A stratified generalized additive model and permutation test for temporal heterogeneity of smoothed bivariate spatial effects

Generalized additive models (GAMs) with bivariate smoothers are frequently used to map geographic disease risks in epidemiology studies. A challenge in identifying health disparities has been the lack of intuitive and computationally feasible methods to assess whether the pattern of spatial effects varies over time. In this research, we accommodate time-stratified smoothers into the GAM framework to estimate time-specific spatial risk patterns while borrowing information from confounding effects across time. A backfitting algorithm for model estimation is proposed along with a permutation testing framework for assessing temporal heterogeneity of geospatial risk patterns across two or more time points. Simulation studies show that our proposed permuted mean squared difference (PMSD) test performs well with respect to type I error and power in various settings when compared with existing methods. The proposed model and PMSD test are used geospatial risk patterns of patent ductus arteriosus (PDA) in the state of Massachusetts over 2003-2009. We show that there is variation over time in spatial patterns of PDA risk, adjusting for other known risk factors, suggesting the presence of potential time-varying and space-related risk factors other than the adjusted ones.

[1]  Henning Omre,et al.  The Bayesian bridge between simple and universal kriging , 1989 .

[2]  Jean Duchon,et al.  Splines minimizing rotation-invariant semi-norms in Sobolev spaces , 1976, Constructive Theory of Functions of Several Variables.

[3]  Daniel L. Gillen,et al.  Mapping Smoothed Spatial Effect Estimates from Individual-Level Data: MapGAM , 2020, R J..

[4]  S. Wood,et al.  Smoothing Parameter and Model Selection for General Smooth Models , 2015, 1511.03864.

[5]  W. Cleveland.  Robust Locally Weighted Regression and Smoothing Scatterplots , 1979 .

[6]  H. Omre.  Bayesian kriging—Merging observations and qualified guesses in kriging , 1987 .

[7]  P. Diggle,et al.  Bayesian Inference in Gaussian Model-based Geostatistics , 2002 .

[8]  S. Wood.  Thin plate regression splines , 2003 .

[9]  N. Cressie.  The origins of kriging , 1990 .

[10]  Yang Liu,et al.  Maternal exposure to traffic-related air pollution and birth defects in Massachusetts. , 2016, Environmental research.

[11]  M. Stein,et al.  A Bayesian analysis of kriging , 1993 .

[12]  Andrew O. Finley,et al.  spBayes for Large Univariate and Multivariate Point-Referenced Spatio-Temporal Data Models , 2013, 1310.8192.

[13]  Thomas Webster,et al.  Method for mapping population-based case-control studies: an application using generalized additive models , 2006, International journal of health geographics.

[14]  A. Stewart Fotheringham,et al.  Geographically Weighted Regression: A Method for Exploring Spatial Nonstationarity , 2010 .

[15]  Al Ozonoff,et al.  Generalized additive models and inflated type I error rates of smoother significance tests , 2011, Comput. Stat. Data Anal..

[16]  Bradley P Carlin,et al.  spBayes: An R Package for Univariate and Multivariate Hierarchical Point-referenced Spatial Models. , 2007, Journal of statistical software.

[17]  S. Wood.  Stable and Efficient Multiple Smoothing Parameter Estimation for Generalized Additive Models , 2004 .

[18]  Sudipto Banerjee,et al.  Bayesian Modeling and Analysis of Geostatistical Data. , 2017, Annual review of statistics and its application.

[19]  S. Wood.  Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models , 2011 .