title: A modified version of Moran's I.
abstract: BACKGROUND
  Investigation of global clustering patterns across regions is very important in spatial data analysis. Moran's I is a widely used spatial statistic for detecting global spatial patterns such as an east-west trend or an unusually large cluster. Here, we intend to improve Moran's I for evaluating global clustering patterns by including the weight function in the variance, introducing a population density (PD) weight function in the statistics, and conducting Monte Carlo simulation for testing. We compare our modified Moran's I with Oden's I*pop for simulated data with homogeneous populations. The proposed method is applied to a census tract data set.
  
  
  METHODS
  We present a modified version of Moran's I which includes information about the strength of the neighboring association when estimating the variance for the statistic. We provide a power analysis on Moran's I, a modified version of Moran's I, and I*pop in a simulation study. Data were simulated under two common spatial correlation scenarios of local and global clustering.
  
  
  RESULTS
  For simulated data with a large cluster pattern, the modified Moran's I has the highest power (43.4%) compared to Moran's I (39.9%) and I*pop (12.4%) when the adjacent weight function is used with 5%, 10%, 15%, 20%, or 30% of the total population as the geographic range for the cluster.For two global clustering patterns, the modified Moran's I (power > 25.3%) performed better than both Moran's I (> 24.6%) and I*pop (> 7.9%) with the adjacent weight function. With the population density weight function, all methods performed equally well.In the real data example, all statistics indicate the existence of a global clustering pattern in a leukemia data set. The modified Moran's I has the lowest p-value (.0014) followed by Moran's I (.0156) and I*pop (.011).
  
  
  CONCLUSIONS
  Our power analysis and simulation study show that the modified Moran's I achieved higher power than Moran's I and I*pop for evaluating global and local clustering patterns on geographic data with homogeneous populations. The inclusion of the PD weight function which in turn redefines the neighbors seems to have a large impact on the power of detecting global clustering patterns. Our methods to improve the original version of Moran's I for homogeneous populations can also be extended to some alternative versions of Moran's I methods developed for heterogeneous populations.
authors.affiliation: Department of Mathematics and Statistics, 4400 Massachusetts Ave NW, American University, Washington, DC 20016, USA. monica@american.edu
authors.forename: Monica C
authors.initials: MC
authors.lastname: Jackson
authors.forename: Lan
authors.initials: L
authors.lastname: Huang
authors.forename: Qian
authors.initials: Q
authors.lastname: Xie
authors.forename: Ram C
authors.initials: RC
authors.lastname: Tiwari
mesh_terms: D003627:Data Interpretation, Statistical; D006801:Humans; D007938:Leukemia / Q000453:epidemiology*; D015233:Models, Statistical; D009010:Monte Carlo Method; D009519:New York City / Q000453:epidemiology; D012680:Sensitivity and Specificity; D013029:Space-Time Clustering