Proportional hazards model

Cox regression is a named after David Cox regression analysis for modeling survival times. It is based on the concept of failure rate.

Model

The approach proposed by Cox regression model is used to study the behavior of the failure rate as a function of environmental influences. Of the model are based influence vectors that can be observed for each individual in the study. The relationship between these factors and the loss function is then the relation

Produced. denotes an unknown loss function (that ) represents the associated loss function in the base case without influences. is an unknown parameter, also q- dimensional. Task of statistics is the estimation of this parameter.

The observations

The observations made ​​in the model of Cox regression of a triple, where as above denotes the influence vector for the individual.

Is ( as in the case of the investigation of censored data common) defined as the minimum of two random variables. In the case of death of an individual are actually observed at the time of death of. In contrast, if only the trial was stopped, specifies the termination date. It is obvious that only an observation of death conclusions regarding the shape of the hazard function can be closed. Therefore, specifies whether the death or the end of the study was observed. in this case denotes the indicator function.

The estimate of

Due to the structure of the problem arises that in intervals without death no conclusions can be drawn on. Finally, it is possible that the unknown baseline hazard function vanishes in this interval and thus a priori can be no deaths. We turn therefore to a trick and considered conditional probabilities.

If only then information can be obtained about when a death has occurred, provides at the time of the death of an individual to calculate the following probability: how likely is it that of all the surviving individuals, of all people die? Formally, they can be used as

Calculate. denotes those individuals living at the time of the death of yet.

In order to find a kind of maximum likelihood estimator for, is now a function of the likelihood function

Maximized. This is supported by the exponentiation of the individual conditional probabilities with the fact that only the observation of a death and not the end of the study provides information about.