A Computational Bayesian Method for Estimating the Number of Knots In Regression Splines

Abstract

To determine the size of the drug-involved offender population that could be served effectively and efficiently by partnerships between courts and treatment in the United States, a synthetic dataset is created by Bhati and Roman (2009). Because of hidden structure and aggregation necessary to create variables, there exists latent variance that can not be explained fully by a normal random effect model. Semiparametric regression is a well-known and frequently used tool to capture the functional dependence between variables with fixed effect parametric and nonlinear regression. A new Gibbs sampler is developed here for the number and positions of knots in regression splines by expressing semiparametric regression as a linear mixed model with a random effect term for the basis functions. Our Gibbs sampler exploits the properties of the multinomial-Dirichlet distribution, and is shown to be an improvement, in terms of operator norm and efficiency, over add/delete one MCMC algorithms. We find that the Dirichlet distribution with small values of the parameters results in a smaller number of knots and, in general, good fit to the data. This approach is shown to reveal previously unseen structures in the synthetic dataset of Bhati and Roman.