Significant effects in screening experiments

Assume that two variables, Xj and X2, have been studied in a 2 factorial design. Assume also that the true response is constant, y = a, in the experimental domain, i.e. the response surface is completely flat and the variables do not influence the 

If the parameters of the model are computed using the model matrix and the 

Factorial and fractional factorial designs are balanced each colunm contains an equal number of (-) and (+) settings of the variables. Due to this, a constant response will be cancelled out when the effects are computed. The estimated effects in such cases will be nothing but different averaged summations of the experimental error. If we assume that the experimental error is normally distributed, the averaged sums will also be normally distributed. 

Let us now have a look at general screening experiments with many variables. Assume that k variables (xj, X2. xJ have been studied by a fractional factorial design and that a response surface model with linear and cross-product interaction terms has been determined. 

If the variables do not have any influence whatsoever on the response, the response surface is completely flat and the true value of all BpS and Byzs is zero. The estimated effects in such cases would be nothing but different average summations of the experimental error. If we have randomized the order of execution of the experiments in the design, and have done all what we can do to avoid systematic errors, the set of estimated model parameters, [ft, b2.b, would be a 

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