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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:

$$y_{j}={\sum}_{k=1}^{K}w_{k}r_{jk}$$

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:

$$\hat{\sigma}_{j}^{2}={\sum}_{k=1}^{K}\frac{W_{k}^{2}}{N_{jk}}d_{jk}$$.