Bayesian Nonlinear Regression with Multivariate Linear Splines

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Proposal ID: DMS-0203215PI: Bani MallickTitle: Bayesian nonlinear regression with multivariate linear splinesThe investigator and his colleagues consider novel, nonparametric modeling of univariate and multivariate non-Gaussian response data. The usual generalized linear models are extended to generalized nonlinear models by modeling the mean function in a flexible way. Data adaptive multivariate smoothing splines are employed to do this, where the number and location of the knot points are treated as random. The posterior model space is explored using a reversible jump Markov chain Monte Carlo (MCMC) sampler. Computational difficulties are partly alleviated by introducing a residual effect in the model that leaves many of the posterior distributions of the model parameters in standard form. The use of the latent residual effect provides a convenient vehicle for modeling correlation in multivariate response data and as such the method can be seen to generalize the seemingly unrelated regression model to non-Gaussian data. In the next part of the project the investigator and his colleagues develop semiparametric Bayesian methods for generalized non-linear models where a predictor is measured with either classical or Berkson error. In the presence of covariate measurement error, estimating usual regression function nonparametrically is extremely difficult, the problem being related todeconvolution. In the case of generalized linear model it is more difficult. Again combinations of spline regression and MCMC techniques are used to handle the problem.Function estimation is an important statistical tool that tries to understand accurately the functional relationships between variables based on data and it has applications in many disciplines for successfully addressing scientific questions. Most of the flexible, nonlinear regression problems are developed when the response is a continuous variable. In important applied problems the response may be count or indicator variable and flexible function estimation is much more harder in these situations. In this proposal the investigator intend to develop the methods that adaptively estimate the functional relationships in these more complicated situations. The area of biotechnology is an especially application for these methods. Scientists now have techniques for measuring gene expression levels for thousands of genes at the same time, allowing the exciting possibility of determining which human genes are involved in a disease such as cancer and heart disease. These methods will be useful to explore nonlinear relationship between gene expression levels and the chance of the disease. Other possible applications are to model correlated multivariate disease or accident counts data where the methods being developed here will improve modeling disease or accidents maps (with uncertainties) which will be useful for disease or transportation risk assessments.