Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Epidemiologic Methods

Edited by faculty of the Harvard School of Public Health

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

1 Issue per year

See all formats and pricing
More options …

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

Hisashi NomaORCID iD: http://orcid.org/0000-0002-2520-9949 / Kengo NagashimaORCID iD: http://orcid.org/0000-0003-4529-9045
Published Online: 2016-05-06 | DOI: https://doi.org/10.1515/em-2015-0004


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.

Keywords: stratified analysis; Mantel-Haenszel estimators; common effect parameters; standardization; model misspecification


  • Breslow, N. E. (1981). Odds ratio estimators when the data are sparse. Biometrika, 68:73–84.Google Scholar

  • Cochran, W. G. (1954). Some methods for strengthening the common chi-square tests. Biometrics, 10:417–451.Google Scholar

  • Fujii, Y., and Yanagimoto, T. (2005). Pairwise conditional score functions: a generalization of the Mantel-Haenszel estimator. Journal of Statistical Planning and Inference, 128:1–12.Google Scholar

  • Godambe, V. P. (1969). An optimum property of regular maximum likelihood estimation. Annals of Mathematical Statistics, 31:1208–1212.Google Scholar

  • Greenland, S. (1982). Interpretation and estimation of summary ratios under heterogeneity. Statistics in Medicine, 1:217–227.Google Scholar

  • Greenland, S. (1987). Interpretation and choice of effect measures in epidemiologic analysis. American Journal of Epidemiology, 125:761–768.Google Scholar

  • Greenland, S., and Maldonado, G. (1994). The interpretation of multiplicative-model parameters as standardized parameters. Statistics in Medicine, 13:989–999.Google Scholar

  • Greenland, S., and Robins, J. (1985). Estimation of a common effect parameter from sparse follow-up data. Biometrics, 41:55–68.Google Scholar

  • Hattori, S., and Henmi, M. (2012). Estimation of treatment effects based on possibly misspecified Cox regression. Lifetime Data Analysis, 18:408–433.Google Scholar

  • Hauck, W. W. (1979). The large sample variance of the Mantel-Haenszel estimator of a common odds ratio. Biometrics, 35:817–819.Google Scholar

  • Higgins, J. P. T., and Green, S. (2008). Cochrane Handbook for Systematic Reviews of Interventions. Chichester: Wiley-Blackwell.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.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.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.Google Scholar

  • Nurminen, M. (1981). Asymptotic efficiency of general noniterative estimators of common relative risk. Biometrika, 68:525–530.Google Scholar

  • Robins, J. M., Breslow, N., and Greenland, S. (1986). Estimators of the Mantel-Haenszel variance consistent in both sparse data and large-strata limiting models. Biometrics, 42:311–323.Crossref

  • Rothman, K. J. (2002). Epidemiology: An Introduction. New York: Oxford University Press.Google Scholar

  • Rothman, K. J., Greenland, G., and Lash, T. L. (2008). Modern Epidemiology. 3rd Edition. Philadelphia: Lippincott Williams & Wilkins.Google Scholar

  • Sato, T. (1989). On variance estimator for the Mantel-Haenszel risk difference. Biometrics, 45:1323–1324.Google Scholar

  • Sato, T. (1990). Confidence intervals for effect parameters common in cancer epidemiology. Environmetal Health Perspectives, 87: 95–101.Google Scholar

  • Sato, T., and Matsuyama, Y. (2003). Marginal structural models as a tool for standardization. Epidemiology, 14:680–686.Google Scholar

  • Tarone, R. E. (1981). On summary estimators of relative risk. Journal of Chronic Diseases, 34:463–468.Google Scholar

  • Walker, A. M. (1985). Small sample properties of some estimators of a common hazard ratio. Applied Statistics, 34:42–48.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.Google Scholar

  • White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50:1–9.Google Scholar

  • Xu, R., and O‘Quigley, J. (2000). Estimating average regression effect under non-proportional hazards. Biostatistics, 1:423–439.Google Scholar

  • Yanagimoto, T. (1990). Combining moment estimates of a parameter common through strata. Journal of Statistical Planning and Inference, 25:187–198.Google Scholar

  • Yi, G. Y., and Reid, N. (2010). A note on mis-specified estimating functions. Statistica Sinica, 20:1749–1769.Google Scholar

About the article

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, ISSN (Online) 2161-962X, ISSN (Print) 2194-9263, DOI: https://doi.org/10.1515/em-2015-0004.

Export Citation

©2016 by De Gruyter. Copyright Clearance Center

Comments (0)

Please log in or register to comment.
Log in