394 Chapter 9: Regression
REMARK
In matrix notation Equation 9.9.1 can be written as
1,291.1
9,549.3
77,758.9
=
10 55 385
55 385 3, 025
385 3,025 25,333
B 0
B 1
B 2
which has the solution
B 0
B 1
B 2
=
10 55 385
55 385 3,025
385 3,025 25,333
− 1
1,291.1
9,549.3
77,758.9
*9.10Multiple Linear Regression
In the majority of applications, the response of an experiment can be predicted more
adequately not on the basis of a single independent input variable but on a collection of
such variables. Indeed, a typical situation is one in which there are a set of, say,kinput
variables and the responseYis related to them by the relation
Y=β 0 +β 1 x 1 +···+βkxk+ewherexj,j=1,...,kis the level of thejth input variable andeis a random error that
we shall assume is normally distributed with mean 0 and (constant) varianceσ^2. The
parametersβ 0 ,β 1 ,...,βkandσ^2 are assumed to be unknown and must be estimated
from the data, which we shall suppose will consist of the values ofY 1 ,...,YnwhereYiis
the response level corresponding to thekinput levelsxi 1 ,...,xi 2 ,...,xik. That is, theYi
are related to these input levels through
E[Yi]=β 0 +β 1 xi 1 +β 2 xi 2 +···+βkxikIf we letB 0 ,B 1 ,...,Bkdenote estimators ofβ 0 ,...,βk, then the sum of the squared
differences between theYiand their estimated expected values is
∑ni= 1(Yi−B 0 −B 1 xi 1 −B 2 xi 2 −···−Bkxik)^2The least squares estimators are those values ofB 0 ,B 1 ,...,Bkthat minimize the foregoing.
* Optional section.