where i indexes the vehicle and j the occupant within the crash. The parameter γi provides a “vehicle-level” estimate of the risk of death that adjusts for all vehicle covariates, observed and unobserved, thus controlling for crash severity. By conditioning on the total number of deaths in the vehicle, each vehicle contributes a multinomial probability:
which calculates the probability that each of the vehicle occupants has the j th outcome.3 As in conditional logistic regression, γi need no longer be estimated, and the behavior of the conditional likelihood estimates for β can rely on the asymptotic properties of the sample size for the number of vehicles, rather than the small sample size within each vehicle. Thus only crashes in which there is variability in xijk among the j occupants provide information about the effect of the k th covariate on risk of death. This implies a vehicle must have at least 2 occupants to contribute any information.
There have been few vehicle-matched analyses in the child passenger injury literature because crashes with multiple child passengers, particularly those where exposures of interest differ, are rare. Rice and Anderson have attempted to avoid this problem by including all passengers regardless of age in the analysis, as long as they were in a vehicle with a child younger than 4 years. This compares children younger than 4 years with older children, teens, and adults, with an adjustment for age via the regression model above. Age is modeled very flexibly (as a quadratic spline with 3 knots), but this seems to require huge extrapolation beyond the data. Adults cannot be in child safety seats, but the model assumes a “child safety seat” effect comparing adults with children after the age adjustment. Also, while the parameter γi is not explicitly estimated, it is assumed to be a meaningful factor common to all persons in the vehicle, which we think is questionable if young children and adults are mixed together in the analysis.