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I keep running into problems in understanding hazard rates. I know, for example, that in a strict sense a hazard rate is not a probability and it is continually mentioned that because of this the hazard rate has no upper bound.

Am I right in understanding that hazard function values can be interpreted as instantaneous failure rates? Given this interpretation, I still have trouble seeing how this could exceed 1.

I've seen derivations of the constant hazard rate $\lambda$ for the exponential distribution, so I see that for $\lambda \geq 1$ my question seems to have an obvious answer. But even then, is $\lambda$ not techincally a rate? I know we often just simplify and say, for instance, $\lambda = 3$. But don't we really mean $\lambda = 3$ per 1000 or 3 per 100? And if that is the case, can we really say the hazard rate is exceeding 1 in this case?

If someone could provide alternative examples where $h(t)\geq 1$ or help me understand my misconception it would be much appreciated.

T Bonnett
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2 Answers2

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The hazard is indeed a rate. It is the expected number of events a person can expect per time unit conditional on being at risk, i.e. not having died before. Say we are studying the time until you get the flu [influenza] , and we measured time in months and we got a hazard rate of .10, that is, a person is expected to get the flu .10 times per month assuming the hazard remains constant during that month. We could just as well measure time in decades (120 months), and we would get a hazard rate of 12, i.e. a person is expected to get flu 12 times per decade. These are just different ways of saying the exact same thing.

This is easy to see with something like the flu, which you can easily get multiple times. It is a bit harder to see when we talk about dying, which typically happens only once. But that is a substantive problem: from a statistical point of view the expectation could still be larger than 1, which means you are unlikely to survive a unit of time.

Maarten Buis
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    So how would you interpret the hazard if we talk about dying? – Meo Sep 07 '18 at 21:17
  • You are very unlikely to survive that period... – Maarten Buis Sep 07 '18 at 22:12
  • If the hazard is 0.02, does that mean a person is expected to die 0.02 times per year? That doesn't make much sense...Or should I interpret it as 2% of the population is expected to die per year? Thank you... – Meo Sep 07 '18 at 22:42
  • The hazard rate is a rate, so the *expected* number of times you die in a time interval. Why do you think that that does not make sense? – Maarten Buis Sep 08 '18 at 18:51
  • It does make sense to me in other situations, but in dying, you can only die once. It makes sense to say "a person is expected to have the flu 0.02 times per year", but it is a bit different to report "a person is expected to die 0.02 times per year". I was just wondering is there a different that you will report/say it if you are explaining to the general population? Thank you so much for your patience! – Meo Sep 09 '18 at 04:31
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This is perhaps best understood by looking at the exponential (time to a single event) distribution with a constant hazard rate $\lambda $, there the probability of an event by time $t$ is $1-e^{-\lambda t}$. So it is always $\in [0,1] $. In contrast the Poisson distribution is the corresponding distribution for recurring events, where the expected number of events seen in time $t $ is $t\times \lambda $.

Björn
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