The CSPATIALREG Procedure
Example 12.6 Comparison of Multiple Models
Data Description and Objective
Example in this section uses the CrimeOH and CrimeWmat data sets that are described in Columbus Crime Data to demonstrate fitting and comparison of multiple models in PROC CSPATIALREG.
Comparison of Multiple Models
The following statements fit multiple models to the CrimeOH and CrimeWmat data sets, which are described in Columbus Crime Data, using the regressors Income and HValue:
/*-- TYPE=AUTO --*/
proc cspatialreg data=mycas.CrimeOH Wmat=mycas.CrimeWmat;
model crime=income hvalue / type=AUTO;
spatialid sid;
run;
In this example, the TYPE=AUTO option is specified in the MODEL statement to fit seven models—SAC, SARMA, SAR, SEM, SMA, CAR, and LINEAR. However, the CAR model is not fitted because the input spatial weights matrix is not symmetric. The parameter estimates for the six remaining models—SAC, SARMA, SAR, SEM, SMA, and LINEAR—are shown in Output 12.6.1. The comparison of these six models is provided in Output 12.6.2. If AIC is used for model comparison, Output 12.6.2 suggests that the SAR model is the best model among the six candidate models, because its AIC value is the smallest.
Output 12.6.1: Parameter Estimates of Multiple Models
| Parameter Estimates | ||||||
|---|---|---|---|---|---|---|
| Model Type | Parameter | DF | Estimate | Standard Error | t Value | Approx Pr > |t| |
| SAC | Intercept | 1 | 47.784041 | 9.279548 | 5.15 | <.0001 |
| income | 1 | -1.025892 | 0.334029 | -3.07 | 0.0021 | |
| hvalue | 1 | -0.281653 | 0.093369 | -3.02 | 0.0026 | |
| _rho | 1 | 0.368060 | 0.181191 | 2.03 | 0.0422 | |
| _lambda | 1 | 0.166684 | 0.298147 | 0.56 | 0.5761 | |
| _sigma2 | 1 | 95.603723 | 19.475524 | 4.91 | <.0001 | |
| SARMA | Intercept | 1 | 48.979253 | 9.603012 | 5.10 | <.0001 |
| income | 1 | -1.016394 | 0.337242 | -3.01 | 0.0026 | |
| hvalue | 1 | -0.287474 | 0.093078 | -3.09 | 0.0020 | |
| _rho | 1 | 0.336163 | 0.204404 | 1.64 | 0.1001 | |
| _lambda | 1 | -0.272194 | 0.426917 | -0.64 | 0.5237 | |
| _sigma2 | 1 | 98.003653 | 21.257696 | 4.61 | <.0001 | |
| SAR | Intercept | 1 | 45.079467 | 7.871241 | 5.73 | <.0001 |
| income | 1 | -1.031616 | 0.328416 | -3.14 | 0.0017 | |
| hvalue | 1 | -0.265927 | 0.088222 | -3.01 | 0.0026 | |
| _rho | 1 | 0.431018 | 0.123619 | 3.49 | 0.0005 | |
| _sigma2 | 1 | 95.494608 | 19.507896 | 4.90 | <.0001 | |
| SEM | Intercept | 1 | 59.893276 | 5.883689 | 10.18 | <.0001 |
| income | 1 | -0.941312 | 0.370277 | -2.54 | 0.0110 | |
| hvalue | 1 | -0.302252 | 0.090553 | -3.34 | 0.0008 | |
| _lambda | 1 | 0.561786 | 0.152423 | 3.69 | 0.0002 | |
| _sigma2 | 1 | 95.574411 | 20.038002 | 4.77 | <.0001 | |
| SMA | Intercept | 1 | 59.253312 | 5.934854 | 9.98 | <.0001 |
| income | 1 | -0.921805 | 0.363491 | -2.54 | 0.0112 | |
| hvalue | 1 | -0.287389 | 0.086881 | -3.31 | 0.0009 | |
| _lambda | 1 | -0.799096 | 0.277870 | -2.88 | 0.0040 | |
| _sigma2 | 1 | 117.732337 | 26.373311 | 4.46 | <.0001 | |
| LINEAR | Intercept | 1 | 68.618863 | 4.588210 | 14.96 | <.0001 |
| income | 1 | -1.597304 | 0.323739 | -4.93 | <.0001 | |
| hvalue | 1 | -0.273931 | 0.099989 | -2.74 | 0.0062 | |
| _sigma2 | 1 | 122.751696 | 24.799493 | 4.95 | <.0001 | |
Output 12.6.2: Comparison of Multiple Models
| Model Comparison | |||||||
|---|---|---|---|---|---|---|---|
| Model Type | Status | Reason | Number of Iterations | Max Abs Gradient | Log Likelihood | AIC | SBC |
| SAC | 0 | Convergence criterion (FCONV=1E-8) satisfied. | 34 | 0.0000 | -182.2 | 376.4695 | 387.8204 |
| SARMA | 0 | Convergence criterion (FCONV=1E-8) satisfied. | 17 | 0.0000 | -182.2 | 376.3127 | 387.6637 |
| SAR | 0 | Convergence criterion (FCONV=1E-8) satisfied. | 16 | 0.0000 | -182.4 | 374.7809* | 384.2400* |
| SEM | 0 | Convergence criterion (FCONV=1E-8) satisfied. | 36 | 0.0000 | -183.4 | 376.7608 | 386.2199 |
| SMA | 0 | Convergence criterion (FCONV=1E-8) satisfied. | 17 | 0.0000 | -183.1 | 376.1446 | 385.6038 |
| LINEAR | 0 | Convergence criterion (ABSGCONV=0.00001) satisfied. | 16 | 0.0000 | -187.4 | 382.7542 | 390.3215 |
| Asterisk (*) denotes the best model in the column. | |||||||