Composite designs based on irregular fractions of factorials.

TLDR: The examination of designs for the estimation of polynomials in which the design is as 'nearly-saturated' as possible is suggested, to center, almost exclusively, on the variances of the predicted responses over the relevant region of factor space.

Abstract: A situation which is frequently encountered in experimental work is that in which a multi-response system is to be explored with the minimum of experimental work. In research and exploratory work, in particular, it is often necessary to ascertain whether there is any region of the factor space for which certain inequalities on the responses hold. This requirement almost inevitably implies polynomial estimation (responses as functions of the factors) together with some suitable graphical display such as that provided by contour plots. Often there is a severe limitation on the amount of experimental work that can be undertaken, although more will certainly be performed as a check if the initial experimentation suggests that a desirable region of factor space appears to exist. The foregoing comments suggest the examination of designs for the estimation of polynomials (specifically, quadratic polynomials in this paper) in which the design is as 'nearly-saturated' as possible. The possibility of providing tests on individual coefficients of the polynomial will not be required of such designs, since their main function is to allow estimation of the coefficients of the polynomial; this polynomnial can then be used as an interpolating function over the appropriate region of factor space. In such cases, our interest will center, almost exclusively, on the variances of the predicted responses over the relevant region of factor space. The general quadratic in n factors is: