Cox regression and the hazard ratio

Cox models the hazard, the instantaneous rate of the event among those still at risk, as a baseline hazard multiplied by a factor depending on covariates. The clever part is that the baseline hazard is never estimated. It cancels out of the partial likelihood, so you get covariate effects without committing to any shape for how risk changes over time.

That is why Cox is called semi-parametric, and why it is the default rather than a parametric survival model. You rarely know the true shape of a baseline hazard, and Cox does not make you guess.

The coefficient is on the log-hazard scale, so the exponentiated coefficient is the hazard ratio. Above 1 means a higher rate of events, below 1 a lower one.

The phrase to avoid is "patients lived 30% longer". A hazard ratio of 0.7 says nothing directly about survival time. A large hazard ratio applied to a tiny baseline hazard is still a tiny absolute effect, which is why it should always be reported alongside absolute measures such as survival at fixed time points.

A cohort of 410 patients, 78 of whom have the event. Four covariates. This is what the software returns and what each column means.

Step 1. Where the hazard ratio comes from

The coefficient is on the log-hazard scale. Exponentiate it.

HR = exp(−0.3567) = 0.700

Treated patients are having events at 70% of the rate of controls, among those still event-free at any given moment.

Step 2. Where the interval comes from

Build it on the log scale, then transform back. Never add and subtract on the hazard ratio itself.

exp(−0.3567 ± 1.96 × 0.1480) = exp(−0.647) to exp(−0.067)
    = 0.524 to 0.936

It excludes 1, consistent with the p-value of 0.016. The asymmetry around 0.700 is expected and correct: ratios are symmetric on the log scale, not the natural one.

Step 3. Continuous covariates need a sensible unit

Age has a hazard ratio of 1.041, which sounds negligible until you notice it is per single year. Rescale it to something a reader can use.

per 10 years: exp(0.0405 × 10) = exp(0.405) = 1.499, 95% CI 1.20 to 1.87

A decade of age raises the hazard by about half. Same model, same data, far more informative presentation. Scale continuous covariates to a meaningful unit before reporting, and say what the unit is.

Step 4. The non-significant term

Sex has a hazard ratio of 1.25 with an interval from 0.90 to 1.74. That is not evidence of no difference, it is a result compatible with anything from a 10% reduction to a 74% increase. With 78 events the study was never going to settle it, and saying "sex was not associated with outcome" overstates what the data can support.

Step 5. Check the events budget before trusting any of it

78 events / 4 covariates = 19.5 events per variable

Comfortably above the usual minimum of ten. Had the model carried twelve covariates, the same 78 events would give 6.5 per variable and every interval above would be narrower than it should be.

A surgical intervention carries early operative risk and long-term benefit. Suppose the truth is:

months 0–12: HR = 1.80  (worse)
months 12+: HR = 0.60  (better)

What a single Cox model reports

A single hazard ratio is roughly an average of the log hazard ratios, weighted by how many events fall in each period. With events split evenly, 39 in each:

weighted log HR = (39 × ln 1.80 + 39 × ln 0.60) / 78 = 0.038
overall HR = exp(0.038) ≈ 1.04

The model reports a hazard ratio of 1.04 with an interval comfortably spanning 1, and the paper concludes no effect. In reality the intervention nearly doubled the early hazard and cut the later one by 40%. Both findings matter clinically, and the single number erased both.

Why the answer depends on follow-up length

Same underlying truth, three different published conclusions, decided entirely by how long the study ran. This is why a hazard ratio is described as averaged over the follow-up actually observed, and why that phrase is not a technicality.

What to report instead

Split follow-up at twelve months and give both hazard ratios, each with an interval, stating that the split was chosen because the operative period is clinically distinct rather than because it maximised anything. Add restricted mean survival time at a fixed horizon, which stays interpretable whether or not hazards are proportional.

Cox assumes the hazard ratio is constant over time. One group may be at higher risk throughout, but the multiple must not change as follow-up proceeds.

Plenty of real situations break this. Surgery carries high early risk and a long-term benefit, so its hazard ratio starts well above 1 and falls below it. A treatment effect that wears off produces a ratio drifting back towards 1. In both cases a single hazard ratio averages opposing effects and can land near 1, reporting "no difference" for an intervention that mattered a great deal at both ends.

Beware the sample size trap in both directions. With a large sample the Schoenfeld test flags trivial departures; with a small one it misses serious ones. Read the plot as well as the p-value.

Restricted mean survival time is underused and worth knowing about. It answers the question clinicians actually ask, which is how much extra event-free time a treatment buys over a stated period.

R

The survival package is the reference implementation and everything else follows its conventions.

library(survival)
fit <- coxph(Surv(time, status) ~ treatment + age + sex + stage, data = df)
summary(fit)

summary() prints coefficients and exponentiated hazard ratios with intervals in one table. Test the assumption with cox.zph(fit), then plot(cox.zph(fit)) to see the residuals against time, which is more informative than the p-value. For a time-varying effect use tt() inside the formula, and for clustered data add + cluster(centre) for robust standard errors or + frailty(centre) for a random effect.

SPSS

Cox regression sits under Analyze, Survival, Cox Regression. The syntax is:

COXREG time /STATUS = status(1)
  /METHOD = ENTER treatment age sex stage
  /PRINT = CI(95)
  /CRITERIA = PIN(.05) POUT(.10).

SPSS reports Exp(B), which is the hazard ratio, with its interval. It has no direct equivalent of cox.zph, so test proportional hazards by adding a time-dependent covariate through Cox Regression with Time-Dependent Covariates, defining an interaction between the variable and T_.

Stata

Declare the survival structure once with stset, then the model needs no time or status arguments.

stset time, failure(status == 1)
stcox treatment age i.sex i.stage
estat phtest, detail

stcox reports hazard ratios by default; add , nohr for coefficients on the log scale. estat phtest, detail gives the Schoenfeld test per covariate and globally. Use , vce(cluster centre) for clustered data and stsplit to create period-specific effects.

If an analysis is already written up and you want it checked before submission, proportional hazards is one of the first things a reviewer will look for. An independent statistical review covers it.