Designs for fitting response surfaces in agricultural experiments

Abstract

Response surface methodology is used (i) to determine and to quantify the relationship between the response and the levels of quantitative factors and (ii) to obtain optimum combinations of levels of various quantitative factors. Response surface methodology has been extensively used in industrial experimentation but appears to be not so popular in experiments pertaining to agricultural, horticultural and allied sciences. This may be due to the fact that the experimental situations in agricultural sciences are different from those in industrial experiments. Broadly five distinctions viz. (i) time and factor range (ii) factor levels (iii) blocking (iv) accuracy of observations’ (v) shape of response surface are identified. In the present investigation, an attempt has been made to remove these bottlenecks so as to make response surface designs suitable for the agricultural experimentation.

Response surface designs for response optimization and slope estimation when various factors have equispaced levels for both symmetric and asymmetric factorial have been obtained. A new characterization property that minimizes the average variance of the predicted response to a reasonable extent has been introduced. The designs satisfying this property have been termed as modified second order response surface designs. Catalogues of the modified and/or rotatable and group divisible rotatable second order response surface designs with number of factor (v) and number of design points (N) satisfying 3<=v<=10 and N<=500 have been developed. To deal with the situations, requiring control of variability in the experimental units, a catalogue of second order rotatable designs with orthogonal blocking for 3<=v<=8 factors each at 3 or 5 equispaced doses along with block contents has been prepared.

The robustness aspects of modified and/or rotatable second order response surface designs for response optimization have been investigated with emphasis on the designs in which various factors have equispaced doses. A new criterion of robustness viz. percent loss in information is introduced. Other criteria used in this investigation are information contained in an observation, D-efficiency and A-efficiency of the resulting design. The response surface designs for qualitative-cum-quantitative experiments have been obtained using response surface designs for various factors with equispaced doses, asymmetric rotatable designs, cylindrically rotatable designs and second order response surface designs with orthogonal blocking.

A comprehensive account of response surface methodology has been given. A computer software has been developed and Statistical Analysis System (SAS) and Statistical Package for Social Sciences (SPSS) codes have been written for fitting of response surfaces in agricultural experiments both with and without intercept, performing canonical analysis and exploration of the response surface in the vicinity of stationary point. The SAS and SPSS codes have been used in the analysis of data of All India Co-ordinated Research Project on Energy Requirement in Agricultural Sector.

One important feature of this study is that some designs obtained during the present investigation have actually been used in the National Agricultural Research System. The experimental situations along with the design are also given for the benefit of the users. This clearly indicates that response surface methodology can be gainfully employed in Agricultural and food processing experiments.