Indirect standardization compares the rates of the study and reference populations by applying the stratum-specific rates in the reference population to the study population, where the stratum-specific rates might not be reliable.
The expected number of events in the study population is
where is the population-time in the jth stratum of the study population and is the rate in the jth stratum of the reference population.
With the expected number of events, , the standardized morbidity ratio or standardized mortality ratio can be expressed as
where is the observed number of events (Breslow and Day, 1987, p. 65).
The ratio indicates that the mortality rate or risk in the study population is larger than the estimate in the reference population, and indicates that the mortality rate or risk in the study population is smaller than the estimate in the reference population.
With the ratio , an indirectly standardized rate for the study population is computed as
where is the overall crude rate in the reference population.
Similarly, to compare the risks of the study and reference populations, the stratum-specific risks in the reference population are used to compute the expected number of events in the study population
where is the number of observations in the jth stratum of the study population and is the risk in the jth stratum of the reference population.
Also, with the standardized morbidity ratio , an indirectly standardized risk for the study population is computed as
where is the overall crude risk in the reference population.
The observed number of events in the study population is , where is the number of events in the jth stratum of the population. For the rate estimate, if has a Poisson distribution, then the variance of the standardized mortality ratio is
For the risk estimate, if has a binomial distribution, then the variance of is
where
By using the method of statistical differentials (Elandt-Johnson and Johnson, 1980, pp. 70–71), the variance of the logarithm of can be estimated by
For the rate estimate,
The confidence intervals for can be constructed based on normal, lognormal, and Poisson distributions.
A confidence interval for based on a normal distribution is given by
where is the quantile of the standard normal distribution.
A test statistic for the null hypothesis is then given by
The test statistic has an approximate standard normal distribution under .
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
A test statistic for the null hypothesis is then given by
The test statistic has an approximate standard normal distribution under .
Denote the quantile for the distribution with degrees of freedom by
Denote the quantiles for the distribution with degrees of freedom by
Then a confidence interval for based on the distribution is given by
A p-value for the test of the null hypothesis is given by
With a rate-standardized mortality ratio , an indirectly standardized rate for the study population is computed as
where is the overall crude rate in the reference population.
The confidence intervals for can be constructed as
where is the confidence interval for .
With a risk-standardized mortality ratio , an indirectly standardized risk for the study population is computed as
where is the overall crude risk in the reference population.
The confidence intervals for can be constructed as
where is the confidence interval for .