The REG Procedure
- Overview
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Getting Started
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Syntax
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DetailsMissing ValuesInput Data SetsOutput Data SetsInteractive AnalysisModel-Selection MethodsCriteria Used in Model-Selection MethodsLimitations in Model-Selection MethodsParameter Estimates and Associated StatisticsPredicted and Residual ValuesModels of Less Than Full RankCollinearity DiagnosticsModel Fit and Diagnostic StatisticsInfluence StatisticsReweighting Observations in an AnalysisTesting for HeteroscedasticityTesting for Lack of FitMultivariate TestsAutocorrelation in Time Series DataComputations for Ridge Regression and IPC AnalysisConstruction of Q-Q and P-P PlotsComputational MethodsComputer Resources in Regression AnalysisDisplayed OutputODS Table NamesODS Graphics
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Examples
- References
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 
multiplied by the true parameters:
![\[ E(\mb {b}) = (\mb {X}’\mb {X})^{-}(\mb {X}’\mb {X})\bbeta \]](images/statug_reg0076.png){kind=small} |
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=RunPulse-RestPulse; 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 | 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.