Balanced arrays were introduced and first studied by I. M. Chakravarti and called partially balanced arrays. J. N. Srivastava and D. V. Chopra have recently made major contributions to the theory and construction of these arrays. They suggested dropping the adjective "partially" and we have followed their lead. Whereas their work has been concerned mainly with arrays of strength four with two symbols, we are here concerned primarily with arrays of strength two with two symbols. However, when the proofs can be carried out as easily for any strength and any number of symbols, we do state the theorems in full generality. We are concerned here with finding bounds on the maximum possible number of rows and with the problem of constructing balanced arrays for given sets of parameters. Analogous to the problem of constructing other combinatorial configurations, we investigated whether some schemes, i.e. some subsets of columns of balanced arrays, could be extended to full balanced arrays. It is shown, analogously to orthogonal arrays, that balanced arrays of even strength, say $2u$ are extendable to arrays of strength $2u + 1$. A new technique of construction of balanced arrays with the maximum number of constraints is also described. It is shown that BIB designs with $\lambda = 1$ can be utilized for constructing balanced arrays with the number of symbols equal to the block size of the BIB design. For completeness we include an example of the analysis of a partially balanced array of strength two with two symbols when it is used as a fractional factorial design. We exhibit in this way an explicit method of estimating main effects when higher ordered interactions are assumed to be negligible.
"Contributions to the Theory and Construction of Balanced Arrays." Ann. Statist. 2 (6) 1256 - 1273, November, 1974. https://doi.org/10.1214/aos/1176342877