Example 30.2: Regression with Mileage Data
A car is tested for gas mileage at various speeds to determine
at what speed the car achieves the greatest gas mileage.
A quadratic model is fit to the experimental data.
The following statements produce
Output 30.2.1 through Output 30.2.4:
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
;
proc glm;
model mpg=mph mph*mph / p clm;
output out=pp p=mpgpred r=resid;
axis1 minor=none major=(number=5);
axis2 minor=none major=(number=8);
symbol1 c=black i=none v=plus;
symbol2 c=black i=spline v=none;
proc gplot data=pp;
plot mpg*mph=1 mpgpred*mph=2 / overlay haxis=axis1 vaxis=axis2;
run;
Output 30.2.1: Standard Regression Analysis Output from PROC GLM
|
| Gasoline Mileage Experiment |
| NOTE: |
Due to missing values, only 7 observations can be used in this analysis. |
|
|
|
| Gasoline Mileage Experiment |
| The GLM Procedure |
| Dependent Variable: mpg |
| 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 R2 of 0.97.
The table of parameter estimates indicates that
the estimated regression equation is
Output 30.2.2: Results of Requesting the P and CLM Options
|
| Gasoline Mileage Experiment |
| 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 |
| * Observation was not used in this analysis |
|
The P and CLM options in the MODEL statement produce the table shown
in Output 30.2.2.
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 30.2.3: Additional Results of Requesting the P and CLM Options
|
| Gasoline Mileage Experiment |
| 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 final portion of output gives some
additional information on the residuals.
The Press statistic gives the sum of squares of
predicted residual errors, as described in Chapter 3, "Introduction to Regression Procedures."
The First Order Autocorrelation and the Durbin-Watson
D statistic, which measures
first-order autocorrelation, are also given.
Output 30.2.4: Plot of Mileage Data
Output 30.2.4 shows the actual
and predicted values for the data. The quadratic relationship
between mpg and mph is evident.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.
