Hazard ratios (HRs) are vital statistical measures widely used in medical research, clinical trials, epidemiology, and other fields involving time-to-event data. They provide an insightful way to compare the risk or rate of events occurring between two groups over time. Whether you are a healthcare professional, researcher, or a student, understanding hazard ratios is crucial for interpreting study results related to survival, disease progression, treatment effects, and more.
This blog post will explain what hazard ratios are, their interpretation, usage, and common pitfalls. To conclude, there is a helpful Q&A section to address frequent queries.
What is a Hazard Ratio?
A hazard ratio is a measure comparing the hazard rates between two groups or conditions over time. The hazard rate represents the instantaneous risk, or probability, that an event (e.g., death, relapse, recovery) will occur at a particular moment, given that the individual has survived up to that time.
Mathematically, the hazard ratio is expressed as:
- An HR of 1 indicates no difference between groups (equal risk).
- An HR greater than 1 indicates a higher hazard in the treatment group (more frequent or faster occurrence of the event).
- An HR less than 1 indicates a lower hazard in the treatment group (reduced risk or slower event occurrence).
How to Interpret Hazard Ratio
Unlike relative risk or odds ratios which compare risk at a single point in time, hazard ratios consider the timing of the event throughout the study period, which adds dynamic insight into how the risk evolves.
Key Points for Interpretation:
- HR = 1: The event hazard is the same in both groups.
- HR > 1: The treatment group experiences the event more frequently or sooner than the control group.
- HR < 1: The treatment is protective, lowering the event’s hazard compared to control.
Example
If a clinical trial reports an HR of 2 for death, it means that at any given time, patients in the treatment group have twice the risk of dying as those in the control group, assuming they have survived up to that point. This does not mean their absolute risk of death is double, but rather the instantaneous rate is twice as high.
Conversely, an HR of 0.5 indicates the treatment halves the hazard of the event (e.g., death, relapse) at any moment compared to control. This might translate into longer survival or delayed disease progression.
The Proportional Hazards Assumption
The Proportional Hazards (PH) Assumption is one of the most important principles in survival analysis, especially in the Cox proportional hazards model. It assumes that the ratio of hazard rates, known as the hazard ratio (HR), between two or more groups remains constant over time. This means that if one group has a higher risk of experiencing the event compared to another group at the start of the study, the same relative difference in risk should hold throughout the entire follow-up period. Mathematically, the hazard function can be expressed as
The baseline hazard function is h0(t), and exp(β) represents the hazard ratio associated with covariates.
This assumption is very useful because it simplifies interpretation: a hazard ratio greater than one consistently indicates an increased risk, while a value less than one indicates a reduced risk, regardless of the time point considered.
If the PH assumption is violated, alternative modeling strategies are needed. Options include stratified Cox models, which allow baseline hazards to vary across groups, adding time-varying covariates, or using accelerated failure time (AFT) models. Verifying and addressing the PH assumption is essential for producing valid, reliable, and interpretable results in survival analysis.
How Hazard Ratios Are Used
Clinical Trials:
Hazard ratios are standard in analyzing time-to-event outcomes such as:
- Overall survival in cancer studies
- Time to disease progression or relapse
- Time to recovery or symptom resolution
For example, in a trial evaluating a cancer drug, the HR might indicate how much the treatment reduces the instantaneous risk of death compared to standard therapy.
Epidemiological Studies:
In large cohort studies, HRs assess effects of risk factors like smoking, air pollution, or diet on disease incidence or mortality over time, adjusting for confounding variables.
Other Applications:
Areas such as cardiology, infectious diseases, and psychology rely on hazard ratios to evaluate interventions affecting timing and likelihood of outcomes.
Limitations and Cautions
- HRs do not provide absolute risk. They are relative measures and should be interpreted alongside baseline risks for context.
- They do not convey how much sooner or later an event happens—only the relative likelihood at any instant.
- Reporting median survival times and other time-based measures in addition to HRs helps clarify practical patient outcomes.
- The proportional hazards assumption must be tested; if violated, HRs may mislead.
Example Calculation of a Hazard Ratio
Imagine two groups in a study:
- Control group hazard rate: 0.05 events per month
- Treatment group hazard rate: 0.025 events per month
The hazard ratio is: HR=0.025/0.05=0.5
This suggests the treatment halves the instantaneous risk of experiencing the event each month.
Conclusion
The hazard ratio is a powerful statistical measure to quantify the effect of treatments or exposures on the timing of an event occurring, offering dynamic insight across the study duration. It is particularly valuable in clinical research and epidemiology for analyzing and comparing survival or time-to-event outcomes.
While useful, hazard ratios require careful interpretation, especially understanding their relative nature and underlying assumptions. Combining HRs with additional time-based statistics improves their real-world relevance and communication. Data Science Blog
Q&A Section
Q1: What is the difference between hazard ratio and relative risk?
A1: Relative risk compares the cumulative probability of an event occurring by the end of a study, while hazard ratio compares the instantaneous risk of event occurrence at any moment during the study period.
Q2: Can hazard ratios be less than 1? What does that mean?
A2: Yes. An HR less than 1 indicates the treatment or exposure reduces the hazard (risk over time) of the event compared to control, suggesting a protective effect.
Q3: Why is the proportional hazards assumption important?
A3: It assumes the hazard ratio between groups remains constant over time, allowing simpler interpretation and valid use of models like the Cox proportional hazards. If violated, HR interpretation can be misleading.
Q4: Does a hazard ratio tell the exact time until an event happens?
A4: No. HR conveys relative risk at any point in time, not actual event timings. Median survival times or other metrics should be used alongside to understand timing.
Q5: How should confidence intervals be interpreted around a hazard ratio?
A5: Confidence intervals provide a range in which the true HR likely lies. If the interval includes 1, the HR is not statistically significant, indicating no clear difference between groups.
Q6: Can hazard ratios be used if some patients drop out or are censored?
A6: Yes. HRs are derived from survival analysis methods that appropriately handle censored data, which is an advantage over some other risk measures.
Understanding hazard ratios enhances your ability to critically evaluate scientific literature and clinical evidence about treatment effects and risks over time. With this knowledge, you can better grasp the implications of studies and improve decision-making in healthcare and research contexts.