Studies in Experimental Design and Data Analysis with Special Reference to Computational Problems

An experimenter, designing or analysing an experiment, frequently finds that not all can be done according to the text-book. Considerations outside his control may have to be taken into account that affect the design; he may not be interested in all his treatments equally but perhaps in some unusual contrasts between them. Again, unforseen circumstances and happenings can afterwards force him to discard some of his results, so upsetting the balance or orthogonality of the design. The aim of this thesis is to help an experimenter in such a situation. The first part is concerned with analysing a block experiment that is in general unbalanced and non-orthogonal. 'Two different methods, one iterative, one non-iterative, are derived for obtaining the analysis, each with its own advantages. The non-iterative method basically is derived from the actual design and produces matrices, which can then operate on any suitable data supplied. The iterative method, however, found in appendix A, is applied directly to the data from the start, to produce the treatment effects directly. Although the iterative method is easier to apply and can also be used with a wider class of design than can the non-iterative method, the inter-block analysis and the analysis of the dual become easier using the non-iterative method. Certain contrasts are related to the design in special ways, and, if known, make the analysis of the design easier. The implications are discussed in chapter 2, which is also concerned with finding the contribution to the sum of squares for these and other, more general, contrasts of interest. The dual of a design is defined as that design formed from the original design when treatments and blocks classifications are interchanged, i.e. treatments become blocks and vice-versa in the new design. It is useful for studying block differences eliminating those due to treatments, which may, for example, be required if the blocking system arose from the possible residual effects of treatments from some previous experiment on the same material. The second part of the thesis, in chapter 3> is concerned with the analysis of the dual. It is shown that there is no need to start again from the beginning when analysing the dual if the original design had already been analysed, because the analysis can provide information about that of the dual. The method is especially easy when the non-iterative method of analysing block designs, discussed in chapter 2, has been used for the original design. The experimenter will often be more interested in some contrasts between treatments than in others and a design can be selected to give more precise information about these contrasts. The construction of such designs is discussed in the third part of the thesis. Various measures can be used to judge which design is best as regards contrasts of interest. Algorithms for finding the optimal design according to these measures are derived and discussed in chapter 4. Listings and flowcharts of a program to carry out the non-iterative analysis of chapter 2 and of a program to construct optimal block designs appear in appendices B and C.