Cox Proportional Hazards Regression

In the analysis of survival data, researchers want to ascertain characteristics of the patient that influence patient survival time. The relationship between a single response variable (survival time) and covariates (patient/disease characteristics) is often inferred through the use of a regression model. Typical regression models, such as linear or logistic regression, do not work when the response variable is survival time, since the time to death may not be recorded for all patients at the time of analysis. If a patient is still alive at the time of analysis or has been lost to follow-up, the patient survival time is said to have been right censored (or simply censored) at the time of the last observed follow-up. If the patient has been lost to follow-up, an important assumption in many survival analytic methods is that the reason a patient is lost to follow-up is unrelated to the risk of death.

In survival analysis, when the survival time, T, is possibly right censored, the Cox proportional hazards model is the predominant regression model. The proportional hazards model is written as

where h(t|X) is the hazard function conditional on a set of patient-specific covariates, which is denoted by the vector X, and b represents the vector of regression coefficients that determines the relationship between the covariates and the risk of death. Covariates in the Cox model are handled using standard regression techniques. Thus, categorical factors may be entered into the model using dummy variables, and interactions may be introduced through the multiplication of two covariates. However, due to complications that stem from censored observations, additional methodology, based on what is called the partial likelihood, is needed for estimation of the regression coefficients b.

The conditional hazard h(t|X) provides the patient-specific risk of death over time. The proportional hazards specification, Equation 1, divides the conditional hazard into two components, a baseline hazard function h0(t) independent of the patient characteristic vector and the patient relative risk function, exp[bTX], independent of time; the relationship between the two components is multiplicative. The baseline hazard function is left unspecified but governs how the patient-specific hazard varies over time. Heuristically, the hazard function is proportional to the probability of death by time t, given the patient has not died prior to time t. For any two patients with characteristics X1 and X2, the ratio of their conditional hazards,

is independent of time. The term proportional hazards refers to the fact that the two conditional hazards are proportional to each other, with the proportionality constant equal to exp[bT(X1 – X2)].

The widespread popularity of the proportional hazards methodology stems from the interpretation of the regression parameter, b, as a relative risk parameter constant with respect to time, the accuracy of the estimate of the relative risk parameter in the presence of censored data, the development of inferential procedures that are easy to implement with available software, and the efficiency of the regression parameters for a wide range of baseline hazard functions.

An alternative specification of the proportional hazards regression model is through the patient specific (conditional) survival

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