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 multiplied by the true parameters:

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 following statements use the fitness data from Example 79.2. 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 79.34 shows how this problem is diagnosed.
data fit2; set fitness; Dif=RunPulseRestPulse; run; proc reg data=fit2; model Oxygen=RunTime Age Weight RunPulse MaxPulse RestPulse Dif; run;
Figure 79.34: Model That Is Not Full Rank: REG Procedure
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  RSquare  0.8487 

Dependent Mean  47.37581  Adj RSq  0.8108 
Coeff Var  4.89057 
Parameter Estimates  

Variable  DF  Parameter Estimate 
Standard Error 
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  .  .  . 
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.