The hare and the tortoise

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Hazard ratio

We all know the fable of the hare and the tortoise. It turns out that, for some reason I cannot understand, someone comes up to make a race with two participants: a tortoise and a hare. Naturally, the favorite of the race is the hare, infinitely faster than the tortoise. It turns out that the hare relies too much and goes to sleep, so when it realizes it can no longer recover the advantage of the tortoise and loses the race against all odds. Moral: Never underestimate others and do not rest on its laurels, or you may spend what happened to the hare.

Sometimes we can think of the clinical trial as a race among participants. This is when the main outcome variable is a time-to-event variable. These variables measure how many participants have the event in question and, more importantly, the time it takes to occur. Time-to-event variables are also called survival variables, although they don’t need to be related to mortality.

Here’s an example. Suppose we want to know the effectiveness of a drug for controlling blood pressure. We give the drug to the intervention group and a placebo to the control group to see how many are better controlled and how long it takes them to improve.

Different meassures of association

One possibility would be to use risk ratios or relative risks. We divide the proportion of patients who improve in the intervention group by the proportion that improve in the control group and get our relative risk. The problem is that we get information about how many more improve in one group than in the other, but we cannot say anything about the temporal aspect. We do not know if they improve sooner or later.

Another possibility is to consider blood pressure control as a dichotomous outcome variable (yes or not) and compute a logistic regression model. With this model will get an odds ratio that will give us similar information to that obtained by risk ratio, but nothing about the temporal aspect of the occurrence of the event.

The appropriate method to analyze this problem would be to establish the dichotomous measure of blood pressure control, but calculating a model of proportional hazards regression or Cox regression. This regression model does take into account the time it takes the event to occur.

Hazard ratio

The Cox regression model calculates the risk of the event in exposed to the intervention compared to unexposed at any given time. To do this, it calculates how much more likely is that the event occurs in the next time interval among subjects who have not suffered the event yet. Taking this measure to the limit, if we proceed to shorten the time interval until zero, we get to the instantaneous risk, which oscillates with time, but the model calculates an average extrapolation with it. This extrapolation is called hazard ratio (HR).

HR can have values between zero and infinity. The neutral value is one, indicating the same risk in both groups. A value less than one indicates lower risk in the exposed group. Finally, a value greater than one indicates higher risk among exposed, the greater the larger the value of HR.

The HR is not a measure of probability but an odds, so its interpretation is similar to that of the odds ratio, but with the addition that also takes into account the temporal aspect. A common misconception is that the HR reports the temporal duration to the event. For example, an HR = 2 does not mean that the event in exposed develop twice as fast, but that those exposed who have not yet presented the event are twice as likely to present it than the unexposed.

If you want information about the rate at which the event occurs, you have to use other index which is the median time in which the 50% of the participants present the event.

Returning to our tale of the race, the HR would tell us who is most likely to win the race, while the median tell us how much benefit would take the winner to the loser.

We’re leaving…

And here we leave the subject of the hare, the tortoise and the proportional hazards regression. We have not talked anything about how to represent the results of the Cox regression model. For this a special type of graphics called survival curves or Kaplan-Meier’s are used. But that is another story…

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