Assessing the Homogeneity of Three Odds Ratios: A Case Study in Small-Sample Inference

  • John B. Carlin
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 162)


In an experiment on the effects of different types of high-frequency ventilation on lung damage in rabbits, six groups of 6 to 8 animals each were compared, using a factorial treatment structure of 3 frequency values crossed with 2 amplitudes. The resulting data were reduced to binary outcomes for each animal, producing a 3 × 2 × 2 contingency table. Although the numbers were small, there appeared to be a large effect of amplitude at the two extreme frequency levels, but there were no failures at either amplitude in the middle frequency group. The question of interest was whether the data provided evidence that the effect of amplitude differed between the 3 frequencies and in particular whether the effect in the middle group was lower than in the two extreme groups. Various models were considered for the 3 odds ratios in question, all seeking to incorporate minimally informative prior assumptions. Because of the small numbers, sensitivity to prior distribution specifications was considerable, and we compared the effect of assuming independent prior distributions on each cell in the 3 × 2 factorial with that of using a more structured prior distribution incorporating exchangeable row, column and interaction effects. Only under rather strong prior assumptions could it be concluded that there was substantial evidence of non-homogeneity. The analysis provides a case study of the sensitivity of inferences in small samples, in an example where the popular exact frequentist approach, based on a null hypothesis of equality of the odds ratios, breaks down.


Posterior Distribution Prior Distribution Middle Frequency Marginal Total Lung Damage 
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  1. Breslow, N.E. and Day, N.E. (1980). Statistical Methods in Cancer Research I: Analysis of Case-Control Studies. Lyon: IARC.Google Scholar
  2. Carlin, J.B. (2000) Meta-analysis: formulating, evaluating, combining, and reporting by S-L. T. Normand [letter]. Statistics in Medicine 19, 753–759.CrossRefGoogle Scholar
  3. Cox, D.R. (1970). Analysis of Binary Data. London: Methuen. [See also 2nd ed. (1989) with E.J. Snell. Chapman & Hall: London.]MATHGoogle Scholar
  4. Cytel Software Corporation (1999). StatXact software package. Cambridge, Massachusetts. (Web address: Scholar
  5. Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. (1995). Bayesian Data Analysis. Chapman & Hall: London.Google Scholar
  6. Leonard, T. (1975) Bayesian estimation methods for two-way contingency tables. JRSS-B, 23–37.Google Scholar
  7. Leonard, T. and Hsu, J.S.J. (1994) The Bayesian analysis of categorical data-a selective review. In Freeman, RR. and Smith, A.F.M. (eds) Aspects of Uncertainty. Chichester: John Wiley.Google Scholar
  8. Mehta, C.R., Patel, N.R., Gray, R. (1985) On computing an exact confidence interval for the common odds ratio in several 2 × 2 contingency tables. JASA 80, 969–973.MathSciNetMATHGoogle Scholar
  9. MRC Biostatistics Unit (1999). WinBUGS software package, version 1.2 (beta). Cambridge, U.K. (Web address Scholar
  10. Zelen, M. (1971). The analysis of several contingency tables. Biometrika 58, 129–137.MathSciNetMATHGoogle Scholar

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© Springer Science+Business Media New York 2002

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  • John B. Carlin

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