The “Symmetrical Complementation Design” which is discussed in this dissertation is intended for those experimental situations where the levels of three factors always sum to the same constant. The levels of the factors, if referred to a common unit of measurement, must be equally spaced. Certain cell entries are omitted to ensure complete interchangeability of the three factors. The usual additive model is assumed.

A detailed study about the types of functions which are estimable in this design is presented in the chapter on linear estimation. The study shows that the number of contrasts that can be estimated is limited. For example, the usual linear contrast, which would lead to the hypothesis of equality of effects of all levels of one factor, is not testable in this design. On the other hand, quadratic and higher-order contrasts are estimable for each factor separately. These contrasts are combined into different hypotheses. Estimable functions in one factor only and in two factors are presented for the general case of p levels.

There are several methods which can be employed in order to obtain estimates of the treatment effects under various constraints. It must be noted, however, that these estimates are rather meaningless quantities. It is only when they are combined in estimable functions that unique results are obtained. Two methods are described in complete detail; the “high-low” method if only estimation is required, and the "modified high-low” method if both estimation and tests of hypotheses are required. The complete inverse matrix required for this latter method, or a method of obtaining this patterned inverse, is presented.

For testing hypotheses, a general technique, based upon the inversion of the matrix in the modified high-low method, is presented. Sums of squares and test statistics are presented for the various hypotheses formulated. Sections are also included which indicate how one might obtain the response for intermediate levels of the factors, and how one might obtain response functions for single factors.

A chapter on extensions is presented, where n observations are available per treatment combination. In this connection, three different cases are considered;

a) the replications are strictly repetitions of the experiment under otherwise identical conditions. In this case, the analysis proceeds in the customary three-way analysis with n replicates per treatment combination;

b) the experiments within a cell represent repetitions over a period of time, during which some kind of trend may be present. In this case the analysis is readily extended into an analysis of covariance;

c) the experiments within a cell represent several experiments with the same experimental units, so that the observations within a cell are dependent. On the assumption that the covariance matrix of observations in a cell is the same for every cell, a multivariate analysis can be performed.

The problem of estimation is essentially the same in these methods. However different methods are necessary for the testing of hypotheses. Special discussion is also presented for the case where the levels of the factors are not equally spaced; and the case where the model is considered as a mixed model.

In conclusion, it has been found that this type of design requires a rather careful consideration of the types of functions that can be estimated and the types of hypotheses that can be tested. Recommendations for interpretation and statement of limitations are made in detail.