On investigation of response surfaces

Abstract

Of major importance to statistical designs of experiments involving quantitative variables or factors is the detection and description of the functional relationship between two or more variables – the relationship such as exists between yield or response on one hand and the levels of different quantitative variables on the other. An empirical investigation of such a relationship requires in taking observations at predetermined levels of the controllable variables, i.e. an experimental design must be selected prior to experimentation. An attempt has been made first to study such response surfaces from the data collected on the usual factorial experiments in agriculture. A method has been evolved for fitting a second order (i.e. a quadratic) surface by expressing the different coefficients involved in this surface as functions of main effects and two factor interactions using data from complete, confounded, and fractionally replicated designs of 3k series. The approach adopted in quite general but for clarity and appreciation, the discussion has been restricted to designs of 3k series. A numerical example for a 32 factorial is also given. Box and Hunter (1957) advanced the criterion of “rotatability” to experimental designs for fitting response surfaces. Such designs permit a response surface to be fitted easily and provide spherical information contours. A second order rotatable design aids the fitting of second order (i.e. a quadratic) surface. Draper (1960) obtained infinite series of second order rotatable designs in four or more factors. These designs contain excessively large number of experimental points. It has been possible, through the present investigation to obtain infinite series of second order rotatable designs in four, five, six, and seven factors with considerably small number of experimental points. A general method for obtaining such designs for any even number of factors has also been indicated. Gardiner et. al. (1957) extended the criterion of rotatability to experimental design for fitting a third order (i.e. a cubic) surface. They derived sequential third order rotatable designs up to four factors. No attempt was made by them to obtain designs in five and six factors, chiefly because the approach pursued required excessively large number of experimental points. It has been possible, through the present investigation, to obtain sequential third order rotatable designs up to eleven factors. These designs possess the desirable property of having small number of experimental points. Non-sequential third order rotatable designs in two and three factors are also derived.