ORTHOGONALITY AND ROTABILITY OF TRIPLE SQUARES AND CUBES

In México, the use for 30 years of double squares and cubes has suggested a line of research that extends the basic designs in use, to the consideration of triple squares and cubes. The laying out of designs that include the properties of orthogonality (associated with accuracy of the estimators) and rotability (associated to the uniformity of the variance on the estimated response) is important. This paper analyzes several families of triple squares and cubes, and four examples of each type are examined. The generated families are obtained either by modifying the coordinates of one of the squares (or cubes) from a certain typical design or by modifying the coordinates of the axial points. The results include the identification of 14 orthogonal triple squares, 8 rotatable triple squares, and 4 orthogonal and 5 rotatable triple cubes. An experiment on fertilizers under the orthogonal triple square design with a=2.58375 was used as an example of these results.