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.

Date of Award	1972
Original language	English