A modified version of Moran's I

Abstract

BackgroundInvestigation 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.MethodsWe 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.ResultsFor 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).ConclusionsOur 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.

Spatial analysis of human population distribution and growth in Marinduque Island, Philippines

“…To evaluate the degree of similarity of observation across space, global indices of spatial autocorrelation are commonly used (Jackson et al, 2010). Moran's I is a global index of spatial autocorrelation widely used and tested for studying spatial autocorrelation over the past 50 years (Bae et al, 2008;Jackson et al, 2010;Young and Jensen, 2012;Zhang and Lin, 2007).…”

“…To evaluate the degree of similarity of observation across space, global indices of spatial autocorrelation are commonly used (Jackson et al, 2010). Moran's I is a global index of spatial autocorrelation widely used and tested for studying spatial autocorrelation over the past 50 years (Bae et al, 2008;Jackson et al, 2010;Young and Jensen, 2012;Zhang and Lin, 2007). On the other hand, while global indices like Moran's I can measure spatial association of the entire data set, use of local indicators for association local spatial clusters (LISA, Anselin, 1995) is needed to identify clusters of high (hot) and low (col) spots across space (Bae et al, 2008;Rossen et al, 2014).…”

Race Matters: Analyzing the Relationship between Colorectal Cancer Mortality Rates and Various Factors within Respective Racial Groups

“…A modified version of Moran’s I (18) was used to test for geographic clustering for the three dependent variables. Modified Moran’s I is defined as: $I_{W}MathClass-rel=\frac{{MathClass-op\sum }_{iMathClass-rel=1}^N{MathClass-op\sum }_{\left\{jMathClass-punc:jMathClass-rel\ne i\right\}}^N{W}_{\mathrm{italicij}}\left({\mathrm{\gamma }}_{i}MathClass-bin-\overline{\mathrm{\gamma }}\right)\left({\mathrm{\gamma }}_{j}MathClass-bin-\overline{\mathrm{\gamma }}\right)}{{MathClass-op\sum }_{1MathClass-rel\le iMathClass-rel<jMathClass-rel\le N}{W}_{\mathrm{italicij}}{\left({\mathrm{\gamma }}_{i}MathClass-bin-{\mathrm{\gamma }}_j\right)}^2}MathClass-punc,$ where γ i and γ j are the colorectal death rates at geographic locations i and j , respectively; $\overline{\mathrm{\gamma }}$ is the expected colorectal death rate using all the data; and N is the total number of geographic units.…”

Spatial autocorrelation analysis in plant population: An overview

“…They have reported that for both α and β tree diversities, the distance of the spatial autocorrelation increased slightly at the scale of 10-15 and 200 km and decreased around 20-40 km. Modified version of Moran's I: Traditional calculation of Moran's I for heterogeneous populations is not working well (Jackson et al, 2010). Therefore, several alternative version of Moran's I have been proposed to account for heterogeneous population, for example Oden (1995);Waldhor (1996); Assuncao and Reis (1999) and Waller et al (2006).…”

“…In order to capture the variability present in the region, Oden includes the first term in the numerator which is used to model the spatial variation in a manner similar to conventional chisquared for heterogeneity rates. The odens's I* pop can be written as Jackson et al (2010) improves the original version of Moran's I by incorporation of (a) a weight function in the variance computation (b) introducing the population density weight function and (c) conducting Monte Carlo simulation. Their weight function is not only included in the differences of the geographic unit's cases from the overall mean, but also in the calculation of the variance.…”