A Comparison of Super-Valid Restricted and Row–Column Randomization

Abstract

In complete block experiments, treatments are often randomized within blocks without any other restrictions. When the blocks are rows of plots and the blocks are laid out in parallel so that also columns of plots are formed, there might be random effects of both rows and columns. In this situation, a row–column design is a natural choice. Super-valid restricted randomization is another option. This article compares these randomization procedures for small complete block experiments (5–10 treatments in 3–6 blocks). Validity of a randomization procedure is defined for mixed-effects models. The two randomization procedures are compared with regard to average variance in pairwise comparisons. Row–column randomization is recommended when either there are at least four replicates, or the number of replicates is three and intercolumn variance is not known to be small. These conclusions assume a model with fixed effects of treatments and random effects of rows and columns, and estimation using the REML method and the Kenward and Roger approximation.

Appendix

Let A and B be any two treatments. According to Azaïs et al. (1993), a randomization procedure is strongly valid (Definition 2) for a complete block experiment with n treatments if the following two conditions are satisfied:

• If plots s and t are two arbitrarily selected plots in the same block, then the probability is $1/(n(n-1))$ that treatment A is randomized to plot s and treatment B to plot t.

• If plots s and t are two arbitrarily selected plots in different blocks, then the probability is $1/n^2$ that treatment A is randomized to plot s and treatment B to plot t; and the probability is $1/n^2$ that treatment A is randomized to both plot s and plot t.

In the following, it is shown that super-valid restricted randomization is strongly valid, since the above two conditions are met. Let $z = C$ denote the event that treatment C is randomized to plot z, and let $z \ne C$ denote the event that C is not randomized to z. First, let plots s and t be two distinct arbitrarily selected plots in the same block. Then $\text {Pr}[(s = A)\cap (t = B)] = \text {Pr}[s = A] \, \text {Pr}[(t = B)|(s = A)] = (1/n)(1/(n-1)) = 1/(n(n-1))$. Next, let plots s and t be two arbitrarily selected plots in different blocks. Then $\text {Pr}[(s = A)\cap (t = B)] = \text {Pr}[(s=A)\cap (t = B)\cap (t \ne A)] = \text {Pr}[s = A] \, \text {Pr}[(t \ne A)|(s = A)] \, \text {Pr}[(t = B)|((s = A)\cap (t \ne A))] = (1/n)(((n - 1))/n)(1/(n-1)) = 1/n^2$, and $\text {Pr}[(s = A)\cap (t=A)] = \text {Pr}[s = A] \text {Pr}[(t = A)|(s = A)] = (1/n)(1/n) = 1/n^2$.