Equal and Proportional Frequency Squares

Abstract

The relationship between the class of Latin, Graeco-Latin and Hyper-Graeco-Latin squares which can be formally classified as orthogonal fractional replicate plans and the totality of Latin, Graeco-Latin and Hyper-Graeco-Latin squares is discussed. A $v \times v$ square for which the $t$ letters of the alphabet that represents the treatments each occur $k$ times in every row and column, so that $v = tk$, is defined to be an equal frequency square. The properties of these squares are discussed and compared with the properties of $k$ Latin, Graeco-Latin or Hyper-Graeco-Latin squares of side $t$. A $v \times v$ square for which the $t < v$ letters occur in each row and column with frequencies proportional to the number of times they occur in the entire square is defined to be a proportional frequency square. The structure of the analysis of variance for several proportional frequency squares is developed and illustrated with an example of the analysis of data from three $5 \times 5$ proportional frequency squares with three treatments