BiostatisticsTier 1Medical Sciences concept

Time-to-event outcome measures (hazard ratio vs relative risk vs incidence rate)

Core concept

Hazard ratio (HR) measures the instantaneous rate of an event occurring at any given moment in one group vs another, across the entire follow-up period -- not a single-timepoint risk comparison
Relative risk (RR) compares cumulative proportions with an event by a fixed, single time point -- ignores when within that period events occurred
Incidence rate = events / person-time at risk -- accounts for varying follow-up duration between subjects (drop-outs, staggered entry)

HR = 0.5 -> at any instant, the event is occurring at half the rate in the treatment group compared to control, averaged over the study -- it is not the same as "50% risk reduction" (that is RRR, a cumulative-risk concept)
HR assumes proportional hazards (the ratio of event rates stays constant over time) -- if hazards cross or diverge over the study (visible on a Kaplan-Meier plot), a single HR summary can be misleading
Trap: quoting an HR from a trial as if it were a relative risk reduction -- they are different quantities, only numerically similar when event rates are low and hazards are proportional

HR is the correct measure when follow-up time varies between participants (typical of survival/time-to-event trials with staggered recruitment and losses to follow-up) -- RR/OR from a fixed endpoint discard this timing information
Choose HR (Cox regression, log-rank test) for oncology/mortality trials; choose RR/OR for outcomes assessed at one fixed follow-up point (e.g. "response at 12 weeks")
Always check the Kaplan-Meier curve shape, not just the HR -- crossing curves signal non-proportional hazards, meaning a single HR oversimplifies a treatment effect that changes over time

Directly complements Kaplan-Meier interpretation - HR is the numerical summary statistic that log-rank testing/Cox regression assigns to a KM curve difference
Incidence rate ratios (comparable logic to HR) are used in cohort studies with variable follow-up, distinct from cumulative incidence-based RR
Statistical test selection: log-rank test compares survival curves; Cox proportional hazards regression estimates the adjusted HR

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