Indirect standardization compares the rates of the study and reference populations by applying the stratumspecific rates in the reference population to the study population, where the stratumspecific rates might not be reliable.
The expected number of events in the study population is

where is the populationtime 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 stratumspecific 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 (ElandtJohnson 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 pvalue for the test of the null hypothesis is given by

With a ratestandardized 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 riskstandardized 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 .