Journal of Statistical Planning and Inference 27 (1991) 75-83 North-Holland 75 Some families of optimal and efficient repeated measurements designs John Stufken Statistical Laboratory and Department of Statistics, Iowa State University, Snedecor Hall, Ames, IA50011, USA Received 1 March 1989; revised manuscript received 14 July 1989 Recommended by C.S. Cheng Abstract: Conditions that ensure simple information matrices for the estimation of direct and residual treatment effects under an additive, homoscedastic model are given. Examples of designs that satisfy these conditions are presented. For the number of periods not exceeding the number of treatments designs that satisfy the conditions are derived from orthogonal arrays of index unity. Their efficiency is considered and some of them, as well as some other designs, are shown to be universally optimal over certain subclasses of designs. AMS Subject Classification: Primary 62K05; secondary 62KlO. Key words and phrases: Direct and residual effects; balanced and strongly balanced designs; orthogonal arrays; orthogonal arrays of Type I; efficient designs; universal optimality. 1. Introduction Repeated measurements designs (RMD’s) have been used in various areas of research for many years. A good source for references is the recent review paper on the subject by Matthews (1988). Studies on optimality of RMD’s are of a much more recent date and have essentially started with the work of Hedayat and Afsarinejad (1975, 1978). This has generated numerous other contributions, that include Cheng and Wu (1980, 1983), Magda (1980), Afsarinejad (1983, 1985), Dey, Gupta and Singh (1983), Mukhopadhyay and Saha (1983), Kunert (1983, 1984a, 1984b, 198.5), Gill and Shukla (1987), Majumdar (1988) and Hedayat and Zhao (1990). The current paper generalizes some of the available results, with emphasis on the situation with the number of periods exceeded by the number of treatments. We will present conditions under which information matrices associated with the model under consideration reduce to matrices that are easy to handle and construct highly efficient designs, some of which are shown to be universally optimal in rather large subclasses of competing designs. 0378-3758/91/$03.50 0 1991-Elsevier Science Publishers B.V. (North-Holland)