Ci/Σ Ci) / (Ni/Σ Ni)

Ci: number of cases of laboratory-confirmed influenza in a given age group

Σ Ci: Sum of cases of laboratory-confirmed influenza in all age groups

Ni: Population in a given age group

Σ Ni: Sum of populations in all age groups.

The above index, which contains 4 ratios, is not included in the Lemaitre’s paper that Sofia_Cividini has suggested. Hence, the Lemaitre’s reference may not provide the formulae to compute 95%CI for a “ratio of 4 ratios”.

I appreciate any suggestions and resources you are willing to share. Thanks!

"We can calculate a bootstrap t confidence interval for any parameter by bootstrapping the corresponding statistic. We don’t need conditions on the population or special knowledge about the sampling distribution of the statistic."

Other type of CIs that can be derived:

1) Bootstrap percentile confidence interval, 2) The bootstrap bias-corrected accelerated (BCa) interval (this is more reliable when the bootstrap distribution isn't Normal and has a large bias estimate, unless the sample size is very large).

                      - Introduction to Practical Statistics by Moore, McCabe and Craig, 8th Ed.

@Boon How Chew

Thank you Dr Chew for your suggestion. I've thought of bootstrap method as I used it in my MSc course many years ago. Now, with your suggestion and reference, I will relook into it. Thanks! 

In addition to Dr Boon How Chew's suggestion of Boostrap method (see above post), I've found an alternative approach applied by other researchers (see below). The researchers applied  Boostrap and Delta methods  to compute confidence intervals for different variables.  Anyone who has experience in these methods, kindly share with the group. Thank you.

Reference: Wang, X. L., Yang, L., Chan, K. H., Chan, K. P., Cao, P. H., Lau, E. H., . . . Wong, C. M. (2015). Age and Sex Differences in Rates of Influenza-Associated Hospitalizations in Hong Kong. Am J Epidemiol, 182(4), 335-344. doi:10.1093/aje/kwv068