Models of Less than Full Rank
If the model is not full rank, there are an infinite
number of leastsquares 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:
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=RunPulseRestPulse 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;
Dif=RunPulseRestPulse;
proc reg data=fit2;
model Oxygen=RunTime Age Weight RunPulse MaxPulse RestPulse Dif;
run;
The REG Procedure 
Model: MODEL1 
Dependent Variable: Oxygen 
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 


NOTE: 
Model is not full rank. Leastsquares 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 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 
. 
. 
. 

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.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.