The Cox proportional hazards model, proposed by David Cox in 1972, is a statistical survival model. The purpose of the model is to simultaneously explore the effects of several explanatory variables on survival.
The definition of the Cox proportional hazard model is:
h(t) = h0(t)exp(βX)
- h (t), the hazard function, is the probability that a subject will experience an event (such as machine failure or death) within the time interval t, given that the subject has survived until the beginning of t.
- h0(t), the baseline or underlying hazard function, is the probability of reaching an event when all covariates have the value 0.
h0(t) is unspecified, but cannot be negative.
- A linear function of a set of k fixed covariates is vectorized in X. ß is a vector of coefficients for X. The product ßX is the exponent of e.
Because h0(t) is unspecified, the Cox model is semiparametric.
For example, if the explanatory variables are age, weight, and treatment group, the hazard (or risk) of dying at time t is:
h(t) = h0(t)exp(β age * X age + β weight + X weight + β group + X group )
The model is called proportional because the hazard for any subject is a fixed proportion of the hazard for any other subject. The ratio of the hazards for individuals i and j is:
hi(t) / hj(t) = exp (β1(Xi1 - Xj1) + … + β1(Xik - Xjk)
h 0(t) cancels out of the numerator and denominator; therefore, the ratio of the hazards is constant over time.
If an event does not occur by time t, the event is right censored.