First, we discuss the common risk ratio and risk difference estimation for stratified analysis in cohort studies. Consider a series of *K* 2 × 2 tables formed by pairs of independent binomial observations $\left({X}_{0k},{X}_{1k}\right)$ with sample sizes $\left({N}_{0k,}{N}_{1k}\right)$ and success probabilities $\left({p}_{0k},{p}_{1k}\right)$ for $k=1,2,\dots ,K$. Under the common effect assumption for the risk ratio, we assume the stratum-specific risk ratio ${\mathrm{\psi}}_{k}={p}_{1k}/{p}_{0k}$ are all equal across the strata, i. e., $\mathrm{\psi}={\mathrm{\psi}}_{1}={\mathrm{\psi}}_{2}=\dots ={\mathrm{\psi}}_{K}$. Similarly, for the risk difference, we assume the common effect for the stratum specific risk difference ${\mathrm{\omega}}_{k}={p}_{1k}-{p}_{0k}$ across the strata, i. e., $\mathrm{\omega}={\mathrm{\omega}}_{1}={\mathrm{\omega}}_{2}=\dots ={\mathrm{\omega}}_{K}$.

The Mantel-Haenszel estimators of the common risk ratio $\mathrm{\psi}$ (Nurminen 1981; Tarone 1981) and the common risk difference $\mathrm{\omega}$ (Cochran 1954) are presented as
${\stackrel{\u02c6}{\mathrm{\psi}}}_{MH}=\frac{{\sum}_{k=1}^{K}{X}_{1k}{N}_{0k}/{N}_{k}}{{\sum}_{k=1}^{K}{X}_{0k}{N}_{1k}/{N}_{k}}=\frac{{\sum}_{k=1}^{K}{w}_{MH,k}{\stackrel{\u02c6}{p}}_{1k}}{{\sum}_{k=1}^{K}{w}_{MH,k}{\stackrel{\u02c6}{p}}_{0k}}=\frac{{\sum}_{k=1}^{K}{w}_{MH,k}{\stackrel{\u02c6}{p}}_{0k}{\stackrel{\u02c6}{\mathrm{\psi}}}_{k}}{{\sum}_{k=1}^{K}{w}_{MH,k}{\stackrel{\u02c6}{p}}_{0k}},$
${\stackrel{\u02c6}{\mathrm{\omega}}}_{MH}=\frac{{\sum}_{k=1}^{K}({X}_{1k}{N}_{0k}-{X}_{0k}{N}_{1k})/{N}_{k}}{{\sum}_{k=1}^{K}{N}_{0k}{N}_{1k}/{N}_{k}}=\frac{{\sum}_{k=1}^{K}{w}_{MH,k}{\stackrel{\u02c6}{p}}_{1k}}{{\sum}_{k=1}^{K}{w}_{MH,k}}-\frac{{\sum}_{k=1}^{K}{w}_{MH,k}{\stackrel{\u02c6}{p}}_{0k}}{{\sum}_{k=1}^{K}{w}_{MH,k}}=\frac{{\sum}_{k=1}^{K}{w}_{MH,k}{\stackrel{\u02c6}{\mathrm{\omega}}}_{k}}{{\sum}_{k=1}^{K}{w}_{MH,k}},$where ${\stackrel{\u02c6}{p}}_{0k}={X}_{0k}/{N}_{0k},{\stackrel{\u02c6}{p}}_{1k}={X}_{1k}/{N}_{1k},{N}_{k}={N}_{1k}+{N}_{0k}$ and ${w}_{MH,k}=\left({N}_{0k}{N}_{1k}/{N}_{k}\right)$. Also, ${\stackrel{\u02c6}{\mathrm{\psi}}}_{k}={\stackrel{\u02c6}{p}}_{1k}/{\stackrel{\u02c6}{p}}_{0k}$ and ${\stackrel{\u02c6}{\mathrm{\omega}}}_{k}={\stackrel{\u02c6}{p}}_{1k}-{\stackrel{\u02c6}{p}}_{0k}$. These estimators are obtained as solutions of the following estimation equations,
$Q\left(\mathrm{\psi}\right)=\sum _{k=1}^{K}{w}_{MH,k}\left[{\stackrel{\u02c6}{p}}_{1k}-\mathrm{\psi}{\stackrel{\u02c6}{p}}_{0k}\right]=0,$
$R\left(\mathrm{\omega}\right)=\sum _{k=1}^{K}{w}_{MH,k}\left[\mathrm{\omega}-\left({\stackrel{\u02c6}{p}}_{1k}-{\stackrel{\u02c6}{p}}_{0k}\right)\right]=0.$Note that these estimating functions are unbiased under the common effects assumptions, such that $E\left[Q\left(\mathrm{\psi}\right)\right]=0$ and $E\left[R\left(\mathrm{\omega}\right)\right]=0$, and consistency of the estimators follow straightforwardly. Here, suppose the common effect assumptions are violated, i. e., ${\mathrm{\psi}}_{1},{\mathrm{\psi}}_{2},\dots ,{\mathrm{\psi}}_{K}$ and ${\mathrm{\omega}}_{1},{\mathrm{\omega}}_{2},\dots ,{\mathrm{\omega}}_{K}$ are possibly heterogeneous. In this case, the target parameters $\mathrm{\psi}$ and $\mathrm{\omega}$ for the above estimating equations are no longer interpreted as the common effects.

For evaluating the asymptotic behaviors of the Mantel-Haenszel estimators, it is useful to formulate two large sample schemes that are common for stratified analyses. The first, denoted as Asymptotic I, is to have a fixed number of strata *K* while ${N}_{0k,}{N}_{1k}\to \mathrm{\infty}$. In the second, denoted as Asymptotic II, $\left({N}_{0k,}{N}_{1k}\right)$ are bounded while $K$ is large. A well-known example of Asymptotic II is the matched designs.

First, denoting ${Q}_{k}\left(\mathrm{\psi}\right)={\stackrel{\u02c6}{p}}_{1k}-\mathrm{\psi}{\stackrel{\u02c6}{p}}_{0k},{R}_{k}\left(\mathrm{\omega}\right)=\mathrm{\omega}-\left({\stackrel{\u02c6}{p}}_{1k}-{\stackrel{\u02c6}{p}}_{0k}\right)$, the asymptotic behaviors of Mantel-Haenszel estimators under Asymptotic I can be characterized as follows.

Under the Asymptotic I, we assume ${N}_{0k}/N\to {\mathrm{\lambda}}_{0k}>0$, ${N}_{1k}/N\to {\mathrm{\lambda}}_{1k}>0$, where $N={N}_{01}+\dots +{N}_{0K}+{N}_{11}+\dots +{N}_{1K}$. Then, the Mantel-Haenszel estimators converge to normal distributions with means equal to
${\mathrm{\psi}}_{\mathrm{I}}^{\ast}=\frac{{\sum}_{k=1}^{K}{\mathrm{\alpha}}_{k}{p}_{1k}}{{\sum}_{k=1}^{K}{\mathrm{\alpha}}_{k}{p}_{0k}}=\frac{{\sum}_{k=1}^{K}{\mathrm{\alpha}}_{k}{p}_{0k}{\mathrm{\psi}}_{k}}{{\sum}_{k=1}^{K}{\mathrm{\alpha}}_{k}{p}_{0k}},$
${\mathrm{\omega}}_{\mathrm{I}}^{\ast}=\frac{{\sum}_{k=1}^{K}{\mathrm{\alpha}}_{k}{p}_{1k}}{{\sum}_{k=1}^{K}{\mathrm{\alpha}}_{k}}-\frac{{\sum}_{k=1}^{K}{\mathrm{\alpha}}_{k}{p}_{0k}}{{\sum}_{k=1}^{K}{\mathrm{\alpha}}_{k}}=\frac{{\sum}_{k=1}^{K}{\mathrm{\alpha}}_{k}{\mathrm{\omega}}_{k}}{{\sum}_{k=1}^{K}{\mathrm{\alpha}}_{k}},$and variances
${V}^{A}\left({\stackrel{\u02c6}{\mathrm{\psi}}}_{MH}\right)=\frac{{\sum}_{k=1}^{K}{\mathrm{\alpha}}_{k}^{2}{V}^{A}\left[{Q}_{k}\left({\mathrm{\psi}}_{\mathrm{I}}^{\ast}\right)\right]}{{\left\{{\sum}_{k=1}^{K}{\mathrm{\alpha}}_{k}{p}_{0k}\right\}}^{2}},{V}^{A}\left({\stackrel{\u02c6}{\mathrm{\omega}}}_{MH}\right)=\frac{{\sum}_{k=1}^{K}{\mathrm{\alpha}}_{k}^{2}{V}^{A}\left[{R}_{k}\left({\mathrm{\omega}}_{\mathrm{I}}^{\ast}\right)\right]}{{\left\{{\sum}_{k=1}^{K}{\mathrm{\alpha}}_{k}\right\}}^{2}},$

where ${\mathrm{\alpha}}_{k}={\mathrm{\lambda}}_{0k}{\mathrm{\lambda}}_{1k}/\left({\mathrm{\lambda}}_{0k}+{\mathrm{\lambda}}_{1k}\right)$ and ${V}^{A}\left[.\right]$ means asymptotic variance.

An outline of proof is provided in Appendix. Note that ${N}^{-1}{w}_{MH,k}$ is an empirical quantity of ${\mathrm{\alpha}}_{k}$. Intuitively, the quantities ${\mathrm{\psi}}_{\mathrm{I}}^{\ast}$ and ${\mathrm{\omega}}_{\mathrm{I}}^{\ast}$ can be roughly interpreted as expected quantities of standardized risk ratio and risk differences with the standard weight ${\mathrm{\alpha}}_{k}$. Since the Mantel-Haenszel risk ratio and risk difference estimators are expressed as standardized estimators with the weight ${w}_{MH,k}$ (Rothman et al. 2008), this result would be intuitive. However, this standard would not have any intrinsic epidemiologic interests. Greenland and Maldonado (1994) inferred the Mantel-Haenszel rate-ratio estimator converges to a similar quantity of ${\mathrm{\psi}}_{\mathrm{I}}^{\ast}$ (see Section 3.1), and it can be expected to approximate a standardized rate ratio on a standard distribution of the stratified factor in the total cohort, under the exposure has null effect. Similar arguments would be composed for the Mantel-Haenszel risk ratio estimator, namely, the Mantel-Haneszel risk ratio estimator might be interpreted to approximate a standardized risk ratio on a traditional standard weight with ${w}_{{N}_{k}}={N}_{k}/\sum _{i=1}^{K}{N}_{i}$ under null effect. Also, the standards ${w}_{{N}_{k}}$ and ${w}_{MH,k}$ are substantially identical when a product of exposure prevalence ${h}_{k}={N}_{1k}/{N}_{k}$ and its complement $\left(1-{h}_{k}\right)$ is a constant across strata, i. e., ${w}_{MH,k}=\left({N}_{k}\left(1-{h}_{k}\right){N}_{k}{h}_{k}\right)/{N}_{k}={N}_{k}{h}_{k}\left(1-{h}_{k}\right)$ is proportional to ${N}_{k}$ when ${h}_{k}\left(1-{h}_{k}\right)$ is constant. The arguments hold for risk difference, similarly. Under this constancy assumption, ${\mathrm{\psi}}_{\mathrm{I}}^{\ast}$ and ${\mathrm{\omega}}_{\mathrm{I}}^{\ast}$ nearly accord to the standardized risk ratio and risk difference with standards ${\mathrm{\alpha}}_{k}$. Although it does not necessarily require an exact constancy of ${h}_{k}$ (the distribution of exposure prevalence), the concordance of ${w}_{{N}_{k}}$ and ${w}_{MH,k}$ mostly depends on the strength of confounding.

In addition, the Mantel-Haenszel estimators can also be interpreted to converge to weighted averages of stratum specific risk ratios ${\mathrm{\psi}}_{k}$’s or risk differences ${\mathrm{\omega}}_{k}$’s, with the weights ${p}_{0k}{\mathrm{\alpha}}_{k}$ and ${\mathrm{\alpha}}_{k}$. The Mante-Haenszel-type weights ${w}_{MH,k}$ would reflect the precision of the stratum specific effect estimators approximately (in particular, under null effects; Sato 1990), ${\mathrm{\psi}}_{\mathrm{I}}^{\ast}$ and ${\mathrm{\omega}}_{\mathrm{I}}^{\ast}$ might also be interpreted intuitive summaries for the stratum-specific effect measures.

Second, we consider Asymptotic II. The limiting model considered here is similar to those employed by Breslow (1981) and Greenland and Robins (1985). We suppose there is a finite number of possible configurations of total sample sizes $\left({N}_{0k,}{N}_{1k}\right)$ among *K* strata, and we denote the number of it as *L*. Then, the weight ${w}_{MH,k}$ is constant within each configuration, and we denote it as ${a}_{l}\left(l=1,2,\dots L\right)$. Also, we assume there is *G* heterogeneous risk ratios and/or risk differences among *K* strata, i. e., ${\mathrm{\psi}}_{1},{\mathrm{\psi}}_{2},\dots ,{\mathrm{\psi}}_{K}$ and ${\mathrm{\omega}}_{1},{\mathrm{\omega}}_{2},\dots ,{\mathrm{\omega}}_{K}$ are possibly heterogeneous, but these are categorized to *G* subsets that the common effect assumptions hold within each subset. We denote these *G* heterogeneous effect parameters as ${\tilde{\mathrm{\psi}}}_{1},{\tilde{\mathrm{\psi}}}_{2},\dots ,{\tilde{\mathrm{\psi}}}_{G}$ and ${\tilde{\mathrm{\omega}}}_{1},{\tilde{\mathrm{\omega}}}_{2},\dots ,{\tilde{\mathrm{\omega}}}_{G}$. We also assume there is ${K}_{lg}$ strata for *l*th configuration and *g*th subset and ${K}_{l,g}/K$ converges to ${\mathrm{\pi}}_{lg}$, which depends on the distribution of exposure across strata (*l* = 1,2,…,*L*; *g* = 1,2,…,*G*). In addition, the nuisance parameter ${p}_{0k}$ is assumed to be sampled from a certain probability distribution ${F}_{0lg}\left(z\right)$ for the *l*th and *g*th subset. The corresponding ${p}_{1k}$ is specified to be ${\tilde{\mathrm{\psi}}}_{g}{p}_{0k}$ or ${\tilde{\mathrm{\omega}}}_{\mathrm{g}}+{p}_{0k}$ then, and we denote the distribution of ${p}_{1k}$ as ${F}_{1lg}\left(z\right)$ formally, here. Also, we denote the means of ${F}_{0lg}\left(z\right)$ and ${F}_{1lg}\left(z\right)$ as ${\mathrm{\rho}}_{0lg}$ and ${\mathrm{\rho}}_{1lg}$. In addition, we denote the conditional variances of stratum specific statistics ${Q}_{k}\left(\mathrm{\psi}\right),{R}_{k}\left(\mathrm{\omega}\right)$ for the *l*th and *g*th subset as ${V}_{Q,lg}\left(\mathrm{\psi}|z\right),{V}_{R,lg}\left(\mathrm{\omega}|z\right)$ conditioning on ${p}_{0k}=z$. After that, the asymptotic behaviors of the Mantel-Haenszel estimators can be characterized as follows.

Under the Asymptotic II, the Mantel-Haenszel estimators converge to normal distribution with means equal to
${\mathrm{\psi}}_{\mathrm{I}\mathrm{I}}^{\ast}=\frac{{\sum}_{l=1}^{L}{\sum}_{g=1}^{G}{\mathrm{\pi}}_{lg}{a}_{l}{\mathrm{\rho}}_{1lg}}{{\sum}_{l=1}^{L}{\sum}_{g=1}^{G}{\mathrm{\pi}}_{lg}{a}_{l}{\mathrm{\rho}}_{0lg}}=\frac{{\sum}_{l=1}^{L}{\sum}_{g=1}^{G}{\mathrm{\pi}}_{lg}{a}_{l}{\mathrm{\rho}}_{0lg}{\tilde{\mathrm{\psi}}}_{g}}{{\sum}_{l=1}^{L}{\sum}_{g=1}^{G}{\mathrm{\pi}}_{lg}{a}_{l}{\mathrm{\rho}}_{0lg}},$
${\mathrm{\omega}}_{\mathrm{I}\mathrm{I}}^{\ast}=\frac{{\sum}_{l=1}^{L}{\sum}_{g=1}^{G}{\mathrm{\pi}}_{lg}{a}_{l}{\mathrm{\rho}}_{1lg}}{{\sum}_{l=1}^{L}{\sum}_{g=1}^{G}{\mathrm{\pi}}_{lg}{a}_{l}}-\frac{{\sum}_{l=1}^{L}{\sum}_{g=1}^{G}{\mathrm{\pi}}_{lg}{a}_{l}{\mathrm{\rho}}_{0lg}}{{\sum}_{l=1}^{L}{\sum}_{g=1}^{G}{\mathrm{\pi}}_{lg}{a}_{l}}=\frac{{\sum}_{l=1}^{L}{\sum}_{g=1}^{G}{\mathrm{\pi}}_{lg}{a}_{l}{\tilde{\mathrm{\omega}}}_{g}}{{\sum}_{l=1}^{L}{\sum}_{g=1}^{G}{\mathrm{\pi}}_{lg}{a}_{l}},$and variances
${V}^{A}\left({\stackrel{\u02c6}{\mathrm{\psi}}}_{MH}\right)=\frac{{\sum}_{l=1}^{L}{\sum}_{g=1}^{G}{\mathrm{\pi}}_{lg}^{2}{a}_{l}^{2}E\left[{V}_{Q,lg}\left({\mathrm{\psi}}_{\mathrm{I}\mathrm{I}}^{\ast}|z\right)\right]}{{\left\{{\sum}_{l=1}^{L}{\sum}_{g=1}^{G}{\mathrm{\pi}}_{lg}{a}_{l}{\mathrm{\rho}}_{0lg}\right\}}^{2}},$
${V}^{A}\left({\stackrel{\u02c6}{\mathrm{\omega}}}_{MH}\right)=\frac{{\sum}_{l=1}^{L}{\sum}_{g=1}^{G}{\mathrm{\pi}}_{lg}^{2}{a}_{l}^{2}E\left[{V}_{R,lg}\left({\mathrm{\omega}}_{\mathrm{I}\mathrm{I}}^{\ast}|z\right)\right]}{{\left\{{\sum}_{l=1}^{L}{\sum}_{g=1}^{G}{\mathrm{\pi}}_{lg}{a}_{l}\right\}}^{2}}.$

An outline of proof is provided in Appendix II. Similar to Asymptotic I, ${a}_{l}$ just accords to the standard Mantel-Haentszel weight ${w}_{MH,k}$, the quantities ${\mathrm{\psi}}_{\mathrm{I}\mathrm{I}}^{\ast}$ and ${\mathrm{\omega}}_{\mathrm{I}\mathrm{I}}^{\ast}$ can also be interpreted as expected quantities of standardized risk ratio and risk differences with the standard weight ${a}_{l}$’s. Only the nuisance parameters ${p}_{0k}$’s are possibly different across strata, but it is substituted to the means within certain substrata ${\mathrm{\rho}}_{0lg}$’s. Thus, similar to discussed above, the Mantel-Haneszel risk ratio and risk difference estimators might be interpreted as an approximate of standardized risk ratio and risk difference on a traditional standard weight with ${w}_{{N}_{k}}={N}_{k}/\sum _{i=1}^{K}{N}_{i}$. In addition, the Mantel-Haenszel estimators can also be interpreted to converge to weighted averages of heterogeneous risk ratios ${\tilde{\mathrm{\psi}}}_{g}$’s or risk differences ${\tilde{\mathrm{\omega}}}_{g}$’s, with the weights ${\mathrm{\pi}}_{lg}{a}_{l}{\mathrm{\rho}}_{0lg}$ and ${\mathrm{\pi}}_{lg}{a}_{l}$. The Mante-Haenszel-type weights ${w}_{MH,k}$ would reflect the precision of the stratum specific effect estimators approximately (in particular, under null effects; Sato 1990), ${\mathrm{\psi}}_{\mathrm{I}\mathrm{I}}^{\ast}$ and ${\mathrm{\omega}}_{\mathrm{I}\mathrm{I}}^{\ast}$ would also be interpreted intuitive summary for the stratum-specific effect measures.

Another concern is the asymptotic variance estimation of ${\stackrel{\u02c6}{\mathrm{\psi}}}_{MH}$ and ${\stackrel{\u02c6}{\mathrm{\omega}}}_{MH}$ under the heterogeneity assumptions. However, as shown in Theorem 1 and 2, the asymptotic variances have the standard sandwich-type forms. So, using the existing sandwich-type asymptotic variance estimators, these quantities can be validly estimated. For example, the standard dually consistent estimators (under both of Asymptotics I and II) for risk ratio of Greenland and Robins (1985) and for risk difference of Sato (1989) are consistent even if the common effect assumptions are violated. In addition, the asymptotic variances can be consistently estimated using the ordinary bootstrap methods. Empirical performances of the same type estimators are evaluated by simulations in latter section.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.