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

• Hisashi Noma and Kengo Nagashima
From the journal Epidemiologic Methods

## 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 Analysis of cohort studies with binary data

### 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 X0k,X1k with sample sizes N0k,N1k and success probabilities p0k,p1k for k=1,2,,K. Under the common effect assumption for the risk ratio, we assume the stratum-specific risk ratio ψk=p1k/p0k are all equal across the strata, i. e., ψ=ψ1=ψ2==ψK. Similarly, for the risk difference, we assume the common effect for the stratum specific risk difference ωk=p1kp0k across the strata, i. e., ω=ω1=ω2==ωK.

The Mantel-Haenszel estimators of the common risk ratio ψ (Nurminen 1981; Tarone 1981) and the common risk difference ω (Cochran 1954) are presented as

ψˆMH=k=1KX1kN0k/Nkk=1KX0kN1k/Nk=k=1KwMH,kpˆ1kk=1KwMH,kpˆ0k=k=1KwMH,kpˆ0kψˆkk=1KwMH,kpˆ0k,
ωˆMH=k=1K(X1kN0kX0kN1k)/Nkk=1KN0kN1k/Nk=k=1KwMH,kpˆ1kk=1KwMH,kk=1KwMH,kpˆ0kk=1KwMH,k=k=1KwMH,kωˆkk=1KwMH,k,

where pˆ0k=X0k/N0k,pˆ1k=X1k/N1k,Nk=N1k+N0k and wMH,k=N0kN1k/Nk. Also, ψˆk=pˆ1k/pˆ0k and ωˆk=pˆ1kpˆ0k. These estimators are obtained as solutions of the following estimation equations,

Qψ=k=1KwMH,kpˆ1kψpˆ0k=0,
Rω=k=1KwMH,kωpˆ1kpˆ0k=0.

Note that these estimating functions are unbiased under the common effects assumptions, such that EQψ=0 and ERω=0, and consistency of the estimators follow straightforwardly. Here, suppose the common effect assumptions are violated, i. e., ψ1,ψ2,,ψK and ω1,ω2,,ωK are possibly heterogeneous. In this case, the target parameters ψ and ω 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 N0k,N1k. In the second, denoted as Asymptotic II, N0k,N1k are bounded while K is large. A well-known example of Asymptotic II is the matched designs.

First, denoting Qkψ=pˆ1kψpˆ0k,Rkω=ωpˆ1kpˆ0k, the asymptotic behaviors of Mantel-Haenszel estimators under Asymptotic I can be characterized as follows.

Theorem 1

Under the Asymptotic I, we assume N0k/Nλ0k>0, N1k/Nλ1k>0, where N=N01++N0K+N11++N1K. Then, the Mantel-Haenszel estimators converge to normal distributions with means equal to

ψI=k=1Kαkp1kk=1Kαkp0k=k=1Kαkp0kψkk=1Kαkp0k,
ωI=k=1Kαkp1kk=1Kαkk=1Kαkp0kk=1Kαk=k=1Kαkωkk=1Kαk,

and variances

VAψˆMH=k=1Kαk2VAQkψIk=1Kαkp0k2,VAωˆMH=k=1Kαk2VARkωIk=1Kαk2,

where αk=λ0kλ1k/λ0k+λ1k and VA. means asymptotic variance.

An outline of proof is provided in Appendix. Note that N1wMH,k is an empirical quantity of αk. Intuitively, the quantities ψI and ωI can be roughly interpreted as expected quantities of standardized risk ratio and risk differences with the standard weight αk. Since the Mantel-Haenszel risk ratio and risk difference estimators are expressed as standardized estimators with the weight wMH,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 ψI (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 wNk=Nk/i=1KNi under null effect. Also, the standards wNk and wMH,k are substantially identical when a product of exposure prevalence hk=N1k/Nk and its complement 1hk is a constant across strata, i. e., wMH,k=Nk1hkNkhk/Nk=Nkhk1hk is proportional to Nk when hk1hk is constant. The arguments hold for risk difference, similarly. Under this constancy assumption, ψI and ωI nearly accord to the standardized risk ratio and risk difference with standards αk. Although it does not necessarily require an exact constancy of hk (the distribution of exposure prevalence), the concordance of wNk and wMH,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 ψk’s or risk differences ωk’s, with the weights p0kαk and αk. The Mante-Haenszel-type weights wMH,k would reflect the precision of the stratum specific effect estimators approximately (in particular, under null effects; Sato 1990), ψI and ωI 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 N0k,N1k among K strata, and we denote the number of it as L. Then, the weight wMH,k is constant within each configuration, and we denote it as all=1,2,L. Also, we assume there is G heterogeneous risk ratios and/or risk differences among K strata, i. e., ψ1,ψ2,,ψK and ω1,ω2,,ωK 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 ψ˜1,ψ˜2,,ψ˜G and ω˜1,ω˜2,,ω˜G. We also assume there is Klg strata for lth configuration and gth subset and Kl,g/K converges to πlg, which depends on the distribution of exposure across strata (l = 1,2,…,L; g = 1,2,…,G). In addition, the nuisance parameter p0k is assumed to be sampled from a certain probability distribution F0lgz for the lth and gth subset. The corresponding p1k is specified to be ψ˜gp0k or ω˜g+p0k then, and we denote the distribution of p1k as F1lgz formally, here. Also, we denote the means of F0lgz and F1lgz as ρ0lg and ρ1lg. In addition, we denote the conditional variances of stratum specific statistics Qkψ,Rkω for the lth and gth subset as VQ,lgψ|z,VR,lgω|z conditioning on p0k=z. After that, the asymptotic behaviors of the Mantel-Haenszel estimators can be characterized as follows.

Theorem 2

Under the Asymptotic II, the Mantel-Haenszel estimators converge to normal distribution with means equal to

ψII=l=1Lg=1Gπlgalρ1lgl=1Lg=1Gπlgalρ0lg=l=1Lg=1Gπlgalρ0lgψ˜gl=1Lg=1Gπlgalρ0lg,
ωII=l=1Lg=1Gπlgalρ1lgl=1Lg=1Gπlgall=1Lg=1Gπlgalρ0lgl=1Lg=1Gπlgal=l=1Lg=1Gπlgalω˜gl=1Lg=1Gπlgal,

and variances

VAψˆMH=l=1Lg=1Gπlg2al2EVQ,lgψII|zl=1Lg=1Gπlgalρ0lg2,
VAωˆMH=l=1Lg=1Gπlg2al2EVR,lgωII|zl=1Lg=1Gπlgal2.

An outline of proof is provided in Appendix II. Similar to Asymptotic I, al just accords to the standard Mantel-Haentszel weight wMH,k, the quantities ψII and ωII can also be interpreted as expected quantities of standardized risk ratio and risk differences with the standard weight al’s. Only the nuisance parameters p0k’s are possibly different across strata, but it is substituted to the means within certain substrata ρ0lg’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 wNk=Nk/i=1KNi. In addition, the Mantel-Haenszel estimators can also be interpreted to converge to weighted averages of heterogeneous risk ratios ψ˜g’s or risk differences ω˜g’s, with the weights πlgalρ0lg and πlgal. The Mante-Haenszel-type weights wMH,k would reflect the precision of the stratum specific effect estimators approximately (in particular, under null effects; Sato 1990), ψII and ωII would also be interpreted intuitive summary for the stratum-specific effect measures.

Another concern is the asymptotic variance estimation of ψˆMH and ωˆMH 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 Nk 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 N11=1215,N01=760,N12=1334,N02=1592, and p01=253/760=0.333,p12=96/1334=0.072,p02=171/1592=0.107. In this case, the risk ratio and the risk difference of the 2nd stratum are p12/p02=0.670 (RR2) and p12p02=0.035 (RD2). Denoting the risk ratio and the risk difference of the 1st stratum as RR1 and RD1, we define heterogeneity factors ζRR=RR1/RR2 and ζRD=RD1RD2. Varying ζRR and ζRD by 0.001, we simulated 25,000 dataset and compared empirical means of the Mantel-Haenszel estimators and the standardized estimators with standards Nk. 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 ψ and ω 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 ψI and ωI. 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 (ζRR=1 and ζRD=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 ζRR2.0 and ζRD0.15). 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).

Lymph node metastasis at surgeryNot lymph node metastasis at surgery
Tamoxifen useNot useTamoxifen useNot use
Recurrence36825396171
Not recurrence8475071,2381,421
Total1,2157601,3341,592
Recurrence proportion0.3030.3330.0720.107
Risk ratio0.9100.670
Risk difference−0.030−0.035
Figure 1:

Results of simulations: Means of 25,000 estimates of the Mantel-Haenszel estimates (ψˆMH and ωˆMH), the standardized estimates (ψˆNk and ωˆNk), and the quantities that the Mantel-Haenszel estimators converge to (ψI and ωI).

## 3 Analysis of cohort studies with person-time data

### 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 Y0k,Y1k with fixed person-time denominators T0k,T1k and means T0kr0k,T1kr1k for k=1,2,,K. Also, r0k,r1k 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 ϕk=r1k/r0k are all equal across the strata, i. e., ϕ=ϕ1=ϕ2==ϕK. In addition, for the rate difference, we assume the common effect for the stratum specific rate difference ξk=r1kr0k across the strata, i. e., ξ=ξ1=ξ2==ξK. The Mantel-Haenszel estimators of the common rate ratio ϕ (Greenland and Robins 1985; Walker 1985) and the common risk difference ξ (Greenland and Robins 1985) are given as

ϕˆMH=k=1KY1kT0k/Tkk=1KY0kT1k/Tk=k=1KuMH,krˆ1kk=1KuMH,krˆ0k=k=1KuMH,krˆ0krˆ1kk=1KuMH,krˆ0k,
ξˆMH=k=1K(Y1kT0kY0kT1k)/Tkk=1KT0kT1k/Tk=k=1KuMH,krˆ1kk=1KuMH,kk=1KuMH,krˆ0kk=1KuMH,k=k=1KuMH,kξˆkk=1KuMH,k,

where rˆ0k=Y0k/T0k,rˆ1k=Y1k/T1k,Tk=T1k+T0k and uMH,k=(T0kT1k/Tk). Also, ϕˆk=rˆ1k/rˆ0k and ξˆk=rˆ1krˆ0k. The Mantel-Haenszel estimators are obtained as the solution of the estimation equations,

Hϕ=k=1KuMH,krˆ1kϕrˆ0k=0,
Uξ=k=1KuMH,k(rˆ1krˆ0k)ξ=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 T0k,T1k. Also, for the sparse data limiting model, denoted as Asymptotic II, T0k,T1k are bounded while K is large. We also denote Hkψ=rˆ1kϕrˆ0k,Ukω=(rˆ1krˆ0k)ξ, here. The asymptotic behaviors of Mantel-Haenszel estimators can be characterized as follows.

Theorem 3

Under the Asymptotic I, we assume T0k/Tν0k>0, T1k/Tν1k>0, where T=T01++T0K+T11++T1K. The Mantel-Haenszel estimators converge to normal distribution with means equal to

ϕI=k=1Kγkr1kk=1Kγkr0k=k=1Kγkr0kϕkk=1Kγkr0k,
ξI=k=1Kγkr1kk=1Kγkk=1Kγkr0kk=1Kγk=k=1Kγkξkk=1Kγk,

and variances

VAϕˆMH=k=1Kγk2VAHkϕIk=1Kγkr0k2,VAξˆMH=k=1Kγk2VAUkξIk=1Kγk2,

where γk=ν0kν1k/ν0k+ν1k.

Also, under Asymptotic II, we suppose there is a finite number of possible configurations of total sample sizes T0k,T1k among K strata, and we denote the number of it as L. Then, the weight uMH,k is formally constant within each configuration, and we denote it as sll=1,2,L. Also, we assume there is G heterogeneous rate ratios and/or rate differences among K strata, i. e., ϕ1,ϕ2,,ϕK and ξ1,ξ2,,ξK 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 ϕ˜1,ϕ˜2,,ϕ˜G and ξ˜1,ξ˜2,,ξ˜G. We also assume there is Klg strata for lth configuration and gth subset and Kl,g/K converges to πlg, which depends on the distribution of exposure across strata (l = 1,2,…,L; g = 1,2,…,G). In addition, the nuisance parameter r0k is assumed to be sampled from a certain probability distribution F0lgz for the lth and gth subset. The corresponding r1k is specified to ϕ˜gr0k or ξ˜g+r0k, then. We denote the distribution of r1k as F1lgz formally, here. Also we denote the means of F0lgz and F1lgz as τ0lg and τ1lg, here. In addition, we denote the conditional variances of stratum specific statistics Hkϕ,Ukτ for the lth and gth subset as VH,lgϕ|z,VU,lgξ|z under r0k=z. After that, the asymptotic behavior of the Mantel-Haenszel estimators can be characterized as follows.

Theorem 4

Under the Asymptotic II, the Mantel-Haenszel estimators converge to normal distribution with means equal to

ϕII=l=1Lg=1Gπlgslτ1lgl=1Lg=1Gπlgslτ0lg=l=1Lg=1Gπlgslτ0lgϕ˜gl=1Lg=1Gπlgslτ0lg,
ξII=l=1Lg=1Gπlgslτ1lgl=1Lg=1Gπlgsll=1Lg=1Gπlgslτ0lgl=1Lg=1Gπlgsl=l=1Lg=1Gπlgslξ˜gl=1Lg=1Gπlgsl,

and variances

VAϕˆMH=l=1Lg=1Gπlg2sl2EVH,lgϕII|zl=1Lg=1Gπlgslτ0lg2,
VAξˆMH=l=1Lg=1Gπlg2sl2EVU,lgξII|zl=1Lg=1Gπlgsl2.

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 γk or uMH,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 uTk=Tk/i=1KTi and uMH,k are substantially identical when a product of exposure prevalence fk=T1k/Tk and its complement 1fk is a constant across strata, i. e., uMH,k=Tk1fkTkfk/Tk=Tkfk1fkTk when fk1fk is constant. Under this constancy assumption, ψI and ωI accord to the standardized rate ratio and rate difference with standard uN,k. Furthermore, the Mantel-Haenszel estimators can also be interpreted to converge to weighted averages of stratum specific rate ratios ϕk’s or rate differences ξk’s, with the weights γkr0k and γk. ϕI and ξI would also be interpreted intuitive summary for the stratum-specific effect measures. ϕII and ξII 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 Tk 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 T11=62119,T01=15763,T12=2744,T02=2780, and r11=196/62119=315.5×105,r01=111/15763=704.2×105,r02=157/2780=5647×105. In this case, the rate ratio and the rate difference of the 1st stratum are r11/r01=0.448 (IRR1) and r11r01=388.7×105 (IRD1). Denoting the rate ratio and the rate difference of the 2nd stratum as IRR2 and IRD2, we define heterogeneity factors ζIRR=IRR2/IRR1 and ζIRD=IRD2IRD1. Varying ζIRR and ζIRD by 0.0001, we simulated 25,000 dataset and compared the Mantel-Haenszel estimators and the standardized estimators with standards Tk. 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 ϕI and ξI 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 ϕ and ξ. Also, as expected, in the settings around ζIRR1,ζIRD0 (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 f11f1=0.161,f21f2=0.215, thus, these discrepancies might be anticipated. Besides, these discrepancies were not so large, for instance, when ζIRR=1.5, the mean of ϕˆMH was 0.574 and that of ϕˆTk was 0.557. Also, when ζIRD=2000×105, the mean of ξˆMH was 125×105 and that of ξˆTk was 184×105. 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 Tk.

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).

Age (years): 10–54Age (years): 55–94
CurrentPastCurrentPast
Deaths196111167157
Person-years62,11915,7636,0852,780
Rate (per 105 person-years)315.5704.22,7445,647
Rate ratio0.4480.486
Rate difference (per 105 person-years)−388.7−2,903
Figure 2:

Results of simulations: Means of 25,000 estimates of the Mantel-Haenszel estimates (ϕˆMH and ξˆMH), the standardized estimates (ϕˆTk and ξˆTk), and the quantities that the Mantel-Haenszel estimators converge to (ϕI and ξI).

## 4 Analysis of case-control studies

### 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 X0k,X1k with sample sizes N0k,N1k and success probabilities p0k,p1k for k=1,2,,K. In case-control studies, N0k and N1k correspond to the total numbers of cases and controls, and X0k and X1k 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 θk=p1k1p0k/p0k1p1k are all equal across the strata, i. e., θ=θ1=θ2==θK. The Mantel-Haenszel estimator of the common odds ratio θ (Mantel and Haenszel 1959) is presented as

θˆMH=k=1KX1kN0kX0k/Nkk=1KX0kN1kX1k/Nk=k=1KwMH,kpˆ1k1pˆ0kk=1KwMH,kpˆ0k1pˆ1k
=k=1KwMH,kpˆ0k1pˆ1kθˆkk=1KwMH,kpˆ0k1pˆ1k,

where θˆk=pˆ1k1pˆ0k/{pˆ0k(1pˆ1k)}. This estimator is also obtained as a solution of the following estimation equation,

Wθ=k=1KwMH,kpˆ1k1pˆ0kθpˆ0k1pˆ1k=0.

Under the common effect assumption, this estimating function is unbiased, i. e., EWθ=0, and consistency of the estimator θˆMH is assured. We consider here the common effect assumption is violated. Under this setting, the target parameter θ 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,

Theorem 5.

Under the Asymptotic I, the Mantel-Haenszel estimator converges to normal distribution with mean equal to

θI=k=1Kαkp1k1p0kk=1Kαkp0k1p1k=k=1Kαkp0k1p1kθkk=1Kαkp0k1p1k.

and variances

VAθˆMH=k=1Kαk2VAWkθIk=1Kαkp0k1p1k2,

where Wkθ=pˆ1k1pˆ0kθpˆ0k1pˆ1k.

Also, under the Asymptotic II, θˆMH converges to normal distribution with mean

θII=l=1Lg=1Gπlgalρ1lg1ρ0lgl=1Lg=1Gπlgalρ0lg1ρ1lg=l=1Lg=1Gπlgalρ0lg1ρ1lgθ˜gl=1Lg=1Gπlgalρ0lg1ρ1lg,

and variances

VAθˆMH=l=1Lg=1Gπlg2al2EVW,lgθII|zl=1Lg=1Gπlgalρ0lg1ρ1lg2,

where θ˜1,θ˜2,,θ˜G are heterogeneous odds ratios among the G strata and VW,lgθ|z is conditional variance of Wkθ when p0k=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 θk’s with the weights αkp0k1p1k or πlgalρ0lg1ρ1lg. The Mantel-Haenszel-type weights wMH,k would reflect the precision of the stratum-specific effect estimators (Fujii and Yanagimoto 2005; Sato 1990) approximately (in particular, under null effects), thus θI and θII 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, θI and θII can be interpreted as a standardized risk ratio under certain conditions as discussions in Section 2. In addition, asymptotic normality of θˆMH 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 Numerical evaluation by simulations

#### 4.2.1 Behaviors of the Mantel-Haenszel estimator

We assessed the empirical properties of the Mantel-Haenszel estimator θˆMH 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 N11=1200,N01=800,N12=1400,N02=1600. We consider the risk ratio of the 2nd stratum (RR2) varying 0.75, 1.00 and set p12=0.050,0.075. Also, we set the risk ratio of the 1st stratum (RR1) as ζRR2, where ζ is the heterogeneity factor (when ζ=1, the common effect assumption is held). We also suppose p02=p01=p12/RR2. Here, varying ζ, 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 Nk 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 Nk 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 (ζ=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 (θˆMH), and the standardized risk ratio estimates for the corresponding cohort studies (ψˆNk).

Figure 4:

Results of simulations under Asymptotic II: Means of 3,600 estimates of the Mantel-Haenszel estimates (θˆMH), and the standardized risk ratio estimates for the corresponding cohort studies (ψˆNk).

#### 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 N11=150,N01=285,N12=160,N02=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 p11 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 (VˆHauck), the Robins-Breslow-Greenland’s estimator (VˆRBG), and the bootstrap variance estimator (Vˆboot) 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 N11=150 and N12=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 p11 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 (VˆRBG), and the bootstrap variance estimator (Vˆboot) 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 (VˆHauck), the Robins-Breslow-Greenland’s estimator (VˆRBG), and the bootstrap variance estimator (Vˆboot) of 3,600 replications.

Actual SEVHauck1/2VRBG1/2Vboot1/2
OR1 = 0.500OR2 = 0.2500.1510.1560.1520.151
OR2 = 0.3750.1530.1510.1500.150
OR2 = 0.5000.1480.1500.1490.150
OR2 = 0.6250.1490.1500.1490.149
OR2 = 0.7500.1500.1510.1490.149
OR2 = 0.8750.1500.1520.1490.149
OR2 = 1.0000.1480.1530.1490.149
OR1 = 0.750OR2 = 0.3750.1510.1540.1500.149
OR2 = 0.5630.1500.1500.1480.149
OR2 = 0.7500.1490.1480.1480.148
OR2 = 0.9380.1460.1480.1470.148
OR2 = 1.1250.1470.1490.1470.147
OR2 = 1.3130.1500.1500.1470.147
OR2 = 1.5000.1460.1510.1470.146
OR1 = 1.000OR2 = 0.5000.1480.1550.1500.150
OR2 = 0.7500.1490.1500.1490.149
OR2 = 1.0000.1490.1490.1480.148
OR2 = 1.2500.1500.1490.1480.148
OR2 = 1.5000.1500.1490.1470.147
OR2 = 1.7500.1470.1500.1470.147
OR2 = 2.0000.1470.1520.1480.146
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 (VˆRBG) and the bootstrap variance estimator (Vˆboot) of 3,600 replications.

1:1 matching1:4 matching
Actual SEVRBG1/2Vboot1/2Actual SEVRBG1/2Vboot1/2
OR1 = 0.500OR2 = 0.2500.1800.1780.1810.1360.1350.136
OR2 = 0.3750.1710.1720.1740.1330.1320.132
OR2 = 0.5000.1710.1690.1710.1310.1300.131
OR2 = 0.6250.1680.1680.1690.1310.1290.130
OR2 = 0.7500.1660.1660.1680.1280.1290.129
OR2 = 0.8750.1660.1660.1670.1300.1290.129
OR2 = 1.0000.1690.1650.1670.1310.1290.129
OR1 = 0.750OR2 = 0.3750.1690.1690.1700.1300.1320.133
OR2 = 0.5630.1700.1650.1660.1290.1290.130
OR2 = 0.7500.1650.1630.1640.1290.1280.129
OR2 = 0.9380.1620.1620.1630.1290.1270.128
OR2 = 1.1250.1620.1620.1630.1270.1270.128
OR2 = 1.3130.1600.1620.1630.1240.1270.128
OR2 = 1.5000.1640.1620.1630.1260.1280.128
OR1 = 1.000OR2 = 0.5000.1650.1660.1680.1290.1320.132
OR2 = 0.7500.1650.1630.1650.1280.1290.129
OR2 = 1.0000.1640.1620.1640.1260.1280.128
OR2 = 1.2500.1640.1620.1630.1260.1280.128
OR2 = 1.5000.1630.1620.1640.1270.1280.128
OR2 = 1.7500.1600.1630.1640.1260.1280.128
OR2 = 2.0000.1600.1630.1650.1250.1280.129

## 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.

Funding statement: 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).

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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 Wθ gives

θˆMHθ1Nk=1KwMH,kWkθθ11Nk=1KwMH,kWkθ

Because of the law of large number, Wkθ/θEWkθ/θ under the limiting model. So, the asymptotic normality of the estimator is deduced by the central limit theorem for Wkθ’s, and the asymptotic mean of θˆMH corresponds the solution of

limNk=1KwMH,kNWkθ=0.

It accords to θI. In addition, the asymptotic variance is expressed as

k=1Kαk2VA[WkθI/k=1KαkEWkθIθ2.

Thus, the asymptotic distribution is derived.

Asymptotic II. For the sparse data limiting model K, Wθ is regarded as a sum of many non-identical but independent quantities, so the asymptotic normality of θˆMH is deduced. Also, Taylor expansion on the Mantel-Haenszel estimating function Wθ provides

θˆMHθ1Kk=1KwMH,kWkθθ11Kk=1KwMH,kWkθ

The first term of the right-hand can be expressed as

1Kk=1KwMH,kWkθθ=l=1Lg=1GKlgKal1KlgkΞlgWkθθl=1Lg=1GπlgalEWkθθ.

where Ξlg is the index set corresponding to l,g. So, the asymptotic mean of θˆMH corresponds the solution of

limKk=1KwMH,kKWkθ=0

The expression of the objective function is

k=1KwMH,kKWk(θ)=l=1Lg=1GKlgKal{1KlgkΞlgWk(θ)}l=1Lg=1Gπlgal{E[p^1k(1p^0k)]θE[p^0k(1p^1k)]}=l=1Lg=1Gπlgal{ρ1k(1ρ0k)θρ0k(1ρ1k)}

Thus, the solution corresponds to θII. In addition, the asymptotic variance is expressed as

l=1Lg=1Gπlg2al2EVW,lgθII|z/l=1Lg=1GπlgalEWkθIIθ2.

Therefore, the asymptotic distribution is derived.

Published Online: 2016-5-6
Published in Print: 2016-12-1