ABSTRACT

In Chapter three the analysis of variance is shown to be a method by which factorial experiments, as introduced in Section 7.1, can be analysed, provided all the factors are qualitative. If all the factors are quantitative, we can regard them as regressors in a Model I multiple regression. This topic is considered in Chapter eight. In applications, factorial experiments with quantitative factors are mostly used to search for the optimum on a response surface. The description of such non-linear relationships can be achieved most simply by a quadratic function of the regressor variables. In this chapter we shall describe the design and analysis of such experiments. See also Myers (1976) and Khuri and Cornell (1987). This chapter combines the ideas from Sections 8.1.1 and 8.2.2. We assume that we have more than one regressor variable and a quadratic regression function. In the case of p = 2 regressors (x1, x2), the complete regression model takes the form (with i.i.d. error terms ej with Eej , V ar(ej) = σ2 for all j):

yj = β0 + β1x1j + β2x2j + β11x 2 1j + β22x

2 2j + β12x1jx2j + ej (9.1)

or for p factors the form

yj = β0 + p∑ i=1

βixij + p∑ i=1

βiix 2 ij +

βilxijxlj + ej ; j = 1, . . . , n (9.2)

or

y = Xβ + e (9.3)

In (9.3) we have j = 1, . . . , n > 1 + p(p+ 3)/2 and

βT = (βa, β1, . . . , βp, β11, . . . , βpp, β12, . . . , βp−1p) and the other vectors as defined in Chapter eight. If any of the β’s on the right hand side of (9.2) are negligibly small, a reduced model can be employed having the corresponding terms omitted. Model (9.1) without the term β12x1jx2j is called a pure quadratic model in x1 and x2 (without an interaction term x1x2). The analysis proceeds as described in Section 8.1.1; we merely adapt X and the vector β accordingly. The expectation E(yj) of yj in (9.2) is called the expected quadratic response surface. Equation (9.2) may be written in an alternative matrix version as

E(yj) = β0 + x T j Axj +B

Txj (j = 1, 2, . . . , n) (9.4)

where xj = (x1j , x2j , . . . , xkj)T is a vector of the k factor-variables for the j-th observation; A is the symmetric k × k matrix of non-linear regressionparameters

A =

 β11 . . . 1 2β1p

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