Analyzing Spatially Distributed Binary Data Using Independent‐Block Estimating Equations

We estimate the relation between binary responses and corresponding covariate vectors, both observed over a large spatial lattice. We assume a hierarchical generalized linear model with probit link function, partition the lattice into blocks, and adopt the working assumption of independence between the blocks to obtain an easily solved estimating equation. Standard errors are obtained using the "sandwich" estimator together with window subsampling (Sherman, 1996, Journal of the Royal Statistical Society, Series B58, 509-523). We apply this to a large data set describing long-term vegetation growth, together with two other approximate-likelihood approaches: pairwise composite likelihood (CL) and estimation under a working assumption of independence. The independence and CL methods give similar point estimates and standard errors, while the independent-block approach gives considerably smaller standard errors, as well as more easily interpretable point estimates. We present numerical evidence suggesting this increased efficiency may hold more generally.

[1]  S. Zeger,et al.  Longitudinal data analysis using generalized linear models , 1986 .

[2]  Bradley P. Carlin,et al.  Hierarchical Spatio-Temporal Mapping of Disease Rates , 1997 .

[3]  S. Zeger,et al.  Marginal Regression Models for Clustered Ordinal Measurements , 1996 .

[4]  Martin Crowder,et al.  On Consistency and Inconsistency of Estimating Equations , 1986, Econometric Theory.

[5]  David J. Nott,et al.  Pairwise likelihood methods for inference in image models , 1999 .

[6]  S. Lele,et al.  A Composite Likelihood Approach to Binary Spatial Data , 1998 .

[7]  David M. Zucker,et al.  Modelling and generating correlated binary variables , 2001 .

[8]  M. Sherman.  Variance Estimation for Statistics Computed from Spatial Lattice Data , 1996 .

[9]  D. Sharon,et al.  THE WINDFIELD AND RAINFALL DISTRIBUTION INDUCED WITHIN A SMALL VALLEY: FIELD OBSERVATIONS AND 2-D NUMERICAL MODELLING , 1997 .

[10]  Carlo Gaetan,et al.  Composite likelihood methods for space-time data , 2006 .

[11]  Anthony N. Pettitt,et al.  Binary probability maps using a hidden conditional autoregressive Gaussian process with an application to Finnish common toad data , 2000 .

[12]  D. Sharon,et al.  The distribution of wind-driven rainfall in a small valley: an empirical basis for numerical model verification , 1997 .

[13]  Karl Pearson,et al.  Mathematical contributions to the theory of evolution. VIII. On the correlation of characters not quantitatively measurable , 1900, Proceedings of the Royal Society of London.

[14]  Y. Carmel,et al.  Effects of grazing and topography on long-term vegetation changes in a Mediterranean ecosystem in Israel , 1999, Plant Ecology.

[15]  K Y Liang,et al.  Longitudinal data analysis for discrete and continuous outcomes. , 1986, Biometrics.

[16]  Gareth O. Roberts,et al.  Robust Markov chain Monte Carlo Methods for Spatial Generalized Linear Mixed Models , 2006 .

[17]  V. Carey,et al.  Working correlation structure misspecification, estimation and covariate design: Implications for generalised estimating equations performance , 2003 .

[18]  W. Pauli.  Mathematical contributions to the theory of Dirac's matrices , 1936 .

[19]  Anthony N. Pettitt,et al.  A Conditional Autoregressive Gaussian Process for Irregularly Spaced Multivariate Data with Application to Modelling Large Sets of Binary Data , 2002, Stat. Comput..

[20]  R. Waagepetersen,et al.  Bayesian Prediction of Spatial Count Data Using Generalized Linear Mixed Models , 2002, Biometrics.

[21]  N. Breslow,et al.  Approximate inference in generalized linear mixed models , 1993 .

[22]  P. Doukhan.  Mixing: Properties and Examples , 1994 .

[23]  Παναγιώτα Τσαλαπάτη.  An evaluation and prioritisation system for improvement suggestions : a QFD application , 2000 .

[24]  T. C. Haas,et al.  Model-based geostatistics. Discussion. Authors' reply , 1998 .

[25]  Patrick J. Heagerty,et al.  Window Subsampling of Estimating Functions with Application to Regression Models , 2000 .

[26]  L. Baxter.  Random Fields on a Network: Modeling, Statistics, and Applications , 1996 .

[27]  Vincent J. Carey,et al.  Unbiased Estimating Equations From Working Correlation Models for Irregularly Timed Repeated Measures , 2004 .

[28]  D. Clayton,et al.  Empirical Bayes estimates of age-standardized relative risks for use in disease mapping. , 1987, Biometrics.

[29]  P. Green.  Discussion of 'Bayesian image restoration with two applications in spatial statistics' by J Besag, J C York & A Mollie , 1991 .

[30]  B. Lindsay,et al.  Improving generalised estimating equations using quadratic inference functions , 2000 .

[31]  N. Rao Chaganty,et al.  Efficiency of generalized estimating equations for binary responses , 2004 .

[32]  P. Diggle,et al.  Model-based geostatistics (with discussion). , 1998 .

[33]  P S Albert,et al.  A generalized estimating equations approach for spatially correlated binary data: applications to the analysis of neuroimaging data. , 1995, Biometrics.

[34]  J. Besag,et al.  Bayesian image restoration, with two applications in spatial statistics , 1991 .