Classification and Analysis of Linked Block Designs

Abstract

In comparative experiments involving a fairly large number of varieties, when for lack of homogeneous experimental units, complete block designs are not available. Balanced Incomplete Block (BIB) designs prove very convenient in two respects. Firstly, the analysis is very simple and secondly, comparison between any two varieties has the same precision. On the other hand, a BIB design requires a large number of experimental units, because balancing is not possible unless the number of blocks is atleast as large as the number of varieties. To obviate this dif ficulty, different types of incomplete block designs have been introduced, of which the most notable is the Partially Balanced Incomplete Block (PBIB) design introduced by Bose and Nair (1939). Another class of incomplete block designs, called Linked Block (LB) designs was introduced by Youden (1951) which he obtained by dualising several BIB designs, that is by taking the varieties and blocks of the BIB design respectively as blocks and varieties in the LB design. The LB designs so constructed by Youden happen to be all PBIB designs, but this is not necessarily true. A great advantage of LB designs is that the analysis can be easily worked out. In this paper, we show how the intra-blockTanalysis of a LB design can be neatly carried out. The efficiency factor of a LB design is obtained. The methods are illustrated with a numerical example. An exhaustive list of all LB designs with ten or less plots per block and involving ten or less replications is given. These designs fall into three groups: (1) symmetrical BIB designs, (2) PBIB designs and (3) Irregular designs For plans of LB designs belonging to group (2) reference is made to the serial number of the two-associate class PBIB designs enumerated by Bose, Clatworthy and Shrikhande (1954). Plans for other designs belonging to groups (2) and (3) are given in detail. Certain theorems are derived which are useful in determining from the parameters of a given two-associate class PBIB design whether the design is of the LB type or not.