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 42.2.1 through Output 42.2.4.
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 42.2.1: 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 
RSquare  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 42.2.2: 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 42.2.2. 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 42.2.3: 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 
DurbinWatson D  2.94425592 
The last portion of the output listing, shown in Output 42.2.3, 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 DurbinWatson D statistic, which measures firstorder autocorrelation, are also given.
Output 42.2.4: Plot of Mileage Data
Finally, the ODS GRAPHICS ON command in the previous statements enables ODS Graphics, which in this case produces the plot
shown in Output 42.2.4 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.