Models of Less than Full Rank

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;
      Dif=RunPulse-RestPulse;
   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	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 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.

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