Semiparametric regression with shape-constrained penalized splines |
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Authors: | Martin L. Hazelton Berwin A. Turlach |
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Affiliation: | a Institute of Fundamental Sciences—Statistics & Bioinformatics, Massey University PN461, Private Bag 11222, Palmerston North, New Zealandb School of Mathematics and Statistics (M019), University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia |
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Abstract: | In semiparametric regression models, penalized splines can be used to describe complex, non-linear relationships between the mean response and covariates. In some applications it is desirable to restrict the shape of the splines so as to enforce properties such as monotonicity or convexity on regression functions. We describe a method for imposing such shape constraints on penalized splines within a linear mixed model framework. We employ Markov chain Monte Carlo (MCMC) methods for model fitting, using a truncated prior distribution to impose the requisite shape restrictions. We develop a computationally efficient MCMC sampler by using a correspondingly truncated multivariate normal proposal distribution, which is a restricted version of the approximate sampling distribution of the model parameters in an unconstrained version of the model. We also describe a cheap approximation to this methodology that can be applied for shape-constrained scatterplot smoothing. Our methods are illustrated through two applications, the first involving the length of dugongs and the second concerned with growth curves for sitka spruce trees. |
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Keywords: | Linear mixed model MCMC Shape constraint Spline Truncated multivariate normal |
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