Prediction in Poisson and other errors in variables models
We want to be able to use information about the traffic flows at road junctions and covariates describing those junctions to predict the number of accidents occurring there. We develop here a Bayesian predictive approach. Initially we considered three simpler but related problems to assess the efficiency of some approximation techniques, namely: (I) Given a treatment with an effect that can be described mathematically as of a multiplicative form, we record Poisson countings before and after the treatment is applied. Then, given a new individual with a known counting before the treatment is used, we want to predict the outcome on that individual after the treatment is applied. (II) After observing the value on an individual before any treatment is applied, we decide, based on that value, which of two treatments to apply, and then register the post- treatment outcome. Given a new individual, with an observed value before he receives any treatment, we aim to derive the predictive distribution for the outcome after one of the treatments is used. (This problem is also considered when several possible treatments are available). (III) We compare the effects of two treatments, through a two-period crossover design. We assume that both the treatment effect and the period effect are of multiplicative forms. Estimative and approximation methods are developed for each of these problems. We use the Gibbs sampling approach, normal asymptotic approximations for the posterior distributions and the Laplace approximations. Examples are presented to compare the efficiency and performance of the different methods. We find that the Laplace method performs well, and has computational advantages over the other methods. Using the knowledge obtained solving these simpler problems we develop solutions for the traffic accidents problem and analyse a real data set. Stepwise procedures for the incorporation of the covariates through the use of Kullback-Leibler measure of divergence are developed. We also consider the three simpler problems assuming that the observations are exponentially and binomially distributed.