Prescott, P. and Mansson, R.A.
Efficiency of pair-wise treatment comparisons in incomplete block experiments subject to the loss of a block of observations.
Communications in Statistics: Theory and Methods, 31, (3), . (doi:10.1081/STA-120002858).
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The robustness of incomplete block designs to the loss of all observations in a block is investigated in terms of the efficiency of the residual design. Previous results use the eigenvalues of the information matrix of the treatment effects to determine the overall efficiency of the implemented design, relative to the complete design, in terms of the average variance of pair-wise treatment comparisons. Here we use a simple generalized inverse of this information matrix to identify the variances of the individual pair-wise treatment comparisons and show the effects of the loss of a block on specific treatment comparisons. We determine results for balanced incomplete block (BIB) designs and Youden square designs and show that, although the overall efficiency remains high when a block is lost, comparisons of two treatments that appear in the missing block can be quite seriously affected. The efficiencies of individual treatment comparisons in a BIB design are shown to depend on the number of treatments, the number of blocks used in the initial design and the number of treatments in a block. However, we also show that, for a single replicate of a Youden square, the efficiencies of individual treatment comparisons depend only on the size of the block that is lost and not on the number of treatments being compared.
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