Asymmetrical response surface designs

Abstract

After the introduction of rotatable design by Box and Hunter (1957) considerable amount of study has been made in the theory of fitting multifactor response surfaces. But till now only designs of the symmetric type i.e. designs with factors all with the same number of levels have been studied. In this thesis we have suitably modified the idea of rotatability and presented two series of asymmetric designs together with a design in four factors which gives rise to an infinite series of designs. some of these designs are found to be yielding more accurate estimates of the response at particular points than that obtainable through rotatable designs. These designs are also available with lesser number of design points. Along with the designs the expressions for estimates of various regression coefficients and their variance has also been presented. To use such designs for agricultural experimentation it is necessary that they are split up into blocks of equal size. In this thesis we have given a number of designs belonging to the asymmetric type with blocking. Along with estimates of regression coefficients and their variances, the method of analysis in case of one or more replications has also been described. The problem of analysis of the design through estimation of a missing value in response surface designs has been considered by Draper (1961). A slightly different approach through minimising the lack of fit component of s.s has been adopted and the estimate of one missing value in case of 3 response surface designs has been given. A comparison of the variances of estimated response at various points when the data are analysed (1) after substitution of missing value and proceeding further as usual and (2) taking into account all consequences of the missing observation, has been made and it has been found that there is considerable amount of difference in the variances of any estimated response.