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 The REG Procedure

## Parameter Estimates and Associated Statistics

The following example uses the fitness data from Example 55.1. Figure 55.27 shows the parameter estimates and the tables from the SS1, SS2, STB, CLB, COVB, and CORRB options:

   proc reg data=fitness;
model Oxygen=RunTime Age Weight RunPulse MaxPulse RestPulse
/ ss1 ss2 stb clb covb corrb;
run;

The procedure first displays an Analysis of Variance table (Figure 55.26). The F statistic for the overall model is significant, indicating that the model explains a significant portion of the variation in the data.

 The REG Procedure Model: MODEL1 Dependent Variable: Oxygen

 Analysis of Variance Source DF Sum ofSquares MeanSquare 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 47.3758 Adj R-Sq 0.8108 Coeff Var 4.89057
Figure 55.27: ANOVA Table

The procedure next displays Parameter Estimates and some associated statistics (Figure 55.27). First, the estimates are shown, followed by their Standard Errors. The next two columns of the table contain the t statistics and the corresponding probabilities for testing the null hypothesis that the parameter is not significantly different from zero. These probabilities are usually referred to as p-values. For example, the Intercept term in the model is estimated to be 102.9 and is significantly different from zero. The next two columns of the table are the result of requesting the SS1 and SS2 options, and they show sequential and partial Sums of Squares (SS) associated with each variable. The Standardized Estimates (produced by the STB option) are the parameter estimates that result when all variables are standardized to a mean of 0 and a variance of 1. These estimates are computed by multiplying the original estimates by the standard deviation of the regressor (independent) variable and then dividing by the standard deviation of the dependent variable. The CLB option adds the upper and lower 95% confidence limits for the parameter estimates; the level can be changed by specifying the ALPHA= option in the PROC REG or MODEL statement.

 The REG Procedure Model: MODEL1 Dependent Variable: Oxygen

 Parameter Estimates Variable DF ParameterEstimate StandardError t Value Pr > |t| Type I SS Type II SS StandardizedEstimate 95% Confidence Limits Intercept 1 102.93448 12.40326 8.30 <.0001 69578 369.72831 0 77.33541 128.53355 RunTime 1 -2.62865 0.38456 -6.84 <.0001 632.90010 250.82210 -0.68460 -3.42235 -1.83496 Age 1 -0.22697 0.09984 -2.27 0.0322 17.76563 27.74577 -0.22204 -0.43303 -0.02092 Weight 1 -0.07418 0.05459 -1.36 0.1869 5.60522 9.91059 -0.11597 -0.18685 0.03850 RunPulse 1 -0.36963 0.11985 -3.08 0.0051 38.87574 51.05806 -0.71133 -0.61699 -0.12226 MaxPulse 1 0.30322 0.13650 2.22 0.0360 26.82640 26.49142 0.52161 0.02150 0.58493 RestPulse 1 -0.02153 0.06605 -0.33 0.7473 0.57051 0.57051 -0.03080 -0.15786 0.11480
Figure 55.28: SS1, SS2, STB, CLB, COVB, and CORRB Options: Parameter Estimates

The final two tables are produced as a result of requesting the COVB and CORRB options (Figure 55.28). These tables show the estimated covariance matrix of the parameter estimates, and the estimated correlation matrix of the estimates.

 The REG Procedure Model: MODEL1 Dependent Variable: Oxygen

 Covariance of Estimates Variable Intercept RunTime Age Weight RunPulse MaxPulse RestPulse Intercept 153.84081152 0.7678373769 -0.902049478 -0.178237818 0.280796516 -0.832761667 -0.147954715 RunTime 0.7678373769 0.1478880839 -0.014191688 -0.004417672 -0.009047784 0.0046249498 -0.010915224 Age -0.902049478 -0.014191688 0.009967521 0.0010219105 -0.001203914 0.0035823843 0.0014897532 Weight -0.178237818 -0.004417672 0.0010219105 0.0029804131 0.0009644683 -0.001372241 0.0003799295 RunPulse 0.280796516 -0.009047784 -0.001203914 0.0009644683 0.0143647273 -0.014952457 -0.000764507 MaxPulse -0.832761667 0.0046249498 0.0035823843 -0.001372241 -0.014952457 0.0186309364 0.0003425724 RestPulse -0.147954715 -0.010915224 0.0014897532 0.0003799295 -0.000764507 0.0003425724 0.0043631674

 Correlation of Estimates Variable Intercept RunTime Age Weight RunPulse MaxPulse RestPulse Intercept 1.0000 0.1610 -0.7285 -0.2632 0.1889 -0.4919 -0.1806 RunTime 0.1610 1.0000 -0.3696 -0.2104 -0.1963 0.0881 -0.4297 Age -0.7285 -0.3696 1.0000 0.1875 -0.1006 0.2629 0.2259 Weight -0.2632 -0.2104 0.1875 1.0000 0.1474 -0.1842 0.1054 RunPulse 0.1889 -0.1963 -0.1006 0.1474 1.0000 -0.9140 -0.0966 MaxPulse -0.4919 0.0881 0.2629 -0.1842 -0.9140 1.0000 0.0380 RestPulse -0.1806 -0.4297 0.2259 0.1054 -0.0966 0.0380 1.0000
Figure 55.29: SS1, SS2, STB, CLB, COVB, and CORRB Options: Covariances and Correlations

For further discussion of the parameters and statistics, see the "Displayed Output" section, and Chapter 3, "Introduction to Regression Procedures."

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