In direct standardization, the derived standardized rates and risks in a study population are the weighted average of the stratum-specific rates and risks in the population, respectively, where the weights are given by the population-time 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 Mantel-Haenszel estimates of this effect can be constructed from directly standardized rates or risks in the two populations, where the weights are constructed from the stratum-specific population-times 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 Mantel-Haenszel difference statistic is
with variance
Under the null hypothesis , the difference statistic has a normal distribution with mean 0.
The Mantel-Haenszel 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 Mantel-Haenszel 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).