© 1970 by Biometrika Trust
Sample size requirements: randomized block designs
Oak Ridge National Laboratory
Computing Technology Center
Now at the Tobacco Institute, Washington, D.C.
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 
= 0.01, 0.05 and ß = 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 and ß 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
yijl = µ + li + bj+(tb)ij + 
ijl'
where ti is the ith treatment effect, bj is the jth block effect, (tb)ij is the interaction effect of treatment i with block j and ijl is the effect due to the lth observation on the ith treatment in block j (i = 1,..., k; j = 1,..., b; l = 1,..., N). Here the ijl are assumed to be independent normal variables with mean zero and variance 
2.
In this situation the sample size, N, necessary to test the absence of a treatment effect(ti = 0 for all i = 1,..., k) for specified values of k, b, and ß 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, , ß, v1 = k – 1 and v2 = 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 = (tmax-tmin)/.
It can easily be shown that T ø{(2k)/(bN)}1/2
with equality if and only if ti = 0, 
ti = 0, for all ti other than tmsx and tmin. Maximum values of the standardized range of the treatment means, ø{(2k)/(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 and ß levels of risk, in a randomized block design containing b blocks and N observations per cell, for a = 0.01, 0.05; ß = 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.