Influence of the experimental design

Each element of the X X) matrix is a summation of products (see Appendix A). A common algebraic representation of the X X) matrix for the straight-line model y, - Po + PiXi,. + r, is 

Let sIq be the estimated variance associated with the parameter estimate bo let be the estimated variance associated with b and let slf (or sl ) represent the estimated covariance between and b,. Then 

From Equation 7.3 it is evident that the experimental design (i.e., the level of each Xj,) has a direct effect on the variances and covariances of the parameter estimates. 

The effect on the variance-covariance matrix of two experiments located at different positions in factor space can be investigated by locating one experiment at X, = 1 and varying the location of the second experiment. The first row of the matrix of parameter coefficients for the model y,- = + p,jc, + r, can be made to 

Let us now locate a series of second experiments from x 2 = -5 to x 2 - +5. The X matrix for this changing experimental design may be represented as 

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