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Notations Based on Age-adjusted Rate


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: 


where \(w_{k}\) is the standard population weight attached to age group \(k\) (

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: 


An unbiased estimate of the variance \(\sigma_{j}^{2}\) is: