Over the holidays my dad posed a two-part question after dinner: If a vehicle goes 200,000 miles without a mechanical failure does that mean that it is more likely to have a failure soon? And does the same hold true for an employee who goes 25 years without a lost-time accident at work?

In answer to the first question I responded “no.” Failure rates of electrical and mechanical devices are typically modeled as an exponential distribution. This distribution is “memoryless” in the sense that the probability of having an accident at time *t=*1 given no accidents up til time *t*=0 is the same as the probability of having an accident at time *t*=2 given no accidents through time *t*=1.

For the employee question the answer is a bit more complex. Here I would assign workers to two types: accident prone or safe. The longer a worker is accident free, the higher the probability that they are in the safe category. (I would use a beta-binomial model with Bayesian updating for those playing at home). A more realistic analysis might include aging effects such as deterioration of eyesight and balance on the one hand, with gaining wisdom and on-the-job experience on the other. For all I know such a study may exist.

At its core this is the same idea as an athlete “coming due” in baseball–false–or having a “hot hand” in basketball”–also false. The interesting part about all this to me is that for two superficially similar problems different models are required by virtue of the fact that humans are involved. Mechanical parts are memoryless; people are not. That makes all the difference.

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Neat examples. It would help, though, to make explicit what you’re comparing in the automobile example. When your dad asks “more likely to have a failure soon,” my question is, “More likely than what?”

If the answer is “more likely than the same vehicle 10 or 50 or 100 thousand miles ago,” the answer is probably yes, not no. I suspect the hazard rate for mechanical failures rises with age because mechanical parts *do* have a memory of sorts, in the form of physical changes resulting from stress. In event-history speak, there may be strong duration dependency in mechanical failure data because stress accumulates.

If, however, the answer is “more likely than a similar vehicle that has had failures before,” then I think your “hot hand” analogy is relevant, and the answer may be “no.” I say maybe, though, because it could be that prior failures increase risks of future failures by imposing extreme stresses (heat, torque) on other parts without quite breaking them.

All of which is a drawn-out way of pointing out how important it is to be clear about exactly what comparison you’re drawing when talking about relative risks.

Good points. It’s been too long since the actual conversation for me to be certain of the implied comparison, but I think your analysis is right in either case.