Proportional hazards model

A proportional hazards model writes $h(t\mid z)=h_0(t)r(z)$, so covariates multiply the baseline hazard by a ratio that does not depend on time.

In a proportional hazards model $h(t\mid z)=h_0(t)e^{{\beta }^{T}z}$, the baseline survival function is $S_0(t)=\exp \{-{\int }_0^t{h}_0(u)du\}$, corresponding to the reference covariate vector $z=0$. Integrating the hazard function gives $S(t\mid z)=S_0(t{)}^{e^{{\beta }^{T}z}}$. Estimated hazard ratios alone therefore do not determine absolute survival probabilities: the baseline hazard or baseline survival function must also be estimated.

When event times are observed only in intervals and the covariate vector is constant within an interval, a proportional hazards model gives $p_{ij}=1-\exp [-\Delta H_{0j}\exp ({\beta }^T{z}_i)]$. Thus the interval event probability has a complementary log-log link, with ${\alpha }_j=\log \Delta H_{0j}$ an interval-specific intercept. The person-period Bernoulli likelihood models several events in one interval directly, without inventing their order. It needs appropriate independent censoring assumptions and a consistent account of interval entry and observation.

In a proportional hazards model $h_i(t)=h_0(t){\varphi }_i$, the hazard multiplier ${\varphi }_i>0$ is the relative hazard against the reference multiplier one. In a Cox proportional-hazards model it is $e^{x_i^T\beta }$. Ratios of two time-constant multipliers give constant hazard ratios; time-dependent multipliers require a corresponding extension.

The baseline hazard is the hazard for the reference covariate value whose hazard multiplier equals one. In a semiparametric Cox model it is left unspecified and estimated after the regression coefficients.

A proportional hazards family consists of event-time distributions whose hazard functions are constant multiples of a common baseline hazard.

Consider independent event times with hazard function $h_i(t)=\lambda (t)e^{{\beta }^T{x}_i}$ and baseline cumulative hazard function $\Lambda (t)={\int }_0^t\lambda (s)ds$. For optimization over $\beta$, first hold $\lambda$ and $\Lambda$ fixed and independent of $\beta$. Under ignorable independent censoring, an observation $(y_i,{\delta }_i)$ has log-likelihood contributionSet ${\mu }_i=\Lambda (y_i)e^{{\beta }^T{x}_i}$. A Poisson regression with response ${\delta }_i$ and log link has log-likelihood contribution ${\delta }_i\log {\mu }_i-{\mu }_i-\log ({\delta }_i!)$, which differs from ${\ell }_i$ only by terms independent of $\beta$. Thus the same $\beta$ maximizes both criteria. The linear predictor is ${\beta }^T{x}_i+\log \Lambda (y_i)$, with $\log \Lambda (y_i)$ a known offset whenever $0<\Lambda (y_i)<\infty$.

For uncensored observations every response is one. This is an equivalence of objective functions: the event indicators need not actually be Poisson counts. A related split-interval construction handles piecewise-constant hazard functions.^[ref]

If the baseline is unspecified, its integrated values are not known offsets; fitting an ordinary generalized linear model as if they were known does not estimate an arbitrary Cox proportional-hazards model. One must estimate the baseline jointly, use a suitable baseline representation, or eliminate it through Cox partial likelihood. Zero exposure also needs separate handling rather than taking the logarithm of zero.

A semiparametric proportional hazards model specifies a finite-dimensional hazard ratio while leaving the baseline hazard function unspecified.