### Abstract

When data can be presented as a series of k 2×2 tables with cell counts (x_{i} ,n_{i}-xi, yi, m_{i}-y_{i}), it is often assumed that x_{i} and y_{i} are binomially distributed. This paper deals with inference for the common odds ratio Ψ when the binomial assumption is invalid. When k increases, the consistency and asymptotic normality of the Mantel & Haenszel (1959) estimator is derived. The conditional maximum likelihood estimator is shown to be inconsistent and the asymptotic bias is computed when either the first-order Markov chain or beta-binomial model is assumed. The Mantel-Haenszel test for testing iΨ = 1 is also shown to be inappropriate through some simulation studies. Two consistent test statistics are proposed and shown to be comparable to each other in terms of size and efficiency. Some possible further work is described.

Original language | English (US) |
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Pages (from-to) | 678-682 |

Number of pages | 5 |

Journal | Biometrika |

Volume | 72 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 1985 |

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### Keywords

- Binary data
- Efficiency
- Mantel-Haenszel estimate and test
- Two by two table

### ASJC Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Agricultural and Biological Sciences (miscellaneous)
- Agricultural and Biological Sciences(all)
- Statistics, Probability and Uncertainty
- Applied Mathematics

### Cite this

*Biometrika*,

*72*(3), 678-682. https://doi.org/10.1093/biomet/72.3.678