Example 41.2 Regression with Mileage Data
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 41.2.1 through Output 41.2.4.
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 41.2.1
Standard Regression Analysis
The GLM Procedure
Dependent Variable: mpg
2 
111.8086183 
55.9043091 
77.96 
0.0006 
4 
2.8685246 
0.7171311 


6 
114.6771429 



0.974986 
3.564553 
0.846836 
23.75714 
1 
85.64464286 
85.64464286 
119.43 
0.0004 
1 
26.16397541 
26.16397541 
36.48 
0.0038 
1 
41.01171219 
41.01171219 
57.19 
0.0016 
1 
26.16397541 
26.16397541 
36.48 
0.0038 
5.985245902 
3.18522249 
1.88 
0.1334 
1.305245902 
0.17259876 
7.56 
0.0016 
0.013098361 
0.00216852 
6.04 
0.0038 
The overall 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 of 0.97. The table of parameter estimates indicates that the estimated regression equation is
Output 41.2.2
Results of Requesting the P and CLM Options

15.40000000 
14.88032787 
0.51967213 
12.69701317 
17.06364257 

20.20000000 
21.38360656 
1.18360656 
20.01727192 
22.74994119 

25.70000000 
25.26721311 
0.43278689 
23.87460041 
26.65982582 

26.20000000 
26.53114754 
0.33114754 
25.44573423 
27.61656085 

26.60000000 
26.53114754 
0.06885246 
25.44573423 
27.61656085 

27.40000000 
26.53114754 
0.86885246 
25.44573423 
27.61656085 
* 
. 
26.18073770 
. 
24.88679308 
27.47468233 

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 41.2.2. 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 41.2.3
Additional Results of Requesting the P and CLM Options
0.00000000 
2.86852459 
0.00000000 
23.18107335 
0.54376613 
2.94425592 
The last portion of the output listing, shown in Output 41.2.3, 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 statistic, which measures firstorder autocorrelation, are also given.
Output 41.2.4
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 41.2.4 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.