The variation of drivers' accident rates between drivers and over time

Accident records of individual drivers in eight large samples from the general driving population have been analysed as nonhomogeneous Poisson processes, i.e. by assuming that each driver's accidents occur randomly with an individual underlying accident rate that varies in time. Previous hypotheses about the form of this variation are reviewed. The moments of the distribution over drivers of individual underlying accident rates averaged over periods of one or more years are estimated; they indicate that most of the distributions are intermediate in form between the gamma and lognormal distributions. The results suggest the presence of an overall annual factor that multiplies all the individual rates; when this is removed the remaining process is approximately stationary (i.e. the mean and standard deviation are approximately constant in time). The coefficient of variation of individual accident rates falls slowly as the length of the period over which they are averaged increases, indicating positive correlation between the underlying accident rates of the same drivers in different periods. These correlations are estimated directly from the accident frequency data, which appears to be a new and useful method of analysis. For one sample, which spans 14 years, the correlation between two adjacent years is about 0.74 and falls approximately linearly to about 0.49 for two years separated by a gap of 12 years. A "switching process" with differing individual means, previously proposed for drivers' accidents by Bartlett, does not appear to fit these results well. A "scaling process," studied by Mandelbrot and previously found to apply in a number of different fields, provides a better fit to the data and could represent the combined effect of a hierarchy of switching processes operating on different time scales, in accordance with what is known or conjectured about the variation of factors affecting accident rates. Although the data analysed are affected by variation in exposure between drivers, the methods and ideas used have implications that may help to resolve the long-standing dispute about individual accident proneness.