Cox proportional hazards model

Proportional hazards models are a class of survival models in statistics. Survival models relate the time that passes, before some event occurs, to one or more covariates that may be associated with that quantity of time. In a proportional hazards model, the unique effect of a unit increase in a covariate is multiplicative with respect to the hazard rate. For example, taking a drug may halve one's hazard rate for a stroke occurring, or, changing the material from which a manufactured component is constructed may double its hazard rate for failure. Other types of survival models such as accelerated failure time models do not exhibit proportional hazards. The accelerated failure time model describes a situation where the biological or mechanical life history of an event is accelerated (or decelerated).

Background

Survival models can be viewed as consisting of two parts: the underlying baseline hazard function, often denoted $\lambda _{0}(t)$ , describing how the risk of event per time unit changes over time at baseline levels of covariates; and the effect parameters, describing how the hazard varies in response to explanatory covariates. A typical medical example would include covariates such as treatment assignment, as well as patient characteristics such as age at start of study, gender, and the presence of other diseases at start of study, in order to reduce variability and/or control for confounding.

The proportional hazards condition^[1] states that covariates are multiplicatively related to the hazard. In the simplest case of stationary coefficients, for example, a treatment with a drug may, say, halve a subject's hazard at any given time $t$ , while the baseline hazard may vary. Note however, that this does not double the lifetime of the subject; the precise effect of the covariates on the lifetime depends on the type of $\lambda _{0}(t)$ . The covariate is not restricted to binary predictors; in the case of a continuous covariate $x$ , it is typically assumed that the hazard responds exponentially; each unit increase in $x$ results in proportional scaling of the hazard.

The Cox model

Introduction

Sir David Cox observed that if the proportional hazards assumption holds (or, is assumed to hold) then it is possible to estimate the effect parameter(s), denoted $\beta _{i}$ below, without any consideration of the full hazard function. This approach to survival data is called application of the Cox proportional hazards model,^[2] sometimes abbreviated to Cox model or to proportional hazards model.^[3] However, Cox also noted that biological interpretation of the proportional hazards assumption can be quite tricky.^[4]^[5]

Let $X i = (X i 1, \dots , X ip)$ be the realized values of the p covariates for subject i. The hazard function for the Cox proportional hazards model has the form

{\begin{aligned}\lambda (t|X_{i})&=\lambda _{0}(t)\exp(\beta _{1}X_{i1}+\cdots +\beta _{p}X_{ip})\\&=\lambda _{0}(t)\exp(X_{i}\cdot \beta )\end{aligned}}

This expression gives the hazard function at time t for subject i with covariate vector (explanatory variables) X_i. Note that between subjects, the baseline hazard

\lambda _{0}(t)

is identical (has no dependency on i). The only difference between subjects' hazards comes from the baseline scaling factor

\exp(X_{i}\cdot \beta )

.

Why it is called "proportional"

To start, suppose we only have a single covariate, $x$ , and therefore a single coefficient, $\beta _{1}$ . Consider the effect of increasing $x$ by 1:

{\begin{aligned}\lambda (t|x+1)&=\lambda _{0}(t)\exp(\beta _{1}(x+1))\\&=\lambda _{0}(t)\exp(\beta _{1}x+\beta _{1})\\&={\Bigl (}\lambda _{0}(t)\exp(\beta _{1}x){\Bigr )}\exp(\beta _{1})\\&=\lambda (t|x)\exp(\beta _{1})\end{aligned}}

We can see that increasing a covariate by 1 scales the original hazard by the constant $\exp(\beta _{1})$ . Rearranging things slightly, we see that:

{\frac {\lambda (t|x+1)}{\lambda (t|x)}}=\exp(\beta _{1})

The right-hand-side is constant over time (no term has a $t$ in it). This relationship, $x/y={\text{constant}}$ , is called a proportional relationship.

More generally, consider two subjects, i and j, with covariates $X_{i}$ and $X_{j}$ respectively. Consider the ratio of their hazards:

{\begin{aligned}{\frac {\lambda (t|X_{i})}{\lambda (t|X_{j})}}&={\frac {\lambda _{0}(t)\exp(X_{i}\cdot \beta )}{\lambda _{0}(t)\exp(X_{j}\cdot \beta )}}\\&={\frac {{\cancel {\lambda _{0}(t)}}\exp(X_{i}\cdot \beta )}{{\cancel {\lambda _{0}(t)}}\exp(X_{j}\cdot \beta )}}\\&=\exp((X_{i}-X_{j})\cdot \beta )\end{aligned}}

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