Revisit Partial Likelihood and Tie Corrections for the Cox Model Using Poisson Binomial Distribution

Oregon Convention Center 

In a Cox model, the partial likelihood (PL), as the product of a series of conditional probabilities, is used to estimate the parameter. In practice, those conditional probabilities are approximated by risk score ratios based on a continuous time model, and thus result in parameter estimates from only an approximate PL. Through a revisit to the original PL idea, we propose an accurate PL computing method for the Cox model, which calculates the exact conditional probability using the Poisson binomial distribution (PBD). We develop new estimating and inference procedures and establish asymptotic theory for the new procedure. Although ties are common in real studies, current theory for the Cox model does not allow ties. In contrast, our new approach includes the theory for grouped data, which allows ties. Our theory for the new method is also valid for continuous data without ties, thus, providing a unified framework for computing PL for data with or without ties. From simulation and real applications in several datasets, we show that the proposed method is superior to current methods in reducing bias, especially when there are ties or when the variability in risk scores is large.

Breslow Estimator

Efron Estimator

Survival Data with/without Ties

Kalbfleisch-Prentice Correction

Proportional Hazards Model 

Lifetime Data Science Section