In a three-way layout with one observation in each cell, the observations Yijk (i = 1, . . . , I; j = 1, . . . , J; k = 1, . . . ,K) are assumed to be independent and normally distributed, with a common variance σ2. Suppose that E(Yijk) = θijk. Show that for every set of numbers θijk, there exists a unique set of numbers μ, αAi , αBj, αCk, βABij, βACik, βBCjk, and γijk (i = 1, . . . , I; j = 1, . . . , J; k = 1, . . . , K) such that
αA+ = αB+ = αC+ = 0,
βABi+ = βAB+j = βACi+ = βAC+k = βBCj+ = βBC+k = 0,
γij+ = γi+k = γ+jk = 0,
and
θijk = μ + αAi + αBj + αCk + βABij + βACik + βBCjk + γijk,
for all values of i, j , and k.