Suppose there is a systematic effect or trend that influences the observations in addition to the block and treatment effects. The problem of experimental designs in the presence of trends was first studied by Cox (1951,1952). Bradley and Yeh (1980) define the concept of trend-free block designs, i.e., the designs in which the analysis of treatment effects are essentially the same whether the trend effects are present or not. If the trend effect within each blocks are the same and linear, Yeh and Bradley (1983) derive a simple necessary condition for designs to be linear trend-free,
　　r_i(k+1)≡0 (mod 2), 1≦i≦v,　　　　　(1)
　　where r_i is the replication of treatment i, for 1≦i≦v, and k is block size.
　　In case where a trend-free version does not exist Yeh et al. (1985) suggest the use of “ nearly trend-free version”. Chai (1995) pays attention to situations where (1) does not hold. He also shows that often, under these circumstances, a nearly linear trend-free design could be constructed.
　　Designs that are derived by extending or deleting m disjoint and binary blocks from BIBD (v,b,k,r,λ)'s are considered. If the resulting designs have linear trend-free versions, by Constantine (1981), they are E-optimal designs with the corresponding classes. When k is even, however, it is impossible to have linear trend-free versions since not all the r_i's are even in such type of designs and (1) is violated. In this paper, we shall convert the designs to be nearly linear trend-free versions of them by permuting the treatment symbols within blocks, and investigate that the resulting designs remain to be E-optimal.

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