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The REG Procedure

Models of Less than Full Rank

If the model is not full rank, there are an infinite number of least-squares solutions for the estimates. PROC REG chooses a nonzero solution for all variables that are linearly independent of previous variables and a zero solution for other variables. This solution corresponds to using a generalized inverse in the normal equations, and the expected values of the estimates are the Hermite normal form of X multiplied by the true parameters:

E(b) = (X'X)^{-}(X'X){\beta}

Degrees of freedom for the zeroed estimates are reported as zero. The hypotheses that are not testable have t tests reported as missing. The message that the model is not full rank includes a display of the relations that exist in the matrix.

The next example uses the fitness data from Example 55.1. The variable Dif=RunPulse-RestPulse is created. When this variable is included in the model along with RunPulse and RestPulse, there is a linear dependency (or exact collinearity) between the independent variables. Figure 55.40 shows how this problem is diagnosed.

   data fit2;
      set fitness;
   proc reg data=fit2;
      model Oxygen=RunTime Age Weight RunPulse MaxPulse RestPulse Dif;

The REG Procedure
Model: MODEL1
Dependent Variable: Oxygen

Analysis of Variance
Source DF Sum of
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    

NOTE: Model is not full rank. Least-squares solutions for the parameters are not unique. Some statistics will be misleading. A reported DF of 0 or B means that the estimate is biased.

NOTE: The following parameters have been set to 0, since the variables are a linear combination of other variables as shown.


Dif = RunPulse - RestPulse
Parameter Estimates
Variable DF Parameter
t Value Pr > |t|
Intercept 1 102.93448 12.40326 8.30 <.0001
RunTime 1 -2.62865 0.38456 -6.84 <.0001
Age 1 -0.22697 0.09984 -2.27 0.0322
Weight 1 -0.07418 0.05459 -1.36 0.1869
RunPulse B -0.36963 0.11985 -3.08 0.0051
MaxPulse 1 0.30322 0.13650 2.22 0.0360
RestPulse B -0.02153 0.06605 -0.33 0.7473
Dif 0 0 . . .
Figure 55.41: Model that is Not Full Rank: REG Procedure

PROC REG produces a message informing you that the model is less than full rank. Parameters with DF=0 are not estimated, and parameters with DF=B are biased. In addition, the form of the linear dependency among the regressors is displayed.

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