Self Modeling Regression (SEMOR) is a flexible, semiparametric approach to fitting multiple curves of the type generated in tumor re-growth experiments. SEMOR is based on the assumption that each of a group of growth curves is a simple parametric transformation of some smooth shape function. This proposal concentrates on incorporating a free-knot spline into SEMOR as the estimate of the shape function and in developing confidence regions and testing procedures for the resulting model. Free-knot splines have not enjoyed popularity as nonparametric estimators because of the computational difficulty of finding least squares estimates of the knots. We have recently developed a penalized estimate of the knot locations which improves the computational properties of the estimation problem without sacrificing significant flexibility in the resulting curve. We propose further work on both penalized and unpenalized free-knot splines including the development of starting value algorithms for the knot locations, confidence regions for the fitted curve, and tests for differences between two fitted curves. In addition, the reversible jump Markov chain Monte Carlo (MCMC) algorithm will be applied to the problem of estimating the number and locations of the knots in a Bayesian setting and its performance compared to (restricted) maximum likelihood estimation. The second group of issues addressed in this proposal relate to the incorporation of penalized free-knot splines into a SEMOR model and the application of the SEMOR model to tumor regrowth data. In previous work we have proposed a nonlinear mixed effects SEMOR model where the parameters that define the individual curve transformations are assumed to follow a distribution in the population of curves. Adding penalized estimation of the shape function to this setting requires careful consideration of the definition of the estimator as well as the optimization algorithm. A number of approaches are proposed including two penalized maximum likelihood estimates and reversible jump MCMC estimation of a Bayesian approach. Bootstrap based confidence bands and testing procedures are proposed. The applicability of theoretical results to small samples will be checked by simulation and all successful algorithms will be implemented in the S statistical language and made publicly available.
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