## What are Hazard Ratios?

A hazard ratio (HR) is the probability of an event in a treatment group relative to the control group probability over a unit of time. This ratio is an effect size measure for time-to-event data. Use hazard ratios to estimate the treatment effect in clinical trials when you want to assess time-to-event.

For example, HRs can determine whether a medical treatment reduces the duration of symptoms or prolongs survival in cancer patients.

A hazard ratio might seem like relative risk ratios (RRs) and odds ratios (ORs). All these measures compare the probabilities of two groups. However, there are critical differences. For instance, RRs and ORs assess risk at one point in time, such as the end of a study.

In contrast, hazard ratios originate from survival analysis studies that record time-to-event data. Regression models derive the HRs from these data, which represent the instantaneous risk at any given point during the study—not just at the end. As you’ll see, this affects the interpretation of a hazard ratio.

Survival analysis and time-to-event studies compare the risk between two groups at multiple time points. Using many time points allows the analysis to include data from subjects who drop out partway through or do not reach a study’s defined endpoint (e.g., not cured) by the end of the study. RRs and ORs must exclude those subjects, potentially producing significant selection bias. Conversely, hazard ratios can incorporate those data, which reduces bias.

In this post, learn about the hazard ratio definition and how to interpret them. I’ll close with some of their more technical details.

**Related posts**: Relative Risk and Odds Ratios

## Time-to-Event Data for Hazard Ratios

An HR is an effect measure for time-to-event data. So, let’s take a look at this data type. By understanding the data and seeing them graphically, you’ll interpret hazard ratios more intuitively.

Clinical trials frequently record the timespan between the subject entering the study and reaching a predefined endpoint related to the disease. These endpoints are the events. Analysts record this time-to-event information for participants in the control and treatment groups, which allows the analysis to calculate the hazard ratio.

The nature of the event depends on the study and can include the following endpoints:

- Symptom resolution.
- Death of subject.
- Infected by disease.

Depending on the nature of the event, you’ll want either long or short times-to-event. For symptom resolution, you’d like short times, while for subject deaths you’re aiming for longer times. This distinction becomes important when interpreting hazard ratios.

Learn more about Experimental Design and Control Groups.

## Hazard Ratios and Kaplan-Meier Curves

Kaplan-Meier curves graphically depict time-to-event data and really bring them to life. Consequently, analysts frequently include them to help with hazard ratio interpretation.

These curves display the proportion of subjects who have not experienced an event (Y-axis) by time intervals (X-axis). As time progresses, events occur, which decreases the proportion who have not experienced the event.

Kaplan-Meier curves display this decrease as a downward slope that depicts the event probability over time, which analysts refer to as the hazard rate. The curves on this chart represent the hazard rates. Steeper slopes indicate higher hazard rates because events happen more frequently, lowering the unaffected proportion more quickly. If the control and treatment groups have different hazard rates, their slopes will differ.

In the Kaplan-Meier plot, there are two curves. The curve with the white dots represents the control group, while the darker dots represent the treatment group. As the days on the X-axis increase, the percentage of patients with pain decreases. Because the percentage in the treatment group decreases more quickly, its hazard rate is greater than the control group, producing a hazard ratio greater than one.

A hazard ratio is the ratio of two hazard rates—represented by the different slopes on the graph. In essence, HRs are a single number that summarizes the magnitude of the difference between Kaplan-Meier curves.

Statisticians frequently use a stratified Cox proportional hazard regression model to estimate hazard ratios and their confidence intervals. In medical settings, these models can evaluate the change in risk associated with a treatment while incorporating patient attributes and risk factors. Read my report about a COVID vaccine analysis that used a Cox proportional hazards model.

## How to Interpret a Hazard Ratio

Keep in mind how Kaplan-Meier curves depict the proportion of subjects who have not experienced the event (i.e., unaffected subjects) at various time points. That depiction clarifies how to interpret hazard ratios.

A hazard ratio tells us whether a subject in the treatment group who is unaffected at any given time has a greater, equal, or lower probability (i.e., hazard rate) of experiencing the event during the next unit of time than an unaffected subject in the control group.

Because we’re dealing with a ratio, the value of 1 becomes critical to interpreting hazard ratios because it indicates that the treatment and control groups have equal hazard rates. As the ratio moves away from one in either direction, the difference between the control and treatment groups increases.

**Hazard Ratio = 1**: An HR equals one when the numerator and denominator are equal. This equivalence occurs when both groups experience the same number of events in a period.

**Hazard Ratio > 1**: The numerator is greater than the denominator in the hazard ratio. Therefore, the treatment group experiences a higher event probability within any given period than the control group.

**Hazard Ratio < 1**: The numerator is less than the denominator in the HR. Consequently, the treatment group experiences a lower event probability during a unit of time than the control group.

## Hazard Ratio Interpretation Example

Let’s interpret an example hazard ratio of 2.

In a medical study, HR = 2 indicates that an unaffected subject in the treatment group has twice the probability of experiencing the event within a time span than someone in the control group.

Is a hazard ratio of 2 good or bad? As I mentioned in a previous section, that depends on the nature of the event and whether you want shorter or longer times to the event. This result indicates that the event happens more frequently in the treatment group per unit of time, which equates to shorter times to events.

If you’re studying a new medicine and the endpoint event is the resolution of symptoms, then an HR of two is good. A subject in the treatment group has twice the probability of symptom resolution than someone in the control group at any given point. Patients feel better more quickly.

Conversely, suppose the event is patient death. In that case, a hazard ratio of two indicates that the probability of dying is double in the treatment group relative to the control group at any point. That’s not good because patients are dying more quickly (i.e., survival times are shorter)!

## Interpretation Cautions

Hazard ratios are an excellent way to quantify relative probabilities for any point in time during the study. The analytical procedures can produce p-values and confidence intervals, allowing you to determine statistical significance.

However, hazard ratio interpretations do not represent a natural time measurement. For instance, an HR of 2 indicates that a patient in the treatment group has 2X the probability of full symptom resolution than a patient in the control group. You know that the treatment group patients will feel better faster but not in an intuitive time-to-event manner.

Consequently, when reporting hazard ratios in statistical results, you should report the median endpoint times and differences between groups to provide a more intuitive interpretation than the HR alone.

For example, the treatment group experienced full symptom resolution in a median of 5 days sooner than the control group. Or survival increased by a median of 90 days. These measures help the reader understand the practical importance of hazard ratio results.

There is no direct calculation from hazard ratios to obtain median times, so you’ll need to calculate them from your data.

**Related post**: Median Definition and Uses

## Hazard Ratio Formula and Calculations

Earlier, I illustrated hazard ratios and rates using Kaplan-Meier curves. Let’s dig deeper into the technical definition of a hazard ratio and its primary assumption of proportional hazards.

A hazard rate is the limit of the ratio of events in an interval to the group size divided by the length of time. The hazard rate formula finds the rate for time intervals approaching zero, producing the instantaneous hazard rate. It is the probability that an unaffected subject experiences the event between time t and t + Δt, where Δt approaches zero.

That hazard ratio definition might sound convoluted, so I’ll use an analogy.

Imagine driving your car and the speedometer displays a rate of 65 miles per hour (MPH). That reading isn’t based on you traveling 65 miles in an actual hour. The speedometer calculates your velocity in an instant of time. That instant of time is the “where Δt approaches zero.” Your speedometer gives you an instantaneous speed.

A hazard rate is an instantaneous probability that a subject who has not experienced the event at time *t* will experience it in the next time interval (Δt) divided by the length of time. This hazard rate applies to any point in the study.

A hazard ratio is a relative hazard for two rates.

Let’s return to the car analogy to understand it. I’m driving at 80MPH while my friend is traveling at 40MPH. The ratio of those two rates is 80/40 = 2. I’m traveling twice as fast as my friend. That’s our relative speed.

### Proportional Hazards

Hazard ratios usually must satisfy the proportional hazards assumption to be interpretable. This assumption states that the hazard ratio remains constant over time. If it varies, you can’t interpret the results. I’ll return our car analogy to explain why.

If my friend and I have a consistent speed ratio of 80 / 40 = 2, I know I’ll travel twice as far as my friend, regardless of the timespan. However, if our speeds vary, that affects the ratio. We might have a ratio of 2 at one point but various ratios at other times. Consequently, we won’t be able to interpret the result meaningfully. We might not travel twice as far for particular travel times.

The same idea applies to hazard ratios. For a hazard ratio of 2, the interpretation is that the probability is double regardless of the timespan. If the underlying hazard rates change during the study, that affects the HR, and we can’t use it to obtain meaningful results. There are methods for testing this assumption, but they go beyond the scope of this post. It can get complicated in Cox Regression, where the ratio must remain constant over time with different values of the independent variables.

## Reference

Kuitunen, I., Ponkilainen, V.T., Uimonen, M.M. *et al.* Testing the proportional hazards assumption in cox regression and dealing with possible non-proportionality in total joint arthroplasty research: methodological perspectives and review. *BMC Musculoskelet Disord* **22**, 489 (2021).

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