Direct standardization uses the weights from a reference population to compute the standardized rate of a study group as the weighted average of stratum-specific rates in the study population. The standardized rate is computed as
where is the rate in the jth stratum of the study population, is the population-time in the jth stratum of the reference population, and is the population-time in the reference population.
Similarly, direct standardization uses the weights from a reference population to compute the standardized risk of a study group as the weighted average of stratum-specific risks in the study population. The standardized risk is computed as
where is the risk in the jth stratum of the study population, is the number of observations in the jth stratum of the reference population, and is the total number of observations in the reference population.
That is, the directly standardized rate and risk of a study population are weighted averages of the stratum-specific rates and risks, respectively, where the weights are the corresponding strata population sizes in the reference population. The direct standardization can be used when the study population is large enough to provide stable stratum-specific rates or risks. When the same reference population is used for multiple study populations, directly standardized rates and risks provide valid comparisons between study populations.
The variances of the directly standardized rate and risk are
By using the method of statistical differentials (Elandt-Johnson and Johnson 1980, pp. 70–71), the variance of the logarithm of directly standardized rate and risk can be estimated by
The confidence intervals for and can be constructed based on normal and lognormal distributions. A gamma distribution confidence interval can also be constructed for .
In the next four subsections, denotes the rate statistic and denotes the risk statistic.
A confidence interval for based on a normal distribution is then 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.
Thus, a confidence interval for based on a lognormal distribution is given by
Fay and Feuer (1997) use the relationship between the Poisson and gamma distributions to derive approximate confidence intervals for the standardized rate based on the gamma distribution. As in the construction of the asymptotic normal confidence intervals, it is assumed that the number of events has a Poisson distribution, and the standardized rate is a weighted sum of independent Poisson random variables. A confidence interval for is then given by
where
and is the maximum .
Tiwari, Clegg, and Zou (2006) propose a less conservative confidence interval for with a different upper confidence limit,
where is the average and is the average .
By using the same reference population, two directly standardized rates or risks from different populations can be compared. Both the difference and ratio statistics can be used in the comparison. Assume that and are directly standardized rates or risks for two populations with variances and , respectively. The difference test assumes that the difference statistic
has a normal distribution with mean 0 under the null hypothesis . The variance is given by
The ratio test assumes that the log ratio statistic,
has a normal distribution with mean 0 under the null hypothesis , or equivalently, . An estimated variance is given by