### Abstract

Length-biased sampling is frequently a convenient technique for the collection of positive-valued or lifetime data. Estimation procedures based on pseudo-likelihood are developed under the proportional hazards model for length-biased data. Bias-adjusted risk set sampling is used for the construction of the pseudo-likelihood and the risk set sampling is replicated to improve estimation performance. The average of the resulting likelihood estimators is taken as the estimator for the regression coefficients. Although the replication procedure is expected to improve estimation accuracy when the sample size is small or moderate, it does not increase the asymptotic efficiency for estimating the regression coefficient. An estimator for the baseline survival function is also presented. It reduces to the maximum likelihood estimator when the regression parameter in the proportional hazards model is zero. Statistical properties of the proposed estimation procedures are developed. Examples are presented to illustrate the methods.

Original language | English (US) |
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Pages (from-to) | 343-354 |

Number of pages | 12 |

Journal | Biometrika |

Volume | 83 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1996 |

### Keywords

- Case-control analysis
- Partial likelihood
- Pseudo-likelihood
- Risk set sampling

### 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

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## Cite this

*Biometrika*,

*83*(2), 343-354. https://doi.org/10.1093/biomet/83.2.343