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CHAPTER 8

MULTIPLE REGRESSION

8.4

For each of two models give the following points

Also, give (10 points) for model comparison. Thus the total points on this problem are (80 points).

As an example of how the problem might be solved, two possible models (using centered predictors CPH and CDAYS) are:

Model 1 (ADDITIVE) (Plots in Figures 1 and 2)

o
Fitted model:
$\begin{displaymath} \widehat{LEAD}=15.97-5.02\cdot CPH+0.56\cdot CDAYS.\end{displaymath}$

o
R²=0.5409, R²_a=0.5235, MSE=25.72.

o
The intercept, 15.97, is the predicted lead level at mean PH and DAYS.

o
The change in predicted lead per unit change in CPH is -5.02, and the change in predicted lead per unit change in CDAYS is 0.56.

o
There is a slight curvature in the plot of residuals versus fitted values. Plots of studentized residuals look reasonable.
Model 2 (INTERACTION) (Plots in Figures 3 and 4)

o
Fitted model:
$\begin{displaymath} \widehat{LEAD}=15.97-5.02\cdot CPH+0.56\cdot CDAYS -0.50\cdot CPH\cdot CDAYS.\end{displaymath}$

o
R²=0.6568, R²_a=0.6370, MSE=19.59.

o
The intercept, 15.97, is the predicted lead level at mean PH and DAYS.

o
The change in predicted lead per unit change in CPH is
$\begin{displaymath} \frac{\partial~ \widehat{LEAD}}{\partial~ CPH}=-5.02-0.50\cdot CDAYS,\end{displaymath}$
and the change in predicted lead per unit change in CDAYS is
$\begin{displaymath} \frac{\partial~ \widehat{LEAD}}{\partial~ CDAYS}=0.56-0.50\cdot CPH.\end{displaymath}$

o
Plots of studentized residuals look reasonable.

Model 2 is preferred because of $R^2~R^2_a~\mbox{ and}~ MSE$ .

8.6

R²=0.6568.

(10 points) 65.68% of the variation in LEAD is explained by the model: R²=SSR/SSTO=1949.8326/2968.7347=0.6568.
(10 points) The model reduces the uncertainty in predicting LEAD by 65.68%: R²=1-SSE/SSTO=1-1018.9021/2968.7347=0.6568.

8.8

The VIFs all equal 1.0, so there is no multicollinearity at all. The previous conclusions are unchanged.

**Figure 1:** *SAS/INSIGHT output, additive model, lead data.*
$\begin{figure} \centerline{ \psfig {file=exsoln8_1.eps,height=7in,width=5in} } \vspace{2ex}\end{figure}$

**Figure 2:** *Studentized residual plots, additive model, lead data.*
$\begin{figure} \centerline{ \psfig {file=exsoln8_2.eps,height=3.5in,width=5in} } \vspace{2ex}\end{figure}$

**Figure 3:** *SAS/INSIGHT output, interaction model, lead data.*
$\begin{figure} \centerline{ \psfig {file=exsoln8_3.eps,height=7in,width=5in} } \vspace{2ex}\end{figure}$

**Figure 4:** *Studentized residual plots, interaction model, lead data.*
$\begin{figure} \centerline{ \psfig {file=exsoln8_4.eps,height=3.5in,width=5in} } \vspace{2ex}\end{figure}$

About this document ...

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Joseph D Petruccelli
12/9/1999