## An Experimental Design for Three Factors at Three Levels |

** Journal/Book: **(Reprinted from Nature Vol. 181 pp. 209-210 January 18 1958) Printed in Great Britain by Fisher Knight & Co. Ltd. St. Albans. 1958;

** Abstract: **American Cyanamid Co. Stamford Connecticut. IN an experiment with three factors at three levels each the experimenter may be willing to sacrifice information an certain components of the two-factor interactions and to ignore the three-factor interactions completely except as an estimate of error1. A possibly useful experimental design for this purpose is detailed below. The experimental arrangement permits independent estimation of and tests of significance an both the ´linear' and ´quadratic' components of the main effects and an the ´linear x linear' components of the three two-factor interactions. As the basic experimental design requires only 16 observations it can be used in ´mixed series' experiments with other factors at two or four levels each in partial replication of the whole (De Baun R. M. unpublished). Moreover quadruplicate replication of one experimental combination permits an estimate of true experimental error. The experimental design and appropriate orthogonal polynomials are detailed in Table 1. We may compare the power of estimation in this design to that of the 27 point 3a factorial design in single replication by taking the variances of the several effects corrected by the ratio of the numbers of observations in each experiment. These variances relative to those of the 3a factorial design are : linear effects 1 333 quadratic effects 0;888 and linear times linear interaction effects 1 x 185. We thus gain information an the quadratic effects while sustaining an information loss an the linear effects and their interactions. The experimental design cannot be further blocked independently of the previously cited effects. However omission of a (111) point permits blocking into three blocks of 5 with the block contrasts partially confounded with the quadratic effects. The basic experimental design can be soon to be a symmetrical fraction of the 3' factorial as is shown in Fig. 1. Geometrically the design may be viewed as a cuboctahedron in single replication with quadruplicate observations at the centre. The analysis of the variance breaks down to nine degrees of freedom for treatments (three linear three quadratic and three interaction) three degrees of freedom for lack of fit that is higher-order interactions and three degrees of freedom for true error if properly randomized. 1 Yates F. lmp. Bur. Soil Sci. Tech. Comm. No. 35 (1937). ___MH

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