{'fraction_non_alphanumeric': 0.02249408050513023, 'fraction_numerical': 0.009865824782951855, 'mean_word_length': 4.909090909090909, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Optimal Fractional Factorial Plans for Asymmetric Factorials

Fractional factorial plans for asymmetric factorial experiments are obtained. These are shown to be universally optimal within the class of all plans involving the same number of runs under a model that includes the mean, all main effects and a specified set of two-factor interactions. Finite projective geometry is used to obtain such plans for experiments wherein the number of levels of each of the factors as also the number of runs is a power of m, a prime or a prime power. Methods of construction of optimal plans under the same model are also discussed for the case where the number of levels as well as the number of runs are not necessarily powers of a prime number. 1. Introduction. The study of optimal fractional factorial plans has received considerable attention in the recent past; see e.g., Dey and Mukerjee ((1999a); Chapters 2, 6 & 7). Many of these results relate to situations where all factorial effects involving the same number of factors are considered equally important and, as such, the underlying model involves the general mean and all factorial effects involving up to a specified number of factors. In practice however, all factorial effects involving the same number of factors may not always be equally important and often, an experimenter is interested in estimating the general mean, all main effects and only a specified set of two-factor interactions, all other interactions being assumed negligible. The issue of estimability and optimality in situations of this kind has been addressed by Hedayat in the context of two-level factorials and, by Dey and Mukerjee (1999b) for arbitrary factorials including the asymmetric ones. Using finite projective geometry, Dey and Suen (2002) recently obtained several families of optimal plans under the stated model for symmetric factorials of the type m n , where m is a prime or a prime power. Continuing with this line of research, in this paper we obtain optimal fractional factorial plans for asymmetric (mixed level) factorials under a model that includes the mean, all main effects and a specified set of two-factor interactions. All other interactions are assumed to be negligible. Throughout, the optimality criterion considered is the universal optimality of Kiefer (1975); see also Sinha and Mukerjee (1982). In Section 2, concepts and results from a finite projective geometry are used to obtain optimal plans for asymmetric factorials, where the levels of the factors as also the …