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He uses statistics as a drunken man uses lamp-postsfor support rather than for illumination.
Andrew Lang
The simple linear regression models discussed so far are restricted in three ways. First, only one predictor variable is allowed. Second, the predictor variables must be quantitative variables. Third, the response must be a linear function of the predictors. Sometimes, these restrictions cannot be met. In some of these cases the regression technique can still be used with a slight modification of the problem. Such special cases are the topic of this chapter. In particular, we describe techniques to handle the following cases:
In addition, the problem of outliers is discussed and a list of common mistakes in regression is presented in this chapter.
A multiple linear regression model allows one to predict a response variable y as a function of k predictor variables x1, x2,...,xk using a linear model of the following form:
y = b0 + b1x1 + b2x2 + ... + bkxk + e
Here, {b0,b1,...,bk} are k + 1 fixed parameters and e is the error term.
Given a sample {(x11,x21,...,xk1,y1),...(x1n,x2n,...,xkn,yn)} of n observations, the model consists of the following n equations:
y1 = b0 + b1x11 + b2x21 + ... + bkxk1 + e1
y2 = b0 + b1x12 + b2x22 + ... + bkxk2 + e2
.
.
.
yn = b0 + b1x1n + b2x2n + ... + bkxkn + en
In vector notation, we have:
or
y = Xb + e
where
Notice that all elements in the first column of X are 1.
The analysis of multiple regression is summarized in Box 15.1. Most of the expressions are similar to those for simple regression and are illustrated by the following example. The only considerations that have not been covered under simple linear regression are those of analysis of variance and the problem of multicollinearity. These are discussed in the sections following the example.
Box 15.1 Multiple Linear Regression
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Example 15.1 Seven programs were monitored to observe their resource demands. In particular, the number of disk I/Os, memory size (in kilo-bytes), and CPU time (in milliseconds) were observed. The data is shown in Table 15.1.
TABLE 15.1 Data on CPU Time, Disk I/Os, and Memory Size | ||
---|---|---|
CPU Time, yi | Disk I/Os, x1i | Memory Size, x2i |
2 | 14 | 70 |
5 | 16 | 75 |
7 | 27 | 144 |
9 | 42 | 190 |
10 | 39 | 210 |
13 | 50 | 235 |
20 | 83 | 400 |
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