On A Comparison between Two Measures of Spatial Association

Transcription

1 Journal of Modern Applied Statistical Methods Volume 9 Issue Article On A Comparison beteen To Measures of Spatial Association Faisal G. Khamis Al-Zaytoonah University of Jordan, faisal_alshamari@yahoo.com Abdul Aziz Jemain Universiti Kebangsaan Malaysia, azizj@pkrisc.cc.ukm.my Kamarulzaman Ibrahim University Kebangsaan Malaysia, kamarulz@ukm.my Follo this and additional orks at: Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Khamis, Faisal G.; Jemain, Abdul Aziz; and Ibrahim, Kamarulzaman (00) "On A Comparison beteen To Measures of Spatial Association," Journal of Modern Applied Statistical Methods: Vol. 9 : Iss., Article 3. DOI: 0.37/jmasm/76870 Available at: This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@ayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@ayneState.

2 Journal of Modern Applied Statistical Methods Copyright 00 JMASM, Inc. May 00, Vol. 9, o., /0/$95.00 On A Comparison beteen To Measures of Spatial Association Faisal G. Khamis Abdul Aziz Jemain Kamarulzaman Ibrahim Al-Zaytoonah University of Jordan, Amman, Jordan University Kebangsaan Malaysia, Selangor, Malaysia To measures of spatial association beteen to variables ere used by many researchers. These are the artenberg (985) and ee (00) measures. Based on simulation for lattice data, the sensitivity of both measures as studied and compared ith different choices of spatial structures, spatial eights and sample sizes using bias and mean square error. Different scenarios are used in terms of assumed numbers and sample sizes. Moran s I is used to examine the spatial autocorrelation of such a variable ith itself. Both the artenberg and ee measures are found to be sensitive, hoever, artenberg s measure is found to be somehat better than ee s measure because it is slightly more sensitive hen sample size is small. Key ords: artenberg and ee measures, simulation study, spatial association, sensitivity, spatial structures, eights. Introduction It is argued that lattice data are spatially correlated. The artenberg (985) and ee (00) are to measures used for investigating the spatial association beteen to or more variables taking into account neighboring information. ee criticized artenberg s measure and suggested to criteria for developing a measure for bivariate spatial association. First, the measure should conform to Pearson s r beteen to variables in terms of direction and magnitude. Second, a bivariate spatial association measure should reflect the degrees of spatial autocorrelation for both variables under investigation. ee developed an index,, that combines Pearson s bivariate correlation ith Moran s spatial autocorrelation measures, to measure spatial association. ee Faisal G. Khamis is an Assistant Professor in the Faculty of Economics and Administrative Sciences. faisal_alshamari@yahoo.com. Abdul Aziz Jemain and Kamarulzaman Ibrahim are Professors in the Faculty of Sciences and Technology in the School of Mathematical Sciences. s: azizj@pkrisc.cc.ukm.my and kamarulz@ukm.my. stated that artenberg s measure is vulnerable to a reverse of the direction of spatial association. Also, ee s measure has the spatial lags of to variables hile artenberg s measure has the spatial lag for one variable. Thus, this study makes a comparison beteen these measures in terms of their sensitivity. The observations for each particular sub-area can be either univariate or multivariate data. hen the data are univariate, Moran s I statistic can be used to describe the spatial autocorrelation of such a variable. If the observations are multivariate then the artenberg and ee measures can be used. Methodology Real data are important for the development of statistical methods and ideally their analysis also stimulates research in statistical theory. Simulated data is also important and has a different role. This role is particularly valuable hen several competing methods are available but little or no theory exists to indicate hich is superior. Simulating spatial data is important because statistical inference for spatial data often relies on randomization tests. The ability to simulate realization of a hypothesized process quickly and efficiently is important to allo a sufficient number of realizations to be produced (Schabenberger & Gotay, 005). The 6

3 KHAMIS, JEMAI & IBRAHIM performance of the artenberg and ee measures is evaluated based on simulated data. The spatial association measures of artenberg and ee can be given respectively as, M = i= j=, ( x x)( y y) i j i= j= 0 ( xi x) ( yi y) i= i= = S ( x x) ( y y) j j i= j= j= ( x x) ( y y) i i i= i= here S0 =, i j, is the sample i= j= size and is the binary spatial eight (, 0). The univariate statistic for spatial association or autocorrelation of Moran s I is defined as (Cliff & Ord, 98) I = S 0 i= j= ( x x)( x x) i= i j ( x x) Because the neighbor structure is the basic structure for the covariance model of lattice data, a careful definition of spatial neighbors is a crucial analysis step (Kaluzny, et al., 998). eighbors may be defined as locations hich border each other or as locations ithin a certain distance of each other. If neighbors are defined as locations bordering each other, then there are several types of spatial neighbors. For example, the first order method (the rook pattern) identifies neighbors as those to left, to the right, or above or belo each location, that is, the rook makes links in four cardinal directions. The diagonal method (the bishop pattern) makes only diagonal links. The second order method (the queen pattern) i includes the first-order neighbors and those diagonally linked, that is, the queen makes links in all eight directions. Figure shos these three types of spatial connectivity. The sensitivity to the choice or the definition of spatial structures of neighbors as studied for both artenberg and ee measures. The simulation study as based on six spatial structures: sharing boundary (rook), sharing boundary (bishop), sharing boundary (queen), distance apart (.5), distance apart (.5) and distance apart (3). If the spatial structure as made based on distance apart, the distances beteen location i and all its surrounding neighbors ill be calculated. These distances ere calculated in the SPUS program based on such distance measures, for example, Euclidian. If the calculated distances ere found ithin for example, distance apart (.5), the surrounded locations ill be considered as neighbors to the location i. Kaluzny, et al., (998) stated the choice of spatial eights beteen such ith location and its neighbors is a crucial step. They recommended that several choices of spatial eights be tried so that the sensitivity of the results can be determined. Hoever, three different spatial eights ere used (, and = = d = d ), here d is the distance beteen location i and location j and hen d is large, the ill be less. This means that for the nearest neighbors ill be higher than that for the farthest neighbors. The bias and mean square error ( MSE ) ere used to decide hich statistic is better. et θ be the parameter of interest, then the MSE of ˆ θ is defined as follos (Garthait, et al., 995) MSE( ˆ θ) = E ˆ θ θ ( ) and the estimated value is calculated using here ˆ ˆ ˆ MSE( θ) = Var( θ) + [ bias( θ)] 7

4 O A COMPARISO BETEE TO MEASURES OF SPATIA ASSOCIATIO Var ( ˆ) = s i= θ ( ˆ θ ) i θ s, bias( ˆ θ) = E( ˆ θ) θ and the estimated value of E( ˆ θ ) is given by and ˆi θ is the estimated bivariate spatial association measure based on simulated data, θ is the actual value of bivariate spatial association measure based on artenberg s or ee s measure and s is the total number of runs (in this study, s = 0,000 ). Figure : Three Different Types of Sharing Boundary Connectivity Rook Bishop Queen The estimate ˆ θ is an unbiased estimator for θ if E( ˆ θ) = θ ; otherise it is biased. The bias of ˆ θ is defined to be (Garthait, et al., 995) ˆ s θi ˆ i= =, θ then the estimated value of s bias ( ˆ θ) = ˆ θ θ. bias( ˆ θ ) is given by Simulation Process The process of the simulation study included several steps hich ere considered somehat complicated. The complication arose from alloing three kinds of spatial correlations before starting the spatial analysis and because the simulation must be made under a randomness assumption. The spatial correlations ere: the bivariate spatial correlation beteen to variables and spatial autocorrelation for each variable. The simulation study as carried out using SPUS programming and accomplished in four steps. First, the original samples for to variables ere designated to act as the population for sampling purposes. In the second step, the original samples ere re-sampled a specified number of times (up to several thousands) to generate a large number of ne samples, here each sample as a random subset of the original sample. In the third step, the bivariate spatial measures of artenberg and ee ere estimated for each ne sample. In the last step, the estimated values of bias and MSE ere calculated using the computed spatial measures found in step 3. During the process of generating ne samples, the simulation program may change certain characteristics of the sample to meet the researcher s objectives. For example, the degree of correlation beteen variables may be varied across the generated samples in some systematic manner. The simulation process as run using 0000 runs, here artenberg s measure, and ee s measure, ere each estimated 5,000 8

5 KHAMIS, JEMAI & IBRAHIM times. The bias and MSE ere then calculated for each measure. The mechanism proposed herein contains certain assumed form of univariate spatial correlation (autocorrelation) for each variable as shon from the distribution of assumed observed values, and hence there is also a bivariate spatial correlation beteen these to autocorrelated variables based on the actual value of artenberg and ee measures. Results To assess the performance of both the artenberg and ee measures, a series of simulations ere conducted. Several scenarios ere studied to investigate the sensitivity of both the artenberg and ee measures based on different choices of spatial structures and spatial eights using bias and MSE. The values of to variables, and, ere generated based on their assumed true means and standard deviation one for each observation. values of each and variables based on Moran s I ere found positive, negative, high or lo because different choices of spatial structures and spatial eights ere used; the nine resulting scenarios follo. Scenario : Sample Size ( 4 4= 6) Figures 3a and 3b sho to study areas ith their assumed true means. Table, shos the autocorrelation values for both variables and based on global Moran s I statistic. Table shos the actual values of the artenberg and ee measures and their bias and MSE using different choices of spatial structures. Figure 3 Proposed Area Divided into 6 Quadrates in a 4 4 attice 3a 3b The numbers in the quadrates are the assumed true means for to different variables and, here is a gradient patch from lo values (in the center) to high values (in the edges) for variable, and (b) is the opposite direction of (a) for variable. Table : for Variables and Based on Moran s I Using Different Types of Spatial Structures Sharing Boundary (Rook) Sharing Boundary (Bishop) Sharing Boundary (Queen) Distance Apart (.5) Distance Apart (.5) Distance Apart (3)

6 O A COMPARISO BETEE TO MEASURES OF SPATIA ASSOCIATIO Table : The bias and MSE of artenberg s Measure () and ee s Measure () ith Actual Values Using Different Types of Spatial Structures Sharing Boundary (Rook) Sharing Boundary (Bishop) Sharing Boundary (Queen) Distance Apart (.5) Distance Apart (.5) Distance Apart (3) Scenario : Sample Size ( 4 4= 6) Figures 4a and 4b sho to study areas ith their assumed true means. Table 3 shos the autocorrelation values for both variables and based on Moran s I statistic. Table 4 shos the actual values of artenberg and ee measures and their bias and MSE using different choices of spatial structures. Figure 4: Proposed Area Divided into 6 Quadrates in a 4 4 attice 4a 4b The numbers in the quadrates are the assumed true means for to different variables and, here both (a) and (b) are a gradient patches from lo values (in the center) to high values (in the edges) for and variables respectively. Table 3: for Variables and Based on Moran s I Using Different Types of Spatial Structures Sharing Boundary (Rook) Sharing Boundary (Bishop) Sharing Boundary (Queen) Distance Apart (.5) Distance Apart (.5) Distance Apart (3)

7 KHAMIS, JEMAI & IBRAHIM Table 4: The bias and MSE of artenberg s Measure () and ee s Measure () ith Actual Values Using Different Types of Spatial Structures Sharing Boundary (Rook) Sharing Boundary (Bishop) Sharing Boundary (Queen) Distance Apart (.5) Distance Apart (.5) Distance Apart (3) Scenario 3: Sample Size ( 4 4= 6) Figures 5a and 5b sho to study areas ith their assumed true means. Table 5 shos the autocorrelation values for both variables and based on Moran s I statistic. Table 6 shos the actual values of artenberg and ee measures and their bias and MSE using different choices of spatial structures. Figure 5: Proposed Area Divided into 6 Quadrates in a 4 4 attice 5a 5b The numbers in the quadrates are the assumed true means for to different variables and, here (a) is a gradient patch from lo values (the bottom left side) to high values (the upper right side) for variable, and (b) is the opposite direction of (a) for variable. Table 5: for Variables and Based on Moran s I Using Different Types of Spatial Structures Sharing Boundary (Rook) Sharing Boundary (Bishop) Sharing Boundary (Queen) Distance Apart (.5) Distance Apart (.5) Distance Apart (3)

8 O A COMPARISO BETEE TO MEASURES OF SPATIA ASSOCIATIO Table 6: The bias and MSE of artenberg s Measure () and ee s Measure () ith Actual Values Using Different Types of Spatial Structures Sharing Boundary (Rook) Sharing Boundary (Bishop) Sharing Boundary (Queen) Distance Apart (.5) Distance Apart (.5) Distance Apart (3) Scenario 4: Sample Size ( 4 4= 6) Figures 6a and 6b sho to study areas ith their assumed true means. Table 7 shos the autocorrelation values for both variables and based on Moran s I statistic. Table 8 shos the actual values of artenberg and ee measures and their bias and MSE using different choices of spatial structures. Figure 6: Proposed Area Divided into 6 Quadrates in a 4 4 attice 6a 6b The numbers in the quadrates are the assumed true means for to different variables and, here both (a) and (b) are a gradient patches from lo values (the bottom left side) to higher values (the upper right side) for and variables respectively. Table 7: for Variables and Based on Moran s I Using Different Types of Spatial Structures Sharing Boundary (Rook) Sharing Boundary (Bishop) Sharing Boundary (Queen) Distance Apart (.5) Distance Apart (.5) Distance Apart (3)

9 KHAMIS, JEMAI & IBRAHIM Table 8: The bias and MSE of artenberg s Measure () and ee s Measure () ith Actual Values Using Different Types of Spatial Structures Sharing Boundary (Rook) Sharing Boundary (Bishop) Sharing Boundary (Queen) Distance Apart (.5) Distance Apart (.5) Distance Apart (3) Scenario 5: Sample Size ( 6 6= 36) Figures 7a and 7b sho to study areas ith their assumed true means. Table 9 shos the autocorrelation values for both variables and based on Moran s I statistic. Table 0 shos the actual values of artenberg and ee measures and their bias and MSE using different choices of spatial structures. Figure 7: Proposed Area Divided into 36 Quadrates in a 6 6 attice 7a 7b The numbers in the quadrates are the assumed true means for to different variables and, here (a) is a gradient patch from lo values (in the center) to high values (in the edges) for variable, and (b) is the opposite direction of (a) for variable. Table 9: for Variables and Based on Moran s I Using Different Types of Spatial Structures Sharing Boundary (Rook) Sharing Boundary (Bishop) Sharing Boundary (Queen) Distance Apart (.5) Distance Apart (.5) Distance Apart (3)

10 O A COMPARISO BETEE TO MEASURES OF SPATIA ASSOCIATIO Table 0: The bias and MSE of artenberg s Measure () and ee s Measure () ith Actual Values Using Different Types of Spatial Structures Sharing Boundary (Rook) Sharing Boundary (Bishop) Sharing Boundary (Queen) Distance Apart (.5) Distance Apart (.5) Distance Apart (3) Scenario 6: Sample Size (8 8= 64) Figures 8a and 8b sho to study areas ith their assumed true means. Table shos the autocorrelation values for both variables and based on Moran s I statistic. Table shos the actual values of artenberg and ee measures and their bias and MSE using different choices of spatial structures. Figure 8: Proposed Area Divided into 64 Quadrates in a 8 8 attice 8a 8b The numbers in the quadrates are the assumed true means for to different variables and, here (a) is a gradient patch from lo values (in the center) to high values (in the edges) for variable, and (b) is the opposite direction of (a) for variable. 4

11 KHAMIS, JEMAI & IBRAHIM Table : for Variables and Based on Moran s I Using Different Types of Spatial Structures Sharing Boundary (Rook) Sharing Boundary (Bishop) Sharing Boundary (Queen) Distance Apart (.5) Distance Apart (.5) Distance Apart (3) Table : The bias and MSE of artenberg s Measure () and ee s Measure () ith Actual Values Using Different Types of Spatial Structures Sharing Boundary (Rook) Sharing Boundary (Bishop) Sharing Boundary (Queen) Distance Apart (.5) Distance Apart (.5) Distance Apart (3) Scenario 7: Sample Size ( 4 4= 6) Figures 9a and 9b sho to study areas ith their assumed true means. The spatial structure is defined using the distance apart (.5). Table 3 shos the autocorrelation values for both variables and based on Moran s I statistic. Table 4 shos the actual values of artenberg and ee measures and their bias and MSE using different choices of spatial eights. Figure 9: Proposed Area Divided into 6 Quadrates in a 4 4 attice 9a 9b The numbers in the quadrates are the assumed true means for to different variables and, here (a) is a gradient patch from lo values (in the center) to high values (in the edges) for variable, and (b) is the opposite direction of (a) for variable. 5

12 O A COMPARISO BETEE TO MEASURES OF SPATIA ASSOCIATIO Table 3: for Variables and Based on Moran s I Using Different Types of Spatial eights Type of Spatial eight = = d = d Table 4: The bias and MSE of artenberg s Measure () and ee s Measure () ith Actual Values Using Different Types of Spatial eights Type of Spatial eight = = d = d Scenario 8: Sample Size ( 6 6= 36) Figures 0a and 0b sho to study areas ith their assumed true means. The spatial structure is defined using the distance apart (.5). Table 5 shos the autocorrelation values for both variables and based on Moran s I statistic. Table 6 shos the actual values of artenberg and ee measures and their bias and MSE using different choices of spatial eights. Figure 0: Proposed Area Divided into 36 Quadrates in a 6 6 attice 0a 0b The numbers in the quadrates are the assumed true means for to different variables and, here (a) is a gradient patch from lo values (in the center) to high values (in the edges) for variable, and (b) is the opposite direction of (a) for variable. 6

13 KHAMIS, JEMAI & IBRAHIM Table 5: for Variables and Based on Moran s I Using Different Spatial eights Type of Spatial eight = = d = d Table 6: The bias and MSE of artenberg s Measure () and ee s Measure () ith Actual Values Using Different Types of Spatial eights Type of Spatial eight = = d = d Scenario 9: Sample Size (8 8= 64) Figures a and b sho to study areas ith their assumed true means. The spatial structure is defined using the distance apart (.5). Table 7 shos the autocorrelation values for both variables and based on Moran s I statistic. Table 8 shos the actual values of artenberg and ee measures and their bias and MSE using different choices of spatial eights. Figure : Proposed Area Divided into 64 Quadrates in a 8 8 attice a b The numbers in the quadrates are the assumed true means for to different variables and, here (a) is a gradient patch from lo values (in the center) to high values (in the edges) for variable, and (b) is the opposite direction of (a) for variable. 7

14 O A COMPARISO BETEE TO MEASURES OF SPATIA ASSOCIATIO Table 7: for Variables and Based on Moran s I Using Different Spatial eights Type of Spatial eight = = d = d Table 8: The bias and MSE of artenberg s Measure () and ee s Measure () ith Actual Values Using Different Types of Spatial eights Type of Spatial eight = = d = d Conclusion The results from these scenarios sho that the artenberg and ee measures differ slightly in terms of their sensitivity to different choices of spatial structures and spatial eights. Results sho that artenberg s measure is somehat more sensitive than ee s measure to the different choices of spatial structures and spatial eights hen the sample size is small; for the large sample sizes the results of both measures are approximately the same. Several techniques in statistics are sensitive - meaning they sometimes provide inaccurate results hen a small sample size is used - because the information in a small sample is less than that of a large sample size. artenberg s equation is vulnerable to a reverse in direction of association as stated by ee. This reverse in direction as found in scenarios and as shon in the column of actual value of artenberg s measure in Tables and 4 respectively. References Cliff, A. D., & Ord, J. K. (98). Spatial processes: Models & Applications. ondon: Page Bros. Garthait, P. H., Jolliffe, I. T., & Jones, B. (995). Statistical Inference. ondon: Prentice Hall. Kaluzny, S. P., Vega, S. C., Cardoso, T. P., & Shelly, A. A. (998). S+ SpatialStats: User s Manual for indos and Unix. e ork: Springer. ee, S-I. (00). Developing a bivariate spatial association measure: An integration of Pearson s r and Moran s I. Journal of Geographical Systems, 3, Schabenberger, O., & Gotay, C. A. (005). Statistical Methods for Spatial Data Analysis. Boca Raton, F: Chapman & Hall. artenberg, D. (985). Multivariate spatial correlation: A method for exploratory geographical analysis. Geographical Analysis, 7(4),