statisticalrethinkin..

274 10. DISTANCE AND DURATION1 component3 components10 components0 100 200 300days until failure0 100 200 300days until failure0 50 100 150 200days until failureFIGURE 10.1. Times to failure for a machine (or organism) in which failureof any one component leads to total failure. In each plot, time to failureis on the horizontal axis and density is on the vertical. Assume that eachcomponent of the machine has an equal chance of failure on any day fromnow until a year from now. Time to failure is the time it took for at least onecomponent to fail. Le: A machine with only one component shows theuniform distribution of the single component’s failure probability. Middle:A machine with 3 components usually fails sooner rather than later. Right:A machine with 10 components exhibits a nearly perfect exponential failuredensity. Comparison to perfect exponential in black.number of machines remaining aer any fixed amount of time follows the exponential. eexpected proportion remaining at time T will be:exp(−λT).e number of failures is highest at the start, because the most machines are still around tofail. Aer a while, there are very few machines le, and so fewer fail per unit time. Still,they are failing at the same rate. Substituting other events for “failure” leads one to noticeexponential processes in many natural systems, from atoms to ecosystems.As with the Gaussian density in Chapter 4, there is a maximum entropy interpretation ofthe exponential density as well. If all we know about a collection of measures is their averagedisplacement from some point, then the least surprising (maximum entropy) distribution forthe measures will be an exponential distribution. 10310.1. Survival analysisAs a fundamental distribution of displacements—distances in space or durations in time—the exponential is very handy for modeling. In this section, I provide a very brief introductionto its use in making GLM’s. is is a good place to start, because the exponential densityhas only one parameter, a rate, that determines both its mean and variance. And so it will beeasier to first encounter some aspects of GLM’s in the context of exponential distributions.On the other hand, the broader subfield that exponential GLM’s are part of, oen calledsurvival analysis (in the biological sciences) or event history analysis (in the social sciences),