Aspects of Optimality of Plans Orthogonal Through Other Factors and Related Multiway Designs

Article Type

Research Article

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Journal of Statistical Theory and Practice


In a blocked main effect plan (MEP), a pair of factors is said to be orthogonal through the block factor if their totals adjusted for the block are uncorrelated, as defined in Bagchi (Technometrics 52:243–249, 2010). This concept is extended here to orthogonality through a set of other factors. We discuss the impact of such an orthogonality on the precision of the estimates as well as on the data analysis. We construct a series of plans in which every pair of factors is orthogonal through a given pair of factors. Next we construct plans orthogonal through the block factors (POTB). We construct the following POTBs for symmetrical experiments. There are an infinite series of E-optimal POTBs with two-level factors and an infinite series of universally optimal plans for three-level factors. We also construct an universally optimal POTB for an st(s+ 1) experiment on blocks of size (s+ 1) / 2 , where s≡3(mod4) is a prime power. Next we study optimality aspects of the “duals” of main effect plans with desirable properties. Here by the dual of a main effect plan we mean a design in a multi-way heterogeneity setting obtained from the plan by interchanging the roles of the block factors and the treatment factors. Specifically, we take up two series of universally optimal POTBs for symmetrical experiments constructed in Morgan and Uddin (Ann Stat 24:1185–1208, 1996). We show that the duals of these plans, as multi-way designs, satisfy M-optimality. Finally, we construct another series of multiway designs, which are also duals of main effect plans, and proved their M-optimality. This result generalizes the result of Bagchi and Shah (J Stat Plan Inf 23:397–402, 1989) for a row–column set-up. It may be noted that M-optimality includes all commonly used optimality criteria like A-, D- and E-optimality.



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