Increasing the average precision or decreasing the aggregate variability among multiple a priori contrasts : two algorithms for allocating sample material

In true experiments in which sample material can be randomly assigned to treatment conditions, most researchers presume that the condition of equal sample sizes is statistically desirable. When one or more a priori contrasts can be identified which represent a few overriding experimental concerns, however, allocating sample material unequally will lead to more powerful tests of certain hypotheses, when the assumptions underlying the analysis of variance are known to be met. The purpose of this study was to develop systematic procedures for assigning a fixed amount of sample material to treatment conditions using either of two optimality criteria: (a) the maximization of the average precision of a given set of contrasts, or (b) the equalization of precision across a set of such contrasts. On the basis of relevant literature, the squared standard error of the estimate was adopted as the appropriate measure to minimize or equalize.

Algorithms were developed mathematically to achieve each of these goals under specified conditions in single factor, fixed effects, analysis of variance applications. Comparisons of the average precision and variability of precision associated with equal allocation schemes and the two developed in the present study were made, using treatment levels of 5, 4, and 6, average cell sizes of 10 and 15, and representative sets of contrasts. It was determined that the algorithm which minimizes the average squared standard error does not, necessarily, result in the greatest average power, since its minimization cannot he equated to the maximization of power. The same algorithm leads to allocations which were found to have appreciable advantages over equal allocation in certain designs; that which equalizes precision across contrasts was found to result in less powerful analyses than either of the other schemes. Recommendations are made regarding situations in which the researcher might consider applying these algorithms in particular, and power analyses in general, when planning experiments.