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Register a new userOptimality of multi-way designs
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Sunanda Bagchi
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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In this paper we study optimality aspects of a certain type of designs in a multi-way heterogeneity setting. These are ``duals of plans orthogonal through the block factor (POTB). Here by the dual of a main effect plan (say $rho$) we mean a design in a multi-way heterogeneity setting obtained from $rho$ 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 (1996). We show that the duals of these plans, as multi-way designs, satisfy M-optimality. Next, we construct another series of multiway designs and proved their M-optimality, thereby generalising the result of Bagchi and Shah (1989). It may be noted that M-optimality includes all commonly used optimality criteria like A-, D- and E-optimality.
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Permutation tests are widely used in statistics, providing a finite-sample guarantee on the type I error rate whenever the distribution of the samples under the null hypothesis is invariant to some rearrangement. Despite its increasing popularity and empirical success, theoretical properties of the permutation test, especially its power, have not been fully explored beyond simple cases. In this paper, we attempt to fill this gap by presenting a general non-asymptotic framework for analyzing the power of the permutation test. The utility of our proposed framework is illustrated in the context of two-sample and independence testing under both discrete and continuous settings. In each setting, we introduce permutation tests based on U-statistics and study their minimax performance. We also develop exponential concentration bounds for permuted U-statistics based on a novel coupling idea, which may be of independent interest. Building on these exponential bounds, we introduce permutation tests which are adaptive to unknown smoothness parameters without losing much power. The proposed framework is further illustrated using more sophisticated test statistics including weighted U-statistics for multinomial testing and Gaussian kernel-based statistics for density testing. Finally, we provide some simulation results that further justify the permutation approach.
A polynomial indicator function of designs is first introduced by Fontana, Pistone and Rogantin (2000) for two-level designs. They give the structure of the indicator function of two-level designs, especially from the viewpoints of the orthogonality of the designs. Based on these structure, they use the indicator functions to classify all the orthogonal fractional factorial designs with given sizes using computational algebraic software. In this paper, generalizing the results on two-level designs, the structure of the indicator functions for multi-level designs is derived. We give a system of algebraic equations for the coefficients of indicator functions of fractional factorial designs with given orthogonality. We also give another representation of the indicator function, a contrast representation, which reflects the size and the orthogonality of the corresponding design directly. The contrast representation is determined by a contrast matrix, and does not depend on the level-coding, which is one of the advantages of it. We use these results to classify orthogonal $2^3times 3$ designs with strength $2$ and orthogonal $2^4times 3$ designs with strength $3$ by computational algebraic software.
The minimum aberration criterion has been frequently used in the selection of fractional factorial designs with nominal factors. For designs with quantitative factors, however, level permutation of factors could alter their geometrical structures and statistical properties. In this paper uniformity is used to further distinguish fractional factorial designs, besides the minimum aberration criterion. We show that minimum aberration designs have low discrepancies on average. An efficient method for constructing uniform minimum aberration designs is proposed and optimal designs with 27 and 81 runs are obtained for practical use. These designs have good uniformity and are effective for studying quantitative factors.
The concept of orthogonality through the block factor (OTB), defined in Bagchi (2010), is extended here to orthogonality through a set (say S) of other factors. We discuss the impact of such an orthogonality on the precision of the estimates as well as on the inference procedure. Concentrating on the case when $S$ is of size two, we construct a series of plans in each of which every pair of other factors is orthogonal through a given pair of factors. Next we concentrate on plans through the block factors (POTB). We construct POTBs for symmetrical experiments with two and three-level factors. The plans for two factors are E-optimal, while those for three-level factors are universally optimal. Finally, we construct POTBs for $s^t(s+1)$ experiments, where $s equiv 3 pmod 4$ is a prime power. The plan is universally optimal.
Supersaturated design (SSD) has received much recent interest because of its potential in factor screening experiments. In this paper, we provide equivalent conditions for two columns to be fully aliased and consequently propose methods for constructing $E(f_{mathrm{NOD}})$- and $chi^2$-optimal mixed-level SSDs without fully aliased columns, via equidistant designs and difference matrices. The methods can be easily performed and many new optimal mixed-level SSDs have been obtained. Furthermore, it is proved that the nonorthogonality between columns of the resulting design is well controlled by the source designs. A rather complete list of newly generated optimal mixed-level SSDs are tabulated for practical use.
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