# Tutorial about Hazard Ratios

Posted on 5th April 2016 by Loai Albarqouni

**You have been asked to run a journal club in your department about heart failure. You followed the key steps of Evidence-Based Practice that you have learned: formulated a clear searchable clinical question, searched the main databases and came across this recent hypothetical article. **

In this hypothetical article, the authors conducted a randomised controlled trial to investigate the effectiveness of a new treatment for improving the survival of patients with heart failure. A total of 105 patients were randomised to the treatment group and a total of 106 were randomised to the control group. The primary outcome was the overall survival rate.

The main result of this trial can be summarised in this paragraph: *“After a median follow-up of 1500 days, the hazard ratio for death in the treatment group, 0.38; 95% confidence interval [CI], 0.28 to 0.53; P<0.0001).” *

In addition, the authors presented the figure below summarizing their main results:

*Figure produced by the author using R software
(Blue line represents treatment group and green line represents control group).*

**You want to critically appraise the article. However, you have some difficulty understanding its main results and have a few questions: **

- What is Hazard Ratio? What do the authors mean by 95% confidence interval? How can you interpret this information?
- How can I read the figure? What are the main findings I can learn from this figure?
- How can I convey this information to my patients?

**In this blog, you may find the answers to these questions.**

## What is a Hazard Ratio & what are Confidence Intervals?

**Hazard ratio** (HR) is a measure of **an effect **of **an intervention **on **an outcome** of interest over time. Hazard ratio is reported most commonly in time-to-event analysis or survival analysis (i.e. when we are interested in knowing how long it takes for a particular event/outcome to occur).

The outcome could be an adverse/negative outcome (e.g. time from treatment/surgery until death/relapse) or a positive outcome (e.g. time to cure/discharge/conceive/heal or disease-free survival).

Hazard Ratio (i.e. the ratio of hazards) = Hazard in the intervention group ÷ Hazard in the control group

Hazard represents the instantaneous event rate, which means the probability that an individual would experience an event (e.g. death/relapse) at a particular given point in time after the intervention, assuming that this individual has survived to that particular point of time without experiencing any event.

**Confidence Interval (CI):** is the range of values that is likely to include the true population value and is used to measure the precision of the study’s estimate (in this case, the precision of the Hazard Ratio). The narrower the confidence interval, the more precise the estimate. (Precision will be affected by the study’s sample size). If the confidence interval includes 1, then the hazard ratio is not significant.

## Interpretation of Hazard Ratio

Because Hazard Ratio is a ratio, then when:

**HR = 0.5: **at any particular time, **half** as many patients in the treatment group are experiencing an event compared to the control group.

**HR = 1: **at any particular time, event rates are the **same** in both groups,

**HR = 2: **at any particular time, **twice** as many patients in the treatment group are experiencing an event compared to the control group.

## Applying this to our hypothetical study results:

__In the results, the authors reported that the hazard ratio for death with the new treatment = 0.38 (95% CI, 0.28-0.53; P<0.0001). What does that mean?__

*Patients in the new treatment group at any time point during the study period were 62% less likely to die than patients in the control group, and we are 95% confident that the true value is lying between 47%-72%. (i.e. we are 95% sure that patients in the new treatment group were between 47% and 72% less likely to die than patients in the control group). *

## Kaplan-Meier curve (or Survival curve)

*Figure produced by the author using R software
(Blue line represents treatment group and green line represents control group).*

**Kaplan-Meier curve: **is a graphical method of displaying survival data or time-to-event analysis (i.e. the proportion of patents surviving against time) and is usually drawn as a step function.

- The
*horizontal axis*represents the time of follow-up starting from enrolment while the*vertical axis*represents the estimated probability of survival. - Each
*downward step*in the lines represents an event (the outcome of interest, e.g. death) experienced by a patient in that corresponding group, while each*small vertical tick*represents a censored observation (i.e. a patient who did not experience the event of interest by the last follow-up. This can be due to being lost to follow-up, the study period ending without an event or because the patient died from an unrelated cause). - The numbers of patients at risk at regular time intervals is shown at the bottom of the figure. With time, fewer people remaining are at risk.
- This graph also allows you to detect the progression of the condition (e.g. early postoperative mortality or a serious adverse event caused by the drug), by the presence of a notable sudden steep drop in survival at a particular time point.

## Interpretation of this figure

The results of this graph can be communicated in various ways:

*Median survival*: half of the patients in the treatment group have survived for 2246 days (median survival rate) compared to 906 days in the control group.*Difference in median survival*: On average, patients in the intervention group survived for 1340 days longer than patients in the control group (2246 – 906 days).*Proportion surviving at a specific time point:*about 45% of patients in the treatment group have survived for 10 years (3650 days, i.e. 10-year survival rate). Another way of saying this is that about 45% of the patients in the treatment group have not experienced the outcome of interest (e.g. death) after 10 years of follow-up.

## Hazard Ratios vs. Risk Ratios (or Relative Risk)

Hazard ratio is frequently interpreted as risk ratio (or relative risk), but they are not technically the same. However, if that helps you to understand hazard ratio then it is OK. But keep in mind HR is not RR.

One of the main differences between risk ratio and hazard ratio is that risk ratio does not care about the *timing* of the event but only about the *occurrence of the event by the end of the study* (i.e. whether they occurred or not: the total number of events by the end of the study period). In contrast, hazard ratio takes account not only of the *total* number of events, but also of the *timing* of each event.

**References & Further Reading**

- Altman D.G., Bland J.M.
*Time to event (survival) data.*BMJ 1998; 317:468 - Bland J.M., Altman D.G.
*Survival probabilities (the Kaplan-Meier method).*BMJ 1998;317:1572 - Spruance S.L., Reid J.E., Grace M., Samore M.
*Hazard ratio in clinical trials.*Antimicrob Agents Chemother 2004; 48: 2787–2792. - Sedgwick P.
*How to read a Kaplan-Meier survival plot*. BMJ 2014; 349: g5608 - Sedgwick P.
*Hazards and hazard ratios*. BMJ 2012; 345: e5980

## No Comments on Tutorial about Hazard Ratios

Alvaro WhittemburyI would recommend to add that the intervention reduced the risk of death by 62% (1-(0.38)*100) when compared to the comparison group.

9th April 2016 at 1:51 amLoai AlbarqouniSure, and i did something similar in the definition of hazard ratio : “Patients in the new treatment group at any time point during the study period were 62% less likely to die than patients in the control group”

11th April 2016 at 11:34 pmAnne AbbottThank you!

20th April 2016 at 4:50 amAprilHi Loai. Could you please clarify this statement:

we are 95% confident that the true value is lying between 47%-72%. (i.e. we are 95% sure that patients in the new treatment group were between 28% and 53% less likely to die than patients in the control group.

Wouldn’t it be (i.e. we are 95% sure that patients in the new treatment group were between 47% and 72%. less likely to die than patients in the control group?

Thanks in advance.

30th April 2016 at 10:55 amSelena Ryan-VigYou’re quite right. Apologies, this was a mistake in the publication process and was not Loai’s error. It has now been amended.

6th May 2016 at 10:19 amMedTrialHi, Loai!

27th May 2017 at 5:30 amI have a question related to this topic, please.

I have 4 groups (let’s call them A, B, C and D) and I am given hazard ratios with confidence intervals for groups B, C and D with A being the comparator. What I really want to do is combine A, B and C ‘s data and have that whole group compared to group D. Possible? How?

Thanks in advance!

Janine KhucHI there, thank you for the explanation it was super helpful. If I had the average age for the treatment and control group, would it be possible to “translate” Hazard Ratios in years of life expectancy following treatment relative to control?

Many Thanks,

17th August 2017 at 4:27 pmJanine

Scott LonningLoai

18th August 2017 at 1:10 pmVery nice explanation on HR. The best I’ve ever seen and much appreciated. I’m sharing with my staff!

Thank you

SimonI am still confused about the difference between HR and RR. You say HR takes into account timing. Does this mean HR is actually a function of time, HR(t)? If so, then is the reported HR basically an average at the end of the study? Doesn’t that make a single HR number quite equivalent to Risk Ratio? Or what? Thanks

25th September 2017 at 6:10 pmGene HeymanHazard ratios are by definition time-dependent and thus must change as a function of time, except in the exponential case. How then can there be a single, summary hazard ratio for two different (non exponential) survival curves? Is there an agreed upon comparison time, say the median for each?

10th November 2017 at 11:47 pmStaceThank you so much for the explanation, so easy to understand, gonna share it with the whole team. Bless you!

15th November 2017 at 7:48 ammayahi there

6th March 2018 at 8:36 amcould you please tell me how to calculate CI from HR & P value in an published article?

it should be noted that i havnt any data about participants in this study.

i need it for my meta analysis article

regards

Richardci IS .25 TO .53 tHEN WHERE IS THE 47 TO 72% COME FROM?

17th May 2018 at 5:02 pmLakshmi AranganWhat does a hazard ratio of 1.75 mean?

9th July 2018 at 1:29 pmjuneThank you !

6th August 2018 at 2:33 pmElliot Dinetzvery informative for M.D.’s thanks

5th September 2018 at 4:29 amRitasman BaisyaNice explanation

18th September 2018 at 5:25 amSilverio Cabellon, Jr.May I commend the clarity in which you have presented Hazard Ratio.

21st September 2018 at 11:56 pmAlice ChuThank you, thank you, thank you for explaining these complex stats so well.

17th October 2018 at 3:55 amAbdullahGood stuff.Thank you Loai

19th November 2018 at 9:44 pmGeorgeIn case anyone is looking for a quick and easy way to calculate a hazard ratio, HR p-value and confidence intervals, I found this free tool: https://www.gigacalculator.com/calculators/hazard-ratio-calculator.php . Real easy to use as well, just copy & paste from a spreadsheet.

17th January 2019 at 11:04 amFunmilayoGreat topic

26th January 2019 at 8:53 pmOmarthank you, great tutorial

4th February 2019 at 9:13 pmCyril Spannwhat about median endpoint ratio

1st April 2019 at 8:11 amEmma CarterA comment from Twitter from Ralph Brinks (Mathematician): Thx for sharing 👍 Not 100% accurate in 2nd part of the def “Hazard [rate] represents the instantaneous event rate, which means the probability that an individual would experience an event”. It’s not a probability (bc can be > 1) but a limit. See Kalbfleisch & Prentice – “The Statistical Analysis of Failure Time Data”.

22nd July 2019 at 10:58 amAkashHi Laoi,

Great job on explanation. Could you help me understand:

Drug A vs placebo: Stable disease in 66.7% vs 37.2% at 12 months p=0.00007 HR 0.34 (95% CI, 0.20-0.59)

How do you verbalize this data?

23rd July 2019 at 6:28 pmEvandroGreat !!! I finally got it!! Thank you and hello from Brazil

18th September 2019 at 2:31 amDr M Kapoorwonderful way to explain ….. worth reading the blog.

15th October 2019 at 5:24 amdefinitely good for medical students and also for teachers.

Girma TeketelewYour tutor is very precise and exellent one. It is clear and easily understandable. I always follow your tutor.

16th January 2020 at 2:19 pmI impressed by this one and I wanted the pdf of this tutor. Would you send me via my email(girmapharm@gmail.com) if it is possible?

I also want to know how the following be interpreted in survival time to death:-

Baseline weightof the patient, HR= 1.03 and age of the patient HR=0.98

assume the weight and age of the patient are continuous.

Please respond soon as possible.

Thank you in advance.

AAndreas BOUNASIt was excellent. Very comprehensive! Thanks and congratulations!

4th April 2020 at 11:30 amMustafaI wish I have seen this website before :(

5th May 2020 at 10:53 pmdr mubasharthank you for this simplicity

20th May 2020 at 1:14 pmMarcia Frenchhow did you get 47% – 72% from a CI of .28-.53?

27th May 2020 at 4:24 pmvictor waluchNice presentation, but a bit abriged. Suggest more examples

23rd July 2020 at 8:22 pmBrooke caruanaHello Loai,

I am a masters student in nursing and very new to research , hence doing as many tutorials as i can. This is one of the best i have seen. Thank you. I am still learning and was hoping you could elaborate on the confidence interval and your comment above regarding if the confidence interval includes one then the HR is not significant? thank you . any further recommendations for tutorials would be welcome as well.

19th August 2020 at 6:48 amJoyceI really appreciate this. Just dropping a message to say Thank You!!!

28th November 2020 at 12:48 pmGulsenThis page has helped me to understand these basic concepts than 2 semesters of Statistics course :) I salute you Loai, many thanks!

8th April 2021 at 9:05 amSmitaHi Loai,

I found your explanation really helpful. I however could not understand the No. at risk section. Does it mean as time goes by less number of individuals are at risk of death because they have died? Does it mean the remaining living people at that point in time? If so it still doesn’t correspond to the figures if we see the 6000 days readings. For treatment group, survival is nearly 35% as per the graph, but the no. at risk shows 2 patients. If we go by 35% of 105, it should mean 37 patients are still alive at 6000 days. I would be really grateful if you could clarify.

Thanks

24th September 2021 at 4:36 pmDaniel EdwardsHi Lori,

Not being in the medical of mathematics fields, can you put these figures in layman’s terms terms for me?

“Among the men, the relative hazard of death from coronary heart disease when one’s twin died of coronary heart disease before the age of 55 years, as compared with the hazard when one’s twin did not die before 55, was 8.1 (95 percent confidence interval, 2.7 to 24.5)”

Thank you very much.

Daniel

Source: https://www.nejm.org/doi/full/10.1056/nejm199404143301503

4th October 2021 at 4:43 pm