Sample size requirements: randomized block designs

Abstract

SUMMARY Techniques described in an earlier paper (Kastenbaum, Hoel Bowman, 1970), are used to generate tables of sample size requirements for testing treatment effects in a randomized block design. Maximum values of the standardized range of the treatment means are tabulated for k = 2, 3, 4, 5, 6 treatments; b = 2, 3, 4, 5 blocks; 1 ≤ N ≤ 5 observations per cell; and α = 0.01, 0.05 and β = 0.005, 0.01, 0.05, 0.1, 0.2, 0.3 levels of risk. To simplify the determination of sample sizes necessary in a one-way analysis of variance, we have presented (Kastenbaum, Hoel & Bowman, 1970) tables of maximum values of the standardized range which apply when the means of k groups, each containing N * observations, are being compared at α and β levels of risk. Use of the standardized range in this situation was first elaborated on by Pearson & Hartley (1951), who also proposed extensions of the procedure to the randomized block design. Their model for the double classification with N observations in each cell is y ijl = μ + l i + b j +(tb) ij + ɛ ijl' where t i is the ith treatment effect, b j is the j th block effect, ( tb ) ij is the interaction effect of treatment i with block j and ɛ ijl is the effect due to the l th observation on the i th treatment in block j ( i = 1,…, k ; j = 1,…, b ; l = 1,…, N ). Here the ɛ ijl are assumed to be independent normal variables with mean zero and variance σ 2 . In this situation the sample size, N , necessary to test the absence of a treatment effect( t i = 0 for all i = 1,…, k ) for specified values of k , b , α and β may be achieved through an iterative process involving Pearson and Hartley's charts of the power function. These charts are given in terms of Tang's noncentrality parameter ø, α, β, v 1 = k − 1 and v 2 = bk ( N − 1). Table 1 provides direct answers to this question without requiring iteration. For the randomized block design the standardized maximum difference between any two treatment means is T = (t max -t min )/σ . It can easily be shown that T≤ ø{(2k)/(bN)} 1/2 with equality if and only if t i = 0, Σt i = 0, for all t i other than t msx and t min . Maximum values of the standardized range of the treatment means, ø{(2 k )/( bN )} ½ , were calculated as described by Kastenbaum et al . (1970) and are tabulated for situations in which k treatment means are being compared at α and β levels of risk, in a randomized block design containing b blocks and N observations per cell, for aα = 0.01, 0.05; β = 0.005, 0.01, 0.05, 0.1, 0.2, 0.3; k 2(1)6; b = 2(1)5; 1 ≤ N ≤ 5. More extensive tables may be obtained from the authors.