The Asymptotic Joint Distribution of Regression and Survival Parameter Estimates in the Cox Regression Model

Abstract

In this paper it is shown that the Cox likelihood (Cox, 1972) may be treated as a standard likelihood, in the sense that its maximizer $\hat{\beta}$ is asymptotically normally distributed with asymptotic covariance matrix equal to $-\{E \partial^2 \log L (\beta)/\partial\beta\partial\beta'\}^{-1}$. In the process, an asymptotic representation of the score function is obtained in terms of functions of the independent observations. This representation may have some uses in itself such as: (1) providing a kind of residual for each observation, censored or uncensored, thereby indicating the relative influence of the observations, and (2) providing some information about the applicability of the asymptotics in a particular small sample. The asymptotic joint distribution of $\hat{\beta}$ and of the cumulative hazard function estimator $\hat{Lambda}_0(t)$ is also derived via a representation of the latter involving an independent increments process. Bailey (1982) shows that the "joint likelihood function" of the regression parameters $\beta$ and of the cumulative hazard jump parameters $\{\Lambda_i\}$ can be used in a natural way to obtain consistent estimates of these joint asymptotic covariances in the case of no ties. This justifies, to some extent, use of the general ML method for joint estimation of $\beta$ and $\Lambda_0(t)$.