In direct standardization, the derived standardized rates and risks in a study population are the weighted average of the stratumspecific rates and risks in the population, respectively, where the weights are given by the populationtime for standardized rate and the number of observations for standardized risk in a reference population.
Assuming that an effect, such as rate difference, rate ratio, risk difference, and risk ratio between two populations, is homogeneous across strata, the MantelHaenszel estimates of this effect can be constructed from directly standardized rates or risks in the two populations, where the weights are constructed from the stratumspecific populationtimes for rate and number of observations for risk of the two populations.
That is, for population k, k=1 and 2, the standardized rate and risk are

where the weights are

for standardized rate, and

for standardized risk.
Denote for rate and for risk. The variance is

The MantelHaenszel difference statistic is

with variance

Under the null hypothesis , the difference statistic has a normal distribution with mean 0.
The MantelHaenszel rate ratio statistic is , and the log ratio statistic is

Under the null hypothesis (or equivalently, ), the log ratio statistic has a normal distribution with mean 0 and variance

where

is the combined rate estimate in stratum j under the null hypothesis of equal rates (Greenland and Robins 1985; Greenland and Rothman 2008, p. 273).
The MantelHaenszel risk ratio statistic is , and the log ratio statistic is

Under the null hypothesis (or equivalently, ), the log ratio statistic has a normal distribution with mean 0 and variance

where

is the combined risk estimate in stratum j under the null hypothesis of equal risks (Greenland and Robins 1985; Greenland and Rothman 2008, p. 275).