An official website of the United States government
Assume there are \(J\) mutually exclusive social groups in the population with \(J\geq {2}\). Let \(r_{jk}\) denote the estimated rate in the kth age group within the jth social group, \(j=1,...J;k=1,...K\). Then \(r_{jk}=\frac{d_{jk}}{N_{jk}}\), where \(d_{jk}\) is the number of events and \(N_{jk}\) is the number of person years in the kth age group within the jth social group. The age-adjusted rate (also called directly standardized rate) estimate for an event (e.g., cancer) in the jth social group is calculated as:
$$y_{j}={\sum}_{k=1}^{K}w_{k}r_{jk}$$
where \(w_{k}\) is the standard population weight attached to age group \(k\) (https://seer.cancer.gov/stdpopulations/.)
Assume that \(d_{jk} \sim Poisson {(N_{jk}\lambda_{jk})},\) where \(\lambda_{jk}\) is the true rate and population size for the kth age category in the jth social group, then the variance of the age-adjusted rate \(y_{j}\) is:
$$\sigma_{j}^{2}={\sum}_{k=1}^{K}\frac{w_{k}^{2}}{N_{jk}}\lambda_{jk}$$
An unbiased estimate of the variance \(\sigma_{j}^{2}\) is:
$$\hat{\sigma}_{j}^{2}={\sum}_{k=1}^{K}\frac{W_{k}^{2}}{N_{jk}}d_{jk}$$.
