Tests of Spatial Randomness Adjusted for an Inhomogeneity

In many applications, it is of interest to test whether a spatial point pattern is randomly generated after adjusting for an underlying spatial inhomogeneity. A great variety of different test statistics have been proposed for this purpose by scientists in different fields; these are reviewed in this article. Despite apparent dissimilarities in terms of their original formulations, most of these statistics can be placed into one general framework of which they are special cases. This makes it easier to see exactly how they relate to one another and also to determine which test to use for a particular application. The general framework can also be used for proposing new tests by combining properties of existing tests, for developing theoretical foundations for these types of test statistics, for doing structured comparative evaluations, and for software development.

[1]  K. Pearson.  On the Criterion that a Given System of Deviations from the Probable in the Case of a Correlated System of Variables is Such that it Can be Reasonably Supposed to have Arisen from Random Sampling , 1900 .

[2]  Stephen Alfred Forbes,et al.  On the Local Distribution of Certain Illinois Fishes: An Essay in Statistical Ecology , 1907 .

[3]  J. Troup,et al.  NOTE ON THE EXTENT TO WHICH THE DISTRIBUTION OF CASES OF DISEASE IN HOUSES IS DETERMINED BY THE LAWS OF CHANCE , 1912 .

[4]  R. Fisher.  On the Interpretation of χ2 from Contingency Tables, and the Calculation of P , 2018, Journal of the Royal Statistical Society Series A (Statistics in Society).

[5]  Jerzy Neyman,et al.  On a New Class of "Contagious" Distributions, Applicable in Entomology and Bacteriology , 1939 .

[6]  O. Lundberg.  On random processes and their application to sickness and accident statistics , 1940 .

[7]  Edgar M. Hoover,et al.  Interstate Redistribution of Population, 1850–1940 , 1941, The Journal of Economic History.

[8]  L. R. Dice.  Measures of the Amount of Ecologic Association Between Species , 1945 .

[9]  Random associations on a lattice , 1947 .

[10]  J. Jahn,et al.  The Measurement of Ecological Segregation , 1947 .

[11]  P. Moran.  The Interpretation of Statistical Maps , 1948 .

[12]  L. C. Cole,et al.  The Measurement of Interspecific Associaton , 1949 .

[13]  E. H. Simpson.  Measurement of Diversity , 1949, Nature.

[14]  The Measurement of Ecological Segregation: Derivation of an Index Based on the Criterion of Reproductibility , 1950 .

[15]  P. Moran.  Notes on continuous stochastic phenomena. , 1950, Biometrika.

[16]  Donald O. Cowgill,et al.  An Index of Segregation Based on Block Statistics , 1951 .

[17]  Robert H. Whittaker,et al.  A Study of Summer Foliage Insect Communities in the Great Smoky Mountains , 1952 .

[18]  P. Greig-Smith,et al.  The Use of Random and Contiguous Quadrats in the Study of the Structure of Plant Communities , 1952 .

[19]  J. G. Skellam,et al.  A New Method for determining the Type of Distribution of Plant Individuals , 1954 .

[20]  P. J. Clark,et al.  Distance to Nearest Neighbor as a Measure of Spatial Relationships in Populations , 1954 .

[21]  R. Geary,et al.  The Contiguity Ratio and Statistical Mapping , 1954 .

[22]  O. D. Duncan,et al.  A METHODOLOGICAL ANALYSIS OF SEGREGATION INDEXES , 1955 .

[23]  M. Dwass.  Modified Randomization Tests for Nonparametric Hypotheses , 1957 .

[24]  M. Choynowski.  Maps Based on Probabilities , 1959 .

[25]  森下.  Measuring of interspecific association and similarity between communities. , 1961 .

[26]  E. C. Pielou.  Segregation and Symmetry in Two-Species Populations as Studied by Nearest- Neighbour Relationships , 1961 .

[27]  R F Potthoff,et al.  Testing for homogeneity. II. The Poisson distribution. , 1966, Biometrika.

[28]  H. S. Horn,et al.  Measurement of "Overlap" in Comparative Ecological Studies , 1966, The American Naturalist.

[29]  R. Levins.  Evolution in Changing Environments: Some Theoretical Explorations. (MPB-2) , 1968 .

[30]  T. Schoener.  The Anolis Lizards of Bimini: Resource Partitioning in a Complex Fauna , 1968 .

[31]  J C Bailar,et al.  A class of permutational and multinomial tests arising in epidemiological research. , 1970, Biometrics.

[32]  E. Pianka.  The structure of lizard communities , 1973 .

[33]  C. Roberts,et al.  A test for space clustering and its application to congenital limb defects in Cardiff. , 1973, British journal of preventive & social medicine.

[34]  S. Walter.  On the detection of household aggregation of disease. , 1974, Biometrics.

[35]  Charles F. Cortese.  Further Consideration on the Methodological Analysis of Segregation Indices. , 1976 .

[36]  Generalisations of two tests for the detection of household aggregation of disease. , 1976, Biometrics.

[37]  P. Diggle.  The detection of random heterogeneity in plant populations , 1977 .

[38]  Robert R. Sokal,et al.  Spatial autocorrelation in biology: 2. Some biological implications and four applications of evolutionary and ecological interest , 1978 .

[39]  K. Aoki,et al.  A test of significance for geographic clusters of disease. , 1979, International journal of epidemiology.

[40]  W. Hines,et al.  The Eberhardt statistic and the detection of nonrandomness of spatial point distributions , 1979 .

[41]  B. Ripley.  Tests of 'Randomness' for Spatial Point Patterns , 1979 .

[42]  D. Burdick,et al.  The Use of Nearest Neighbor Frequency Analyses in Studies of Association , 1980 .

[43]  R. Grimson,et al.  Searching for hierarchical clusters of disease: spatial patterns of sudden infant death syndrome. , 1981 .

[44]  M. Klauber,et al.  Cancer clustering around point sources of pollution: assessment by a case-control methodology. , 1981, Environmental research.

[45]  J. Keith Ord,et al.  Spatial Processes Models and Applications , 1981 .

[46]  Bernard W. Silverman,et al.  Methods for Analysing Spatial Processes of Several Types of Points , 1982 .

[47]  Valerie Isham,et al.  A Bivariate Spatial Point Pattern of Ants' Nests , 1983 .

[48]  Per‐Anders Esseen.  Ecology of lichens in boreal coniferous forests with reference to spatial and temporal patterns , 1983 .

[49]  Clustering of disease in population units: an exact test and its asymptotic version. , 1983, American journal of epidemiology.

[50]  Peter J. Diggle,et al.  Statistical analysis of spatial point patterns , 1983 .

[51]  R C Grimson,et al.  Clustering of rare events. , 1983, Biometrics.

[52]  A. Abbott,et al.  On "The Measurement of Spatial Segregation" , 1984, American Journal of Sociology.

[53]  Una Maclean.  Atlas of Cancer in Scotland 1975-1980. Incidence and Epidemiological Perspective , 1985, IARC scientific publications.

[54]  C. Bird,et al.  Spatial patterns in electoral wards with high lymphoma incidence in Yorkshire health region. , 1987, British Journal of Cancer.

[55]  Geographical clusters and common patterns in cancer mortality of the Federal Republic of Germany. , 1987, Archives of environmental health.

[56]  Martin Charlton,et al.  A Mark 1 Geographical Analysis Machine for the automated analysis of point data sets , 1987, Int. J. Geogr. Inf. Sci..

[57]  A. Whittemore,et al.  A test to detect clusters of disease , 1987 .

[58]  S. Selvin,et al.  Transformations of maps to investigate clusters of disease. , 1988, Social science & medicine.

[59]  R F Raubertas,et al.  Spatial and temporal analysis of disease occurrence for detection of clustering. , 1988, Biometrics.

[60]  R. A. Stone.  Investigations of excess environmental risks around putative sources: statistical problems and a proposed test. , 1988, Statistics in medicine.

[61]  D W Merrill,et al.  Density equalized map projections: a method for analysing clustering around a fixed point. , 1988, Statistics in medicine.

[62]  M Smans,et al.  Analysis of spatial aggregation. , 1989, Recent results in cancer research. Fortschritte der Krebsforschung. Progres dans les recherches sur le cancer.

[63]  M. Hills,et al.  Statistical methods used in assessing the risk of disease near a source of possible environmental pollution: a review , 1989 .

[64]  D. Massey,et al.  The Ecology of Inequality: Minorities and the Concentration of Poverty, 1970-1980 , 1990, American Journal of Sociology.

[65]  T Tango,et al.  An index for cancer clustering. , 1990, Environmental health perspectives.

[66]  J. Cuzick,et al.  Spatial clustering for inhomogeneous populations , 1990 .

[67]  P. Diggle.  A point process modeling approach to raised incidence of a rare phenomenon in the vicinity of a prespecified point , 1990 .

[68]  S Davis,et al.  Point pattern analysis of the spatial proximity of residences prior to diagnosis of persons with Hodgkin's disease. , 1990, American journal of epidemiology.

[69]  B. Turnbull,et al.  Monitoring for clusters of disease: application to leukemia incidence in upstate New York. , 1990, American journal of epidemiology.

[70]  W. Vach,et al.  The Reconstruction of “Genetic Kinship” in Prehistoric Burial Complexes — Problems and Statistics , 1991 .

[71]  R C Grimson,et al.  A Versatile Test for Clustering and a Proximity Analysis of Neurons , 1991, Methods of Information in Medicine.

[72]  Julian Besag,et al.  The Detection of Clusters in Rare Diseases , 1991 .

[73]  P J Diggle,et al.  Second-order analysis of spatial clustering for inhomogeneous populations. , 1991, Biometrics.

[74]  B. Turnbull,et al.  Chronic disease surveillance and testing of clustering of disease and exposure: Application to leukemia incidence and TCE‐contaminated dumpsites in upstate New York , 1992 .

[75]  A. Lawson.  On the analysis of mortality events associated with a prespecified fixed point. , 1993, Journal of the Royal Statistical Society. Series A,.

[76]  R. Grimson.  Disease clusters, exact distributions of maxima, and P-values. , 1993, Statistics in medicine.

[77]  Robert Haining,et al.  Statistics for spatial data: by Noel Cressie, 1991, John Wiley & Sons, New York, 900 p., ISBN 0-471-84336-9, US $89.95 , 1993 .

[78]  Mike Rees,et al.  5. Statistics for Spatial Data , 1993 .

[79]  Cr Palmer,et al.  Probability of recurrence of extreme data: an aid to decision-making , 1993, The Lancet.

[80]  Werner Vach,et al.  Locally optimal tests on spatial clustering , 1994 .

[81]  S D Walter.  A simple test for spatial pattern in regional health data. , 1994, Statistics in medicine.

[82]  P. Dixon.  Testing spatial segregation using a nearest-neighbor contingency table , 1994 .

[83]  M Kulldorff,et al.  Spatial disease clusters: detection and inference. , 1995, Statistics in medicine.

[84]  A B Lawson,et al.  The power of focused tests to detect disease clustering. , 1995, Statistics in medicine.

[85]  J. Bithell.  The choice of test for detecting raised disease risk near a point source. , 1995, Statistics in medicine.

[86]  Efficient execution of Stone's likelihood ratio tests for disease clustering , 1995 .

[87]  Annibale Biggeri,et al.  Case-control designs for the detection of spatial clusters of diseases , 1995 .

[88]  B W Turnbull,et al.  Detection and assessment of clusters of disease: an application to nuclear power plant facilities and childhood leukaemia in Sweden. , 1995, Statistics in medicine.

[89]  Statistical analysis of the distribution pattern of natural regeneration in forests , 1995 .

[90]  M Martuzzi,et al.  Estimating the degree of heterogeneity between event rates using likelihood. , 1995, American journal of epidemiology.

[91]  I. Terashima,et al.  Effects of Plant Density on Frequency Distributions of Plant Height in Chenopodium album Stands: Analysis Based on Continuous Monitoring of Height-growth of Individual Plants , 1995 .

[92]  Muirhead Cr,et al.  Methods for investigating localized clustering of disease. Testing for over-dispersion using an adapted form of the Potthoff-Whittinghill method. , 1996 .

[93]  G. Rushton,et al.  Exploratory spatial analysis of birth defect rates in an urban population. , 1996, Statistics in medicine.

[94]  L A Waller.  Statistical power and design of focused clustering studies. , 1996, Statistics in medicine.

[95]  Methods for investigating localized clustering of disease. Testing for over-dispersion using an adapted form of the Potthoff-Whittinghill method. , 1996, IARC scientific publications.

[96]  A. Lawson,et al.  Adjusting Moran's I for population density. , 1996, Statistics in medicine.

[97]  Methods for investigating localized clustering of disease. Analysing the spatial distribution of disease using a method of constructing geographical areas of approximately equal population size. , 1996, IARC scientific publications.

[98]  N Oden,et al.  Realistic power simulations compare point- and area-based disease cluster tests. , 1996, Statistics in medicine.

[99]  T. Britton.  Tests to detect clustering of infected individuals within families. , 1997, Biometrics.

[100]  D. M. Titterington,et al.  Some Methods for Investigating Spatial Clustering, with Epidemiological Applications , 1997 .

[101]  Vijayan N. Nair,et al.  Monitoring wafer map data from integrated circuit fabrication processes for spatially clustered defects , 1997 .

[102]  Robert Haining,et al.  Spatial Statistics and the Analysis of Health Data , 1998 .

[103]  J B Swartz,et al.  An entropy-based algorithm for detecting clusters of cases and controls and its comparison with a method using nearest neighbours. , 1998, Health & place.

[104]  T. Carpenter,et al.  Spatial Aspects of Foot-Pad Dermatitis in Swedish Broilers , 1998, Acta Veterinaria Scandinavica.

[105]  L. Conradt.  Measuring the degree of sexual segregation in group‐living animals , 1998 .

[106]  J. Perry.  Measures of spatial pattern and spatial association for insect counts , 1998 .

[107]  F. C. Mencken,et al.  Murder, Nonnegligent Manslaughter, and Spatial Autocorrelation in Mid-South Counties , 1999 .

[108]  R. Assunção,et al.  A new proposal to adjust Moran's I for population density. , 1999, Statistics in medicine.

[109]  E. Lesaffre,et al.  Disease mapping and risk assessment for public health. , 1999 .

[110]  Lindsay A. Turnbull,et al.  Seed mass and the competition/colonization trade‐off: a sowing experiment , 1999 .

[111]  P. Diggle,et al.  Case-control isotonic regression for investigation of elevation in risk around a point source. , 1999, Statistics in medicine.

[112]  Martin Kulldorff,et al.  An Isotonic Spatial Scan Statistic for Geographical Disease Surveillance (特集 疾病地図の推定、疾病の集積性の検出に関する最近の進展) , 1999 .

[113]  D. Lai.  Statistical analysis on spatial and temporal patterns of the Chinese elderly population. , 1999, Archives of gerontology and geriatrics.

[114]  Peter A. Rogerson,et al.  The Detection of Clusters Using a Spatial Version of the Chi‐Square Goodness‐of‐Fit Statistic , 1999 .

[115]  T. Tango,et al.  A test for spatial disease clustering adjusted for multiple testing. , 2000, Statistics in medicine.

[116]  M. Bonetti.  A new geometric approach to data analysis using the Minkowski polytope , 2000 .

[117]  J. Viel,et al.  Soft-tissue sarcoma and non-Hodgkin's lymphoma clusters around a municipal solid waste incinerator with high dioxin emission levels. , 2000, American journal of epidemiology.

[118]  Massimo Coppola,et al.  Experiments in Parallel Clustering with DBSCAN , 2001, Euro-Par.

[119]  M. Clayton,et al.  A weighted average likelihood ratio test for spatial clustering of disease , 2001, Statistics in medicine.

[120]  Yifei Sun,et al.  Determining the size of spatial clusters in focused tests: Comparing two methods by means of simulation in a GIS , 2002, J. Geogr. Syst..

[121]  Peter J. Park,et al.  Power comparisons for disease clustering tests , 2003, Comput. Stat. Data Anal..

[122]  Ganapati P. Patil,et al.  Geographic and Network Surveillance via Scan Statistics for Critical Area Detection , 2003 .

[123]  Martin Kulldorff,et al.  Power evaluation of disease clustering tests , 2003, International journal of health geographics.

[124]  Howard J. Hamilton,et al.  DBRS: A Density-Based Spatial Clustering Method with Random Sampling , 2003, PAKDD.

[125]  Yi Zhang,et al.  Efficient Density Clustering Method for Spatial Data , 2003, PKDD.

[126]  G. P. Patil,et al.  Upper level set scan statistic for detecting arbitrarily shaped hotspots , 2004, Environmental and Ecological Statistics.

[127]  Andrew W. Moore,et al.  Rapid detection of significant spatial clusters , 2004, KDD.

[128]  Eric Renshaw,et al.  Spectral tests of randomness for spatial point patterns , 2001, Environmental and Ecological Statistics.

[129]  Gabriella Schoier,et al.  A Clustering Method for Large Spatial Databases , 2004, ICCSA.

[130]  Renato Assunção,et al.  A Simulated Annealing Strategy for the Detection of Arbitrarily Shaped Spatial Clusters , 2022 .

[131]  Martin Kulldorff,et al.  Geographical clustering of prostate cancer grade and stage at diagnosis, before and after adjustment for risk factors , 2005, International Journal of Health Geographics.

[132]  T. Tango,et al.  International Journal of Health Geographics a Flexibly Shaped Spatial Scan Statistic for Detecting Clusters , 2005 .

[133]  Andrew B Lawson,et al.  Scale and shape issues in focused cluster power for count data , 2005, International journal of health geographics.

[134]  Martin Kulldorff,et al.  Tango's maximized excess events test with different weights , 2005, International journal of health geographics.

[135]  Al Ozonoff,et al.  Power comparisons for an improved disease clustering test , 2005, Comput. Stat. Data Anal..

[136]  T. Sheehan,et al.  International Journal of Health Geographics Open Access a Space-time Analysis of the Proportion of Late Stage Breast Cancer , 2022 .

[137]  Marcelo Azevedo Costa,et al.  A fair comparison between the spatial scan and the Besag–Newell Disease clustering tests , 2005, Environmental and Ecological Statistics.

[138]  Marcello Pagano,et al.  The interpoint distance distribution as a descriptor of point patterns, with an application to spatial disease clustering , 2005, Statistics in medicine.

[139]  M. Kulldorff,et al.  An elliptic spatial scan statistic , 2006, Statistics in medicine.

[140]  R. Assunção,et al.  Fast detection of arbitrarily shaped disease clusters , 2006, Statistics in medicine.

[141]  M. Kulldorff,et al.  Evaluation of Spatial Scan Statistics for Irregularly Shaped Clusters , 2006 .

[142]  Lance A Waller,et al.  The geography of power: statistical performance of tests of clusters and clustering in heterogeneous populations , 2006, Statistics in medicine.

[143]  Martin Kulldorff,et al.  Likelihood based tests for spatial randomness , 2006, Statistics in medicine.

[144]  Toshiro Tango,et al.  An extended power of cluster detection tests , 2006, Statistics in medicine.

[145]  Peter A Rogerson,et al.  Statistical methods for the detection of spatial clustering in case–control data , 2006, Statistics in medicine.

[146]  Tony E. Smith,et al.  Alternative financial service providers and the spatial void hypothesis , 2008 .