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 .