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STATISTICAL SCIENCE Volume 31, Number 3 August 2016

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STATISTICAL SCIENCE Volume 31, Number 3 August 2016 STATISTICAL SCIENCE Volume 31, Number 3 August 2016 A Review of Nonparametric Hypothesis Tests of Isotropy Properties in Spatial Data ...................................................Zachary D. Weller and Jennifer A. Hoeting 305 Rank Tests from Partially Ordered Data Using Importance and MCMC Sampling Methods .........................................................Debashis Mondal and Nina Hinrichs 325 On Negative Outcome Control of Unobserved Confounding as a Generalization of Difference-in-Differences................Tamar Sofer, David B. Richardson, Elena Colicino, Joel Schwartz and Eric J. Tchetgen Tchetgen 348 Quantum Annealing with Markov Chain Monte Carlo Simulations and D-Wave Quantum Computers...........................................Yazhen Wang, Shang Wu and Jian Zou 362 MarkovChainsasModelsinStatisticalMechanics.............................Eugene Seneta 399 Fractional Imputation in Survey Sampling: A Comparative Review ...............................................................Shu Yang and Jae Kwang Kim 415 AConversationwithMichaelWoodroofe............Moulinath Banerjee and Bodhisattva Sen 433 AConversationwithArthurCohen...............................................Joseph Naus 442 AConversationwithEstateV.Khmaladze................... Hira L. Koul and Roger Koenker 453 Statistical Science [ISSN 0883-4237 (print); ISSN 2168-8745 (online)], Volume 31, Number 3, August 2016. Published quarterly by the Institute of Mathematical Statistics, 3163 Somerset Drive, Cleveland, OH 44122, USA. Periodicals postage paid at Cleveland, Ohio and at additional mailing offices. POSTMASTER: Send address changes to Statistical Science, Institute of Mathematical Statistics, Dues and Subscriptions Office, 9650 Rockville Pike—Suite L2310, Bethesda, MD 20814-3998, USA. Copyright © 2016 by the Institute of Mathematical Statistics Printed in the United States of America EDITOR Peter Green University of Bristol and University of Technology, Sydney ASSOCIATE EDITORS Steve Buckland Byron Morgan Eric Tchetgen Tchetgen University of St. Andrews University of Kent Harvard School of Public Jiahua Chen Peter Müller Health University of British Columbia University of Texas Yee Whye Teh Rong Chen Sonia Petrone University of Oxford Rutgers University Bocconi University Jon Wakefield Rainer Dahlhaus Nancy Reid University of Washington University of Heidelberg University of Toronto Guenther Walther Peter J. Diggle Glenn Shafer Stanford University Lancaster University Rutgers Business Jon Wellner Robin Evans School–Newark and University of Washington University of Oxford New Brunswick Martin Wells Michael Friendly Royal Holloway College, Cornell University York University University of London Tong Zhang Edward I. George Michael Stein Rutgers University University of Pennsylvania University of Chicago MANAGING EDITOR T. N. Sriram University of Georgia PRODUCTION EDITOR Patrick Kelly EDITORIAL COORDINATOR Kristina Mattson PAST EXECUTIVE EDITORS Morris H. DeGroot, 1986–1988 Morris Eaton, 2001 Carl N. Morris, 1989–1991 George Casella, 2002–2004 Robert E. Kass, 1992–1994 Edward I. George, 2005–2007 Paul Switzer, 1995–1997 David Madigan, 2008–2010 Leon J. Gleser, 1998–2000 Jon A. Wellner, 2011–2013 Richard Tweedie, 2001 Statistical Science 2016, Vol. 31, No. 3, 305–324 DOI: 10.1214/16-STS547 © Institute of Mathematical Statistics, 2016 A Review of Nonparametric Hypothesis Tests of Isotropy Properties in Spatial Data Zachary D. Weller and Jennifer A. Hoeting Abstract. An important aspect of modeling spatially referenced data is ap- propriately specifying the covariance function of the random field. A practi- tioner working with spatial data is presented a number of choices regarding the structure of the dependence between observations. One of these choices is to determine whether or not an isotropic covariance function is appropri- ate. Isotropy implies that spatial dependence does not depend on the direc- tion of the spatial separation between sampling locations. Misspecification of isotropy properties (directional dependence) can lead to misleading infer- ences, for example, inaccurate predictions and parameter estimates. A re- searcher may use graphical diagnostics, such as directional sample vari- ograms, to decide whether the assumption of isotropy is reasonable. These graphical techniques can be difficult to assess, open to subjective interpreta- tions, and misleading. Hypothesis tests of the assumption of isotropy may be more desirable. To this end, a number of tests of directional dependence have been developed using both the spatial and spectral representations of random fields. We provide an overview of nonparametric methods available to test the hypotheses of isotropy and symmetry in spatial data. We discuss important considerations in choosing a test, provide recommendations for implement- ing a test, compare several of the methods via a simulation study, and propose a number of open research questions. Several of the reviewed methods can be implemented in R using our package spTest, available on CRAN. Key words and phrases: Isotropy, symmetry, nonparametric spatial covari- ance. REFERENCES MR3316189 BANERJEE,S.,CARLIN,B.P.andGELFAND, A. E. (2014). Hi- BACZKOWSKI, A. J. (1990). A test of spatial isotropy. In Compstat erarchical Modeling and Analysis for Spatial Data. CRC Press, 277–282. Springer, Berlin. Boca Raton, FL. BACZKOWSKI,A.J.andMARDIA, K. V. (1987). Approxi- BORGMAN,L.andCHAO, L. (1994). Estimation of a multidimen- mate lognormality of the sample semivariogram under a Gaus- sional covariance function in case of anisotropy. Math. Geol. 26 sian process. Comm. Statist. Simulation Comput. 16 571–585. 161–179. MR1267007 MR0903164 BOWMAN,A.W.andAZZALINI, A. (2014). R package sm: Non- BACZKOWSKI,A.J.andMARDIA, K. V. (1990). A test of spa- parametric smoothing methods (version 2.2-5.4), Univ. Glas- tial symmetry with general application. Comm. Statist. Theory gow, UK and Univ. di Padova, Italia. Methods 19 555–572. MR1067588 BOWMAN,A.W.andCRUJEIRAS, R. M. (2013). Inference for BANDYOPADHYAY,S.,LAHIRI,S.N.andNORDMAN,D.J. variograms. Comput. Statist. Data Anal. 66 19–31. MR3064021 (2015). A frequency domain empirical likelihood method for CABAÑA, E. M. (1987). Affine processes: A test of isotropy based irregularly spaced spatial data. Ann. Statist. 43 519–545. on level sets. SIAM J. Appl. Math. 47 886–891. MR0898839 Zachary D. Weller is Ph.D. Candidate, Department of Statistics, Colorado State University, Fort Collins, Colorado 80523-1877, USA (e-mail: [email protected]). Jennifer A. Hoeting is Professor, Department of Statistics, Colorado State University, Fort Collins, Colorado 80523-1877, USA (e-mail: [email protected]). CHORTI,A.andHRISTOPULOS, D. T. (2008). Nonparametric JONA-LASINIO, G. (2001). Modeling and exploring multivariate identification of anisotropic (elliptic) correlations in spatially spatial variation: A test procedure for isotropy of multivariate distributed data sets. IEEE Trans. Signal Process. 56 4738– spatial data. J. Multivariate Anal. 77 295–317. MR1838690 4751. MR2517209 JOURNEL,A.G.andHUIJBREGTS, C. J. 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