## HR interpretation for 3 different scenarios

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Posts: 301

# HR interpretation for 3 different scenarios

Treatment:

HR=2,  For this particular example, HR could be interpreted in terms of probability: P=HR/(1+HR)x100=67% chance of the treated patients progressed to metastasis
than Placebo group at the next point in time.

Gender:

HR=0.937, in % (0.937-1)*100=-6.3 % the risk of death in females decrase compared to males.

Year:

HR=1.069, in % (1.069-1)=6.9 % the risk of death increase every year:

Why when HR=2, we use % in termns of probability, it should be (2-1)*100=100%? but it doesnt make sense.

Anyone can give me consistent explanation for these 3 cases?

Why HR=2 interprete in terms of the probabilities, and not the other cases?

Thanks.

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Posts: 301

## Re: HR interpretation for 3 different scenarios

You can not interpret HR in terms of probability. At least not so directly as you suggest.

Example, if your control Group has a baseline hazardrate of 0.5 per year, then probability for failure within a year is 1-exp(-0.5*1) ~ 0.0393

If HR=2 in an exposed patientgroup relative to the baseline then the probability among the exposed is 1-exp(-0.5*2*1) ~ 0.632.

The ratio between the probabilities is not the same as the HR.

Only, if you look in very tiny intervals, then the HR is the same as the ratio between probabilities of event in these tiny intervals.

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Posts: 301

## Re: HR interpretation for 3 different scenarios

Thnaks, but can you explain me this from wikipedia:

The hazard ratio and survival

Hazard ratios are often treated as a ratio of death probabilities.[2] For example, a hazard ratio of 2 is thought to mean that a group has twice the chance of dying than a comparison group. In the Cox-model, this can be shown to translate to the following relationship between group survival functions: $S_1(t) = S_0(t)^r$ (where r is the hazard ratio).[2] Therefore, with a hazard ratio of 2, if $S_0(t) = .2$ (20% survived at time t), $S_1(t) = .2^2 = .04$ (4% survived at t). The corresponding death probabilities are .8 and .96.[9] It should be clear that the hazard ratio is a relative measure of effect and tells us nothing about absolute risk.[11]

While hazard ratios allow for hypothesis testing, they should be considered alongside other measures for interpretation of the treatment effect, e.g. the ratio of median times (median ratio) at which treatment and control group participants are at some endpoint. If the analogy of a race is applied, the hazard ratio is equivalent to the odds that an individual in the group with the higher hazard reaches the end of the race first. The probability of being first can be derived from the odds, which is the probability of being first divided by the probability of not being first: HR = P/(1 − P); P = HR/(1 + HR). In the previous example, a hazard ratio of 2 corresponds to a 67% chance of an early death. The hazard ratio does not convey information about how soon the death will occur.[1]

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Posts: 301

## Re: HR interpretation for 3 different scenarios

In other words:

I have the next result applying PHREG:

phreg data=test;

model time*censor(0)=age drug / risklimits;

'Cox model for TEST data-- ties';

;

run;

age HR=1.096

Treatment(drug use) HR=2.563

I should understand it as:

9.6% increase in hazard for every 1-year increase in age, controlling for the effect of IV drug use; and 156.3% increase in hazard for using IV drugs, controlling for age....but I think this 156.3% doesnt have many sense.

Thanks.

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Posts: 301

## Re: HR interpretation for 3 different scenarios

Rates is not the same as probabilities. therefore, hazard rate ratios is not the same as ratios between probabilities.

if you have the equation, $S_1(t) = S_0(t)^r$, where r is the hazard ratio and S1 and S0 is the probaility of survivil at time t for the exposed and unexposed group, then you will have r=log(S1(t))/log((S0(t)).That is not the same expression as the probability ratio S1(t)/S0(t). It is often seen in medical papers that rate ratios are called relative risk, but this is misleading. The confusion is partly explained by the common abreviation (RR).

Super Contributor
Posts: 301

## Re: HR interpretation for 3 different scenarios

Thanks for the explanation.more clear now...but

My question is, in terms of HR=2.563, can we say something about the percentage 156.3% increase in hazard for using IV drugs controlling for age? Or how you interprete it?

Thanks.

Super Contributor
Posts: 301

## Re: HR interpretation for 3 different scenarios

You can understand it as a 156.3% increase in rate of infection. You can also call it "hazard rate" or "incidence rate", but not just "hazard". It is not obvious for all that "hazard" means a rate.

I better like to say the rate has a 2.55 fold increase among exposed persons, but that is also just words.

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Posts: 301

## Re: HR interpretation for 3 different scenarios

Hi Jacob, thanks for answer, I appreciate your interest in this discussion...

For me the different between RR and HR is clear

Hazard ratios differ from relative risk (or risk ratio not relative risk rations) in that risk ratios are cumulative over an entire study (you must use a defined endpoint which is pre-selected during design phase), while hazard ratios represent instantaneous risk at some particular time period during study, or a subset thereof. Hazard ratios suffer somewhat less from selection bias with respect to the endpoints chosen, and can indicate risks that happen before the endpoint.

But...still I can see in the literature papers talking about the HR in terms of probabilities, and still I want to understand it, because there not only one paper saying that, i have seen many, maybe we are missing something here:

Please can you have a look in these two ones and let me know if they are still wrong considering HR as a probability?

This one:

What does a 13% increased risk of death mean? | Understanding Uncertainty

where the author saying..

An extremely elegant mathematical result, proved here , says that if we assume a hazard ratio h is kept up throughout their lives, the odds that Mike dies before Sam is precisely h. Now odds is defined as p/(1p), wherep is the chance that Mike dies before Sam. Hence

p=h/(1+h)=0.53.

and looks like the mathematical prove is shown in this other one:

Can we say whether a drug would have enabled someone to live longer? Sadly not. | Understanding Unce...

Please, can you have a look on these two articles, and let me know what do you think?

Thanks,

.

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Posts: 301

## Re: HR interpretation for 3 different scenarios

I agree that the probability between that an exposed person lives longer than an unexposed 1/(1+λ) with assumption of proportional hazards. So, if λ=2 then the probability that the exposed person lives longest is 1/3. When the hazard increase the probability decrease, which intuitively makes sense. Also, observe the probability always will be between 0 and 1. I must confess I had forgot that result, and btw I disagree that it is common to use that interpretation in medical papers.

If the HR is 1.13 it means the rate is increased by 13%. This can be understand also be understand as a probabability. The probability that an exposed person dies with in the next tiny interval is increased by 13% compared to unexposed. This should be understand as the limit when the the length of the interval goes to 0.

Super Contributor
Posts: 301

## Re: HR interpretation for 3 different scenarios

Hi Jacob, thenare you agree with this paragraph when hazard ratio=2 or 3?

For any randomly selected pair of patients, one from the treatment group and one from the control group, the hazard ratio is the odds that the time to healing is less in the patient from the treatment group than in the patient from the control group. With the following equation, the probability of healing first can easily be derived from the odds of healing first, which is the probability of healing first divided by the probability of not healing first: hazard ratio (HR) = odds = P/(1 − P); P = HR/(1 + HR). A hazard ratio of 2 therefore corresponds to a 67% chance of the treated patient's healing first, and a hazard ratio of 3 corresponds to a 75% chance of healing first.

Super Contributor
Posts: 301

## Re: HR interpretation for 3 different scenarios

yes, I agree. Though,  taking odds into this only help to confuse readers.

When HR=2 then the probability that the treated will have longest time to event is P=1/(1+HR)=1/3.

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