An event risk of a population over a specified time period can be defined as the number of new events in the follow-up time period divided by the event-free population size at the beginning of the time period,
where is the population size.
For a general population, the subsets (strata) might not be homogeneous enough to have a similar risk. Thus, the risk for each stratum should be computed separately to reflect this discrepancy. For a population that consists of K homogeneous strata (such as different age groups), the stratum-specific risk for the jth stratum in a population is computed as
where is the population size in the jth stratum of the population.
Assuming the number of events, , has a binomial distribution, then a variance estimate of is
By using the method of statistical differentials (Elandt-Johnson and Johnson 1980, pp. 70–71), the variance of the logarithm of risk can be estimated by
A confidence interval for based on a normal distribution is given by
where is the quantile of the standard normal distribution.
A confidence interval for based on a normal distribution is given by
where is the quantile of the standard normal distribution and the variance .
Thus, a confidence interval for based on a lognormal distribution is given by
For rate estimates from two independent samples, and , a confidence interval for the risk difference is
where is the quantile of the standard normal distribution and the variance
For rate estimates from two independent samples, and , a confidence interval for the log risk ratio statistic is
where is the quantile of the standard normal distribution and the variance
Thus, a confidence interval for the risk ratio statistic is given by
At stratum j, a stratum-specific standardized morbidity/mortality ratio is
where is the expected number of events.
With the risk
SMR can be expressed as
Thus, a confidence interval for is given by
where is a confidence interval for the risk .