Efficient Design of Experiments for Quality Agricultural Research : Part I

Abstract

Alpha designs are essentially resolvable block designs. In a resolvable block design, the blocks can be grouped such that in each group, every treatment appears exactly once. Resolvable block designs allow performing an experiment with one replication at a time. For example, field trials with large number of crop varieties cannot always be laid out in a single location or a single season. Therefore, it is desired that variation due to location or time periods may also be controlled along with controlling within location or time period variation. This can be handled by using resolvable block designs. Here, locations or time periods may be taken as replications and the variation within a location or a time period can be taken care of by blocking. In an agricultural field experiment, the land may be divided into a number of large areas corresponding to the replications and then each area is subdivided into blocks. These designs are also quite useful for varietal trials conducted in the National Agricultural Research System (NARS) and will help in improving the precision of treatment comparisons. A critical look at the experimentation in the NARS reveals that alpha designs have not found much favour from the experimenters. It may possibly be due to the fact that the experimenters find it difficult to lay their hands on alpha designs. The construction of these designs is not easy. An experimenter has to get associated with a statistician to get a randomized layout of this design. For the benefit of the experimenters, a comprehensive catalogue of alpha designs upto 150 treatments has been prepared along with lower bounds to A- and D- efficiencies and generating arrays. The layout of these designs along with block contents has also been prepared. The designs obtained have been compared with corresponding square lattice, rectangular lattice, resolvable two-associate class partially balanced incomplete block (PBIB (2)) designs and the -designs obtainable from basic arrays given by Patterson, Williams and Hunter (1978, J. Agric. Sci., 90, 395-499). Eleven designs are more efficient than the corresponding resolvable PBIB (2) designs (S11, S38, S69, S114, LS8, LS30, LS54, LS76, LS89, LS126 and LS140). It is interesting to note here that for the PBIB (2) designs based on Latin square association scheme, the concurrences of the treatments were 0 or 2 and for singular group divisible designs the concurrences are either 1 or 5. Further all the designs LS8, LS30, LS54, LS76, LS89, LS126 and LS140 were obtained by taking two copies of a design with 2-replications. 10 designs were found to be more efficient than the designs obtainable from basic arrays. 48 designs (29 with k = 4 and 19 with k = 3) are more efficient than the designs obtainable by dualization of basic arrays. 25 designs have been obtained for which no corresponding resolvable solution of PBIB(2) designs is available in the literature. The list of corresponding resolvable PBIB(2) designs is S28, S86, SR18, SR41, SR52, SR58, SR66, SR75, SR80, R42, R70, R97, R109, R139, T14, T16, T20, T44, T48, T49, T72, T73, T86, T87 and M16. Here X # denotes the design of type X at serial number # in Clatworthy, W. H. (1973, Table of two-associate partially balanced designs. NBS Applied Maths Series No. 63. Washington D.C.). In some experimental situations, the user may be interested in getting designs outside the above parametric range. To circumvent such situations, a beta Version of user friendly software module for the generation of -designs has been developed. This module generates the alpha array along with lower bounds to A and D-efficiency. The -array and the design is generated once the user enter the number of treatments (v), number of replications (r) and the block size (k). The module generates the design for any v, k, r provided v is a multiple of k. It also gives the block contents of the design generated. A nested block design is defined as two systems of blocks such that the second system of blocks is nested within the first system of blocks. These designs are quite useful in many experimental situations. For example, consider a field experiment conducted using a block design and harvesting is done block wise. Harvested samples are to be analyzed for their contents either by different technicians at same time or by a technician over different periods of time. The variation due to technicians or time periods may be controlled by another blocking system. Technicians or time periods form a system of blocks that are nested within blocks. Such experimental situations are also common in post harvest value addition of horticultural and vegetable crops. Nested block designs are also quite useful in agricultural field experiments where the plots with similiar fertility occur in patches rather than in a uniform direction. Preece, D.A. (1967 Biometrika, 54, 479-486) was the first to introduce nested block designs and termed them as nested balanced incomplete block (NBIB) designs. In a NBIB design block classification ignoring sub-blocks is a balanced incomplete block (BIB) design and sub-block classification ignoring blocks is also a BIB design. We have prepared a complete catalogue of NBIB designs with number of replications . The catalogue contains a total of 299 designs. Out of 299 designs, 8 designs are non-existent. A new method of construction of NBIB designs has been obtained. Using this method and trial and error solutions, block layouts of 22 new NBIB designs have been obtained. The layout of 199 designs with block contents has been completed. The solution for the block layout for remaining 92 designs is unknown and the statisticians need to develop methods of construction of these NBIB designs. The designs catalogued have also been identified for 1-resolvable and 2-resolvable sets. A NBIB design may not exist for all parametric combinations or even if it exists may require a large number of replications, which the experimenter may not be able to afford. To deal with such situations, nested partially balanced incomplete block (NPBIB) designs have been introduced in the literature. Some new methods of construction of NPBIB designs based on group divisible association scheme have been given using these methods of construction, 31 new NPBIB designs based on group divisible association scheme with r <=15 have been obtained. Nested block (NBIB and NPBIB) designs are useful for experimental situations where the experimenter is interested in making all possible pairwise treatment comparisons with as high a precision as possible. However, there do occur experimental situations where the experimenter is interested in comparing several new treatments (called test treatments) with existing practice (a control treatment) with high precision and the comparisons among the test treatments are not of much importance. To deal with such situations, nested balanced treatment incomplete block (NBTIB) designs have been introduced. Some new methods of construction of NBTIB designs making use of NBIB designs, initial block solutions, etc. have been developed. A new method of construction of nested block designs for making test treatments-control treatment comparisons has been developed which yields minimally connected designs with respect to sub-blocks. The design with respect to bigger blocks is a group divisible treatment design. A new method of construction of efficient block designs for making test treatments-control treatment comparisons by making use of triangular association scheme has been developed. The number of replications of test treatments developed through this method is always 2. A new method of construction of semi-Latin squares based on initial column solution has been developed. This method yields semi-regular group divisible designs after ignoring the row and column classifications. Preece and Freeman (1983, J. Royal. Statist. Soc., 28, 154-163) reported that for could not be obtained by rearrangement in Semi-regular group divisible designs. These three semi-Latin squares can be obtained from the proposed method of construction. A catalogue of block designs with n = v+b-1+i, i=1, 2, 3 observations has been prepared, where v is the number of treatments; b the block size; k is the block size and n is the total number of experimental units. Block contents along with lower bounds to A- and D-efficiencies are also given. The lower bounds to A-efficiencies for making test treatments-control treatment comparisons are being obtained. The design resources server has been initiated and launched on the web site of the Institute. The main objective of this design resources server is to develop a WEB DESIGN in NARS. At present material on binary balanced block designs and designs for making test treatments- control treatment comparisons along with Electronic Book on Design and Analysis of Agricultural Experiments are available on this site. A discussion board has been created. Advisory services have been pursued rigorously in the NARS. Alpha designs, second order response surface designs have been recommended. The sophisticated statistical analytical procedures like contrast analysis and variance component estimation have also been recommended. Fertilizer response ratios were obtained using the data pertaining to an experiment conducted to find out the response of N, P and K under different sub agro-ecological zones/NARP zones under the aegis of Project Directorate of Cropping Systems Research, Modipuram 1999-2000.