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# Epidemiologic Methods

### Edited by faculty of the Harvard School of Public Health

Ed. by Tchetgen Tchetgen, Eric J / VanderWeele, Tyler J. / Daniel, Rhian

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Volume 5, Issue 1

# A Note on the Mantel-Haenszel Estimators When the Common Effect Assumptions Are Violated

Hisashi Noma
/ Kengo Nagashima
Published Online: 2016-05-06 | DOI: https://doi.org/10.1515/em-2015-0004

## Abstract

The Mantel-Haenszel estimators for the common effect parameters of stratified 2×2 tables have been widely adopted in epidemiological and clinical studies for controlling the effects of confounding factors. Although the Mantel-Haenszel estimators are simple and effective estimating methods, the correctness of the common effect assumptions cannot be justified in general practices. Also then, the targeted “common effect parameters” do not exist. Under these settings, even if the Mantel-Haenszel estimators have desirable properties, it is quite uncertain what they estimate and how the estimates are interpreted. In this article, we conducted theoretical evaluations for their asymptotic behaviors when the common effect assumptions are violated. We explicitly showed that the Mantel-Haenszel estimators converge to weighted averages of stratum-specific effect parameters and they can be interpreted as intuitive summaries of the stratum-specific effect measures. Also, the Mantel-Haenszel estimators correspond to the standardized effect measures on standard distributions of stratification variables to be the total cohort, approximately. In addition, the ordinary sandwich-type variance estimators are still valid for quantifying variabilities of the Mantel-Haenszel estimators. We implemented numerical studies based on two epidemiologic studies of breast cancer and schizophrenia for evaluating empirical properties of these estimators, and confirmed general validities of these theoretical results.

## 1 Introduction

In the analysis of epidemiologic and clinical studies, the Mantel-Haenszel estimators (Mantel and Haenszel 1959; Rothman, Greenland and Lash 2008) for the common effect parameters of stratified 2×2 tables have been widely adopted for controlling the effects of confounding factors. Due to their simplicity and highly efficiency, these estimators are preferred by epidemiologists and have also been one of the standard methods in meta-analysis (Higgins and Green 2008). Although the Mantel-Haenszel estimators are effective estimating methods for the common effect parameters, the common effect assumptions cannot be justified rigorously, in practice (Greenland 1982; Mantel et al. 1977). When the common effect assumptions are violated, the targeted parameters estimated by the Mantel-Haenszel methods are quite uncertain and it is not clear what they estimate. Greenland and Maldonado (1994) inferred that the Mantel-Haenszel rate ratio estimator is approximated by the standardized rate ratio on a standard distribution of stratification variables to be the total cohort. They also showed its general correctness through numerical studies, although there were not sufficient theoretical justifications.

The violation of the common effect assumptions can be regarded as one of model misspecification problems. In theoretical studies, the model misspecification problems have been widely researched mainly for the maximum likelihood estimators based on the landmark paper of White (1982). Although its generalization to the estimating equation theory (Godambe 1969) has not been found until recent studies, Yi and Reid (2010) provided generalized results of White (1982)’s asymptotic results for the behavior of maximum likelihood estimators under misspecified models. Since it has been well known that the Mantel-Haenszel estimators can be regarded as local efficient estimators for the common effect parameters under null effects (the exposure effects are zero) through the estimating equation theory (Fujii and Yanagimoto 2005; Sato 1990; Yanagimoto 1990), the asymptotic behaviors can be assessed using the Yi and Reid (2010)’s results.

In this article, we evaluate asymptotic behaviors of the Mantel-Haenszel estimators when the common effect assumptions are violated. We show the Mantel-Haenszel estimators can be approximately interpreted as estimators for average exposure effect under the heterogeneity of effects across strata. We would show that the average effects are generally viewed as good approximations to the standardized estimators under certain conditions. In addition, we would discuss validities of ordinary variance estimators of the Mantel-Haenszel estimators under the heterogeneous settings. We also assess their empirical properties through numerical studies based on two epidemiologic studies of breast cancer and schizophrenia.

## 2.1 Mantel-Haenszel risk ratio and risk difference estimators

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 ${\stackrel{ˆ}{\mathrm{\psi }}}_{MH}=\frac{{\sum }_{k=1}^{K}{X}_{1k}{N}_{0k}/{N}_{k}}{{\sum }_{k=1}^{K}{X}_{0k}{N}_{1k}/{N}_{k}}=\frac{{\sum }_{k=1}^{K}{w}_{MH,k}{\stackrel{ˆ}{p}}_{1k}}{{\sum }_{k=1}^{K}{w}_{MH,k}{\stackrel{ˆ}{p}}_{0k}}=\frac{{\sum }_{k=1}^{K}{w}_{MH,k}{\stackrel{ˆ}{p}}_{0k}{\stackrel{ˆ}{\mathrm{\psi }}}_{k}}{{\sum }_{k=1}^{K}{w}_{MH,k}{\stackrel{ˆ}{p}}_{0k}},$ ${\stackrel{ˆ}{\mathrm{\omega }}}_{MH}=\frac{{\sum }_{k=1}^{K}\left({X}_{1k}{N}_{0k}-{X}_{0k}{N}_{1k}\right)/{N}_{k}}{{\sum }_{k=1}^{K}{N}_{0k}{N}_{1k}/{N}_{k}}=\frac{{\sum }_{k=1}^{K}{w}_{MH,k}{\stackrel{ˆ}{p}}_{1k}}{{\sum }_{k=1}^{K}{w}_{MH,k}}-\frac{{\sum }_{k=1}^{K}{w}_{MH,k}{\stackrel{ˆ}{p}}_{0k}}{{\sum }_{k=1}^{K}{w}_{MH,k}}=\frac{{\sum }_{k=1}^{K}{w}_{MH,k}{\stackrel{ˆ}{\mathrm{\omega }}}_{k}}{{\sum }_{k=1}^{K}{w}_{MH,k}},$where ${\stackrel{ˆ}{p}}_{0k}={X}_{0k}/{N}_{0k},{\stackrel{ˆ}{p}}_{1k}={X}_{1k}/{N}_{1k},{N}_{k}={N}_{1k}+{N}_{0k}$ and ${w}_{MH,k}=\left({N}_{0k}{N}_{1k}/{N}_{k}\right)$. Also, ${\stackrel{ˆ}{\mathrm{\psi }}}_{k}={\stackrel{ˆ}{p}}_{1k}/{\stackrel{ˆ}{p}}_{0k}$ and ${\stackrel{ˆ}{\mathrm{\omega }}}_{k}={\stackrel{ˆ}{p}}_{1k}-{\stackrel{ˆ}{p}}_{0k}$. These estimators are obtained as solutions of the following estimation equations, $Q\left(\mathrm{\psi }\right)=\sum _{k=1}^{K}{w}_{MH,k}\left[{\stackrel{ˆ}{p}}_{1k}-\mathrm{\psi }{\stackrel{ˆ}{p}}_{0k}\right]=0,$ $R\left(\mathrm{\omega }\right)=\sum _{k=1}^{K}{w}_{MH,k}\left[\mathrm{\omega }-\left({\stackrel{ˆ}{p}}_{1k}-{\stackrel{ˆ}{p}}_{0k}\right)\right]=0.$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 ${Q}_{k}\left(\mathrm{\psi }\right)={\stackrel{ˆ}{p}}_{1k}-\mathrm{\psi }{\stackrel{ˆ}{p}}_{0k},{R}_{k}\left(\mathrm{\omega }\right)=\mathrm{\omega }-\left({\stackrel{ˆ}{p}}_{1k}-{\stackrel{ˆ}{p}}_{0k}\right)$, 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 ${V}^{A}\left({\stackrel{ˆ}{\mathrm{\psi }}}_{MH}\right)=\frac{{\sum }_{k=1}^{K}{\mathrm{\alpha }}_{k}^{2}{V}^{A}\left[{Q}_{k}\left({\mathrm{\psi }}_{\mathrm{I}}^{\ast }\right)\right]}{{\left\{{\sum }_{k=1}^{K}{\mathrm{\alpha }}_{k}{p}_{0k}\right\}}^{2}},{V}^{A}\left({\stackrel{ˆ}{\mathrm{\omega }}}_{MH}\right)=\frac{{\sum }_{k=1}^{K}{\mathrm{\alpha }}_{k}^{2}{V}^{A}\left[{R}_{k}\left({\mathrm{\omega }}_{\mathrm{I}}^{\ast }\right)\right]}{{\left\{{\sum }_{k=1}^{K}{\mathrm{\alpha }}_{k}\right\}}^{2}},$

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 ${\stackrel{˜}{\mathrm{\psi }}}_{1},{\stackrel{˜}{\mathrm{\psi }}}_{2},\dots ,{\stackrel{˜}{\mathrm{\psi }}}_{G}$ and ${\stackrel{˜}{\mathrm{\omega }}}_{1},{\stackrel{˜}{\mathrm{\omega }}}_{2},\dots ,{\stackrel{˜}{\mathrm{\omega }}}_{G}$. We also assume there is ${K}_{lg}$ strata for lth configuration and gth 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 lth and gth subset. The corresponding ${p}_{1k}$ is specified to be ${\stackrel{˜}{\mathrm{\psi }}}_{g}{p}_{0k}$ or ${\stackrel{˜}{\mathrm{\omega }}}_{\mathrm{g}}+{p}_{0k}$ 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 lth and gth 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 ${\mathrm{\psi }}_{\mathrm{I}\mathrm{I}}^{\ast }=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}{\mathrm{\rho }}_{1lg}}{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}{\mathrm{\rho }}_{0lg}}=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}{\mathrm{\rho }}_{0lg}{\stackrel{˜}{\mathrm{\psi }}}_{g}}{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}{\mathrm{\rho }}_{0lg}},$ ${\mathrm{\omega }}_{\mathrm{I}\mathrm{I}}^{\ast }=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}{\mathrm{\rho }}_{1lg}}{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}}-\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}{\mathrm{\rho }}_{0lg}}{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}}=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}{\stackrel{˜}{\mathrm{\omega }}}_{g}}{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}},$and variances ${V}^{A}\left({\stackrel{ˆ}{\mathrm{\psi }}}_{MH}\right)=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}^{2}{a}_{l}^{2}E\left[{V}_{Q,lg}\left({\mathrm{\psi }}_{\mathrm{I}\mathrm{I}}^{\ast }|z\right)\right]}{{\left\{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}{\mathrm{\rho }}_{0lg}\right\}}^{2}},$ ${V}^{A}\left({\stackrel{ˆ}{\mathrm{\omega }}}_{MH}\right)=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}^{2}{a}_{l}^{2}E\left[{V}_{R,lg}\left({\mathrm{\omega }}_{\mathrm{I}\mathrm{I}}^{\ast }|z\right)\right]}{{\left\{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}\right\}}^{2}}.$

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 ${\stackrel{˜}{\mathrm{\psi }}}_{g}$’s or risk differences ${\stackrel{˜}{\mathrm{\omega }}}_{g}$’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 ${\stackrel{ˆ}{\mathrm{\psi }}}_{MH}$ and ${\stackrel{ˆ}{\mathrm{\omega }}}_{MH}$ 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.

## 2.2 Illustration: Tamoxifen use and recurrence of breast cancer

Table 1 presents parts of the results of a cohort study to assess the risk of second primary cancers after adjuvant tamoxifen therapy for breast cancer (Matsuyama et al. 2000; Sato and Matsuyama 2003). Nearly null effect of tamoxifen was observed for the unstratified analysis (crude risk ratio: 1.011, crude risk difference: 0.002). However, stratifying by lymph node metastasis at surgery, possible preventive effects were observed in each stratum (the stratum-specific risk ratios: 0.910 and 0.670, the risk differences: −0.030 and −0.035). Although there would be hardly effect modification for the risk differences, that for the risk ratios would exist. We suppose the heterogeneous setting under Asymptotic I. The Mantel-Haenszel risk ratio estimator was 0.830 and the Mantel-Haenszel risk difference estimator was −0.033. Besides, the standardized risk ratio and risk difference with standards ${N}_{k}$ are 0.832 and −0.033, respectively. As usual analysis of epidemiologic studies, although it cannot be confirmed the common effect assumption rigorously, the Mantel-Haenszel estimates and the standardized estimates were approximately identical.

In addition, we conducted simulation studies for investigating empirical properties of the Mantel-Haenszel estimators under heterogeneity. We consider several scenarios based on the stratified dataset of Table 1, such as ${N}_{11}=1215,{N}_{01}=760,{N}_{12}=1334,$ ${N}_{02}=1592$, and ${p}_{01}=253/760=0.333,{p}_{12}=96/1334=0.072,{p}_{02}=171/1592$ $=0.107$. In this case, the risk ratio and the risk difference of the 2nd stratum are ${p}_{12}/{p}_{02}=0.670$ (RR2) and ${p}_{12}-{p}_{02}=-0.035$ (RD2). Denoting the risk ratio and the risk difference of the 1st stratum as RR1 and RD1, we define heterogeneity factors ${\mathrm{\zeta }}_{RR}=\mathrm{R}\mathrm{R}1/\mathrm{R}\mathrm{R}2$ and ${\mathrm{\zeta }}_{RD}=\mathrm{R}\mathrm{D}1-\mathrm{R}\mathrm{D}2$. Varying ${\mathrm{\zeta }}_{RR}$ and ${\mathrm{\zeta }}_{RD}$ by 0.001, we simulated 25,000 dataset and compared empirical means of the Mantel-Haenszel estimators and the standardized estimators with standards ${N}_{k}$. Figure 1 presents the mean of 25,000 estimates for each scenario of the Mantel-Haenszel estimates (blue lines), the standardized estimates (black lines). We also plot ${\mathrm{\psi }}^{\ast }$ and ${\mathrm{\omega }}^{\ast }$ for each scenario (gray lines).

As the results, in the all scenarios, means of the distributions of the Mantel-Haenszel estimates mostly accord to the asymptotic mean of the distributions of the Mantel-Haenszel estimators ${\mathrm{\psi }}_{\mathrm{I}}^{\ast }$ and ${\mathrm{\omega }}_{\mathrm{I}}^{\ast }$. Following these results, the large sample results in the previous section are considered to be valid in the comprehensive finite sample settings. Besides, as expected, in the common effect settings (${\mathrm{\zeta }}_{RR}=1$ and ${\mathrm{\zeta }}_{RD}=0$), means of the Mantel-Haenszel estimates and those of the standardized estimates were accurately identical. Furthermore, under heterogeneity of effects across strata, means of the Mantel-Haenszel estimates and those of the standardized estimators were nearly identical even under large heterogeneity exists (such like ${\mathrm{\zeta }}_{RR}\ge 2.0$ and ${\mathrm{\zeta }}_{RD}\ge 0.15$). These trends were also observed when the qualitative interaction existed (RR1 > 1, RD1 > 0). These results would be somewhat unexpected, but might indicate that the Mantel-Haenszel estimators can be interpreted as approximates of the average exposure effects under heterogeneity, in particular as the standardized estimators.

Table 1:

Results of a cohort study for evaluating the risk of second primary cancers after adjuvant tamoxifen therapy for breast cancer (Matsuyama et al. 2000; Sato and Matsuyama 2003).

Figure 1:

Results of simulations: Means of 25,000 estimates of the Mantel-Haenszel estimates (${\stackrel{ˆ}{\mathrm{\psi }}}_{MH}$ and ${\stackrel{ˆ}{\mathrm{\omega }}}_{MH}$), the standardized estimates (${\stackrel{ˆ}{\mathrm{\psi }}}_{{N}_{k}}$ and ${\stackrel{ˆ}{\mathrm{\omega }}}_{{N}_{k}}$), and the quantities that the Mantel-Haenszel estimators converge to (${\mathrm{\psi }}_{\mathrm{I}}^{\ast }$ and ${\mathrm{\omega }}_{\mathrm{I}}^{\ast }$).

## 3.1 Mantel-Haenszel rate ratio and rate difference estimators

We consider estimating the common rate ratio and rate difference for stratified person-time data of cohort studies. Suppose a series of K strata constructed by independent Poisson observations $\left({Y}_{0k},{Y}_{1k}\right)$ with fixed person-time denominators $\left({T}_{0k,}{T}_{1k}\right)$ and means $\left({T}_{0k}{r}_{0k},{T}_{1k}{r}_{1k}\right)$ for $k=1,2,\dots ,K$. Also, $\left({r}_{0k},{r}_{1k}\right)$ are the instantaneous incidence rates of the exposed and the unexposed populations. Under the common effect assumption of the rate ratio, we assume the stratum-specific rate ratio ${\mathrm{\varphi }}_{k}={r}_{1k}/{r}_{0k}$ are all equal across the strata, i. e., $\mathrm{\varphi }={\mathrm{\varphi }}_{1}={\mathrm{\varphi }}_{2}=\dots ={\mathrm{\varphi }}_{K}$. In addition, for the rate difference, we assume the common effect for the stratum specific rate difference ${\mathrm{\xi }}_{k}={r}_{1k}-{r}_{0k}$ across the strata, i. e., $\mathrm{\xi }={\mathrm{\xi }}_{1}={\mathrm{\xi }}_{2}=\dots ={\mathrm{\xi }}_{K}$. The Mantel-Haenszel estimators of the common rate ratio $\mathrm{\varphi }$ (Greenland and Robins 1985; Walker 1985) and the common risk difference $\mathrm{\xi }$ (Greenland and Robins 1985) are given as ${\stackrel{ˆ}{\mathrm{\varphi }}}_{MH}=\frac{{\sum }_{k=1}^{K}{Y}_{1k}{T}_{0k}/{T}_{k}}{{\sum }_{k=1}^{K}{Y}_{0k}{T}_{1k}/{T}_{k}}=\frac{{\sum }_{k=1}^{K}{u}_{MH,k}{\stackrel{ˆ}{r}}_{1k}}{{\sum }_{k=1}^{K}{u}_{MH,k}{\stackrel{ˆ}{r}}_{0k}}=\frac{{\sum }_{k=1}^{K}{u}_{MH,k}{\stackrel{ˆ}{r}}_{0k}{\stackrel{ˆ}{r}}_{1k}}{{\sum }_{k=1}^{K}{u}_{MH,k}{\stackrel{ˆ}{r}}_{0k}},$ ${\stackrel{ˆ}{\mathrm{\xi }}}_{MH}=\frac{{\sum }_{k=1}^{K}\left({Y}_{1k}{T}_{0k}-{Y}_{0k}{T}_{1k}\right)/{T}_{k}}{{\sum }_{k=1}^{K}{T}_{0k}{T}_{1k}/{T}_{k}}=\frac{{\sum }_{k=1}^{K}{u}_{MH,k}{\stackrel{ˆ}{r}}_{1k}}{{\sum }_{k=1}^{K}{u}_{MH,k}}-\frac{{\sum }_{k=1}^{K}{u}_{MH,k}{\stackrel{ˆ}{r}}_{0k}}{{\sum }_{k=1}^{K}{u}_{MH,k}}=\frac{{\sum }_{k=1}^{K}{u}_{MH,k}{\stackrel{ˆ}{\mathrm{\xi }}}_{k}}{{\sum }_{k=1}^{K}{u}_{MH,k}},$where ${\stackrel{ˆ}{r}}_{0k}={Y}_{0k}/{T}_{0k},{\stackrel{ˆ}{r}}_{1k}={Y}_{1k}/{T}_{1k},{T}_{k}={T}_{1k}+{T}_{0k}$ and ${u}_{MH,k}=\left({T}_{0k}{T}_{1k}/{T}_{k}\right)$. Also, ${\stackrel{ˆ}{\mathrm{\varphi }}}_{k}={\stackrel{ˆ}{r}}_{1k}/{\stackrel{ˆ}{r}}_{0k}$ and ${\stackrel{ˆ}{\mathrm{\xi }}}_{k}={\stackrel{ˆ}{r}}_{1k}-{\stackrel{ˆ}{r}}_{0k}$. The Mantel-Haenszel estimators are obtained as the solution of the estimation equations, $H\left(\mathrm{\varphi }\right)=\sum _{k=1}^{K}{u}_{MH,k}\left[{\stackrel{ˆ}{r}}_{1k}-\mathrm{\varphi }{\stackrel{ˆ}{r}}_{0k}\right]=0,$ $U\left(\mathrm{\xi }\right)=\sum _{k=1}^{K}{u}_{MH,k}\left[\left({\stackrel{ˆ}{r}}_{1k}-{\stackrel{ˆ}{r}}_{0k}\right)-\mathrm{\xi }\right]=0.$It should be noted that these estimating functions are unbiased under the common effects assumptions, and thus, consistency of the estimators follow.

Here, we consider similar limiting models the previous section. The large-strata limiting model, Asymptotic I, is to have a fixed number of strata K while ${T}_{0k,}{T}_{1k}\to \mathrm{\infty }$. Also, for the sparse data limiting model, denoted as Asymptotic II, $\left({T}_{0k,}{T}_{1k}\right)$ are bounded while $K$ is large. We also denote ${H}_{k}\left(\mathrm{\psi }\right)={\stackrel{ˆ}{r}}_{1k}-\mathrm{\varphi }{\stackrel{ˆ}{r}}_{0k},{U}_{k}\left(\mathrm{\omega }\right)=\left({\stackrel{ˆ}{r}}_{1k}-{\stackrel{ˆ}{r}}_{0k}\right)-\mathrm{\xi }$, here. The asymptotic behaviors of Mantel-Haenszel estimators can be characterized as follows.

Under the Asymptotic I, we assume ${T}_{0k}/T\to {\mathrm{\nu }}_{0k}>0$, ${T}_{1k}/T\to {\mathrm{\nu }}_{1k}>0$, where $T={T}_{01}+\dots +{T}_{0K}+{T}_{11}+\dots +{T}_{1K}$. The Mantel-Haenszel estimators converge to normal distribution with means equal to ${\mathrm{\varphi }}_{\mathrm{I}}^{\ast }=\frac{{\sum }_{k=1}^{K}{\mathrm{\gamma }}_{k}{r}_{1k}}{{\sum }_{k=1}^{K}{\mathrm{\gamma }}_{k}{r}_{0k}}=\frac{{\sum }_{k=1}^{K}{\mathrm{\gamma }}_{k}{r}_{0k}{\mathrm{\varphi }}_{k}}{{\sum }_{k=1}^{K}{\mathrm{\gamma }}_{k}{r}_{0k}},$ ${\mathrm{\xi }}_{\mathrm{I}}^{\ast }=\frac{{\sum }_{k=1}^{K}{\mathrm{\gamma }}_{k}{r}_{1k}}{{\sum }_{k=1}^{K}{\mathrm{\gamma }}_{k}}-\frac{{\sum }_{k=1}^{K}{\mathrm{\gamma }}_{k}{r}_{0k}}{{\sum }_{k=1}^{K}{\mathrm{\gamma }}_{k}}=\frac{{\sum }_{k=1}^{K}{\mathrm{\gamma }}_{k}{\mathrm{\xi }}_{k}}{{\sum }_{k=1}^{K}{\mathrm{\gamma }}_{k}},$and variances ${V}^{A}\left({\stackrel{ˆ}{\mathrm{\varphi }}}_{MH}\right)=\frac{{\sum }_{k=1}^{K}{\mathrm{\gamma }}_{k}^{2}{V}^{A}\left[{H}_{k}\left({\mathrm{\varphi }}_{\mathrm{I}}^{\ast }\right)\right]}{{\left\{{\sum }_{k=1}^{K}{\mathrm{\gamma }}_{k}{r}_{0k}\right\}}^{2}},{V}^{A}\left({\stackrel{ˆ}{\mathrm{\xi }}}_{MH}\right)=\frac{{\sum }_{k=1}^{K}{\mathrm{\gamma }}_{k}^{2}{V}^{A}\left[{U}_{k}\left({\mathrm{\xi }}_{\mathrm{I}}^{\ast }\right)\right]}{{\left\{{\sum }_{k=1}^{K}{\mathrm{\gamma }}_{k}\right\}}^{2}},$

where ${\mathrm{\gamma }}_{k}={\mathrm{\nu }}_{0k}{\mathrm{\nu }}_{1k}/\left({\mathrm{\nu }}_{0k}+{\mathrm{\nu }}_{1k}\right)$.

Also, under Asymptotic II, we suppose there is a finite number of possible configurations of total sample sizes $\left({T}_{0k,}{T}_{1k}\right)$ among K strata, and we denote the number of it as L. Then, the weight ${u}_{MH,k}$ is formally constant within each configuration, and we denote it as ${s}_{l}\left(l=1,2,\dots L\right)$. Also, we assume there is G heterogeneous rate ratios and/or rate differences among K strata, i. e., ${\mathrm{\varphi }}_{1},{\mathrm{\varphi }}_{2},\dots ,{\mathrm{\varphi }}_{K}$ and ${\mathrm{\xi }}_{1},{\mathrm{\xi }}_{2},\dots ,{\mathrm{\xi }}_{K}$ 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 ${\stackrel{˜}{\mathrm{\varphi }}}_{1},{\stackrel{˜}{\mathrm{\varphi }}}_{2},\dots ,{\stackrel{˜}{\mathrm{\varphi }}}_{G}$ and ${\stackrel{˜}{\mathrm{\xi }}}_{1},{\stackrel{˜}{\mathrm{\xi }}}_{2},\dots ,{\stackrel{˜}{\mathrm{\xi }}}_{G}$. We also assume there is ${K}_{lg}$ strata for lth configuration and gth 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 ${r}_{0k}$ is assumed to be sampled from a certain probability distribution ${F}_{0lg}\left(z\right)$ for the lth and gth subset. The corresponding ${r}_{1k}$ is specified to ${\stackrel{˜}{\mathrm{\varphi }}}_{g}{r}_{0k}$ or ${\stackrel{˜}{\mathrm{\xi }}}_{\mathrm{g}}+{r}_{0k}$, then. We denote the distribution of ${r}_{1k}$ 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 ${\mathrm{\tau }}_{0lg}$ and ${\mathrm{\tau }}_{1lg}$, here. In addition, we denote the conditional variances of stratum specific statistics ${H}_{k}\left(\mathrm{\varphi }\right),{U}_{k}\left(\mathrm{\tau }\right)$ for the lth and gth subset as ${V}_{H,lg}\left(\mathrm{\varphi }|z\right),{V}_{U,lg}\left(\mathrm{\xi }|z\right)$ under ${r}_{0k}=z$. After that, the asymptotic behavior 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 ${\mathrm{\varphi }}_{\mathrm{I}\mathrm{I}}^{\ast }=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{s}_{l}{\mathrm{\tau }}_{1lg}}{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{s}_{l}{\mathrm{\tau }}_{0lg}}=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{s}_{l}{\mathrm{\tau }}_{0lg}{\stackrel{˜}{\mathrm{\varphi }}}_{g}}{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{s}_{l}{\mathrm{\tau }}_{0lg}},$ ${\mathrm{\xi }}_{\mathrm{I}\mathrm{I}}^{\ast }=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{s}_{l}{\mathrm{\tau }}_{1lg}}{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{s}_{l}}-\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{s}_{l}{\mathrm{\tau }}_{0lg}}{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{s}_{l}}=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{s}_{l}{\stackrel{˜}{\mathrm{\xi }}}_{g}}{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{s}_{l}},$and variances ${V}^{A}\left({\stackrel{ˆ}{\mathrm{\varphi }}}_{MH}\right)=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}^{2}{s}_{l}^{2}E\left[{V}_{H,lg}\left({\mathrm{\varphi }}_{\mathrm{I}\mathrm{I}}^{\ast }|z\right)\right]}{{\left\{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{s}_{l}{\mathrm{\tau }}_{0lg}\right\}}^{2}},$ ${V}^{A}\left({\stackrel{ˆ}{\mathrm{\xi }}}_{MH}\right)=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}^{2}{s}_{l}^{2}E\left[{V}_{U,lg}\left({\mathrm{\xi }}_{\mathrm{I}\mathrm{I}}^{\ast }|z\right)\right]}{{\left\{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{s}_{l}\right\}}^{2}}.$

These results can be obtained as the same way with Theorem 1 and 2 (see Appendix). Therefore, similarly to the binomial cases in Section 2, when the common effect assumptions are violated, these quantities can be interpreted as expected quantities of standardized rate ratio and rate differences with the standard weight ${\mathrm{\gamma }}_{k}$ or ${u}_{MH,k}$. This standard would also not have any intrinsic epidemiologic interests. Although Greenland and Maldonado (1994) inferred the Mantel-Haenszel rate-ratio estimator converges to similar quantities, it could be theoretically justified, here. As Greenland and Maldonado (1994) discussed, 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. Besides, as like the discussion in Section 2, under Asymptotic I, the standards ${u}_{{T}_{k}}={T}_{k}/\sum _{i=1}^{K}{T}_{i}$ and ${u}_{MH,k}$ are substantially identical when a product of exposure prevalence ${f}_{k}={T}_{1k}/{T}_{k}$ and its complement $\left(1-{f}_{k}\right)$ is a constant across strata, i. e., ${u}_{MH,k}=\left({T}_{k}\left(1-{f}_{k}\right){T}_{k}{f}_{k}\right)/{T}_{k}={T}_{k}{f}_{k}\left(1-{f}_{k}\right)\approx {T}_{k}$ when ${f}_{k}\left(1-{f}_{k}\right)$ is constant. Under this constancy assumption, ${\mathrm{\psi }}_{\mathrm{I}}^{\ast }$ and ${\mathrm{\omega }}_{\mathrm{I}}^{\ast }$ accord to the standardized rate ratio and rate difference with standard ${u}_{N,k}$. Furthermore, the Mantel-Haenszel estimators can also be interpreted to converge to weighted averages of stratum specific rate ratios ${\mathrm{\varphi }}_{k}$’s or rate differences ${\mathrm{\xi }}_{k}$’s, with the weights ${\mathrm{\gamma }}_{k}{r}_{0k}$ and ${\mathrm{\gamma }}_{k}$. ${\mathrm{\varphi }}_{\mathrm{I}}^{\ast }$ and ${\mathrm{\xi }}_{\mathrm{I}}^{\ast }$ would also be interpreted intuitive summary for the stratum-specific effect measures. ${\mathrm{\varphi }}_{\mathrm{I}\mathrm{I}}^{\ast }$ and ${\mathrm{\xi }}_{\mathrm{I}\mathrm{I}}^{\ast }$ are also be similarly interpreted with the binomial case of Section 2. The asymptotic variances can be estimated by the ordinary sandwich variance-type estimators such those proposed by Greenland and Robins (1985), also through the bootstrap methods.

## 3.2 Illustration: Mortality rates for clozapine users

Table 2 present a result of a study of mortality rates among current users and past users of clozapine that was used to treat schizophrenia (Rothman 2002; Walker et al. 1997). Clozapine uses were thought to be associated to mortality for current users, therefore the past users were used as their controls. Stratifying by two age groups (10–54 years old, and 55–95 years old), although possible protective effects were observed in both strata (the stratum-specific rate ratios: 0.448 and 0.486, the rate differences: −388.7 and −2903 per 105 person-years). In this study, there would be hardly effect modification for the rate ratios, although a certain effect modification would exist for the rate difference. We also consider the large-strata limiting model, here. The Mantel-Haenszel rate ratio estimator was 0.469 and the Mantel-Haenszel risk difference estimator was −710.7 per 105 person-years. Besides, the standardized rate ratio and rate difference with ${T}_{k}$ are 0.466 and −633.2 per 105 person-years, respectively. As for the breast cancer example in Section 2, in this case, the Mantel-Haenszel estimates and the standardized estimates are approximately identical.

Here, we also conducted simulation experiments for investigating empirical properties of the Mantel-Haenszel estimators under heterogeneity. We consider several scenarios based on the stratified dataset of Table 2, such as ${T}_{11}=62119,{T}_{01}=15763,{T}_{12}=2744,$ ${T}_{02}=2780$, and ${r}_{11}=196/62119=315.5×{10}^{5},{r}_{01}=111/15763=704.2×{10}^{5},{r}_{02}=157/2780=5647×{10}^{5}$. In this case, the rate ratio and the rate difference of the 1st stratum are ${r}_{11}/{r}_{01}=0.448$ (IRR1) and ${r}_{11}-{r}_{01}=-388.7×{10}^{5}$ (IRD1). Denoting the rate ratio and the rate difference of the 2nd stratum as IRR2 and IRD2, we define heterogeneity factors ${\mathrm{\zeta }}_{IRR}=\mathrm{I}\mathrm{R}\mathrm{R}2/\mathrm{I}\mathrm{R}\mathrm{R}1$ and ${\mathrm{\zeta }}_{IRD}=\mathrm{I}\mathrm{R}\mathrm{D}2-\mathrm{I}\mathrm{R}\mathrm{D}1$. Varying ${\mathrm{\zeta }}_{IRR}$ and ${\mathrm{\zeta }}_{IRD}$ by 0.0001, we simulated 25,000 dataset and compared the Mantel-Haenszel estimators and the standardized estimators with standards ${T}_{k}$. Figure 2 presents the mean of 25,000 estimates for each scenario of the Mantel-Haenszel estimates (blue lines), the standardized estimates (black lines). We also show ${\mathrm{\varphi }}_{\mathrm{I}}^{\ast }$ and ${\mathrm{\xi }}_{\mathrm{I}}^{\ast }$ for each scenario (gray lines).

In the all settings, means of the distributions of the Mantel-Haenszel estimates mostly accord to the asymptotic mean of the distributions of the Mantel-Haenszel estimators ${\mathrm{\varphi }}^{\ast }$ and ${\mathrm{\xi }}^{\ast }$. Also, as expected, in the settings around ${\mathrm{\zeta }}_{IRR}\approx 1,{\mathrm{\zeta }}_{IRD}\approx 0$ (nearly under the common effect assumptions), means of the Mantel-Haenszel estimates and those of the standardized estimates were approximately identical. However, when the heterogeneity between the strata got to large, these two estimators got to have certain differences. In these settings, the proportions of total person-times of current clozapine users are not substantially identical, and ${f}_{1}\left(1-{f}_{1}\right)=0.161,{f}_{2}\left(1-{f}_{2}\right)=0.215$, thus, these discrepancies might be anticipated. Besides, these discrepancies were not so large, for instance, when ${\mathrm{\zeta }}_{IRR}=1.5$, the mean of ${\stackrel{ˆ}{\mathrm{\varphi }}}_{MH}$ was 0.574 and that of ${\stackrel{ˆ}{\mathrm{\varphi }}}_{{T}_{k}}$ was 0.557. Also, when ${\mathrm{\zeta }}_{IRD}=2000×{10}^{-5}$, the mean of ${\stackrel{ˆ}{\mathrm{\xi }}}_{MH}$ was $125×{10}^{-5}$ and that of ${\stackrel{ˆ}{\mathrm{\xi }}}_{{T}_{k}}$ was $184×{10}^{-5}$. These results might indicate that the Mantel-Haenszel estimators cannot be necessarily interpreted as approximate of the average exposure effect under heterogeneity, but their discordance were not seriously large. In any cases, the uses of the Mantel-Haenszel estimators would not be recommended under strong heterogeneity across strata. However, under small heterogeneity, they can be interpreted as an approximate of the average exposure effect, i. e., the standardized estimators with standards ${T}_{k}$.

Table 2:

Results of a cohort study: Mortality rates for current and past clozapine users (Walker et al. 1997); Data from Rothman (2002, p. 154).

Figure 2:

Results of simulations: Means of 25,000 estimates of the Mantel-Haenszel estimates (${\stackrel{ˆ}{\mathrm{\varphi }}}_{MH}$ and ${\stackrel{ˆ}{\mathrm{\xi }}}_{MH}$), the standardized estimates (${\stackrel{ˆ}{\mathrm{\varphi }}}_{{T}_{k}}$ and ${\stackrel{ˆ}{\mathrm{\xi }}}_{{T}_{k}}$), and the quantities that the Mantel-Haenszel estimators converge to (${\mathrm{\varphi }}_{\mathrm{I}}^{\ast }$ and ${\mathrm{\xi }}_{\mathrm{I}}^{\ast }$).

## 4.1 Mantel-Haenszel odds ratio estimator

We discuss the common odds ratio estimation for stratified analyses in case-control studies. Consider the same setting with Section 2, 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 $k=1,2,\dots ,K$. In case-control studies, ${N}_{0k}$ and ${N}_{1k}$ correspond to the total numbers of cases and controls, and ${X}_{0k}$ and ${X}_{1k}$ are the numbers of exposed cases and controls. This setting can be adapted commonly for cohort and case-control studies through exchanging rows and columns, thus we here adopted the same notation. Under the common effect assumption, we assume the stratum-specific odds ratio ${\mathrm{\theta }}_{k}={p}_{1k}\left(1-{p}_{0k}\right)/\left\{{p}_{0k}\left(1-{p}_{1k}\right)\right\}$ are all equal across the strata, i. e., $\mathrm{\theta }={\mathrm{\theta }}_{1}={\mathrm{\theta }}_{2}=\dots ={\mathrm{\theta }}_{K}$. The Mantel-Haenszel estimator of the common odds ratio $\mathrm{\theta }$ (Mantel and Haenszel 1959) is presented as ${\stackrel{ˆ}{\mathrm{\theta }}}_{MH}=\frac{{\sum }_{k=1}^{K}{X}_{1k}\left({N}_{0k}-{X}_{0k}\right)/{N}_{k}}{{\sum }_{k=1}^{K}{X}_{0k}\left({N}_{1k}-{X}_{1k}\right)/{N}_{k}}=\frac{{\sum }_{k=1}^{K}{w}_{MH,k}{\stackrel{ˆ}{p}}_{1k}\left(1-{\stackrel{ˆ}{p}}_{0k}\right)}{{\sum }_{k=1}^{K}{w}_{MH,k}{\stackrel{ˆ}{p}}_{0k}\left(1-{\stackrel{ˆ}{p}}_{1k}\right)}$ $=\frac{{\sum }_{k=1}^{K}{w}_{MH,k}{\stackrel{ˆ}{p}}_{0k}\left(1-{\stackrel{ˆ}{p}}_{1k}\right){\stackrel{ˆ}{\mathrm{\theta }}}_{k}}{{\sum }_{k=1}^{K}{w}_{MH,k}{\stackrel{ˆ}{p}}_{0k}\left(1-{\stackrel{ˆ}{p}}_{1k}\right)},$where ${\stackrel{ˆ}{\mathrm{\theta }}}_{k}={\stackrel{ˆ}{p}}_{1k}\left(1-{\stackrel{ˆ}{p}}_{0k}\right)/\left\{{\stackrel{ˆ}{p}}_{0k}\left(1-{\stackrel{ˆ}{p}}_{1k}\right)\right\}$. This estimator is also obtained as a solution of the following estimation equation, $W\left(\mathrm{\theta }\right)=\sum _{k=1}^{K}{w}_{MH,k}\left[{\stackrel{ˆ}{p}}_{1k}\left(1-{\stackrel{ˆ}{p}}_{0k}\right)-\mathrm{\theta }{\stackrel{ˆ}{p}}_{0k}\left(1-{\stackrel{ˆ}{p}}_{1k}\right)\right]=0.$Under the common effect assumption, this estimating function is unbiased, i. e., $E\left[W\left(\mathrm{\theta }\right)\right]=0$, and consistency of the estimator ${\stackrel{ˆ}{\mathrm{\theta }}}_{MH}$ is assured. We consider here the common effect assumption is violated. Under this setting, the target parameter $\mathrm{\theta }$ for the above estimating equation is not certainly defined as the common odds ratio. However, as the risk ratio estimation of prospective cohort studies, the asymptotic behavior can be characterized as follows. Using the same notation with Section 2,

Under the Asymptotic I, the Mantel-Haenszel estimator converges to normal distribution with mean equal to ${\mathrm{\theta }}_{\mathrm{I}}^{\ast }=\frac{{\sum }_{k=1}^{K}{\mathrm{\alpha }}_{k}{p}_{1k}\left(1-{p}_{0k}\right)}{{\sum }_{k=1}^{K}{\mathrm{\alpha }}_{k}{p}_{0k}\left(1-{p}_{1k}\right)}=\frac{{\sum }_{k=1}^{K}{\mathrm{\alpha }}_{k}{p}_{0k}\left(1-{p}_{1k}\right){\mathrm{\theta }}_{k}}{{\sum }_{k=1}^{K}{\mathrm{\alpha }}_{k}{p}_{0k}\left(1-{p}_{1k}\right)}.$and variances ${V}^{A}\left({\stackrel{ˆ}{\mathrm{\theta }}}_{MH}\right)=\frac{{\sum }_{k=1}^{K}{\mathrm{\alpha }}_{k}^{2}{V}^{A}\left[{W}_{k}\left({\mathrm{\theta }}_{\mathrm{I}}^{\ast }\right)\right]}{{\left\{{\sum }_{k=1}^{K}{\mathrm{\alpha }}_{k}{p}_{0k}\left(1-{p}_{1k}\right)\right\}}^{2}},$

where ${W}_{k}\left(\mathrm{\theta }\right)={\stackrel{ˆ}{p}}_{1k}\left(1-{\stackrel{ˆ}{p}}_{0k}\right)-\mathrm{\theta }{\stackrel{ˆ}{p}}_{0k}\left(1-{\stackrel{ˆ}{p}}_{1k}\right)$.

Also, under the Asymptotic II, ${\stackrel{ˆ}{\mathrm{\theta }}}_{MH}$ converges to normal distribution with mean ${\mathrm{\theta }}_{\mathrm{I}\mathrm{I}}^{\ast }=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}{\mathrm{\rho }}_{1lg}\left(1-{\mathrm{\rho }}_{0lg}\right)}{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}{\mathrm{\rho }}_{0lg}\left(1-{\mathrm{\rho }}_{1lg}\right)}=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}{\mathrm{\rho }}_{0lg}\left(1-{\mathrm{\rho }}_{1lg}\right){\stackrel{˜}{\mathrm{\theta }}}_{g}}{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}{\mathrm{\rho }}_{0lg}\left(1-{\mathrm{\rho }}_{1lg}\right)},$and variances ${V}^{A}\left({\stackrel{ˆ}{\mathrm{\theta }}}_{MH}\right)=\frac{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}^{2}{a}_{l}^{2}E\left[{V}_{W,lg}\left({\mathrm{\theta }}_{\mathrm{I}\mathrm{I}}^{\ast }|z\right)\right]}{{\left\{{\sum }_{l=1}^{L}{\sum }_{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}{\mathrm{\rho }}_{0lg}\left(1-{\mathrm{\rho }}_{1lg}\right)\right\}}^{2}},$

where ${\stackrel{˜}{\mathrm{\theta }}}_{1},{\stackrel{˜}{\mathrm{\theta }}}_{2},\dots ,{\stackrel{˜}{\mathrm{\theta }}}_{G}$ are heterogeneous odds ratios among the G strata and ${V}_{W,lg}\left(\mathrm{\theta }|z\right)$ is conditional variance of ${W}_{k}\left(\mathrm{\theta }\right)$ when ${p}_{0k}=z$.

Therefore, similar to Sections 2 and 3, the Mantel-Haenszel odds ratio estimator can also be interpreted to converge to a weighted average of stratum specific odds ratios ${\mathrm{\theta }}_{k}$’s with the weights ${\mathrm{\alpha }}_{k}{p}_{0k}\left(1-{p}_{1k}\right)$ or ${\mathrm{\pi }}_{lg}{a}_{l}{\mathrm{\rho }}_{0lg}\left(1-{\mathrm{\rho }}_{1lg}\right)$. The Mantel-Haenszel-type weights ${w}_{MH,k}$ would reflect the precision of the stratum-specific effect estimators (Fujii and Yanagimoto 2005; Sato 1990) approximately (in particular, under null effects), thus ${\mathrm{\theta }}_{\mathrm{I}}^{\ast }$ and ${\mathrm{\theta }}_{\mathrm{I}\mathrm{I}}^{\ast }$ might also be interpreted as an intuitive summary for the stratum-specific effect measures. Although the odds ratio cannot be interpreted as an effect measure by itself, when the incidence risks are low, it can be interpreted as a good approximation of risk ratio (Greenland 1987; Rothman et al. 2008). Therefore, under the rare-disease assumption, ${\mathrm{\theta }}_{\mathrm{I}}^{\ast }$ and ${\mathrm{\theta }}_{\mathrm{I}\mathrm{I}}^{\ast }$ can be interpreted as a standardized risk ratio under certain conditions as discussions in Section 2. In addition, asymptotic normality of ${\stackrel{ˆ}{\mathrm{\theta }}}_{MH}$ is assured under heterogeneity along with the results of Section 2. Also, the dually consistent variance estimator of Robins, Breslow and Greenland (1986) is still consistent when the common effect assumption is violated. Hauck (1979)’s variance estimator is also consistent under Asymptotic I, because it simply derived through combination of the stratum-specific variance estimators. The bootstrap variance estimator is another valid one.

## 4.2.1 Behaviors of the Mantel-Haenszel estimator

We assessed the empirical properties of the Mantel-Haenszel estimator ${\stackrel{ˆ}{\mathrm{\theta }}}_{MH}$ via simulation studies. At first, we generated 2 × 2 tables (K = 2) cohort data mimicked to the breast cancer research (Matsuyama et al. 2000; Sato and Matsuyama 2003) in Section 2.2, such as ${N}_{11}=1200,{N}_{01}=800,{N}_{12}=1400,$ ${N}_{02}=1600$. We consider the risk ratio of the 2nd stratum (RR2) varying 0.75, 1.00 and set ${p}_{12}=0.050,0.075$. Also, we set the risk ratio of the 1st stratum (RR1) as $\mathrm{\zeta }\cdot \mathrm{R}\mathrm{R}2$, where $\mathrm{\zeta }$ is the heterogeneity factor (when $\mathrm{\zeta }=1$, the common effect assumption is held). We also suppose ${p}_{02}={p}_{01}={p}_{12}/\mathrm{R}\mathrm{R}2$. Here, varying $\mathrm{\zeta }$, we simulated 3,600 cohort datasets. Then, we generated case-control dataset, through sampling all cases and 10 % non-cases randomly from the cohort dataset (sampling probabilities were identical across the two strata) supposing the settings of Asymptotic I. In the generated case-control datasets, we calculated the Mantel-Haenszel odds ratio estimates and the maximum likelihood estimates. Also, we computed the standardized risk ratio estimates with standards ${N}_{k}$ for the original cohort data. Figure 3 presents the mean of 3,600 estimates for each scenario of the Mantel-Haenszel estimates (blue lines) and the standardized estimates (black lines). We also considered the Asymptotic II settings. We generated matched case-control datasets, through sampling all cases and 1:2 matched controls matching with the lymph node metastasis status. In both of the generated case-control datasets, we calculated the Mantel-Haenszel odds ratio estimates. Also, we computed the standardized risk ratio estimates with standards ${N}_{k}$ for the original cohort data. Figures 3 and 4 present the means of 3,600 estimates for each scenario of the Mantel-Haenszel estimates (blue lines) and the standardized estimates (black lines).

In the results of the all settings, as expected, in the common effect settings ($\mathrm{\zeta }=1$), means of the Mantel-Haenszel odds ratio estimates and those of the standardized risk ratio estimates in the cohort were approximately identical because the rare-disease assumption hold. Further, under the common effect assumption was violated, these estimates were also nearly identical even under large heterogeneity exists. These results could be anticipated by the numerical results of Section 2.2, because the risk ratio and odds ratio is approximately identical when the rare-disease assumption is held. These results would indicate that the Mantel-Haenszel estimator of common odds ratio can be interpreted as approximate of the average exposure effect across strata, i. e., standardized risk ratio estimator, under the rare-disease assumption.

Figure 3:

Results of simulations under Asymptotic I: Means of 3,600 estimates of the Mantel-Haenszel estimates (${\stackrel{ˆ}{\mathrm{\theta }}}_{MH}$), and the standardized risk ratio estimates for the corresponding cohort studies (${\stackrel{ˆ}{\mathrm{\psi }}}_{{N}_{k}}$).

Figure 4:

Results of simulations under Asymptotic II: Means of 3,600 estimates of the Mantel-Haenszel estimates (${\stackrel{ˆ}{\mathrm{\theta }}}_{MH}$), and the standardized risk ratio estimates for the corresponding cohort studies (${\stackrel{ˆ}{\mathrm{\psi }}}_{{N}_{k}}$).

## 4.2.2 Variance estimation

We also assessed validity of the variance estimators. Settings were roughly mimicked the case-control datasets generated in the previous simulations. At first, for the large-strata settings, we generated 2 × 2 tables (K = 2) such as ${N}_{11}=150,{N}_{01}=285,{N}_{12}=160,$ ${N}_{02}=190$. We denote the odds ratios of the 1st and 2nd strata as OR1 and OR2. We set OR1 to be 0.500, 0.750, 1.000, and OR2 to be 0.500, 0.750, 1.000, 1.250, 1.500, 1.750, 2.000 times of OR1. We also set ${p}_{11}$ to 0.450. We simulated 3,600 case-control datasets, and evaluated actual standard error (SE) of the Mantel-Haenszel estimator and means of squared roots of variance estimates by the Hauck’s estimator ($\stackrel{ˆ}{V}$Hauck), the Robins-Breslow-Greenland’s estimator ($\stackrel{ˆ}{V}$RBG), and the bootstrap variance estimator ($\stackrel{ˆ}{V}$boot) of 3,600 replications. The number of bootstrap resampling was set to 5000. Results of the simulations were presented in Table 3. Under all of the settings considered, the three estimators validly quantified the actual SE, as a whole.

Second, for the sparse data settings, we generated 1:1 and 1:4 matched case-control datasets under possibly heterogeneous two populations. We divided the case datasets to ${N}_{11}=150$ and ${N}_{12}=160$, and considered the true odds ratios between the two strata were possibly heterogeneous. We denote the odds ratios of the 1st and 2nd strata as OR1 and OR2, too. We set OR1 to be 0.500, 0.750, 1.000, and OR2 to be 0.500, 0.750, 1.000, 1.250, 1.500, 1.750, 2.000 times of OR1. We also set ${p}_{11}$ to 0.450. We simulated 3,600 matched case-control datasets under each setting, and evaluated actual standard error (SE) of the Mantel-Haenszel estimator and means of squared roots of variance estimates by the Robins-Breslow-Greenland’s estimator ($\stackrel{ˆ}{V}$RBG), and the bootstrap variance estimator ($\stackrel{ˆ}{V}$boot) of 3,600 replications. The number of bootstrap resampling was set to 5,000. Results of the simulations were presented in Table 4. For the sparse data settings, the variance estimators validly quantified the actual SE under all of the settings considered. The existing variance estimators would be generally valid for quantifying SE of the Mantel-Haenszel estimators even when the common effect assumptions are violated.

Table 3:

Simulations results under Asymptotic I: Actual SE of the Mantel-Haenszel estimator and means of squared roots of variance estimates by the Hauck’s estimator (${\stackrel{ˆ}{V}}_{Hauck}$), the Robins-Breslow-Greenland’s estimator (${\stackrel{ˆ}{V}}_{RBG}$), and the bootstrap variance estimator (${\stackrel{ˆ}{V}}_{boot}$) of 3,600 replications.

Table 4:

Simulations results under Asymptotic II: Actual SE of the Mantel-Haenszel estimator and means of squared roots of variance estimates by the Robins-Breslow-Greenland’s estimator (${\stackrel{ˆ}{V}}_{RBG}$) and the bootstrap variance estimator (${\stackrel{ˆ}{V}}_{boot}$) of 3,600 replications.

## 5 Concluding remarks

The Mantel-Haenszel estimators have been widely applied in epidemiological and clinical researches involving meta-analysis due to their simplicity and efficiency. However, correctness of the common effect assumptions cannot be justified in general practice, and the targeted “common effect parameter” does not exist, then. Under this setting, even if the Mantel-Haenszel estimators have desirable properties, it is uncertain what they estimate and how the estimates are interpreted. However, many epidemiologists and statisticians would anticipate that they might be interpreted as an average exposure effect in some kinds of means, although there were not certain theoretical reasons. In this study, we provided theoretical evaluations of the Mantel-Haenszel estimators under the common effect assumptions are violated, and showed the intuitions are mostly correct. These results also correspond to the anticipations of Greenland and Maldonado (1994). We also showed these large sample results are valid under realistic situations with finite samples by a series of numerical studies.

As related recent theoretical works, Xu and O’Quigley (2000) and Hattori and Henmi (2012) showed the partial likelihood estimator of the Cox regression model can be interpreted as an average hazard ratio estimator even when the proportional hazard assumption was violated. According to the results of this study, the Mantel-Haenszel estimators are also interpreted as (i) when the common effect assumption is correct (as the best scenario), they are nearly efficient estimators of the common effect parameters, and (ii) when the common effect assumption is incorrect, they can be interpreted as the average exposure effect estimators across strata. Obviously, when a strong effect modification exists, it would not be recommended synthesizing the stratum-specific effect measures as a common effect (Greenland 1982; Mantel et al. 1977). The uses of the common effect estimators are appropriate, at least, for the settings that moderate effect modification are. In both ways, these theoretical and numerical evidences of the Mantel-Haenszel estimators would be a meaningful information for practices in epidemiological and clinical researches.

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## Appendix

In this appendix, we describe outline of proofs of derivation of asymptotic distributions of the Mantel-Haenszel estimators. Because the functional forms of the Mantel-Haenszel estimating functions are common, the rationale of proofs is basically common. Here, we briefly describe that of the odds ratio case.

Asymptotic I. Taylor expansion on the Mantel-Haenszel estimating function $W\left(\mathrm{\theta }\right)$ gives $\left({\stackrel{ˆ}{\mathrm{\theta }}}_{MH}-\mathrm{\theta }\right)\approx \left\{-{\left[\frac{1}{N}\sum _{k=1}^{K}{w}_{MH,k}\left(\frac{\mathrm{\partial }{W}_{k}\left(\mathrm{\theta }\right)}{\mathrm{\partial }\mathrm{\theta }}\right)\right]}^{-1}\left[\frac{1}{N}\sum _{k=1}^{K}{w}_{MH,k}{W}_{k}\left(\mathrm{\theta }\right)\right]\right\}$

Because of the law of large number, $\mathrm{\partial }{W}_{k}\left(\mathrm{\theta }\right)/\mathrm{\partial }\mathrm{\theta }\approx E\left[\mathrm{\partial }{W}_{k}\left(\mathrm{\theta }\right)/\mathrm{\partial }\mathrm{\theta }\right]$ under the limiting model. So, the asymptotic normality of the estimator is deduced by the central limit theorem for ${W}_{k}\left(\mathrm{\theta }\right)$’s, and the asymptotic mean of ${\stackrel{ˆ}{\mathrm{\theta }}}_{MH}$ corresponds the solution of $\underset{N\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\sum _{k=1}^{K}\frac{{w}_{MH,k}}{N}{W}_{k}\left(\mathrm{\theta }\right)=0.$It accords to ${\mathrm{\theta }}_{\mathrm{I}}^{\ast }$. In addition, the asymptotic variance is expressed as $\left[\sum _{k=1}^{K}{\mathrm{\alpha }}_{k}^{2}{V}_{A}\left[{W}_{k}\left({\mathrm{\theta }}_{\mathrm{I}}^{\ast }\right)\right]/{\left[\sum _{k=1}^{K}{\mathrm{\alpha }}_{k}E\left[\frac{\mathrm{\partial }{W}_{k}\left({\mathrm{\theta }}_{\mathrm{I}}^{\ast }\right)}{\mathrm{\partial }\mathrm{\theta }}\right]\right]}^{2}.$Thus, the asymptotic distribution is derived.

Asymptotic II. For the sparse data limiting model $K\to \mathrm{\infty }$, $W\left(\mathrm{\theta }\right)$ is regarded as a sum of many non-identical but independent quantities, so the asymptotic normality of ${\stackrel{ˆ}{\mathrm{\theta }}}_{MH}$ is deduced. Also, Taylor expansion on the Mantel-Haenszel estimating function $W\left(\mathrm{\theta }\right)$ provides $\left({\stackrel{ˆ}{\mathrm{\theta }}}_{MH}-\mathrm{\theta }\right)\approx \left\{-{\left[\frac{1}{K}\sum _{k=1}^{K}{w}_{MH,k}\left(\frac{\mathrm{\partial }{W}_{k}\left(\mathrm{\theta }\right)}{\mathrm{\partial }\mathrm{\theta }}\right)\right]}^{-1}\left[\frac{1}{K}\sum _{k=1}^{K}{w}_{MH,k}{W}_{k}\left(\mathrm{\theta }\right)\right]\right\}$The first term of the right-hand can be expressed as $\frac{1}{K}\sum _{k=1}^{K}{w}_{MH,k}\left(\frac{\mathrm{\partial }{W}_{k}\left(\mathrm{\theta }\right)}{\mathrm{\partial }\mathrm{\theta }}\right)=\sum _{l=1}^{L}\sum _{g=1}^{G}\frac{{K}_{lg}}{K}{a}_{l}\left\{\frac{1}{{K}_{lg}}\sum _{k\in {\mathrm{\Xi }}_{lg}}\left(\frac{\mathrm{\partial }{W}_{k}\left(\mathrm{\theta }\right)}{\mathrm{\partial }\mathrm{\theta }}\right)\right\}\to \sum _{l=1}^{L}\sum _{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}E\left[\frac{\mathrm{\partial }{W}_{k}\left(\mathrm{\theta }\right)}{\mathrm{\partial }\mathrm{\theta }}\right].$where ${\mathrm{\Xi }}_{lg}$ is the index set corresponding to $\left(l,g\right)$. So, the asymptotic mean of ${\stackrel{ˆ}{\mathrm{\theta }}}_{MH}$ corresponds the solution of $\underset{K\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\sum _{k=1}^{K}\frac{{w}_{MH,k}}{K}{W}_{k}\left(\mathrm{\theta }\right)=0$The expression of the objective function is $\begin{array}{l}\sum _{k=1}^{K}\frac{{w}_{MH,k}}{K}{W}_{k}\left(\theta \right)=\sum _{l=1}^{L}\sum _{g=1}^{G}\frac{{K}_{lg}}{K}{a}_{l}\left\{\frac{1}{{K}_{lg}}\sum _{k\in {\text{Ξ}}_{lg}}{W}_{k}\left(\theta \right)\right\}\\ \to \sum _{l=1}^{L}\sum _{g=1}^{G}{\pi }_{lg}{a}_{l}\left\{E\left[{\stackrel{^}{p}}_{1k}\left(1-{\stackrel{^}{p}}_{0k}\right)\right]-\theta E\left[{\stackrel{^}{p}}_{0k}\left(1-{\stackrel{^}{p}}_{1k}\right)\right]\right\}\\ =\sum _{l=1}^{L}\sum _{g=1}^{G}{\pi }_{lg}{a}_{l}\left\{{\rho }_{1k}\left(1-{\rho }_{0k}\right)-\theta {\rho }_{0k}\left(1-{\rho }_{1k}\right)\right\}\end{array}$Thus, the solution corresponds to ${\mathrm{\theta }}_{\mathrm{I}\mathrm{I}}^{\ast }$. In addition, the asymptotic variance is expressed as

$\left[\sum _{l=1}^{L}\sum _{g=1}^{G}{\mathrm{\pi }}_{lg}^{2}{a}_{l}^{2}E\left[{V}_{W,lg}\left({\mathrm{\theta }}_{\mathrm{I}\mathrm{I}}^{\ast }|z\right)\right]\right]/{\left[\sum _{l=1}^{L}\sum _{g=1}^{G}{\mathrm{\pi }}_{lg}{a}_{l}E\left[\frac{\mathrm{\partial }{W}_{k}\left({\mathrm{\theta }}_{\mathrm{I}\mathrm{I}}^{\ast }\right)}{\mathrm{\partial }\mathrm{\theta }}\right]\right]}^{2}.$Therefore, the asymptotic distribution is derived.

Published Online: 2016-05-06

Published in Print: 2016-12-01

Funding: This work was supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan (Grant numbers: 25280008, 15K15954).

Citation Information: Epidemiologic Methods, Volume 5, Issue 1, Pages 19–35, ISSN (Online) 2161-962X, ISSN (Print) 2194-9263,

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