Methods of Constructing One‐Way and Factorial Designs Balanced for Trend

This paper is a sequel to a previous one on magic squares (Phillips, 1964) and illustrates the use of other magic figures (rectangles, parallelepipeda and similar higher dimensional constructions) for balancing‐assessing linear and, occasionally, quadratic trend in certain general classes of one‐way and factorial designs. Constructional methods from the literature are given for the cases of: one‐way designs with either an even number of observations per level, or the numbers of observations and levels both odd; factorial designs having more than one observation per cell, with analogous conditions on the numbers of cells and observations; and factorial designs with one observation per cell, the numbers of levels of the variables either all even or all odd. The underlying assumptions of the designs, and their selection and randomization, are briefly discussed.