### Risk

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 #### Normal Distribution Confidence Interval for Risk

A confidence interval for based on a normal distribution is given by where is the quantile of the standard normal distribution.

#### Lognormal Distribution Confidence Interval for Risk

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 #### Confidence Interval for Risk Difference Statistic

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 #### Confidence Interval for Risk Ratio Statistic

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 #### Confidence Interval for Risk SMR

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 .