Can hazards increase over time and still be proportional?

Yes, if their ratio remains constant for the profiles being compared.

Does a hazard ratio of two mean twice the event probability?

Not generally. The worked example shows different cumulative risk ratios under the same hazard ratio.

Does every Cox model require proportional hazards for every variable?

The usual fixed-coefficient specification does for modeled comparisons, but stratification and time-varying extensions alter the structure.

AI Glossary / Statistics and probability / Intermediate

Proportional hazards

Definition

Proportional hazards is an assumption that the hazard functions for two groups or covariate profiles differ by a constant multiplicative factor over time. Their individual hazards may rise or fall, but their hazard ratio remains constant. The assumption is central to the usual Cox proportional hazards model. A hazard describes the instantaneous event rate among units that have not yet experienced the event, rather than the probability of the event by a fixed time.

What it means

The assumption concerns a relative rate throughout follow-up. If one group's hazard is twice another's at each time, the hazards are proportional with ratio two. That does not mean its cumulative event probability is always twice as large, nor that its typical event time is automatically halved.

Time-to-event analysis applies beyond medicine, including equipment failure or customer departure. The event, time origin, and units under observation must be defined carefully. In a model with several predictors, proportionality is a conditional statement for the profiles being compared, holding the relevant covariates fixed.

How it works

The standard Cox form writes the hazard as h(t given x) = h0(t) exp(x-transpose beta). The baseline hazard h0(t) can vary with time. For two fixed covariate profiles, dividing their hazards cancels the common baseline and leaves exp((x1 − x0)-transpose beta), a constant ratio.

In an ordinary continuous-time, single-event setting, proportional hazards imply a corresponding relationship between survival functions. If the hazard ratio is c, the comparison group's survival is the reference survival raised to power c. This relationship helps distinguish hazards from cumulative risks.

Assess proportionality using subject-matter knowledge and suitable diagnostics, including time patterns in scaled Schoenfeld residuals for fitted Cox models. A diagnostic test alone does not establish the assumption. If effects change with time, consider a justified alternative such as time-varying coefficients or stratification. Censoring and competing events require their own assumptions and analysis choices.

Three practical examples

A reliability analyst models time to equipment failure while assessing whether operating-condition hazard ratios remain stable.
A customer analytics team studies time to departure and checks whether an association weakens as tenure increases.
A researcher compares time-to-event models with constant and time-varying relative hazards before interpreting one summary ratio.

Think of it this way

Think of two speedometers whose readings always maintain the same ratio even while both vehicles accelerate or slow down. That ratio describes their instantaneous speeds, not the ratio of distances already traveled. Similarly, a hazard ratio concerns instantaneous event rates among remaining units, not cumulative event probabilities.

Worked example

In an original single-event model, suppose group B has twice group A's hazard at every time. The hazard ratio is therefore two. At a particular time, group A's survival probability is 0.80.

Under proportional hazards, group B's survival is 0.80² = 0.64. Their event probabilities by that time are 1 − 0.80 = 0.20 and 1 − 0.64 = 0.36. The cumulative risk ratio is 0.36/0.20 = 1.80, not two.

At a later time, suppose group A's survival is 0.50. Group B's survival is then 0.50² = 0.25. Their cumulative event probabilities are 0.50 and 0.75, giving a risk ratio of 1.50.

Thus a constant hazard ratio can coexist with a changing cumulative risk ratio. These are assumed survival relationships for a teaching model, not estimates from real equipment or people. Competing-risk settings need additional care because a cause-specific hazard does not alone determine cumulative incidence.

Illustrative case study

Fictional case study: Cedar Reliability fits a Cox model to component failures and reports one hazard ratio for a demanding operating condition. Diagnostic review suggests that the association is strongest early and weaker later.

The analysts investigate possible time-varying effects and compare models using the intended prediction horizon. They present survival probabilities at meaningful times alongside the model assumptions. This fictional case illustrates checking and communication, not evidence of a particular failure mechanism or a recommendation about real maintenance decisions.

Common misconceptions and mistakes

Proportional hazards does not mean constant hazards. Both groups can have changing event rates while maintaining a constant ratio. It also does not mean proportional survival probabilities or proportional cumulative risks.

Another mistake is treating a nonsignificant diagnostic test as proof of proportionality, especially with limited events. A statistically significant departure may likewise need assessment for practical importance. A fitted hazard ratio is not automatically causal, even after adjustment. Finally, the comparison is among units still event-free at each time, so its interpretation should not be casually replaced with an unconditional statement about all original units.

Business use

The assumption can simplify time-to-event modeling into a compact relative-effect summary. It is useful when that summary adequately describes the relevant follow-up period and supports the intended business analysis.

For operational reporting, state the event, time scale, comparison profiles, follow-up range, and censoring treatment. Show absolute predictions where decisions require them. If the effect changes materially over time, a single hazard ratio may conceal important behavior. An AI-generated explanation should preserve the distinction between instantaneous rates and event probabilities at a specified horizon.

Check your understanding

Two groups have hazards 0.02 and 0.04 at one time, and 0.05 and 0.10 at another. Are these values compatible with proportional hazards?

Answer: Yes. The ratio is two at both times, despite changing hazards. Two time points alone, however, do not establish proportionality throughout follow-up.

Related terms

Hazard ratio
Cox model
Survival function
Censoring
Cumulative incidence
Hazard function
Schoenfeld residual
time-varying effect
competing risks

Learn next

Learn survival functions and hazard functions before interpreting hazard ratios. Then study the Cox model, censoring assumptions, and proportionality diagnostics. Compare absolute event probabilities and consider time-varying effects when a constant relative rate is not a suitable summary.

Sources

Harrell's statistical glossary defines proportional hazards and explains Cox models and survival concepts: https://hbiostat.org/glossary/

The R survival documentation describes proportional-hazards diagnostics using scaled Schoenfeld residuals: https://stat.ethz.ch/R-manual/R-devel/library/survival/html/cox.zph.html

The survival calculations and fictional reliability case are original. The entry is a statistical explanation, not a clinical or maintenance recommendation.

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