Statistical Science

Confounding and Collapsibility in Causal Inference

Sander Greenland, Judea Pearl, and James M. Robins

Full-text: Open access

Abstract

Consideration of confounding is fundamental to the design and analysis of studies of causal effects. Yet, apart from confounding in experimental designs, the topic is given little or no discussion in most statistics texts. We here provide an overview of confounding and related concepts based on a counterfactual model for causation. Special attention is given to definitions of confounding, problems in control of confounding, the relation of confounding to exchangeability and collapsibility, and the importance of distinguishing confounding from noncollapsibility.

Article information

Source
Statist. Sci. Volume 14, Number 1 (1999), 29-46.

Dates
First available in Project Euclid: 24 December 2001

Permanent link to this document
http://projecteuclid.org/euclid.ss/1009211805

Digital Object Identifier
doi:10.1214/ss/1009211805

Zentralblatt MATH identifier
02068896

Citation

Greenland, Sander; Robins, James M.; Pearl, Judea. Confounding and Collapsibility in Causal Inference. Statistical Science 14 (1999), no. 1, 29--46. doi:10.1214/ss/1009211805. http://projecteuclid.org/euclid.ss/1009211805.


Export citation

References

  • Aldrich, J. (1995). Correlations genuine and spurious in Pearson and Yule. Statist. Sci. 10 364-376.
  • Asmussen, S. and Edwards, D. (1983). Collapsibility and response variables in contingency tables. Biometrika 70 567- 578.
  • Balke, A. and Pearl, J. (1994). Counterfactual probabilities: computational methods, bounds, and applications. In Uncertainty in Artificial Intelligence 10 (R. Mantaras and D. Poole, eds.) 46-54. Morgan Kaufmann, San Francisco.
  • Becher, H. (1992). The concept of residual confounding in regression models and some applications. Statistics in Medicine 11 1747-1758.
  • Berkane, M., ed. (1997). Latent Variable Modeling and Applications to Causality. Springer, New York.
  • Bishop, Y. M. M., Fienberg, S. E. and Holland, P. W. (1975). Discrete Multivariate Analysis: Theory and Practice. MIT Press.
  • Bross, I. D. J. (1967). Pertinency of an extraneous variable. J. Chronic Disease 20 487-495.
  • Clayton, D. and Hills, M. (1993). Statistical Models in Epidemiology. Oxford Univ. Press.
  • Clogg, C. C., Petkova, E. and Shihadeh, E. S. (1992). Statistical methods for analyzing collapsibility in regression models. J. Educ. Statist. 17 51-74.
  • Clogg, C. C., Petkova, E. and Haritou, A. (1995). Statistical methods for comparing regression coefficients between models (with discussion). Amer. J. Sociololgy 100 1261-1305.
  • Cohen, M. R. and Nagel, E. (1934). An Introduction to Logic and the Scientific Method. Harcourt Brace, New York.
  • Copas, J. B. (1973). Randomization models for matched and unmatched 2 × 2 tables. Biometrika 60 467-476.
  • Copas, J. B. and Li, H. G. (1997). Inference for non-random samples (with discussion). J. Roy. Statist. Soc. Ser. B 59 55-95.
  • Cornfield, J. (1976). Recent methodological contributions to clinical trials. Amer. J. Epidemiol. 104 408-421. Cornfield, J., Haenszel, W., Hammond, W. C., Lilienfeld,
  • A. M., Shimkin, M. B. and Wynder, E. L. (1959). Smoking and lung cancer: recent evidence and a discussion of some questions. J. Nat. Cancer Inst. 22 173-203.
  • Cox, D. R. (1958). The Planning of Experiments. Wiley, New York.
  • Dawid, A. P. (2000). Causal inference without counterfactuals. J. Amer. Statist. Assoc. To appear.
  • Ducharme, G. R. and LePage, Y. (1986). Testing collapsibility in contingency tables. J. Roy. Statist. Soc. Ser. B 48 197-205.
  • Feynman, R. P. (1963). Lectures on Physics. Addison-Wesley, Reading, MA.
  • Fisher, R. A. (1918). The causes of human variability. Eugenics Rev. 10 213-220.
  • Fisher, R. A. (1935). The Design of Experiments. Oliver and Boyd, Edinburgh.
  • Frydenberg, M. (1990). Marginalization and collapsibility in graphical statistical models. Ann. Statist. 18 790-805.
  • Gail, M. H. (1986). Adjusting for covariates that have the same distribution in exposed and unexposed cohorts. In Modern Statistical Methods in Chronic Disease Epidemiology (S. H. Moolgavkar, and R. L. Prentice, eds.) 3-18. Wiley, New York.
  • Gail, M. H., Wieand, S. and Piantadosi, S. (1984). Biased estimates of treatment effect in randomized experiments with nonlinear regressions and omitted covariates. Biometrika 71 431-444.
  • Galles, D. and Pearl, J. (1998). An axiomatic characterization of causal counterfactuals. Found. Sci. 3 151-182.
  • Geng, Z. (1989). Algorithm AS 299. Decomposability and collapsibility for log-linear models. J. Roy. Stat. Soc. Ser. C 38 189-197.
  • Geng, Z. (1992). Collapsibility of relative risk in contingency tables with a response variable. J. Roy Statist. Soc. Ser. B 54 585-593.
  • Goetghebeur, E. and van Houwelingen, H., eds. (1998). Analyzing noncompliance in clinical trials. Statististic in Medicine 17 247-389.
  • Greenland, S. (1987). Interpretation and choice of effect measures in epidemiologic analyses. Amer. J. Epidemiol. 125 761-768.
  • Greenland, S. (1990). Randomization, statistics, and causal inference. Epidemiology 1 421-429.
  • Greenland, S. and Mickey, R. M. (1988). Closed-form and dually consistent methods for inference on collapsibility in 2 × 2 × K and 2 × J × K tables. J. Roy. Statist. Soc. Ser. C 37 335-343.
  • Greenland, S. and Robins, J. M. (1986). Identifiability, exchangeability, and epidemiological confounding. Internat. J. Epidemiol. 15 413-419.
  • Greenland, S. and Robins, J. M. (1988). Conceptual problems in the definition and interpretation of attributable fractions. Amer. J. Epidemiol. 128 1185-1197.
  • Greenland, S., Pearl J. and Robins, J. M. (1999). Causal diagrams for epidemiologic research. Epidemiology 10 37-48.
  • Groves, E. R. and Ogburn, W. F. (1928). American Marriage and Family Relationships 160-164. Holt, New York.
  • Guo, J. and Geng, Z. (1995). Collapsibility of logistic regression coefficients. J. Roy Statist. Soc. Ser. B 57 263-267.
  • Halloran, M. E. and Struchiner, C. J. (1995). Causal inference for infectious diseases. Epidemiol. 6 142-151.
  • Hamilton, M. A. (1979). Choosing a parameter for 2 × 2 table or 2 × 2 × 2 table analysis. Amer. J. Epidemiol. 109 362-375. Hauck, W. W., Neuhas, J. M., Kalbfleisch, J. D. and Ander
  • son, S. (1991). A consequence of omitted covariates when estimating odds ratios. J. Clin. Epidemiol. 44 77-81.
  • Hausman, J. (1978). Specification tests in econometrics. Econometrica 46 1251-1271.
  • Heckman, J. J. and Hotz, V. J. (1989). Choosing among alternative nonexperimental methods for estimating the impact of social programs: The case of manpower training (with discussion). J. Amer. Statist. Assoc. 84 862-874.
  • Holland, P. W. (1986). Statistics and causal inference (with discussion). J. Amer. Statist. Assoc. 81 945-970.
  • Hume, D. (1739). A Treatise of Human Nature. Oxford Univ. Press. (Reprinted 1888.)
  • Hume, D. (1748). An Enquiry Concerning Human Understanding. Open Court Press, LaSalle. (Reprinted 1888.)
  • Kalbfleisch, J. D. and Prentice, R. L. (1980). The Statistical Analysis of Failure-Time Data. Wiley, New York. Kelsey, J. L., Whittemore, A. S., Evans, A. S. and Thompson,
  • W. D. (1996). Methods in Observational Epidemiology, 2nd ed. Oxford Univ. Press.
  • Kitagawa, E. M. (1955). Components of a difference between two rates. J. Amer. Statist. Assoc. 50 1168-1194.
  • Lauritzen, S. L. (1996). Graphical Models. Clarendon Press, Oxford. Lewis, D. (1973a). Causation. J. Philos. 70 556-567. Lewis, D. (1973b). Counterfactuals. Blackwell, Oxford.
  • MacMahon, B. and Pugh, T. F. (1967). Causes and entities of disease. In Preventive Medicine (D. W. Clark and B. MacMahon, eds.) 11-18. Little Brown, Boston.
  • McKechnie, J. L. (ed.) (1979). Webster's New Twentieth Century Dictionary. Simon and Schuster, New York.
  • Miettinen, O. S. (1972). Components of the crude risk ratio. Amer. J. Epidemiol. 96 168-172.
  • Miettinen, O. S. and Cook, E. F. (1981). Confounding: essence and detection. Amer. J. Epidemiol. 114 593-603.
  • Mill, J. S. (1843). A System of Logic, Ratiocinative and Inductive. (Reprinted 1956 by Longmans, Green, London.)
  • Mill, J. S. (1862). A System of Logic, Ratiocinative and Inductive, 5th ed. Parker, Bowin, London.
  • Neuhaus, J. M., Kalbfleisch, J. D. and Hauck, W. W. (1991). A comparison of cluster-specific and population-averaged approaches for analyzing correlated binary data. Internat. Statist. Rev. 59 25-35.
  • Neyman, J. (1923). Sur les applications de la thar des probabilities aux experiences Agaricales: Essay des principle. [English translation of excerpts (1990) by D. Dabrowska and T. Speed, it Statist. Sci. 5 463-472.]
  • Neyman, J. (1935). Statistical problems in agricultural experimentation (with discussion). J. Roy. Statist. Soc. Suppl. 2 107-180.
  • Pearl, J. (1995). Causal diagrams for empirical research. Biometrika 82 669-710.
  • Pearl, J. (1997). On the identification of nonparametric structural models. In Latent Variable Modeling with Application to Causality (M. Berkane, ed.) 29-68. Springer, New York.
  • Pearl, J. and Robins, J. M. (1995). Probabilitic evaluation of sequential plans from causal model with hidden variables. In Uncertainty in Artificial Intelligence (P. Besnard and S. Hanks, eds.) 11 444-453. Morgan-Kaufman, San Francisco.
  • Prentice, R. L. and Kalbfleisch, J. D. (1988). Author's reply. Biometrics 44 1205.
  • Robins, J. M. (1986). A new approach to causal inference in mortality studies with a sustained exposure period-application to control of the healthy worker survivor effect. Math. Modeling 7 1393-1512. Robins, J. M. (1987a). Addendum to "A new approach to causal inference in mortality studies with a sustained exposure period-application to control of the healthy worker survivor effect." Computers Math. Appl. 14 923-945. Robins, J. M. (1987b). A graphical approach to the identification and estimation of causal parameters in mortality studies with sustained exposure periods. J. Chronic Dis. 40 (suppl. 2) 139s-161s.
  • Robins, J. M. (1988). Confidence intervals for causal parameters. Statistics in Medicine 7 773-785. Robins, J. M. (1995a). Discussion of "Causal diagrams for empirical research" by J. Pearl. Biometrika 82 695-698. Robins, J. M. (1995b). An analytic method for randomized trials with informative censoring. Lifetime Data Analysis 1 241- 254.
  • Robins, J. M. (1997). Causal inference from complex longitudinal data. In Latent Variable Modeling with Applications to Causality (M. Berkane, ed.) Springer, New York, 69-117.
  • Robins, J. M. (1998). Correction for non-compliance in equivalence trials. Statistics in Medicine 17 269-302.
  • Robins, J. M. and Greenland, S. (1986). The role of model selection in causal inference from nonexperimental data. Amer. J. Epidemiol. 123 393-402.
  • Robins, J. M. and Greenland, S. (1989). The probability of causation under a stochastic model for individual risks. Biometrics 46 1125-1138.
  • Robins, J. M. and Greenland, S. (1992). Identifiability and exchangeability for direct and indirect effects. Epidemiology 3 143-155.
  • Robins, J. M. and Greenland, S. (1994). Adjusting for differential rates of prophylaxis therapy for PCP in high versus low dose AZT treatment arms in an AIDS randomized trial. J. Amer. Statist. Assoc. 89 737-749.
  • Robins, J. M. and Morgenstern, H. (1987). The mathematical foundations of confounding in epidemiology. Computers Math. Appl. 14 869-916.
  • Robins, J. M. and Ritov, Y. (1997). Toward a curse-of-dimensionality appropriate (CODA) asymptotic theory for semiparametric models. Statistics in Medicine 16 285-319.
  • Robins, J. M., Rotnitzky, A. and Scharfstein, D. O. (1999). Sensitivity analysis for selection bias and unmeasured confounding in missing data and causal inference models. In Statistical Models in Epidemiology (E. Halloran, ed.) Springer, New York.
  • Robinson, L. D. and Jewell, N. P. (1991). Some surprising results about covariate adjustment in logistic regression. Int. Statist. Rev. 59 227-240.
  • Rosenbaum, P. R. (1995). Observational Studies. Springer, New York.
  • Rosenbaum, P. R. and Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika 70 41-55.
  • Rothman, K. J. (1977). Epidemiologic methods in clinical trials. Cancer 39 1771-1775.
  • Rothman, K. J. and Greenland, S. (1998). Modern Epidemiology, 2nd ed. Lippincott-Raven, Philadelphia.
  • Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. J. Educ. Psych. 66 688-701.
  • Rubin, D. B. (1978). Bayesian inference for causal effects: the role of randomization. Ann. Statist. 6 34-58.
  • Rubin, D. B. (1990). Comment on "Neyman (1923) and causal inference in experiments and observational studies." Statist. Sci. 5 472-480.
  • Rubin, D. B. (1991). Practical implications of modes of statistical inference for causal effects and the critical role of the assignment mechanism. Biometrics 47 1213-1234.
  • Senn, S. (1989). Covariate imbalance and random allocation in clinical trials. Statistics in Medicine 8 467-475.
  • Simon, H. A. and Rescher, N. (1966). Cause and counterfactual. Philos. Sci. 33 323-340.
  • Simpson, E. H. (1951). The interpretation of interaction in contingency tables. J. Roy. Statis. Soc. Ser. B 13 238-241. [Reprinted in (1987) The Evolution of Epidemiologic Ideas (S. Greenland, ed.) 103-107. ERI Press, Chestnut Hill, MA.]
  • Slud, E. V., Byar, D. P. and Schatzkin, D. P. (1988). Dependent competing risks and the latent-failure model. Biometrics 44 1203-1204.
  • Sobel, M. E. (1995). Causal inference in the social and behavioral sciences. In Handbook of Statistical Modeling for the Social and Behavioral Sciences (G. Arminger, C. C. Clogg, and M. E. Sobel, eds.) Plenum Press, New York.
  • Stalnaker, R. C. (1968). A theory of conditionals. In Studies in Logical Theory (N. Rescher, ed.) Blackwell, Oxford.
  • Stone, R. (1993). The assumptions on which causal inference rest. J. Roy. Statist. Soc. Ser. B 55 455-466.
  • Wermuth, N. (1987). Parametric collapsibility and lack of moderating effects in contingency tables with a dichotomous response variable. J. Roy. Statist. Soc. Ser. B 49 353-364.
  • Wermuth, N. (1989). Moderating effects of subgroups in linear models. Biometrika 76 81-92.
  • White, H. A. (1994). Estimation, Inference, and Specification Analysis. Cambridge Univ. Press.
  • Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Wiley, New York.
  • Whittemore, A. S. (1978). Collapsing multidimensional contingency tables. J. Roy. Statist. Soc. Ser. B 40 328-340.
  • Wickramaratne, P. and Holford, T. (1987). Confounding in epidemiologic studies: the adequacy of the control group as a measure of confounding. Biometrics 43 751-765.
  • Yule, G. U. (1903). Notes on the theory of association of attributes in statistics. Biometrika 2 121-134.