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

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=p1k−p0k 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

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

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=pˆ1k−ψpˆ0k,Rkω=ω−pˆ1k−pˆ0k, the asymptotic behaviors of Mantel-Haenszel estimators under Asymptotic I can be characterized as follows.

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

An outline of proof is provided in Appendix. Note that N−1wMH,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 1−hk is a constant across strata, i. e., wMH,k=Nk1−hkNkhk/Nk=Nkhk1−hk is proportional to Nk when hk1−hk 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.

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 p12−p02=−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=RD1−RD2. 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 ζRR≥2.0 and ζRD≥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.

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∗).

3.1 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=r1k−r0k 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

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

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=rˆ1k−ϕrˆ0k,Ukω=(rˆ1k−rˆ0k)−ξ, here. The asymptotic behaviors of Mantel-Haenszel estimators can be characterized as follows.

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.

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 1−fk is a constant across strata, i. e., uMH,k=Tk1−fkTkfk/Tk=Tkfk1−fk≈Tk when fk1−fk 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 r11−r01=−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=IRD2−IRD1. 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 ζIRR≈1,ζIRD≈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 f11−f1=0.161,f21−f2=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=2000×10−5, the mean of ξˆMH was 125×10−5 and that of ξˆTk 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 Tk.

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∗).

4.1 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=p1k1−p0k/p0k1−p1k 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

where θˆk=pˆ1k1−pˆ0k/{pˆ0k(1−pˆ1k)}. This estimator is also obtained as a solution of the following estimation equation,

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,

where Wkθ=pˆ1k1−pˆ0k−θpˆ0k1−pˆ1k.

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

and variances

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.