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Note: Moran's I calculations in the SAS variogram procedure is experimental in version 9.2 and unavailable in earlier versions of SAS.
Moran's I is a measure of spatial autocorrelation--how related the values of a variable are based on the locations where they were measured. Using an autocorrelation option in proc variogram, we can generate Moran's I in SAS.Let's look at an example. Our dataset, ozone.sas7bdat, contains ozone measurements from thirty-two locations in the Los Angeles area aggregated over one month. The dataset includes the station number (Station), the latitude and longitude of the station (Lat and Lon), and the average of the highest eight hour daily averages (Av8top). This data, and other spatial datasets, can be downloaded from the University of Illinois's Spatial Analysis Lab. We can look at a summary of our location variables to see the range of locations under consideration.
proc print data = ozone (obs = 10); run; Obs Station Av8top Lat Lon 1 60 7.22581 34.1358 -117.924 2 69 5.89919 34.1761 -118.315 3 72 4.05289 33.8236 -118.188 4 74 7.18145 34.1994 -118.535 5 75 6.07661 34.0669 -117.751 6 84 3.15726 33.9292 -118.210 7 85 5.20161 34.0150 -118.060 8 87 4.71774 34.0672 -118.226 9 88 6.53226 34.0833 -118.107 10 89 7.54032 34.3875 -118.535
To calculate Moran's I using proc variogram, we use the autocorrelation option in the compute line. In parentheses, we indicate if we wish for our weights matrix to contain distance values or binary values. In the coordinates line, we provide the variable names of our x- and y-coordinate variables. Finally, in the var line, we indicate which variable we are interested in testing for spatial autocorrelation. In this example, we will look at the Av8top variable.
proc variogram data=ozone;
compute novar autoc (weights=distance);
coordinates xc=Lon yc=Lat;
var Av8top;
run;
The VARIOGRAM Procedure
Dependent Variable: Av8top
Number of Observations Read 32
Number of Observations Used 32
Pairs Information
Number of Lags 11
Lag Distance 0.25
Maximum Data Distance in Lon 2.30
Maximum Data Distance in Lat 1.06
Maximum Data Distance 2.53
Pairwise Distance Intervals
Number
Lag of Percentage
Class ---------Bounds--------- Pairs of Pairs
0 0.00 0.13 13 2.62%
1 0.13 0.38 106 21.37%
2 0.38 0.63 122 24.60%
3 0.63 0.89 101 20.36%
4 0.89 1.14 71 14.31%
5 1.14 1.39 41 8.27%
6 1.39 1.65 16 3.23%
7 1.65 1.90 13 2.62%
8 1.90 2.15 10 2.02%
9 2.15 2.41 3 0.60%
10 2.41 2.66 0 0.00%
Autocorrelation Statistics
Assumption Coefficient Observed Expected Std Dev Z Pr > |Z|
Normality Moran's I 0.0484 -0.0323 0.0085 9.49 <.0001
Normality Geary's c 0.9392 1.0000 0.0263 -2.31 0.0206
Based on the p-value of the reported Moran's I, we can reject the null hypothesis that there is zero spatial autocorrelation in the values of Av8top.
The distance matrix used in the above calculations is a 32x32 matrix where each off-diagonal entry [i, j] in the matrix is equal to 1/(1+distance between point i and point j). Note that this is just one of several ways in which we can calculate an inverse distance matrix. If there exists some threshold distance d such that pairs with distances less than d are "connected" or "close" and pairs with distances greater than d are not, you can indicate to SAS that binary weights should be used. This is done in the code below for d = .75. In the compute line, we indicate that our weights are binary and give a lagdistance of .75.
proc variogram data=ozone;
compute novar lagdistance = .75 autoc (weights=binary);
coordinates xc=Lon yc=Lat;
var Av8top;
run;
The VARIOGRAM Procedure
Dependent Variable: Av8top
Number of Observations Read 32
Number of Observations Used 32
Pairs Information
Number of Lags 11
Lag Distance 0.25
Maximum Data Distance in Lon 2.30
Maximum Data Distance in Lat 1.06
Maximum Data Distance 2.53
Pairwise Distance Intervals
Number
Lag of Percentage
Class ---------Bounds--------- Pairs of Pairs
0 0.00 0.13 13 2.62%
1 0.13 0.38 106 21.37%
2 0.38 0.63 122 24.60%
3 0.63 0.89 101 20.36%
4 0.89 1.14 71 14.31%
5 1.14 1.39 41 8.27%
6 1.39 1.65 16 3.23%
7 1.65 1.90 13 2.62%
8 1.90 2.15 10 2.02%
9 2.15 2.41 3 0.60%
10 2.41 2.66 0 0.00%
Autocorrelation Statistics
Assumption Coefficient Observed Expected Std Dev Z Pr > |Z|
Normality Moran's I 0.188 -0.0323 0.0323 6.82 <.0001
Normality Geary's c 0.794 1.0000 0.0851 -2.42 0.0156
This change in distance measure does not change our interpretation. Based on the p-value of the reported Moran's I, we can reject the null hypothesis that there is zero spatial autocorrelation in the values of Av8top.
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