A car is tested for gas mileage at various speeds to determine at what speed the car achieves the highest gas mileage. A quadratic model is fit to the experimental data. The following statements produce Output 44.2.1 through Output 44.2.4.
title 'Gasoline Mileage Experiment'; data mileage; input mph mpg @@; datalines; 20 15.4 30 20.2 40 25.7 50 26.2 50 26.6 50 27.4 55 . 60 24.8 ;
ods graphics on; proc glm; model mpg=mph mph*mph / p clm; run; ods graphics off;
Output 44.2.1: Standard Regression Analysis
Gasoline Mileage Experiment |
Number of Observations Read | 8 |
---|---|
Number of Observations Used | 7 |
Gasoline Mileage Experiment |
Source | DF | Sum of Squares | Mean Square | F Value | Pr > F |
---|---|---|---|---|---|
Model | 2 | 111.8086183 | 55.9043091 | 77.96 | 0.0006 |
Error | 4 | 2.8685246 | 0.7171311 | ||
Corrected Total | 6 | 114.6771429 |
R-Square | Coeff Var | Root MSE | mpg Mean |
---|---|---|---|
0.974986 | 3.564553 | 0.846836 | 23.75714 |
Source | DF | Type I SS | Mean Square | F Value | Pr > F |
---|---|---|---|---|---|
mph | 1 | 85.64464286 | 85.64464286 | 119.43 | 0.0004 |
mph*mph | 1 | 26.16397541 | 26.16397541 | 36.48 | 0.0038 |
Source | DF | Type III SS | Mean Square | F Value | Pr > F |
---|---|---|---|---|---|
mph | 1 | 41.01171219 | 41.01171219 | 57.19 | 0.0016 |
mph*mph | 1 | 26.16397541 | 26.16397541 | 36.48 | 0.0038 |
Parameter | Estimate | Standard Error | t Value | Pr > |t| |
---|---|---|---|---|
Intercept | -5.985245902 | 3.18522249 | -1.88 | 0.1334 |
mph | 1.305245902 | 0.17259876 | 7.56 | 0.0016 |
mph*mph | -0.013098361 | 0.00216852 | -6.04 | 0.0038 |
The overall F statistic is significant. The tests of mph
and mph
*mph
in the Type I sums of squares show that both the linear and quadratic terms in the regression model are significant. The
model fits well, with an R square of 0.97. The table of parameter estimates indicates that the estimated regression equation
is
Output 44.2.2: Results of Requesting the P and CLM Options
Observation | Observed | Predicted | Residual | 95% Confidence Limits for Mean Predicted Value |
||
---|---|---|---|---|---|---|
1 | 15.40000000 | 14.88032787 | 0.51967213 | 12.69701317 | 17.06364257 | |
2 | 20.20000000 | 21.38360656 | -1.18360656 | 20.01727192 | 22.74994119 | |
3 | 25.70000000 | 25.26721311 | 0.43278689 | 23.87460041 | 26.65982582 | |
4 | 26.20000000 | 26.53114754 | -0.33114754 | 25.44573423 | 27.61656085 | |
5 | 26.60000000 | 26.53114754 | 0.06885246 | 25.44573423 | 27.61656085 | |
6 | 27.40000000 | 26.53114754 | 0.86885246 | 25.44573423 | 27.61656085 | |
7 | * | . | 26.18073770 | . | 24.88679308 | 27.47468233 |
8 | 24.80000000 | 25.17540984 | -0.37540984 | 23.05954977 | 27.29126990 |
The P and CLM options in the MODEL statement produce the table shown in Output 44.2.2. For each observation, the observed, predicted, and residual values are shown. In addition, the 95% confidence limits for
a mean predicted value are shown for each observation. Note that the observation with a missing value for mph
is not used in the analysis, but predicted and confidence limit values are shown.
Output 44.2.3: Additional Results of Requesting the P and CLM Options
Sum of Residuals | -0.00000000 |
---|---|
Sum of Squared Residuals | 2.86852459 |
Sum of Squared Residuals - Error SS | -0.00000000 |
PRESS Statistic | 23.18107335 |
First Order Autocorrelation | -0.54376613 |
Durbin-Watson D | 2.94425592 |
The last portion of the output listing, shown in Output 44.2.3, gives some additional information about the residuals. The Press statistic gives the sum of squares of predicted residual errors, as described in Chapter 4: Introduction to Regression Procedures. The First Order Autocorrelation and the Durbin-Watson D statistic, which measures first-order autocorrelation, are also given.
Finally, the ODS GRAPHICS ON command in the previous statements enables ODS Graphics, which in this case produces the plot
shown in Output 44.2.4 of the actual and predicted values for the data, as well as a band representing the confidence limits for individual predictions.
The quadratic relationship between mpg
and mph
is evident.