A Method of Constrained Randomization for $p_1^{n_1}p_2^{n_2} \cdots p_k^{n_k}$ Factorials

Abstract

The statistical analysis which is carried out in conducting a $p_1^{n_1}p_2^{n_1} \cdots p_k^{n_k}$ factorial experiment in blocked designs requires that the treatment combinations be randomly arranged for each treatment run. When the nature of the process under investigation restricts the number of factor levels which can be changed from treatment combination to treatment combination, the usual technique of full randomization cannot be carried out. This paper presents methods of constrained randomization for $p_1^{n_1}p_2^{n_2} \cdots p_k^{n_k}$ factorial experiments in blocked designs when the requirement on adjacent treatment combinations is that the number, $\Delta$, of factor levels which can be changed is less than $t$, where $t = \sum^k_{i = 1} n_i(p_i - 1)$. If $\Delta = t$ this is ordinary full randomization. The method of constrained randomization contained in this paper requires the construction of an operational sequence in which $\Delta = 1$ for the first and last treatment combinations in the sequence as well as for all adjacent treatment combinations. The existence and construction of such operational sequences present interesting graph theory problems whose formulations and solutions are found in this paper. This method of constrained randomization provides a basis for a statistical analysis utilizing the randomization model, which results in unbiased estimates of treatment effects and an unbiased estimate of experimental error.