Research Designs and Causal Inferences: On Lord’s Paradox

Abstract

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.

The preparation of this paper was supported in part by the Program Statistics Research Project, Educational Testing Service, Princeton, New Jersey 08541. This chapter was prepared for the Festschrift in honor of Frederick M. Lord, May 22–23, 1982, and was the basis of a presentation at the Social Science Research Council’s Seminar, Designing Research With Scarce Resources, November 11–12, 1982, Washington, D. C.