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Research method

Cox Proportional Hazards Model

16 studies

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The Cox proportional-hazards model estimates how a covariate shifts the instantaneous rate of an event — death, cancer recurrence, incident depression, a cardiovascular composite — without having to specify the baseline hazard. The headline number is a hazard ratio: HR 0.80 means about 20% lower hazard in that group, holding the model's other covariates fixed, provided the proportional-hazards assumption is close enough to true. Time-to-event methods also let investigators keep people who are censored (lost to follow-up, end of study) in the risk set until they leave.

Researchers reach for Cox models when the outcome is not a yes/no at a fixed week but whether and when something happens in a cohort or a trial. It answers 'is this exposure associated with a different event rate over follow-up?' Its main limitation is that a hazard ratio from an observational cohort is not a randomised treatment effect, and when events are rare even a large trial can return an HR near 1 that is hard to read as proof of equivalence.

Studies

16

Findings

4

7 supporting · 0 challenging · 0 qualifying citations

Open tensions

2

Currently

What we know

  1. In observational cancer and sleep cohorts, Cox models turn deprivation, education and disease severity into excess-hazard or event-rate contrasts. ONS Longitudinal Study analyses found less-deprived areas had lower excess hazard for prostate cancer (EHR 0.80) and that absolute net-survival gaps between extreme individual SES profiles were 29.6% in the most deprived areas versus 9.1% in the least; an OSA sleep-lab cohort of 10,149 patients had 11.5% composite events over a median 68 months, with higher AHI linked to higher hazard (e.g. univariate HR 1.49 for high versus low AHI).
  2. Diet and alcohol papers in this set report HRs for incident depression or CVD that are threshold-like or beverage-specific rather than strictly linear. Low-to-moderate alcohol (>5–15 g/day) had HR 0.72 for incident depression among 5,505 older Mediterranean adults; each extra 10 g/day olive oil in PREDIMED participants had HR 0.87 for major CVD events; higher diet-quality quintiles had depression HRs around 0.74–0.78.
  3. A randomised trial can still use Cox (or a closely related survival contrast) as the primary analysis. Patients undergoing curative breast-cancer surgery randomised to propofol versus sevoflurane had five-year overall survival around 92% in both arms, with a hazard ratio near 1 and no statistically significant difference in intention-to-treat or per-protocol analyses.
  4. When depression is not the trial's original primary endpoint, a Cox HR below 1 is easy to over-read. PREDIMED's merged Mediterranean-diet HR for incident depression was 0.85 and not significant; a nuts-arm benefit appeared in a diabetes subgroup after 224 cases over a median 5.4 years.

Largest unresolved question

The same modelling family is asked to do two different jobs: describe inequalities and exposures in cohorts, versus test a randomised intervention. The SES-cancer and OSA papers cannot name a single causal mechanism or a treatment that would close the HR gap, whereas the propofol–sevoflurane trial can say the assigned anaesthetic did not shift 5-year survival under its protocol. Reading every HR as if it were that trial overclaims the observational papers and underclaims the trial's design.

Procedure

How it works

The main stages of the method. Specific protocols and analysis choices vary between studies.

  1. Define the event and the time origin

    The analyst names what counts as an event — death, recurrence, incident depression, a cardiovascular composite — and when the clock starts. Changing the origin or the composite changes the hazard ratio even on the same people.

  2. Censor people who leave without the event

    Those lost to follow-up or still event-free at study end stay in the risk set until they leave. Censoring is not deletion; it is the reason time-to-event methods exist.

  3. Fit a proportional-hazards model

    A partial-likelihood model estimates how covariates shift the instantaneous hazard without specifying the baseline. Other covariates are held fixed only in the sense the model writes down.

  4. Report a hazard ratio

    HR 0.80 means about 20% lower hazard in that group under those covariates. Rare events can still return an HR near 1 that is hard to read as proof of equivalence, even in a large trial.

  5. Check whether the hazard ratio stays proportional

    Diagnostics ask whether the HR is roughly constant over follow-up. If hazards cross, a single HR is a compromise, not a description of the whole period.

  6. Do not read an observational HR as a randomised effect

    Cox answers whether this exposure is associated with a different event rate over follow-up. An observational cancer-survival or diet HR is not a randomised treatment effect; a trial HR near 1 is not automatic proof the arms are the same.

Common misconceptions

  • A hazard ratio is the same thing as a relative risk at the end of follow-up.

    It is a ratio of instantaneous event rates over time, under a proportional-hazards assumption. Absolute gaps can look very different from the HR, as in the cancer-survival paper's 29.6% versus 9.1% net-survival gaps by SES profile despite a prostate EHR of 0.80 for area deprivation.

  • If a large trial's HR is about 1, the two treatments have been proven equivalent.

    The propofol versus sevoflurane breast-cancer trial analysed 1,670 patients and still could not prove equivalence for a rare mortality outcome; opioid co-analgesia also differed by arm.

  • An HR below 1 from a food or alcohol cohort means that exposure was shown to prevent the disease.

    Those analyses remain observational (olive oil within a trial cohort; alcohol and diet-quality scores in follow-up studies) and the PREDIMED depression analysis was not even the trial's original primary endpoint.

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