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 83.2. 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 83.34 shows how this problem is diagnosed.
data fit2; set fitness; Dif=RunPulse-RestPulse; run; proc reg data=fit2; model Oxygen=RunTime Age Weight RunPulse MaxPulse RestPulse Dif; run;
Figure 83.34: 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 | 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| |
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.