# Confidence intervals for Ratios of Correlated (nested) Age-Adjusted Rates

I am interested in calculating the incidence of disease in an ethnic subpopulation (denoted $X$) and comparing this with the incidence in the entire population ($\Omega$) (where $X$ is a true subset of $\Omega$).

The age distribution of the subpopulation is quite different to the whole population, and thus I intend to calculate age-adjusted rates using the following formula

$$R_X = \sum\limits_{j=1}^Jw_j \frac{d_{Xj}}{n_{Xj}}$$

where $w_j$ are known standards normalized to sum to 1 over the $J$ age-groups; $d_Xj$ is the number of new disease cases and $n_Xj$ the size of the population (followed over one year) for a specific agegroup.

By 'compare' I mean that I want to calculate the incidence rate ratio: $\theta = \frac{R_X}{R_\Omega}$ with 95% confidence intervals. This is straight forward for independent samples, but less so for correlated samples where (as I understand) the covariance needs to be taken into account.

Tiwari, Li and Zou (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3279758/) suggest two methods for nested geographical regions (eg. California versus the US), but require the assumption of a proportional age-distribution (the ratio of the size of the subpopulation to the overall population is approximately the same across all age-groups, ie. $\frac{n_X}{n_\Omega}$ is the same for all levels of $J$). This assumption clearly does not hold in my example, where an ethnic subpopulation has a much younger age-distribution that the overall population.

Does anyone know how important this assumption is? Can anyone suggest any other approaches that I don't know about? Potentially bootstrapping might be feasible, but I don't really know how to implement it when I have summary data recording the number of cases and population by age-group at a national level?