Experimental Designs and Analysis of Data for Mixtures

Abstract

The problem of the design and analysis of experiments involving mixtures, where a restriction exists due to the fact that the sum of the proportions must be unity, was, as far as the writer is aware first mentioned in the literature by Claringbold*. Scheffe seems to have been the first to go into the problem in any great detail and he obtained equations for analysing such experiments and also considered some of the ramifications such as process variables, fractionation and restrictions on composition which make it necessary to use pseudocomponents. The equations which Scheffe derived are examined and criticised and their use illustrated by examples from experiments in which the writer has been involved. Scheffe's method of dealing with process variables is also illustrated by a synthetic example. Concentric shell designs recently developed by Doehlert are outlined and their relationship to balanced incomplete blocks, Scheffe's designs and Hadamard matrices is given. Draper and Lawrence have developed more sophisticated designs using techniques worked out by Box and Draper for Minimising variance and bias errors in regression equations. A critical comparison is made between their designs and Scheffe's, Following Box's method of Evolutionary Operation, equations have been derived to enable the technique to be applied to production processes where mixtures are involved. This is illustrated by a Simulated production process and some suggestions are made to overcome a difficulty encountered in this simulation which could easily occur in a real situation. The Appendix contains original tables to enable confidence limits to be calculated when using Scheffe's designs for the special, but usual, case when the replicates are of equal size. * Claringbold,P.J., "The Use of Simplex Designs in the Study of the Joint Action of Related Hormones", Biometrics, 1955, 11(2), 174.