What Is a Hazard Ratio of 0.7 the 0.7 Of? — How to Read a Survival Curve

The story begins with data from two clinical trials involving patients with colorectal cancer that had spread to other organs. Both trials divided patients into two groups by a random draw. A trial that leaves who goes into which group to chance like this is called a randomized trial. One group received standard chemotherapy alone, and the other group received the drug panitumumab added to it. The researchers watched until the cancer progressed or the patients died, and calculated the difference between the two groups in two ways.

One number was 0.82, and the other number was 0.99.

The first number is the hazard ratio, the main subject of this article. A hazard ratio is a value dividing the speed at which events occur between two groups, and its exact meaning is untangled step by step below. The second number is the risk ratio covered in "An Odds Ratio Is Not a Risk Ratio — What the 'Times More Likely' Is a Multiple Of". A risk ratio is a value dividing the proportion of people who experienced the event when followed through to the end between two groups. The hazard ratio says "the drug worked," and the risk ratio says "there is no difference." These are two numbers from the same patients, the same data, and the same paper.

When the authors looked closer at the same data, it was like this. By the end of follow-up, the proportion of people who had experienced the event was nearly identical in the two groups. However, the group with the added drug took longer to experience that event. If you divide the period, up until about 1 year and 9 months, progression and death occurred more slowly in the group with the added drug, and after that, they occurred rather more quickly (PMID 39212064).

The two numbers were answering different questions. The risk ratio measures how many people experienced the event, and the hazard ratio measures when they experienced it. So both are correct.

The problem arises right after. If you ask someone to translate 0.82 into plain human language, most people say "the risk decreased by 18%." This article covers why that translation is inaccurate, and what the accurate translation is.

There are seven new terms to learn in this article, and each term is explained where it first appears. Once you master the seven, you can read a paper's survival curve and the numbers beneath it on your own.

⚠️ Several drug and trial names appear below, and all of them appear solely as examples to show how to read the numbers. This is not a judgment on whether any treatment is good or bad, and each number has meaning only within the design, participants, and time point of that study.

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When Time Enters the Axis, the Question Changes

The simplest way to find out if a medicine works is to count people. You count how many are alive 5 years later. If one side has 70 people and the other side has 60, that is a difference of 10 people. "An Odds Ratio Is Not a Risk Ratio — What the 'Times More Likely' Is a Multiple Of" and "What Does "50% Lower Risk" Mean in People? — Reading Relative Risk, Absolute Risk, and NNT" covered how to count this way.

However, this approach has two awkward problems.

First, you cannot watch every person for the full 5 years. People do not enter a study all at once. Some people entered 2 years ago and some entered last month. You cannot ask someone who entered last month, "Were you alive 5 years later?" Yet if you drop that person, you throw away the information of the people who entered late altogether.

Second, the 5-year time point is something someone picked arbitrarily. Even if the 5-year survival rates of the two groups are the same, if one side died off in large numbers all at once around year 3 and the other side died off gradually from around 4 and a half years, something different happened in the two groups. The colorectal cancer case earlier was just that kind of case.

Therefore, there is a separate analysis that deals with when events happen. This analysis asks "when" instead of "how many." And it accepts the fact that the follow-up period differs for each person as part of the data. All of the concepts that follow are tools made to do these two things.

Survival Curves — Why Do They Step Down?

A survival curve is a graph showing what percentage of people have not yet experienced an event as time passes. It starts at 100% in the top left and steps down toward the right. It is also called a Kaplan-Meier curve, named after two statisticians named Kaplan and Meier.

The vertical axis is the proportion of people who have not had the event up to that point, and the horizontal axis is time. Here, the "event" is a pre-defined outcome in the study, which could be death, recurrence, or readmission.

The reason the curve is a staircase lies in how it is drawn. The curve steps down by one notch only at the exact moment an event happens to someone, and runs flat during periods when nothing happens. So each individual step represents a person (or multiple deep steps if several people on the same day). The shape of the curve does not follow time evenly, but follows the points in time when events occur.

Risk Set — The Number of Remaining People Supporting the Curve

How deep a single stair is is determined by the number of people remaining at that point. The risk set is the people still remaining at a given point in time—meaning those who have not yet experienced the event and whose observation has not been cut off. If the risk set at the 12-month mark is 340 people, it means there are 340 candidates to whom the event can happen at that moment. This is the most practical concept in this article.

When 1,000 people remain, an event happening to one person barely moves the curve. When 12 people remain, an event happening to one person noticeably bends it. How much a single event shakes the curve differs at each point in time.

That is why the right-hand end of a survival curve is much more unstable than the left. In the figure, it is a line of the same thickness, but hundreds of people support the left side while a dozen or so support the right. Looking at the line alone, that difference is not visible. Papers show this difference by writing the size of the risk set for each time point in numbers below the curve, which is written in English as the number at risk.

However, this number is frequently omitted. Among 138 original papers (papers reporting new data for the first time) dealing with the timing of events in sports science and medicine, over 90% of the studies did not report the size of the risk set at each time point (2013–2023, PMID 41368207). In the same data, the median survival time (the time it took for half of the participants to experience the event) was reported in 20% of cases where it applied. Reports of the reasons why observations were cut off were also commonly incomplete.

This number is omitted even in large cancer trials. Phase 3 trials are the final stage of trials involving humans, and are usually large. Scanning 1,400 recent Phase 3 oncology trials, 28% of the studies that included survival curves did not report the risk set size. This review originally set out to create guidelines for how to compare crossing survival curves. The authors wrote that without the risk set size, it becomes nearly impossible to reconstruct data from the curve or to check whether the results are valid (PMID 35094686).

Therefore, the order for reading a curve is simple. If the right-hand end of the curve splits dramatically, check the size of the risk set at that point first. If that number is in the double digits, that split is the work of a few people.

Censoring — The People Whose Observation Stopped Without Knowing the End

Let us say we follow 500 people for 5 years. On the day the study ends, 300 people are alive without any events. What we know about these 300 people is up to "there was no event for at least 5 years," and we do not know what happened in the 6th year. This state, where there was no event up to the point observation ended and what comes after is unknown, is called censoring. It happens because the study ends, or because people move away or lose touch.

Censoring exists in almost every study. This is because almost no studies wait until every single participant experiences an event. Therefore, the calculation method of survival curves is built to handle censoring from the start. Censored people stay in the risk set until their observation is cut off, and quietly leave the risk set after that. Doing this means we do not throw away existing information, nor do we invent information we do not have.

However, one premise comes attached to this handling. The future of the person whose observation was cut off must be the same as the future of the people who remain. When this premise breaks, the curve itself skews to one side. When and how it breaks is a major topic, so it is covered separately in "When Did the Clock Start for These People? — How Observational Studies Manufacture Effects".

It is also rare for this handling to be properly written in papers. When 242 survival analysis papers published in 4 major Chinese oncology journals in 2013 were evaluated across 16 items, 139 papers (57.44%) did not define what was measured as survival time. Among 100 papers reporting that there were dropouts from follow-up—meaning people who could no longer be watched because they lost touch—only 11 wrote down how that dropout was handled in the analysis (PMID 29390340). If it is not written when counting started and how the people who disappeared were handled, readers have no way to check whether that curve was drawn correctly.

Hazard — The Speed at Which Events Happen at This Very Moment

The hazard is the speed at which an event happens in the very next moment to people who have not had the event up to a certain point. If the survival curve is likened to a balance showing the proportion remaining so far, the hazard is a speedometer showing how fast that balance is currently shrinking.

One sentence from a colorectal cancer paper was a story about hazard. It was the sentence: "It was low until the first year and nine months, and high thereafter." This sentence cannot be written by looking at the survival curve alone. That is because the curve only shows what percentage has survived so far, not how fast it is shrinking at that moment.

Hazard has two properties. First, the value of the hazard differs at each time point. The hazard the day after surgery and the hazard three years after surgery are naturally different. Second, the hazard is calculated only among the people who have survived up to that point. The denominator is not the initial 500 people, but the risk set at that point.

This second property is half of the answer to the question in the title. As time goes on, the groups compared by the hazard change from the two groups divided by a lottery at the beginning, to the two groups that have survived until then.

Hazard Ratios — Dividing the Speeds of Two Groups

A hazard ratio is the hazard in the treatment group divided by the hazard in the control group. It is the reading of one speedometer divided by the reading of the other.

• A hazard ratio of 1 — the speeds of the two groups are the same.
• A hazard ratio of 0.7 — events occur in the treatment group at 0.7 times the speed of the control group.
• A hazard ratio of 1.4 — events occur faster in the treatment group.

Up to this point, it is not difficult. What is difficult is translating this number into plain English.

Why "30% Lower Risk" Is an Inaccurate Translation

When people encounter a hazard ratio of 0.7, they mostly translate it as "risk is reduced by 30%." This translation is inaccurate, and multiple papers point this out directly.

One paper places its main point right in the title. A hazard ratio should not be described as a relative risk, but as a relative rate of occurrence. The authors explain it as follows. The direction of a hazard ratio tells you whether the risk went down or up. But the magnitude of a hazard ratio alone cannot tell you how much the risk went down. The authors show in a table how widely the risk ratio can vary even for the same hazard ratio value. They also point out that although some journals have guidelines telling authors not to report hazard ratios as risk ratios, these guidelines are frequently ignored (PMID 29239842). In English literature, risk ratio is written as relative risk, and when translated directly it is also called relative risk.

A short commentary reaches the same conclusion. One minus the hazard ratio must not be interpreted as "risk reduction" in the conventional sense (PMID 28314839).

Another paper suggests an alternative, arguing that "relative risk" should not be used as a synonym for "hazard ratio." It is the probability that a treated person survives longer than an untreated person, all else being equal. The authors write that this probability is obtained by a simple transformation of the hazard ratio, and that the transformation clearly reveals the limits of interpreting a hazard ratio (PMID 30623465).

A colorectal cancer case illustrates this critique with numbers. Translating a hazard ratio of 0.82 as an "18% reduction in risk" clashes directly with the risk ratio of 0.99 obtained from the same data. This is because the proportion of people who experienced the event by the end of follow-up was nearly identical in the two groups.

Then What Is 0.7 the 0.7 Of?

A hazard ratio also has a precise meaning that can be expressed in plain English. However, this is only true when the condition that the ratio of hazards between the two groups does not change over time is met. This condition is the proportional hazards that will be covered in the next section.

That translation uses the word odds. Odds are the probability that an event occurs divided by the probability that it does not occur. In a study where proportional hazards hold, the hazard ratio is equal to a set of odds (PMID 40697630). It is the odds that, if you randomly pick one person from the treatment group and one person from the control group and pair them up, the person from the treatment group will experience the event first.

When you convert these odds into a probability, the meaning becomes tangible. With a hazard ratio of 0.7, the probability that the person picked from the treatment group survives longer than the person picked from the control group is about 59%. Since a coin flip is 50%, what that striking number of 0.7 actually promises is only slightly better than a coin toss. This 59% is the definition from the paper above translated into arithmetic, and it is correct only when proportional hazards hold. It is the exact same quantity as the "probability of surviving longer" that appeared just above.

"A 30% reduction in risk" and "if you pick one person from each group and pair them up, in about six out of ten pairs the treatment group person survives longer" give completely different impressions. The accurate one is the latter.

Proportional Hazards — Can We Summarize with Just One Number?

Proportional hazards is the premise that the hazard ratio of two groups remains constant over time. Even if both speedometers speed up and slow down, the ratio between them stays at 0.7, for example. Because the hazard changes at every point in time, this premise is necessary to summarize the divided hazard of two groups into a single number. The standard procedure for calculating a hazard ratio under this premise is the Cox model, which usually appears in papers as the "Cox proportional hazards model."

If the premise holds, summarizing with a single number loses nothing. The problem is that this premise is frequently broken. A hepatocellular carcinoma trial shows what that looks like.

When the results of this trial are recalculated by dividing them into periods, the hazard ratio for the period before 12 months was 0.59 (95% confidence interval 0.43–0.73) and for the period after 12 months was 1.12 (0.88–1.37). A confidence interval is the range where the true value is likely to lie. This means there was a benefit in the earlier period, but no benefit in the later period. However, the interim analysis of this trial—an analysis peeked at once before the trial ended—was reported as showing a benefit.

This trial came from data reviewing 20 phase 3 hepatocellular carcinoma trials (2008–2024). Among them, proportional hazards were broken in 4 trials (20%), and all 4 were immunotherapy trials. Immunotherapy is a drug that wakes up the body's immune system to attack instead of directly attacking cancer cells. The numbers by period above belong to one of those four trials. The authors conclude that a proportional hazards violation contributed to the discrepancy between the interim and final analysis results in this trial, and suggest that trials should not be terminated early before meeting the minimum follow-up duration or minimum number of events (PMID 40992574). It is a warning that a benefit visible only in the earlier period can remain as the conclusion of the entire trial.

Such violations are not rare. Re-examining 341 two-arm randomized phase 3 oncology trials (357 comparisons, roughly 290,000 registered patients) whose results were published up to 2020, proportional hazards violations appeared in 85 cases (23.8%). The confirmation method was reconstructing individual patient data from the published survival curves. Among the 85 cases where violations occurred, only 25 cases (29.4%) had pre-specified statistical plans to handle them. In 14 trials where violations occurred, alternative analysis methods gave different signals than the original analysis, and 10 of those were trials originally reported as "no effect" (PMID 39133081). This means some of the trials wrapped up as "no effect" could be read differently when viewed with other methods.

Papers stating that they checked proportional hazards are also rare. Looking at 66 trials where the primary outcome was time-to-event among 446 original articles published in 4 highly cited medical journals in the first half of 2017, 98% used survival curves and 97% used the Cox model. However, among the trials reporting Cox model results, only 11% wrote anything about checking whether proportional hazards held (PMID 31096924).

When proportional hazards are broken, a single hazard ratio is even difficult to call an "average." That is because it is unclear what it is an average of. One commentary calculated with hypothetical data to show that the hazard ratio produced by standard procedures at this time is difficult to interpret even as an "average treatment effect." This commentary throws out a question as its title. How do when patients were enrolled and how long they were followed change that value (PMID 30201741)? If it changes, it means the effect size moves simply because the data was watched longer, making it difficult to call that number "the effect of this treatment."

Three Common Misunderstandings

① "A Hazard Ratio of 0.7 Means 30% Fewer Deaths"

This is a widespread misreading in the field. The hazard ratio is the ratio of the speeds at which events occur, not the ratio of the number of people who experience the event. In the colorectal cancer example, the ratio of speeds was 0.82 and the ratio of the number of people was 0.99 (PMID 39212064). The accurate translation is closer to "they last longer with odds slightly better than a coin flip" (PMID 40697630, PMID 30623465).

② "The Difference in Median Survival Time Is the Size of the Benefit"

The median survival time is the time point where the survival curve crosses 50%, in other words, the time it took for half of the participants to experience the event. That is the number written in papers as "median overall survival 12.4 months vs. 9.8 months" (a fabricated value just to show the format).

While this value is intuitive and useful, the difference between the two groups is not the size of the effect. This is because the median survival time is a value read from a single point on the curve, whereas the hazard ratio comes from the entire curve. If two curves touch only at the 50% mark and diverge everywhere else, the difference in median survival time can be 0 while the hazard ratio is far from 1.

In real papers, the two values usually agree in direction but vary wildly in magnitude. Out of 106 pairs of hazard ratios and median survival time ratios (the median survival time of the two groups divided) extracted from 54 cancer survival papers published in the New England Journal of Medicine between 2000 and 2010, 99 pairs (93.4%) moved in the same direction. On average, the magnitudes also matched well (mean ratio 1.01, 95% confidence interval 0.95–1.06).

What failed to match was not the average, but the spread. Down at the level of individual studies, the median survival time ratio could be up to 50% larger or 50% smaller than the hazard ratio. Therefore, the authors concluded that while the average value can be used to roughly estimate sample sizes, the two metrics cannot be used interchangeably, and they noted that current practices of reporting only the hazard ratio are inadequate (PMID 25063554). This means that guessing the hazard ratio based solely on the difference in median survival time in a single paper can lead to a massive miss.

③ "The Curves Diverged, So There Is an Effect"

When two lines spread apart in a figure, it is visually persuasive. However, where that divergence came from matters.

If the separation happened at the far right end of the curve, it might have been created by just a handful of people. That is because in a section where the risk set has dwindled down to a dozen or so people, a single person's event can violently shake the curve. The number to judge this by is the size of the risk set below the curve. Yet in 138 original sports science and medicine papers, over 90% of studies failed to include this number (PMID 41368207), and among recent phase 3 oncology trials featuring survival curves, 28% also omitted it (PMID 35094686).

Whether the divergence is sustained must also be checked separately. In the cancer immunotherapy trial we saw earlier, the separation before 12 months did not hold up afterward, and the conclusions of the interim analysis and the final analysis conflicted (PMID 40992574).

Honest Unresolved Issues

① There is no consensus on what to replace proportional hazards with when it breaks. Many methods that do not assume proportional hazards have been proposed, but there is no consensus on which is best. This gap was the starting point for a review that systematically searched for and organized relevant methods. The authors scanned 907 abstracts to fill that gap, organized 211 papers, and went as far as extracting situational guidelines from them (PMID 38592333).

② Whether checking proportional hazards is even necessary is itself under debate. On one side, there are demands to check and report it. On the other side is a more radical argument. It is that the proportional hazards assumption is implausible to begin with in most medical studies, especially randomized trials, and simultaneously unnecessary—so methods that do not rely on that assumption should be used by default (PMID 39756420).

③ Hazard ratios are difficult to pool across trials. When there is no trial directly pitting drug A against drug B, it is common to indirectly compare the two by combining hazard ratios from different trials. However, one paper shows that hazard ratios generally lack the properties needed for such pooling, and recommends other metrics (PMID 41992905).

④ "Causal hazard ratios" are still at the theoretical stage. What a hazard ratio compares are two groups that have survived up to that point. Even if these two groups were initially divided at random, over time they are usually no longer groups that can be fairly compared. There are attempts to define a hazard ratio that can be interpreted as a difference caused by treatment—namely, a causal hazard ratio—but that definition relies on assumptions that cannot be verified with data (PMID 32654089).

To Go Deeper

This article goes only as far as reading a single survival curve and a single number beneath it. To take apart the methods section of a paper yourself, there are several primer papers. An introductory primer showing step-by-step procedures with cardiovascular nursing data (PMID 36748198), an overview that skims censoring methods and curve comparison methods while attaching a checklist of reporting guidelines (PMID 32658651), and a tutorial that carries you from the basic concepts using a brain metastasis surgery case (PMID 35505548) are examples of these.

There are also two places this article intentionally left blank. One is how these numbers are created in observational studies. When follow-up starts being counted, and what happens if the reasons observations stop are tangled up with the outcome, belong here. The other is what to look at instead when the proportional hazards assumption is broken. Metrics that measure benefit in units of time, and cases where events other than death happen first, belong here. The former is handled by "When Did the Clock Start for These People? — How Observational Studies Manufacture Effects", and the latter by "When People Don't Stay in Line — The Ones Who Switched, and the Ones Who Left for Another Reason".

Source Papers

Where the Hazard Ratio and Risk Ratio Measure Different Things

• "Hazard Ratios and Alternative Effect Measures: An Applied Illustration." (2024) — PMID 39212064 — Data from two randomized trials of colorectal cancer patients with metastasis to other organs. Hazard ratio 0.82 (0.71–0.93) versus risk ratio 0.99 (0.96–1.02). The proportion of people who experienced the event by the end of follow-up was similar, but the treatment group took longer to reach the event, and the proportional hazards assumption was clearly violated (the direction reversed around 1.75 years).
• "Relative rates not relative risks: addressing a widespread misinterpretation of hazard ratios." (2018) — PMID 29239842 — Hazard ratios must be described as relative occurrence rates, and while the direction can be used, the magnitude alone cannot be used to state the size of the risk. A table showing the variation range of risk ratios by value is presented.
• "On the Interpretation of the Hazard Ratio and Communication of Survival Benefit." (2017) — PMID 28314839 — 1−HR should not be interpreted as risk reduction in the conventional sense.
• "On the interpretation of the hazard ratio in Cox regression." (2020) — PMID 30623465 — Do not use "relative risk" as a synonym. As an alternative, it proposes "the probability that an event occurs later in the treatment group."
• "The Hazard Ratio Is Equivalent to an Odds Under the Assumption of Proportional Hazards." (2025) — PMID 40697630 — When proportional hazards are justified, the hazard ratio is equivalent to the odds that a person randomly drawn from the numerator group experiences the event before a person randomly drawn from the denominator group.

Numbers Beneath the Curve and Reporting Practices

• "Methodological quality and reporting using time-to-event data in sports science and medicine: a scoping review." (2025) — PMID 41368207 — 138 papers (2013–2023). Over 90% did not report risk set sizes by time point, 75% did not address model assumptions, and 20% reported median survival time.
• "Which test for crossing survival curves? A user's guideline." (2022) — PMID 35094686 — Reviewed 1,400 phase 3 oncology trials, with 15 passing data reconstruction requirements. 28% of studies featuring survival curves failed to report risk set sizes.
• "Reporting and methodological quality of survival analysis in articles published in Chinese oncology journals." (2017) — PMID 29390340 — 242 papers. 139 papers (57.44%) left survival time undefined, and out of 100 reporting loss to follow-up, 11 described how it was handled.

What Real Violations of Proportional Hazards Look Like and How Often They Occur

• "Strategies to address non-proportional hazards between survival curves — Lessons from phase III trials in hepatocellular carcinoma." (2026) — PMID 40992574 — Out of 20 phase III trials in hepatocellular carcinoma (2008–2024), 4 trials (20%) violated proportional hazards, all involving immunotherapy. IMbrave050 piecewise hazard ratios were 0.59 (0.43–0.73) versus 1.12 (0.88–1.37). Early termination boundaries at interim analysis.
• "Proportional Hazards Violations in Phase III Cancer Clinical Trials: A Potential Source of Trial Misinterpretation." (2024) — PMID 39133081 — 341 trials, 357 comparisons, and roughly 290,000 people. 85 violations (23.8%). 25 pre-specified statistical plans (29.4%). Of 14 cases where signals conflicted with other methods, 10 were negative reporting trials.
• "Are non-constant rates and non-proportional treatment effects accounted for in the design and analysis of randomised controlled trials?" (2019) — PMID 31096924 — 66 relevant trials out of 446 original articles across 4 high-impact journals. Survival curves used in 98% and Cox models in 97%. Proportional hazards check results reported in 11%.
• "How Do the Accrual Pattern and Follow-Up Duration Affect the Hazard Ratio Estimate When the Proportional Hazards Assumption Is Violated?" (2019) — PMID 30201741 — Calculated using simulated data, showing that when proportional hazards are broken, the standard Cox procedure's hazard ratio is difficult to interpret even as an "average treatment effect." The relationship with accrual patterns and follow-up duration is the question posed by the commentary's title.

Median Survival Time

• "A hazard ratio was estimated by a ratio of median survival times, but with considerable uncertainty." (2014) — PMID 25063554 — 54 cancer survival papers and 106 pairs from NEJM. Direction matched 93.4%, and quantitative agreement was good on average (mean ratio 1.01, 0.95–1.06). However, because the spread is large, in individual studies the ratio of median survival times can be up to 50% larger or smaller than the hazard ratio. While estimating sample size with the average value is possible, the two metrics are not interchangeable. The practice of reporting only hazard ratios is insufficient.

Limitations and Debates

• "Methods for non-proportional hazards in clinical trials: A systematic review." (2024) — PMID 38592333 — Included 211 out of 907 abstracts. "No consensus on the best approach" is the gap the authors set out to fill, and the paper's conclusion is providing an overview of methods plus deriving guidance from the identified literature.
• "Why use methods that require proportional hazards?" (2025) — PMID 39756420 — The argument that the proportional hazards assumption is implausible and unnecessary in most medical research.
• "The hazards of using hazard ratios from proportional hazard models in indirect treatment comparisons." (2026) — PMID 41992905 — Hazard ratios generally lack the properties required for indirect comparisons between trials.
• "Subtleties in the interpretation of hazard contrasts." (2020) — PMID 32654089 — Hazard ratios compare two groups that have survived a certain period of time, and those two groups usually become no longer comparable even in randomized trials. Causal hazard ratios rely on empirically unverifiable assumptions.

Introductory & Tutorial

• "Survival analysis 101: an easy start guide to analysing time-to-event data." (2023) — PMID 36748198
• "A Practical Overview and Reporting Strategies for Statistical Analysis of Survival Studies." (2020) — PMID 32658651
• "The Proper Use and Reporting of Survival Analysis and Cox Regression." (2022) — PMID 35505548