Efficiency of experimental designs for estimating common-row-and-column (CRAC) effects
Authors: Connolly J. 
Published: March 2009

Two factor designs are considered in which the levels of the two factors are linked (e.g. certain diallel cross and reciprocal transplant experiments). A common-row-and-column (CRAC) effect is defined as a systematic effect in the response when the treatment combination has the same level of the two factors. The efficiency of designs for estimating CRAC effects is discussed. Square designs with various levels of extra replication for CRAC treatment combinations and strand designs, based on omitting certain treatment combinations for which the two factors do not have the same level of the two factors, are defined and their efficiency examined.

In the design for optimal estimation of a single CRAC effect in square or strand designs, half the total replication is in CRAC treatment combinations and half in the other combinations. This gives a major increase in design efficiency, particularly where the number of levels of the factors is five or greater. Designs with equal replication of all treatment combinations are generally very inefficient. The optimal square and strand designs have equal efficiency for estimating the CRAC effect, but the strand designs are more flexible in practice, requiring fewer plots to make a complete block and this relative flexibility increases with the number of levels of the factors.

Most of the square or strand designs allow the estimation of n separate CRAC effects where n is the number of levels of the factors. Optimal square designs for estimation of separate CRAC effects give replication of CRAC combinations as ((n–1)(n–2)/2)½ times that of other combinations, which is different from the optimum value for estimating the single CRAC effect. However, for many designs, the difference is not of major importance because optimal designs for the estimation of a single CRAC effect are relatively insensitive to small deviations from the optimal ratio.

When the replication of CRAC combinations differs from that of the other treatment combinations, the designs are less efficient in estimating the factor effects. Where estimation of factor effects is also of relevance, a trade-off between increased efficiency in the estimation of CRAC effects and of factor effects may be desirable.

Strand designs are not as efficient as the optimal square design for estimating separate CRAC effects. For three-strand designs of even order examined, the CRAC effects are confounded with factor effects and are not all separately estimable. When factor effects and/or separate CRAC effects are of interest, square are preferable to strand designs.

Some of the results generalize readily to three linked factors. There is a three-factor CRAC effect and three two-factor CRAC effects. The replication allocation for CRAC relative to other combinations, to optimize the estimation of the three-factor CRAC effect is (n–1)(n–2)/2 for the three-factor CRAC combinations and (n–2)/2 for the two-factor CRAC combinations. Selection to optimize the estimation of the two-factor CRAC effects gives very different optimal allocation.