Abstract
The MantelHaenszel 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 MantelHaenszel 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 MantelHaenszel 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 MantelHaenszel estimators converge to weighted averages of stratumspecific effect parameters and they can be interpreted as intuitive summaries of the stratumspecific effect measures. Also, the MantelHaenszel estimators correspond to the standardized effect measures on standard distributions of stratification variables to be the total cohort, approximately. In addition, the ordinary sandwichtype variance estimators are still valid for quantifying variabilities of the MantelHaenszel 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 MantelHaenszel 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 metaanalysis (Higgins and Green 2008). Although the MantelHaenszel 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 MantelHaenszel methods are quite uncertain and it is not clear what they estimate. Greenland and Maldonado (1994) inferred that the MantelHaenszel 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 MantelHaenszel 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 MantelHaenszel estimators when the common effect assumptions are violated. We show the MantelHaenszel 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 MantelHaenszel 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 MantelHaenszel 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
The MantelHaenszel estimators of the common risk ratio
For evaluating the asymptotic behaviors of the MantelHaenszel 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
First, denoting
Under the Asymptotic I, we assume
where
An outline of proof is provided in Appendix. Note that
In addition, the MantelHaenszel estimators can also be interpreted to converge to weighted averages of stratum specific risk ratios
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
Under the Asymptotic II, the MantelHaenszel estimators converge to normal distribution with means equal to
An outline of proof is provided in Appendix II. Similar to Asymptotic I,
Another concern is the asymptotic variance estimation of
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 stratumspecific 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 MantelHaenszel risk ratio estimator was 0.830 and the MantelHaenszel risk difference estimator was −0.033. Besides, the standardized risk ratio and risk difference with standards
In addition, we conducted simulation studies for investigating empirical properties of the MantelHaenszel estimators under heterogeneity. We consider several scenarios based on the stratified dataset of Table 1, such as
As the results, in the all scenarios, means of the distributions of the MantelHaenszel estimates mostly accord to the asymptotic mean of the distributions of the MantelHaenszel estimators
Lymph node metastasis at surgery  Not lymph node metastasis at surgery  

Tamoxifen use  Not use  Tamoxifen use  Not use  
Recurrence  368  253  96  171 
Not recurrence  847  507  1,238  1,421 
Total  1,215  760  1,334  1,592 
Recurrence proportion  0.303  0.333  0.072  0.107 
Risk ratio  0.910  0.670  
Risk difference  −0.030  −0.035 
Figure 1:
3 Analysis of cohort studies with persontime data
3.1 MantelHaenszel rate ratio and rate difference estimators
We consider estimating the common rate ratio and rate difference for stratified persontime data of cohort studies. Suppose a series of K strata constructed by independent Poisson observations
Here, we consider similar limiting models the previous section. The largestrata limiting model, Asymptotic I, is to have a fixed number of strata K while
Under the Asymptotic I, we assume
where
Also, under Asymptotic II, we suppose there is a finite number of possible configurations of total sample sizes
Under the Asymptotic II, the MantelHaenszel estimators converge to normal distribution with means equal to
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
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 stratumspecific rate ratios: 0.448 and 0.486, the rate differences: −388.7 and −2903 per 10^{5} personyears). 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 largestrata limiting model, here. The MantelHaenszel rate ratio estimator was 0.469 and the MantelHaenszel risk difference estimator was −710.7 per 10^{5} personyears. Besides, the standardized rate ratio and rate difference with
Here, we also conducted simulation experiments for investigating empirical properties of the MantelHaenszel estimators under heterogeneity. We consider several scenarios based on the stratified dataset of Table 2, such as
In the all settings, means of the distributions of the MantelHaenszel estimates mostly accord to the asymptotic mean of the distributions of the MantelHaenszel estimators
Age (years): 10–54  Age (years): 55–94  

Current  Past  Current  Past  
Deaths  196  111  167  157 
Personyears  62,119  15,763  6,085  2,780 
Rate (per 10^{5} personyears)  315.5  704.2  2,744  5,647 
Rate ratio  0.448  0.486  
Rate difference (per 10^{5} personyears)  −388.7  −2,903 
Figure 2:
4 Analysis of casecontrol studies
4.1 MantelHaenszel odds ratio estimator
We discuss the common odds ratio estimation for stratified analyses in casecontrol studies. Consider the same setting with Section 2, a series of K 2 × 2 tables formed by pairs of independent binomial observations
Under the Asymptotic I, the MantelHaenszel estimator converges to normal distribution with mean equal to
where
Also, under the Asymptotic II,
where
Therefore, similar to Sections 2 and 3, the MantelHaenszel odds ratio estimator can also be interpreted to converge to a weighted average of stratum specific odds ratios
4.2 Numerical evaluation by simulations
4.2.1 Behaviors of the MantelHaenszel estimator
We assessed the empirical properties of the MantelHaenszel estimator
In the results of the all settings, as expected, in the common effect settings (
Figure 3:
Figure 4:
4.2.2 Variance estimation
We also assessed validity of the variance estimators. Settings were roughly mimicked the casecontrol datasets generated in the previous simulations. At first, for the largestrata settings, we generated 2 × 2 tables (K = 2) such as
Second, for the sparse data settings, we generated 1:1 and 1:4 matched casecontrol datasets under possibly heterogeneous two populations. We divided the case datasets to
Actual SE 





OR_{1} = 0.500  OR_{2} = 0.250  0.151  0.156  0.152  0.151 
OR_{2} = 0.375  0.153  0.151  0.150  0.150  
OR_{2} = 0.500  0.148  0.150  0.149  0.150  
OR_{2} = 0.625  0.149  0.150  0.149  0.149  
OR_{2} = 0.750  0.150  0.151  0.149  0.149  
OR_{2} = 0.875  0.150  0.152  0.149  0.149  
OR_{2} = 1.000  0.148  0.153  0.149  0.149  
OR_{1} = 0.750  OR_{2} = 0.375  0.151  0.154  0.150  0.149 
OR_{2} = 0.563  0.150  0.150  0.148  0.149  
OR_{2} = 0.750  0.149  0.148  0.148  0.148  
OR_{2} = 0.938  0.146  0.148  0.147  0.148  
OR_{2} = 1.125  0.147  0.149  0.147  0.147  
OR_{2} = 1.313  0.150  0.150  0.147  0.147  
OR_{2} = 1.500  0.146  0.151  0.147  0.146  
OR_{1} = 1.000  OR_{2} = 0.500  0.148  0.155  0.150  0.150 
OR_{2} = 0.750  0.149  0.150  0.149  0.149  
OR_{2} = 1.000  0.149  0.149  0.148  0.148  
OR_{2} = 1.250  0.150  0.149  0.148  0.148  
OR_{2} = 1.500  0.150  0.149  0.147  0.147  
OR_{2} = 1.750  0.147  0.150  0.147  0.147  
OR_{2} = 2.000  0.147  0.152  0.148  0.146 
1:1 matching  1:4 matching  

Actual SE 


Actual SE 



OR_{1} = 0.500  OR_{2} = 0.250  0.180  0.178  0.181  0.136  0.135  0.136 
OR_{2} = 0.375  0.171  0.172  0.174  0.133  0.132  0.132  
OR_{2} = 0.500  0.171  0.169  0.171  0.131  0.130  0.131  
OR_{2} = 0.625  0.168  0.168  0.169  0.131  0.129  0.130  
OR_{2} = 0.750  0.166  0.166  0.168  0.128  0.129  0.129  
OR_{2} = 0.875  0.166  0.166  0.167  0.130  0.129  0.129  
OR_{2} = 1.000  0.169  0.165  0.167  0.131  0.129  0.129  
OR_{1} = 0.750  OR_{2} = 0.375  0.169  0.169  0.170  0.130  0.132  0.133 
OR_{2} = 0.563  0.170  0.165  0.166  0.129  0.129  0.130  
OR_{2} = 0.750  0.165  0.163  0.164  0.129  0.128  0.129  
OR_{2} = 0.938  0.162  0.162  0.163  0.129  0.127  0.128  
OR_{2} = 1.125  0.162  0.162  0.163  0.127  0.127  0.128  
OR_{2} = 1.313  0.160  0.162  0.163  0.124  0.127  0.128  
OR_{2} = 1.500  0.164  0.162  0.163  0.126  0.128  0.128  
OR_{1} = 1.000  OR_{2} = 0.500  0.165  0.166  0.168  0.129  0.132  0.132 
OR_{2} = 0.750  0.165  0.163  0.165  0.128  0.129  0.129  
OR_{2} = 1.000  0.164  0.162  0.164  0.126  0.128  0.128  
OR_{2} = 1.250  0.164  0.162  0.163  0.126  0.128  0.128  
OR_{2} = 1.500  0.163  0.162  0.164  0.127  0.128  0.128  
OR_{2} = 1.750  0.160  0.163  0.164  0.126  0.128  0.128  
OR_{2} = 2.000  0.160  0.163  0.165  0.125  0.128  0.129 
5 Concluding remarks
The MantelHaenszel estimators have been widely applied in epidemiological and clinical researches involving metaanalysis 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 MantelHaenszel 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 MantelHaenszel 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 MantelHaenszel 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 stratumspecific 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 MantelHaenszel estimators would be a meaningful information for practices in epidemiological and clinical researches.
References
Breslow, N. E. (1981). Odds ratio estimators when the data are sparse. Biometrika, 68:73–84. Search in Google Scholar
Cochran, W. G. (1954). Some methods for strengthening the common chisquare tests. Biometrics, 10:417–451. Search in Google Scholar
Fujii, Y., and Yanagimoto, T. (2005). Pairwise conditional score functions: a generalization of the MantelHaenszel estimator. Journal of Statistical Planning and Inference, 128:1–12. Search in Google Scholar
Godambe, V. P. (1969). An optimum property of regular maximum likelihood estimation. Annals of Mathematical Statistics, 31:1208–1212. Search in Google Scholar
Greenland, S. (1982). Interpretation and estimation of summary ratios under heterogeneity. Statistics in Medicine, 1:217–227. Search in Google Scholar
Greenland, S. (1987). Interpretation and choice of effect measures in epidemiologic analysis. American Journal of Epidemiology, 125:761–768. Search in Google Scholar
Greenland, S., and Maldonado, G. (1994). The interpretation of multiplicativemodel parameters as standardized parameters. Statistics in Medicine, 13:989–999. Search in Google Scholar
Greenland, S., and Robins, J. (1985). Estimation of a common effect parameter from sparse followup data. Biometrics, 41:55–68. Search in Google Scholar
Hattori, S., and Henmi, M. (2012). Estimation of treatment effects based on possibly misspecified Cox regression. Lifetime Data Analysis, 18:408–433. Search in Google Scholar
Hauck, W. W. (1979). The large sample variance of the MantelHaenszel estimator of a common odds ratio. Biometrics, 35:817–819. Search in Google Scholar
Higgins, J. P. T., and Green, S. (2008). Cochrane Handbook for Systematic Reviews of Interventions. Chichester: WileyBlackwell. Search in Google Scholar
Mantel, N., Brown, C., and Byar, D. P. (1977). Tests for homogeneity of effect in an epidemiologic investigation. American Journal of Epidemiology, 106:125–129. Search in Google Scholar
Mantel, N., and Haenszel, W. H. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute, 22:719–748. Search in Google Scholar
Matsuyama, Y., Tominaga, T., Nomura, Y., et al. (2000). Second cancers after adjuvant tamoxifen therapy for breast cancer in Japan. Annals of Oncology, 11:1537–1543. Search in Google Scholar
Nurminen, M. (1981). Asymptotic efficiency of general noniterative estimators of common relative risk. Biometrika, 68:525–530. Search in Google Scholar
Robins, J. M., Breslow, N., and Greenland, S. (1986). Estimators of the MantelHaenszel variance consistent in both sparse data and largestrata limiting models. Biometrics, 42:311–323. Search in Google Scholar
Rothman, K. J. (2002). Epidemiology: An Introduction. New York: Oxford University Press. Search in Google Scholar
Rothman, K. J., Greenland, G., and Lash, T. L. (2008). Modern Epidemiology. 3rd Edition. Philadelphia: Lippincott Williams & Wilkins. Search in Google Scholar
Sato, T. (1989). On variance estimator for the MantelHaenszel risk difference. Biometrics, 45:1323–1324. Search in Google Scholar
Sato, T. (1990). Confidence intervals for effect parameters common in cancer epidemiology. Environmetal Health Perspectives, 87: 95–101. Search in Google Scholar
Sato, T., and Matsuyama, Y. (2003). Marginal structural models as a tool for standardization. Epidemiology, 14:680–686. Search in Google Scholar
Tarone, R. E. (1981). On summary estimators of relative risk. Journal of Chronic Diseases, 34:463–468. Search in Google Scholar
Walker, A. M. (1985). Small sample properties of some estimators of a common hazard ratio. Applied Statistics, 34:42–48. Search in Google Scholar
Walker, A. M., Lanza, L. L., Arellano, F., and Rothman, K. J. (1997). Mortality in current and former users of clozapine. Epidemiology, 8:671–677. Search in Google Scholar
White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50:1–9. Search in Google Scholar
Xu, R., and O‘Quigley, J. (2000). Estimating average regression effect under nonproportional hazards. Biostatistics, 1:423–439. Search in Google Scholar
Yanagimoto, T. (1990). Combining moment estimates of a parameter common through strata. Journal of Statistical Planning and Inference, 25:187–198. Search in Google Scholar
Yi, G. Y., and Reid, N. (2010). A note on misspecified estimating functions. Statistica Sinica, 20:1749–1769. Search in Google Scholar
Appendix
In this appendix, we describe outline of proofs of derivation of asymptotic distributions of the MantelHaenszel estimators. Because the functional forms of the MantelHaenszel 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 MantelHaenszel estimating function
Because of the law of large number,
Asymptotic II. For the sparse data limiting model
© 2016 by De Gruyter