Cox Proportional Hazards

Cox Proportional Hazards Model in Health Economic Evaluation

The Cox proportional hazards model, or Cox regression, published by Cox in 1972, is the standard method for estimating how treatment and patient characteristics change the rate of death, progression or another event in a clinical study, and many of the hazard ratios quoted in trial reports and health technology assessment submissions come from it. Covariate effects are estimated from the order of events among people still at risk, so no distribution of survival times is assumed. Bradburn and colleagues called it the most commonly used multivariate approach for survival data in medical research. For a health economist the model matters because its estimates feed lifetime survival extrapolations, yet the fitted model by itself describes survival only within observed follow-up.

How covariates scale an unspecified baseline hazard

The model links the hazard rate, the event rate among people still event-free, to explanatory variables such as treatment arm, age or disease stage. Cox's 1972 paper takes the hazard to be a function of the explanatory variables and unknown regression coefficients multiplied by an arbitrary and unknown function of time. The model is

$$h(t\mid x)=h_0(t)\exp(\beta_1x_1+\beta_2x_2+\dots+\beta_px_p),$$

where $h(t\mid x)$ is the hazard at time $t$ for a person with covariate values $x_1,\dots,x_p$, $h_0(t)$ is the baseline hazard for a person whose covariates are all zero, and $\beta_1,\dots,\beta_p$ are the regression coefficients. Bradburn and colleagues liken it to a multiple linear regression of the log hazard on the covariates, with the baseline hazard as an intercept that varies with time.

The exponentiated coefficient $\exp(\beta_j)$ is the hazard ratio for a one-unit increase in $x_j$ with the other covariates held fixed. Because $h_0(t)$ cancels when two people are compared, their hazard ratio does not depend on time, which is the proportional hazards assumption. The model is called semi-parametric because the covariate part has a fixed algebraic form while the time part, $h_0(t)$, is left free. A constant $h_0(t)$ would instead give the constant hazard exponential model.

Estimation from risk sets with the partial likelihood

Cox showed that the coefficients can be estimated without ever estimating $h_0(t)$. At each event time it asks how likely it was that the person who had the event, rather than anyone else at risk, was the one to have it. Multiplying these conditional probabilities over all event times gives the partial likelihood

$$L(\beta)=\prod_{i=1}^{D}\frac{\exp(x_i^{\mathsf T}\beta)}{\sum_{j\in R(t_i)}\exp(x_j^{\mathsf T}\beta)},$$

where $D$ is the number of distinct event times, $t_i$ is the $i$-th event time, $x_i$ is the covariate vector of the person who had the event at $t_i$, $R(t_i)$ is the risk set of people still under observation and event-free just before $t_i$, and $\exp(x^{\mathsf T}\beta)$ is a person's risk score. The baseline hazard appears in every numerator and denominator and cancels, so the coefficients are found by maximising $L(\beta)$ in the same way as in maximum likelihood estimation.

Only the order of event times matters, not the gaps. A censored person stays in each risk set until leaving observation, which is valid only when censoring is unrelated to the event process given the covariates.

Event times recorded in days or months often tie. The R survival package documentation describes the Breslow approximation as the first to be programmed, the Efron approximation as more accurate when ties are many (and the package default), and the exact partial likelihood as suited to a small set of discrete times. Without ties all three agree; with many ties the method should be reported.

Worked example with four patients in a small trial

The numbers are illustrative only. Four trial patients are followed for overall survival, with $x=1$ for the new treatment and $x=0$ for the comparator.

Writing $r=\exp(\beta)$, a comparator patient has risk score 1 and a treated patient has risk score $r$. At 2 months all four are at risk and the summed score is $S_2=2+2r$; at 3 months patients 2, 3 and 4 remain, so $S_3=2r+1$; at 5 months patients 3 and 4 remain, so $S_5=1+r$. The partial likelihood is

$$L(\beta)=\frac{1}{2+2r}\times\frac{r}{2r+1}\times\frac{1}{1+r},$$

where each fraction is the risk score of the patient who died divided by the summed scores of the risk set. With no treatment effect ($\beta=0$, so $r=1$) the value is $\tfrac14\times\tfrac13\times\tfrac12=\tfrac{1}{24}\approx0.0417$, the probability that each death happens to fall on the observed patient when every patient at risk is equally likely to die.

Setting the derivative of $\log L$ with respect to $\beta$ to zero gives

$$1-\frac{2r}{1+r}-\frac{2r}{2r+1}=0, \qquad\text{equivalently}\qquad 4r^2+r-1=0,$$

where the left-hand expression is the slope of the log partial likelihood. The positive root is $r=(\sqrt{17}-1)/8\approx0.390$, so $\hat\beta=\ln 0.390\approx-0.941$ and the estimated hazard ratio is about 0.39.

The baseline cumulative hazard comes from the same risk sets. Therneau's validation note for the survival package gives the increment at each event time as the number of events divided by the summed risk scores of those at risk. Survival for the comparator is $\exp\{-H_0(t)\}$ and survival for the treatment arm is that value raised to the power $r$.

Three events are far too few for inference, so the example shows mechanics only. Both estimated curves are step functions that change only at 2, 3 and 5 months, stay flat until the last patient is censored at 7 months, and say nothing about survival after 7 months. A lifetime cost-effectiveness model needs survival over many years, which this fitted model cannot supply on its own.

From a fitted Cox model to survival in an economic model

Predicted survival for any covariate profile follows from the baseline curve. Absolute outcomes such as life years need that curve. Bradburn and colleagues give the link as

$$S(t\mid x)=S_0(t)^{\exp(x^{\mathsf T}\beta)},$$

where $S(t\mid x)$ is the probability of remaining event-free beyond time $t$ for a person with covariates $x$, $S_0(t)$ is the baseline survival function, and $\exp(x^{\mathsf T}\beta)$ is that person's risk score. They add that, where a suitable distribution can be found, a parametric model is more informative than the Cox model because predicted survival times are straightforward to obtain from it.

For health economic evaluation this is the main limitation. NICE DSU Technical Support Document 14 (TSD14) notes that a Cox model bases inferences only upon observed data, which is reasonable only when data are almost complete, as otherwise mean survival will be inaccurate. Lifetime mean survival, and the quality-adjusted life years built on it, therefore usually come from parametric survival models or flexible parametric models used for extrapolation within a partitioned survival model or a state-transition model.

TSD14 adds further cautions for Cox output. With patient-level data it is unnecessary to rely on proportional hazards; the assumption should be tested, and separate parametric models for each arm may be preferable. When a proportional hazards model is used, only the hazard ratio from the chosen parametric model should be applied to that model's control-arm curve, and TSD14 calls applying a Cox hazard ratio theoretically incorrect. With only published summary data, the origin of a quoted hazard ratio may be unclear.

Assumptions and choices that change the estimate

The model rests on proportional hazards, a log-linear effect of each covariate and independent censoring. The NICE technology appraisal manual (PMG36, section 4.6.22) states that the proportional hazards assumption should always be assessed. Bradburn and colleagues cite a simulation study in which tests based on scaled Schoenfeld residuals were among the most powerful checks.

When a covariate's effect changes over follow-up, an interaction with time lets the hazard ratio vary. In the Bradburn ovarian cancer example, an interaction with log time gave a hazard ratio for the absence of ascites of $\exp(-0.466+0.233\ln 2)\approx0.74$ at 2 years and $\exp(-0.466+0.233\ln 5)\approx0.91$ at 5 years, an effect that weakened over time. A time-varying covariate, whose value changes during follow-up, is a different idea.

Other specification choices also matter.

• Stratification. A stratified Cox model fits a separate baseline hazard for each stratum, according to the survival package documentation. Bradburn and colleagues note that the stratifying covariate must be categorical and receives no estimated effect, so the approach suits covariates that are not of primary interest.
• Number of events. Bradburn and colleagues cite simulation work suggesting at least 10 events for each covariate, with fewer giving biased coefficients.
• Clustering. For patients grouped within centres, the survival package computes a sandwich variance when a cluster term is given; a frailty model is the alternative.

Interpretation and use in technology appraisal

A Cox hazard ratio is a relative measure conditional on the covariates in the model. Adjusted and unadjusted hazard ratios are therefore different quantities, and the adjustment set should be reported. In observational data a causal reading also requires confounding to be addressed.

PMG36 sets out how such estimates move into appraisal. Section 4.6.23 states that if the proportional hazards assumption holds within the trial and is clinically plausible during extrapolation, hazard ratios may be pooled using standard code for treatment differences. Section 4.6.24 states that alternative methods described in NICE DSU Technical Support Document 21 should be considered when the assumption does not hold in some studies. TSD21 covers flexible parametric methods using splines or fractional polynomials, cure and mixture models, and piecewise approaches.

How the Cox model differs from neighbouring survival methods

The Cox model sits between fully non-parametric description and fully parametric modelling. The choice depends on whether the task is adjustment or projection beyond the data.

Common misunderstandings

Most errors with Cox output come from treating a relative, within-trial summary as an absolute or long-term one. Several recur in economic models.

• Reading the hazard ratio as a ratio of risks or survival times.
• Taking a non-significant proportionality test as proof of proportional hazards when few events limit the power of the test.

Sources

• Bradburn MJ, Clark TG, Love SB, Altman DG. Survival analysis part II: multivariate data analysis, an introduction to concepts and methods. British Journal of Cancer. 2003;89(3):431-436. https://doi.org/10.1038/sj.bjc.6601119
• Bradburn MJ, Clark TG, Love SB, Altman DG. Survival analysis part III: multivariate data analysis, choosing a model and assessing its adequacy and fit. British Journal of Cancer. 2003;89(4):605-611. https://doi.org/10.1038/sj.bjc.6601120
• Cox DR. Regression models and life-tables. Journal of the Royal Statistical Society Series B (Methodological). 1972;34(2):187-202. https://doi.org/10.1111/j.2517-6161.1972.tb00899.x
• Latimer NR. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2011 (last updated March 2013). https://www.sheffield.ac.uk/nice-dsu/tsds/survival-analysis
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36), section 4.6. London: NICE; 2022 (last updated 13 October 2026). https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• Rutherford MJ, Lambert PC, Sweeting MJ, Pennington B, Crowther MJ, Abrams KR, Latimer NR. NICE DSU Technical Support Document 21: Flexible methods for survival analysis. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2020. https://www.sheffield.ac.uk/nice-dsu/tsds/flexible-methods-survival-analysis
• Therneau TM. coxph: Fit proportional hazards regression model. survival: Survival Analysis, R package documentation (accessed 11 October 2026). https://CRAN.R-project.org/package=survival
• Therneau TM. Software validation. Vignette for the R survival package; updated December 2022. https://cran.r-project.org/web/packages/survival/vignettes/validate.pdf