On the construction of balanced and partially balanced factorial experiments

Satisfactory systems of confounding for symmetrical factorial experiments can be constructed oy the familiar methods, using the. theory of Galois fields. Although these methods can be extended to asymmetrical factorial experiments· (White and Hultquist, 1965; Raktoe, 1969) the actual construction of designs becomes much mor:e complicated for the general case and does not always lead to satisfactory plans. A different approach to this problem is to consider balanced factorial experiments (BFE), due to Shah (1958, 1960). Such BFE have a one-to-one relationship to EGD-PBIB designs given by Hinkelmann (1964). The problem of constructing BFE is then equivalent to constructing EGD-PBIB designs. A new method is proposed here to construct such designs. This method is based upon the so-called (1,1, ...,1)th-associate matrix and the operations symbolic direct product (SDP), generalized symbolic direct product (GSDP), symbolic direct multiplication (SDM), and generalized symbolic direct multiplication (GSDM). Let A₁ , A₂, ... , A_n be n factors in a factorial experiment, with A_i having t_i levels (i = 1, 2, ... , n). It is shown that an EGD-PBIB design with blocks of size t_i can be constructed, provided that t_iᵢ ≠ max ( t₁ , t₂, . . . , t_n ). This method is more general and more flexible than the method of Aggarwal (1974) in that any two treatment combinations can be γ-th associates where γ has at least two unity components, and it can be shown the number of possible candidates for such is 2^{n-i l} -1 for blocks of size t_i (i = 1, 2, .. , n -1), where t₁ < t₂ <...< t_n. This method is also more general than the Kronecker product method due to Vartak (1955}.

Two types of PBIB designs· are used for reducing the numbers of associa,te classes in EGD-PBIB designs. When the t_i (i = 1, 2, ... , n) are equal, then some EGD-PBIB designs can be reduced to a hypercubic design. The EGD-PBIB designs with block size π [below jεA] t_j, where A is an arbitrary subset of the set {1, 2, ... , n} can be reduced to newly introduced F_A⁽ⁿ⁾-type PBIR designs.

Since BFE results very often in designs with a large number of blocks, the notion of partial balanced factorial experiment (PBFE) has been introduced. It is investigated how such designs can be constructed and related to PBIB-designs similar to that between BFE and EGD-PBIB designs. Two new types of PBIB designs have been introduced in this context.