Free-knot Splines with Rjmcmc for Logistic Models and Threshold Selection

In medical statistics, the logistic model is a popular choice for the analysis of the dependence between a response variable and one or more explanatory variables. The response variable is the log odds and it is a linear function of explanatory variables. This type of modeling is restrictive, as the behaviour of the log odds can be best represented by a smooth non-linear function. Thus, we use a representation B-spline, where the number and location of knots are seen as free variables, is used to improve the fitting. For a piecewise linear spline, knots are points where the slope is changing in the shape of the function. Therefore, a quick change of slope allows to interpret the knot location as a threshold value. The use of MCMC simulation techniques is a very important computational tool in Bayesian statistics. These methods belong to a class of algorithms for sampling from target distributions on a space of fixed dimension. The Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm, allows simulations from target distributions on spaces of M. DENIS and N. MOLINARI 2 varying dimension. One of the main purposes of the present investigation is to use this RJMCMC method for modeling the log odds by a B-spline representation with an unknown number of knots at unknown locations. The method is illustrated with simulations and a real data set from an in vitro fertilization program.