Abstract
Generalized linear mixed models for spatial processes are widely used in applied statistics. In many applications of the spatial generalized linear mixed model (SGLMM), the goal is to obtain inference about regression coefficients while achieving optimal predictive ability. When implementing the SGLMM, multicollinearity among covariates and the spatial random effects can make computation challenging and influence inference. We present a Bayesian group lasso prior with a single tuning parameter that can be chosen to optimize predictive ability of the SGLMM and jointly regularize the regression coefficients and spatial random effect. We implement the group lasso SGLMM using efficient Markov chain Monte Carlo (MCMC) algorithms and demonstrate how multicollinearity among covariates and the spatial random effect can be monitored as a derived quantity. To test our method, we compared several parameterizations of the SGLMM using simulated data and two examples from plant ecology and disease ecology. In all examples, problematic levels multicollinearity occurred and influenced sampling efficiency and inference. We found that the group lasso prior resulted in roughly twice the effective sample size for MCMC samples of regression coefficients and can have higher and less variable predictive accuracy based on out-of-sample data when compared to the standard SGLMM.
Supplementary materials accompanying this paper appear online.
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Acknowledgements
We would like to acknowledge Dennis Heisey for his early contributions to development of this research endeavor. We thank Jun Zhu and two anonymous reviewers for valuable insight and discussions about this work. We thank the staff of the Wisconsin Department of Natural Resources for their collaboration in obtaining deer tissue samples and the Wisconsin hunters who provided them. In particular, we thank Erin Larson for maintaining the CWD sample data base. Funding for this project was provided by the USGS National Wildlife Health Center via Grant G14AC00366. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government.
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Supplementary Materials: Full-conditional distributions and MCMC algorithms are presented in Appendix A. R code to implement MCMC algorithms for the spatial, group lasso, and non-spatial probit models is given in Appendix B. Data, R code, and details of implementation for the species distribution example (Appendix C), disease risk factor analysis example (Appendix D), and the simulated data example (Appendix E). (pdf4453KB)
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Hefley, T.J., Hooten, M.B., Hanks, E.M. et al. The Bayesian Group Lasso for Confounded Spatial Data. JABES 22, 42–59 (2017). https://doi.org/10.1007/s13253-016-0274-1
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DOI: https://doi.org/10.1007/s13253-016-0274-1