Collinearity Diagnostics |
When a regressor is nearly a linear combination of other regressors in the model, the affected estimates are unstable and have high standard errors. This problem is called collinearity or multicollinearity. It is a good idea to find out which variables are nearly collinear with which other variables. The approach in PROC REG follows that of Belsley, Kuh, and Welsch (1980). PROC REG provides several methods for detecting collinearity with the COLLIN, COLLINOINT, TOL, and VIF options.
The COLLIN option in the MODEL statement requests that a collinearity analysis be performed. First, is scaled to have 1s on the diagonal. If you specify the COLLINOINT option, the intercept variable is adjusted out first. Then the eigenvalues and eigenvectors are extracted. The analysis in PROC REG is reported with eigenvalues of rather than singular values of . The eigenvalues of are the squares of the singular values of .
The condition indices are the square roots of the ratio of the largest eigenvalue to each individual eigenvalue. The largest condition index is the condition number of the scaled matrix. Belsey, Kuh, and Welsch (1980) suggest that, when this number is around 10, weak dependencies might be starting to affect the regression estimates. When this number is larger than 100, the estimates might have a fair amount of numerical error (although the statistical standard error almost always is much greater than the numerical error).
For each variable, PROC REG produces the proportion of the variance of the estimate accounted for by each principal component. A collinearity problem occurs when a component associated with a high condition index contributes strongly (variance proportion greater than about 0.5) to the variance of two or more variables.
The VIF option in the MODEL statement provides the variance inflation factors (VIF). These factors measure the inflation in the variances of the parameter estimates due to collinearities that exist among the regressor (independent) variables. There are no formal criteria for deciding if a VIF is large enough to affect the predicted values.
The TOL option requests the tolerance values for the parameter estimates. The tolerance is defined as .
For a complete discussion of the preceding methods, refer to Belsley, Kuh, and Welsch (1980). For a more detailed explanation of using the methods with PROC REG, refer to Freund and Littell (1986).
This example uses the COLLIN option on the fitness data found in Example 76.2. The following statements produce Figure 76.35.
proc reg data=fitness; model Oxygen=RunTime Age Weight RunPulse MaxPulse RestPulse / tol vif collin; run;
Analysis of Variance | |||||
---|---|---|---|---|---|
Source | DF | Sum of Squares |
Mean Square |
F Value | Pr > F |
Model | 6 | 722.54361 | 120.42393 | 22.43 | <.0001 |
Error | 24 | 128.83794 | 5.36825 | ||
Corrected Total | 30 | 851.38154 |
Root MSE | 2.31695 | R-Square | 0.8487 |
---|---|---|---|
Dependent Mean | 47.37581 | Adj R-Sq | 0.8108 |
Coeff Var | 4.89057 |
Parameter Estimates | |||||||
---|---|---|---|---|---|---|---|
Variable | DF | Parameter Estimate |
Standard Error |
t Value | Pr > |t| | Tolerance | Variance Inflation |
Intercept | 1 | 102.93448 | 12.40326 | 8.30 | <.0001 | . | 0 |
RunTime | 1 | -2.62865 | 0.38456 | -6.84 | <.0001 | 0.62859 | 1.59087 |
Age | 1 | -0.22697 | 0.09984 | -2.27 | 0.0322 | 0.66101 | 1.51284 |
Weight | 1 | -0.07418 | 0.05459 | -1.36 | 0.1869 | 0.86555 | 1.15533 |
RunPulse | 1 | -0.36963 | 0.11985 | -3.08 | 0.0051 | 0.11852 | 8.43727 |
MaxPulse | 1 | 0.30322 | 0.13650 | 2.22 | 0.0360 | 0.11437 | 8.74385 |
RestPulse | 1 | -0.02153 | 0.06605 | -0.33 | 0.7473 | 0.70642 | 1.41559 |
Collinearity Diagnostics | |||||||||
---|---|---|---|---|---|---|---|---|---|
Number | Eigenvalue | Condition Index |
Proportion of Variation | ||||||
Intercept | RunTime | Age | Weight | RunPulse | MaxPulse | RestPulse | |||
1 | 6.94991 | 1.00000 | 0.00002326 | 0.00021086 | 0.00015451 | 0.00019651 | 0.00000862 | 0.00000634 | 0.00027850 |
2 | 0.01868 | 19.29087 | 0.00218 | 0.02522 | 0.14632 | 0.01042 | 0.00000244 | 0.00000743 | 0.39064 |
3 | 0.01503 | 21.50072 | 0.00061541 | 0.12858 | 0.15013 | 0.23571 | 0.00119 | 0.00125 | 0.02809 |
4 | 0.00911 | 27.62115 | 0.00638 | 0.60897 | 0.03186 | 0.18313 | 0.00149 | 0.00123 | 0.19030 |
5 | 0.00607 | 33.82918 | 0.00133 | 0.12501 | 0.11284 | 0.44442 | 0.01506 | 0.00833 | 0.36475 |
6 | 0.00102 | 82.63757 | 0.79966 | 0.09746 | 0.49660 | 0.10330 | 0.06948 | 0.00561 | 0.02026 |
7 | 0.00017947 | 196.78560 | 0.18981 | 0.01455 | 0.06210 | 0.02283 | 0.91277 | 0.98357 | 0.00568 |