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Abstract

We expand logistic regression models to accommodate spatial structure in a functional covariate process, direct spatial structure in responses and a geographically structured random effect. Our proposed model builds upon conditionally specified models, where the joint distribution corresponding to a specified full conditional can be identified up to an unknown constant of proportionality. To tackle the challenges of estimation and inference, we employ two separate double Metropolis–Hastings procedures in the Bayesian framework. Furthermore, we suggest methods for making modelling decisions to distinguish the sources of spatial structure through data-driven diagnostics and model assessments. Finally, we apply the model to a real-world problem involving unemployment rates over time in counties of the Midwestern United States as a functional covariate, poverty rates in those counties at a given time as response variables and state-level random effects.

Keywords

Functional data analysis mixed-effects logistic regression spatial statistics

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References

and Cabral

(2020) Bayesian estimation of the functional spatial lag model. Journal of Time Series Econometrics , 12, 20190047.

Besag

(1974) Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society: Series B (Methodological) , 36, 192–225.

Caragea

and Kaiser

(2009) Autologistic models with interpretable parameters. Journal of Agricultural, Biological, and Environmental Statistics , 14, 281–300.

Crainiceanu

, Ruppert

, Carroll

, Joshi

and Goodner

(2007) Spatially adaptive Bayesian penalized splines with heteroscedastic errors. Journal of Computational and Graphical Statistics , 16, 265–88.

Cressie

and Lele

(1992) New models for Markov random fields. Journal of Applied Probability , 29, 877–84.

Gelman

(2006) Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian Analysis , 1, 515–34.

Hanks

, Schliep

, Hooten

and Hoeting

(2015) Restricted spatial regression in practice: geostatistical models, confounding, and robustness under model misspecification. Environmetrics , 26, 243–54.

Hodges

and Reich

(2010) Adding spatially-correlated errors can mess up the fixed effect you love. The American Statistician , 64, 325–34.

Huang

, Saporta

, Wang

and Wang

(2021) A robust spatial autoregressive scalar-on-function regression with t-distribution. Advances in Data Analysis and Classification , 15, 57–81.

10.

Hughes

and Haran

(2013) Dimension reduction and alleviation of confounding for spatial generalized linear mixed models. Journal of the Royal Statistical Society Series B: Statistical Methodology , 75, 139–59.

11.

James

(2002) Generalized linear models with functional predictors. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 64, 411–32.

12.

Kaiser

and Caragea

(2009) Exploring dependence with data on spatial lattices. Biometrics , 65, 857–65.

13.

Kaiser

and Cressie

(2000) The construction of multivariate distributions from Markov random fields. Journal of Multivariate Analysis , 73, 199–220.

14.

Kaiser

, Lahiri

and Nordman

(2012) Goodness of fit tests for a class of Markov random field models. The Annals of Statistics , 40, 104–30.

15.

Kang

, Jung

and Park

(2023) Fast Bayesian functional regression for non-Gaussian spatial data. Bayesian Analysis , 1, 1–32.

16.

Kaplan

, Kaiser

, Lahiri

and Nordman

(2020) Simulating Markov random fields with a conclique-based Gibbs sampler. Journal of Computational and Graphical Statistics , 29, 286–96.

17.

Kaplan

, Nordman

and Vardeman

(2020) On the S-instability and degeneracy of discrete deep learning models. Information and Inference: A Journal of the IMA , 9, 627–55.

18.

Kim

, Kaiser

and Dai

(2024) Generalized linear models with spatial dependence and a functional covariate. arXiv preprint, arXiv:2402.13472.

19.

Lee

, Kaiser

and Cressie

(2001) Multiway dependence in exponential family conditional distributions. Journal of Multivariate Analysis , 79, 171–90.

20.

Liang

, Zhang

, Chang

and Huang

(2021) Modeling and regionalization of China’s PM2.5 using spatial-functional mixture models. Journal of the American Statistical Association , 116, 116–32.

21.

Liang

(2010) A double Metropolis–Hastings sampler for spatial models with intractable normalizing constants. Journal of Statistical Computation and Simulation , 80, 1007–22.

22.

Meyer

, Coull

, Versace

, Cinciripini

and Morris

(2015) Bayesian function-on-function regression for multilevel functional data. Biometrics , 71, 563–74.

23.

Moran

, Blei

and Ranganath

(2024) Holdout predictive checks for Bayesian model criticism. Journal of the Royal Statistical Society Series B: Statistical Methodology , 86, 194–214.

24.

Mousavi

and Sorensen

(2018) Functional logistic regression: A comparison of three methods. Journal of Statistical Computation and Simulation , 88, 250–68.

25.

Müller

and Stadtmüller

(2005) Generalized functional linear models. The Annals of Statistics , 33, 774–805.

26.

Pineda-Ríos

, Giraldo

and Porcu

(2019) Functional SAR models: With application to spatial econometrics. Spatial Statistics , 29, 145–59.

27.

Porter

, Holan

, Wikle

and Cressie

(2014) Spatial Fay–Herriot models for small area estimation with functional covariates. Spatial Statistics , 10, 27–42.

28.

Reich

, Hodges

and Zadnik

(2006) Effects of residual smoothing on the posterior of the fixed effects in disease-mapping models. Biometrics , 62, 1197–1206.

29.

Reiss

and Ogden

(2010) Functional generalized linear models with images as predictors. Biometrics , 66, 61–9.

30.

Schweinberger

(2011) Instability, sensitivity, and degeneracy of discrete exponential families. Journal of the American Statistical Association , 106, 1361–70.

31.

Watanabe

(2013) WAIC and WBIC are information criteria for singular statistical model evaluation. In Proceedings of the Workshop on Information Theoretic Methods in Science and Engineering , pages 90–94.

32.

W-H

Yang

, Wikle

, Holan

, Myers

and Sudduth

(2015) Bayesian analysis of spatially-dependent functional responses with spatially-dependent multi-dimensional functional predictors. Statistica Sinica , 25, 205–23.

33.

Zhang

, Baladandayuthapani

, Zhu

, Baggerly

, Majewski

, Czerniak

and Morris

(2016) Functional CAR models for large spatially correlated functional datasets. Journal of the American Statistical Association , 111, 772–86.

34.

Zhao

, Staudenmayer

, Coull

and Wand

(2006) General design Bayesian generalized linear mixed models. Statistical Science , 21, 35–51.

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Bayesian analysis of binary Markov random field models with functional covariates and random effects

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