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Open Access Research article

The transitive fallacy for randomized trials: If A bests B and B bests C in separate trials, is A better than C?

Stuart G Baker1* and Barnett S Kramer2

Author Affiliations

1 Division of Cancer Prevention, National Cancer Institute

2 Offices of Disease Prevention and Medical Applications of Research, National Institutes of Health

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BMC Medical Research Methodology 2002, 2:13  doi:10.1186/1471-2288-2-13

Published: 13 November 2002

Abstract

Background

If intervention A bests B in one randomized trial, and B bests C in another randomized trial, can one conclude that A is better than C? The problem was motivated by the planning of a randomized trial, where A is spiral-CT screening, B is x-ray screening, and C is no screening. On its surface, this would appear to be a straightforward application of the transitive principle of logic.

Methods

We extended the graphical approach for omitted binary variables that was originally developed to illustrate Simpson's paradox, applying it to hypothetical, but plausible scenarios involving lung cancer screening, treatment for gastric cancer, and antibiotic therapy for clinical pneumonia.

Results

Graphical illustrations of the three examples show different ways the transitive fallacy for randomized trials can arise due to changes in an unobserved or unadjusted binary variable. In the most dramatic scenario, B bests C in the first trial, A bests B in the second trial, but C bests A at the time of the second trial.

Conclusion

Even with large sample sizes, combining results from a previous randomized trial of B versus C with results from a new randomized trial of A versus B will not guarantee correct inference about A versus C. A three-arm trial of A, B, and C would protect against this problem and should be considered when the sequential trials are performed in the context of changing secular trends in important omitted variables such as therapy in cancer screening trials.