The Hazard Ratio Is Not a Risk Ratio: Proportional Hazards, Restricted Mean Survival Time and Model Inputs

อ่านฉบับภาษาไทย (Thai version)

Abstract

The hazard ratio compares instantaneous event rates among people who remain event-free. It does not directly describe their cumulative chance of an event. A baseline survival curve connects relative effects to absolute outcomes under proportional hazards. Our illustrative example starts with control survival of 60.00% at five years and a hazard ratio of 0.700. The treated risk is 30.06%, compared with 40.00% in controls. The risk ratio is 0.752, and the absolute reduction is 9.94 percentage points. The number needed to treat rounds upward to 11. Assuming constant hazards, restricted mean survival increases by 105 days within five years. This area between survival curves measures average time gained within the chosen horizon. It remains useful when proportional hazards fails. Censoring and competing events still require attention. Economic models need compatible survival curves or cycle probabilities, with uncertainty carried forward. Extrapolation and quality adjustments need separate justification.

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Read the denominator first

A hazard is an instantaneous event rate among people still event-free. Those people form the risk set. A risk is the probability of an event by a stated time.

Here the event is all-cause death, measured from treatment allocation. Survival $S(t)$ is the probability of remaining alive beyond time $t$; death risk is $1-S(t)$. Name the endpoint and horizon before interpreting any effect.

A Cox model estimates relative hazards while leaving the baseline hazard unspecified. Its treatment coefficient alone cannot supply absolute survival. Keep these denominators distinct: [[incidence-prevalence-and-the-right-denominator]].

Connect the ratio to a baseline curve

Proportional hazards (PH) means the treatment-to-control hazard ratio stays constant over time. Each hazard may still change. With compatible baseline survival, $S_T(t)=S_C(t)^{HR}$.

The cumulative risk ratio is $[1-S_C(t)^{HR}]/[1-S_C(t)]$. For a protective HR, it moves toward unity as baseline survival approaches zero. At entry, both risks are zero and their ratio is undefined.

Match the endpoint, population and time origin before transferring an effect. An adjusted HR needs a compatible conditional baseline curve. Population-average predictions require averaging over the target population.

# Writes: stdout only; illustrative five-year death risks.
control_survival <- 0.60; hazard_ratio <- 0.70; horizon <- 5
control_risk <- 1 - control_survival; treated_risk <- 1 - control_survival^hazard_ratio
absolute_reduction <- control_risk - treated_risk
cat("Horizon (years):", horizon, "\n")
cat("Control survival (%):", formatC(100 * control_survival, digits=2, format="f"), "\n")
cat("Hazard ratio:", formatC(hazard_ratio, digits=3, format="f"), "\n")
cat("Control death risk (%):", formatC(100 * control_risk, digits=2, format="f"), "\n")
cat("Treated death risk (%):", formatC(100 * treated_risk, digits=2, format="f"), "\n")
cat("Risk ratio:", formatC(treated_risk / control_risk, digits=3, format="f"), "\n")
cat("Absolute reduction (points):", formatC(100 * absolute_reduction, digits=2, format="f"), "\n")
cat("NNT (rounded up):", ceiling(1 / absolute_reduction), "\n")

Horizon (years): 5
Control survival (%): 60.00
Hazard ratio: 0.700
Control death risk (%): 40.00
Treated death risk (%): 30.06
Risk ratio: 0.752
Absolute reduction (points): 9.94
NNT (rounded up): 11

Read the area as time gained

Restricted mean survival time (RMST) is average time alive before a chosen cutoff $\tau$. It equals the area under survival: $RMST(\tau)=\int_0^\tau S(t)\,dt$. The area between curves is the average survival gain.

Constant hazards are 0.102 and 0.072 per year for control and treatment. Integrating their exponential curves gives RMST of 3.92 and 4.20 years. Subtract before rounding: the gain is 0.29 years, or 105 days, within five years.

RMST does not require PH. It does require valid survival estimates and a common cutoff supported by follow-up. Choose that cutoff before comparing treatment effects.

Calculate restricted survival time

# Writes: stdout only; illustrative restricted survival time.
control_survival <- 0.60; hazard_ratio <- 0.70; horizon <- 5
control_hazard <- -log(control_survival) / horizon; treated_hazard <- hazard_ratio * control_hazard
control_rmst <- (1 - control_survival) / control_hazard
treated_rmst <- (1 - control_survival^hazard_ratio) / treated_hazard
gain <- treated_rmst - control_rmst
cat("Control hazard (per year):", formatC(control_hazard, digits=3, format="f"), "\n")
cat("Treated hazard (per year):", formatC(treated_hazard, digits=3, format="f"), "\n")
cat("Control RMST (years):", formatC(control_rmst, digits=2, format="f"), "\n")
cat("Treated RMST (years):", formatC(treated_rmst, digits=2, format="f"), "\n")
cat("RMST gain (years):", formatC(gain, digits=2, format="f"), "\n")
cat("RMST gain (days):", round(gain * 365.25), "\n")

Control hazard (per year): 0.102
Treated hazard (per year): 0.072
Control RMST (years): 3.92
Treated RMST (years): 4.20
RMST gain (years): 0.29
RMST gain (days): 105

Check what the curves assume

Censoring ends observation before an event time is known. Standard estimates need censoring independent of future event time, possibly conditional on measured predictors. Investigate loss related to worsening illness.

A competing event prevents the outcome, such as death before readmission. Cause-specific hazards describe people still free of every event. Subdistribution hazards use a modified risk set retaining those with competing events.

Neither hazard ratio is a cumulative event probability. Use cumulative incidence, which accounts for competing causes. Censoring competing deaths and using the Kaplan-Meier complement generally overstates readmission risk.

Inspect survival curves and log-minus-log survival plots for changing effects. Schoenfeld residuals compare observed covariates at event times with risk-set expectations. Systematic time trends challenge a constant coefficient.

Even randomization does not keep later survivors comparable. Treatment changes who remains event-free, so period-specific HRs can reflect selection. Interpret them alongside survival or RMST contrasts from the original groups.

Send survival quantities into the model

A state-transition model moves people between health states over successive cycles. Its death probability conditions on being alive at cycle entry: $p_C=1-S_C(t+\Delta)/S_C(t)$. Under PH throughout the cycle, use $p_T=1-(1-p_C)^{HR}$. Multiplying the control probability by HR is generally incorrect.

Competing transitions need a joint calculation from all relevant rates. A partitioned-survival model instead derives state occupancy from survival curves. Choose the structure that represents the treatment pathway: [[decision-trees-markov-models-and-psa]].

Extrapolation predicts beyond observed follow-up. Compare plausible survival tails against clinical expectations and external evidence. RMST measures survival time; quality-adjusted life-years also require health-state utility weights.

Our fixed illustrative inputs cannot supply confidence intervals or p-values. With fitted data, report the estimate, confidence interval and p-value where appropriate. Propagate uncertainty in both the baseline curve and treatment effect.

The series, in learning order

1. Health Economics Series Guide: Read the Evidence Before the Costs
2. Incidence, Prevalence and the Right Denominator: The Epidemiology Beneath Cost-of-Illness and Markov Models [[incidence-prevalence-and-the-right-denominator]]
3. Design, Model and Estimand: What Each Study Design Can Actually Estimate [[study-design-estimand-and-effect-measures]]
4. Risk Ratio, Odds Ratio, Risk Difference and NNT: One 2×2 Table, Four Answers
5. The Hazard Ratio Is Not a Risk Ratio: Proportional Hazards, Restricted Mean Survival Time and Model Inputs (this article)
6. Chance, Bias and Confounding: Why an Association Is Not Yet a Treatment Effect
7. Beyond the Evidence Pyramid: GRADE Certainty and Choosing Effectiveness Inputs for Economic Models
8. From Sensitivity to Cost per Case Detected: Predictive Values, Prevalence and the Economics of Screening
9. Whose Cost, in Which Year? Perspective, Opportunity Cost, Inflation and Discounting
10. QALY, DALY and the Utilities Behind Them: What Is Actually Being Measured
11. ICER, Dominance and the Cost-Effectiveness Plane: Reading the Thai Threshold of 160,000 Baht per QALY
12. Decision Trees, Markov Cohorts and Probabilistic Sensitivity Analysis: How a Model Turns Evidence into a Decision
13. Cost-Effective Is Not Affordable, and Efficient Is Not Fair: Budget Impact and Equity

Related in the wiki: Cause-Specific Cox or Fine-Gray: Two Hazards, Two Questions · Restricted Mean Survival Time: How Much Event-Free Time Did Treatment Buy?