The relationship between material failures and flight hours

If material failures follow a Poisson distribution, then the expected number of failures is exactly proportional to flight hours. However, this article demonstrates that proportionality will not be revealed by simple correlation or regression analysis between monthly flight hours and the number of monthly failures. To test for proportionality, one must instead test the underlying hypothesis that the data follow a Poisson distribution. This article presents three simple tests that may be used for this purpose. The Poisson distribution requires that the mean and variance of the number of failures be equal. This article suggests several alternative models that may be used for samples in which the variance exceeds the mean. First, the mean of the Poisson distribution may itself be randomly distributed across the observational units according to a gamma distribution. If so, the number of failures will have a negative binomial distribution. Second, the mean of the Poisson distribution may depend systematically upon a set of observable explanatory variables. In this case, the Poisson regression model is appropriate. Finally, the mean of the Poisson distribution may contain both a systematic component that depends upon observable variables and a random component. This situation yields a generalized Poisson regression model.