Some Families of Designs for Multistage Experiments: Mutually Balanced Youden Designs when the Number of Treatments is Prime Power or Twin Primes. I

The concepts of balance for ordered and for unordered pairs of treatments are introduced. Methods for constructing multistage experimental designs which are Youden designs at each stage, are given. In the construction of these designs an attempt was made to accommodate as much orthogonality and balance (both in our sense and in the classical sense) as is possible. These constructions are presented in several theorems. In one theorem, we give a uniform method of converting $t$ mutually orthogonal Latin squares of order $n$ into a $t$-stage balanced $(n - 1) \times n$ Youden designs. Since the known methods of constructing orthogonal Latin squares of order $n = 4t + 2$, for $t > 1$, are not uniform, a uniform method for constructing two-stage $(n - 1) \times n$ Youden designs for all even $n$ has been developed. In another theorem, a method of constructing $(2\lambda + 1)$-stage balanced $(2\lambda + 1) \times (4\lambda + 3)$ Youden designs, for $4\lambda + 3$ a prime power, is presented. A method of constructing $(p^\alpha - 1)$-stage balanced $(\nu - 1)/2 \times \nu$ Youden designs is given in another theorem for the case when $v = 4\lambda + 3$ and is the product of twin primes, i.e., $\nu = p^\alpha q^\beta, q^\beta = p^\alpha + 2$. Difference sets based on the elements of Galois fields were utilized for these constructions. Other miscellaneous results are given.