The following example uses the fitness data from Example 97.2. Figure 97.31 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 97.30). The F statistic for the overall model is significant, indicating that the model explains a significant portion of the variation in the data.
Figure 97.30: ANOVA Table
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 |
The procedure next displays parameter estimates and some associated statistics (Figure 97.31). 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.
Figure 97.31: SS1, SS2, STB, CLB, COVB, and CORRB Options: Parameter Estimates
Parameter Estimates | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Variable | DF | Parameter Estimate |
Standard Error |
t Value | Pr > |t| | Type I SS | Type II SS | Standardized Estimate |
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 |
The final two tables are produced as a result of requesting the COVB and CORRB options (Figure 97.32). These tables show the estimated covariance matrix of the parameter estimates, and the estimated correlation matrix of the estimates.
Figure 97.32: SS1, SS2, STB, CLB, COVB, and CORRB Options: Covariances and Correlations
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 |
For further discussion of the parameters and statistics, see the section Displayed Output, and Chapter 4: Introduction to Regression Procedures.