Proportional Hazards Model
The proportional hazards model (Cox regression) is a semi-parametric survival regression framework that relates multiple covariates to the instantaneous hazard rate without specifying a parametric distribution for the underlying baseline hazard.
Proportional hazards model (most commonly implemented as the Cox proportional hazards regression model) is a semi-parametric regression model in biostatistics and clinical epidemiology that quantifies the multiplicative effect of one or more explanatory variables (covariates) on the instantaneous rate of an event over time—the hazard function h(t). Introduced by British statistician Sir David Roxbee Cox in his seminal 1972 paper Regression Models and Life-Tables in the Journal of the Royal Statistical Society, the Cox model revolutionized medical statistics and prospective cohort studies by factoring the hazard function into an arbitrary, unspecified non-parametric baseline hazard h0(t) and a parametric exponential linear predictor exp(βTX). While univariate Kaplan–Meier estimators compare survival curves across discrete strata, Cox regression enables simultaneous adjustment for continuous and categorical confounders via partial likelihood estimation.
Semi-Parametric Hazard Architecture
Definition of the Instantaneous Hazard Function
For a continuous time-to-event random variable T with probability density function f(t) and survival function S(t) = Pr(T > t), the instantaneous hazard rate h(t) is the limiting conditional rate of failure in the infinitesimal interval [t, t + Δt) given survival up to time t:
Notice that h(t) is a rate (with units of reciprocal time, such as events per person-year), not a probability; it can take any non-negative value from 0 to ∞.
The Cox Multiplicative Specification and Hazard Ratio (HR)
For an individual i with covariate vector Xi = (Xi1, Xi2, …, Xip)T, the Cox proportional hazards model specifies:
where:
- h0(t) is the baseline hazard function—the instantaneous hazard at time t for a reference individual whose covariates all equal zero (X = 0). Crucially, h0(t) is left unspecified (non-parametric), allowing the model to accommodate Weibull, Gompertz, bathtub-shaped, or fluctuating surgical post-operative hazard trajectories without distributional misspecification.
- exp(βT Xi) is the parametric relative risk component, chosen on the exponential exp(·) scale so that the predicted hazard remains strictly positive for any real coefficient vector β.
When comparing two individuals with covariate vectors Xi and Xj, the baseline hazard h0(t) cancels out in the ratio:
Because h0(t) cancels out, the hazard ratio (HR) between any two fixed covariate profiles is constant over time t—the defining proportional hazards assumption. A one-unit increase in covariate Xk holding all other variables constant multiplies the instantaneous hazard by HRk = exp(βk).
Cox Partial Likelihood Estimation and Tied Event Times
Eliminating the Nuisance Baseline Hazard h0(t)
In 1972 and 1975, Cox showed that β can be estimated without estimating h0(t) by conditioning on the ordered distinct event times t1 < t2 < ··· < tD. Suppose an event occurs at time tm to individual (m), and let R(tm) denote the risk set of all subjects still at risk and uncensored just prior to tm. Given that one failure occurs at tm among the members of R(tm), the conditional probability that individual (m) is the one who fails is:
Multiplying these conditional probabilities over all D observed failure times gives the Cox partial likelihood Lp(β) = ∏m=1D Lm(β), and taking its logarithm yields the partial log-likelihood maximized via Newton–Raphson iteration:
Notice that only the rank ordering of failure and censoring times enters ℓp(β); monotone transformations of the time axis leave β̂ unchanged. When incidence-density sampling is applied to a nested case–control study, the conditional logistic likelihood is identical in functional form to the Cox partial likelihood.
Handling Tied Failure Times and Diagnostic Extensions
| Analytical Feature or Diagnostic | Mathematical Mechanism | Primary Epidemiological Application |
|---|---|---|
| Breslow & Efron Tie Approximations | Adjusts the risk-set denominator ∑ exp(βTXj) when dm > 1 events occur at the same recorded time tm. | Discrete clinical follow-up intervals; Efron's approximation (1977) provides much higher accuracy than Breslow's when ties are frequent. |
| Schoenfeld Residuals (rmk) | rmk = X(m)k − X̄k(tm), comparing the failing subject's covariate to the risk-set weighted mean at tm. | Grambsch–Therneau test for non-proportional hazards: a non-zero slope of scaled Schoenfeld residuals against time t indicates time-varying βk(t). |
| Stratified Cox Regression | hs(t | X) = h0s(t) exp(βTX), allowing a separate baseline hazard h0s(t) per stratum s while pooling β. | Controls for categorical variables (e.g., clinic center or birth cohort on a Lexis diagram) that violate proportional hazards. |
| Time-Varying Covariates (X(t)) | h(t | X(t)) = h0(t) exp[βTX(t)] via Andersen–Gill counting-process intervals (tstart, tstop]. | Prevents immortal time bias when modeling post-baseline treatments, organ transplants, or dynamic biomarker trajectories. |
Frequently Asked Questions
How do you check whether the proportional hazards assumption is violated?
Epidemiologists test the proportional hazards assumption via three complementary methods: (1) plotting log-cumulative hazard curves ln[−ln Ŝ(t)] against ln(t) for categorical groups to verify that the curves remain parallel; (2) regressing scaled Schoenfeld residuals against functions of time g(t) using the Grambsch–Therneau test (1994); and (3) adding explicit covariate-by-time interaction terms Xk × ln(t) to test whether their coefficients differ significantly from zero.
What is the difference between a hazard ratio (HR) and a relative risk (risk ratio, RR)?
A hazard ratio (HR) compares instantaneous event rates h1(t) / h0(t) at any point in time, whereas a risk ratio (RR) compares cumulative proportions of subjects who have experienced the event by a specified follow-up horizon t, [1 − S1(t)] / [1 − S0(t)]. As cumulative incidence approaches 100% over long follow-up horizons, RR is compressed toward 1.0 while HR remains invariant to follow-up duration under proportional hazards.
References
- Cox, D. R. (1972). "Regression models and life-tables." Journal of the Royal Statistical Society: Series B (Methodological), 34(2), 187–220. https://doi.org/10.1111/j.2517-6161.1972.tb00899.x
- Cox, D. R. (1975). "Partial likelihood." Biometrika, 62(2), 269–276. https://doi.org/10.1093/biomet/62.2.269
- Efron, B. (1977). "The efficiency of Cox's likelihood function for censored data." Journal of the American Statistical Association, 72(359), 557–565. https://doi.org/10.1080/01621459.1977.10480613
- Grambsch, P. M., & Therneau, T. M. (1994). "Proportional hazards tests and diagnostics based on weighted residuals." Biometrika, 81(3), 515–526. https://doi.org/10.1093/biomet/81.3.515
- Andersen, P. K., & Gill, R. D. (1982). "Cox's regression model for counting processes: a large sample study." The Annals of Statistics, 10(4), 1100–1120. https://doi.org/10.1214/aos/1176345976