Research Designs and Causal Inferences: On Lord’s Paradox

  • Paul W. Holland
  • Donald B. Rubin
Part of the Lecture Notes in Statistics book series (LNS, volume 38)


Lord’s Paradox is analyzed in terms of a simple mathematical model for causal inference. The resolution of Lord’s Paradox from this perspective has two aspects. First, the descriptive, non-causal conclusions of the two hypothetical statisticians are both correct. They appear contradictory only because they describe quite different aspects of the data. Second, the causal inferences of the statisticians are neither correct nor incorrect since they are based on different assumptions that our mathematical model makes explicit, but neither assumption can be tested using the data set that is described in the example. We identify these differing assumptions and show how each may be used to justify the differing causal conclusions of the two statisticians. In addition to analyzing the classic “diet” example which Lord used to introduce his paradox, we also examine three other examples that appear in the three papers where Lord discusses the paradox and related matters.


Causal Effect Experimental Manipulation Causal Inference Causal Statement Intact Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anderson, S. B. et al. 1973. Encyclopedia of Educational Evaluation. San Francisco, CA: Jossey-Bass.Google Scholar
  2. Evans, S. H. and Anastasio, E. J. 1968. Misuse of analysis of covariance when treatment effect and covariate are confounded. Psychological Bulletin, 69:225–234.CrossRefGoogle Scholar
  3. Games, P. A. 1976. Limitations of analysis of covariance on intact group quasi-experimental designs. Journal of Experimental Education, 44:51–54.Google Scholar
  4. Holland, P. W. and Rubin, D. B. 1980. Causal inference in case-control studies. Jerome Cornfield Memorial Lecture, American Statistical Association Meetings, Houston, August.Google Scholar
  5. Lindley, D. V. and Novick, M. R. 1981. The role of exchangeability in inference. Annals of Statistics 9:45–58.MathSciNetMATHCrossRefGoogle Scholar
  6. Lord, F. M. 1967. A paradox in the interpretation of group comparisons. Psychological Bulletin 68:304–305.CrossRefGoogle Scholar
  7. Lord, F. M. 1968. Statistical adjustments when comparing preexisting groups. Psychological Bulletin 72:336–337.CrossRefGoogle Scholar
  8. Lord, F. M. 1973. Lord’s paradox. In Encyclopedia of Educational Evaluation. Anderson, S.B. et al. San Francisco: Jossey-Bass.Google Scholar
  9. Rosenbaum, P. R. and Rubin, D. B. 1982. The central role of the propensity score in observational studies. Biometrika.Google Scholar
  10. Rubin, D. B. 1974. Estimating causal effect of treatments in randomized and non-randomized studies. Journal of Educational Psychology 66:688–701.CrossRefGoogle Scholar
  11. Rubin, D. B. 1977. Assignment of treatment group on the basis of a covariate. Journal of Educational Statistics 2:1–26.CrossRefGoogle Scholar
  12. Rubin, D. B. 1978. Bayesian inference for causal effects: The role of randomization. The Annals of Statistics 7:34–58.CrossRefGoogle Scholar
  13. Rubin, D. B. 1980. Discussion of “Randomization analysis of experimental data in the Fisher randomization test,” by Basu. The Journal of the American Statistical Association 75:591–593.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Paul W. Holland
  • Donald B. Rubin

There are no affiliations available

Personalised recommendations