Inference: Estimating equations

The results of this chapter and, for the most, all of the succeeding chapters, are based on an elementary and central theorem. We call this theorem the main theorem of proportional hazards regression. Its development is essentially that of O'Quigley (2003) which generalizes earlier results of Schoenfeld (1980), O'Quigley and Flandre (1994) and Xu and O'Quigley (2000). The theorem has several immediate corollaries and we can use these to write down estimating equations upon which we can then construct suitable inferential procedures for our models. While a particular choice of estimating equation can result in high efficiency when model assumptions are correct or close to being correct, other equations may be less efficient but still provide estimates which can be interpreted when model assumptions are incorrect. For example, when the regression function β(t) might vary with time we are able to construct an estimating equation, the solution of which provides an estimate of β,in the case where β(t) is a constant β,and E{β(T)},the average effect, in the case where β(t) changes through time. It is worth underlining that the usual partial likelihood estimate fails to achieve this.