Monograph on alpha designs

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 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 experiment, for example, 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 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 may find it difficult to lay their hands on these 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 for 6 <=v(= sk) <= 150, 2 <= r <= 5, 3 <= k <= 10 and 2 <= s <= 15 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 Partially Balanced Incomplete Block (PBIB (2)) designs and the alpha-designs obtainable from basic arrays given by Patterson et al. (1978). 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 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(1973).