### Mantel-Haenszel Effect Estimation

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.

#### Rate and Risk Difference Statistics

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.

#### Rate Ratio Statistic

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).

#### Risk Ratio Statistic

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).