Sankhy¯a: The Indian Journal of Statistics 1998, Volume 60, Series B, Pt. 3, pp. 496-510 A-OPTIMAL BLOCK DESIGNS UNDER A MIXED MODEL FOR MAKING TEST TREATMENTS - CONTROL COMPARISONS By V.K. GUPTA ARCHANA PANDEY and RAJENDER PARSAD Indian Agricultural Statistics Research Institute, New Delhi SUMMARY. The purpose of this paper is to study the A-optimality of block designs for making test treatments-control comparisons under a mixed effects model. The A-optimality and the A-efficiency of Balanced Treatment Incomplete Block designs has been investigated for different values of ρ, a function of intra and inter block variances. A catalogue of A-optimal and A-efficient designs has been prepared. The designs are robust against the value of ρ studied. 1. Introduction We consider the problem of obtaining optimal designs for comparing v test treatments with a control treatment in incomplete blocks. The (v +1) treatments are indexed as 0, 1, 2,...,v with 0 denoting the control treatment and 1, 2,...,v the test treatments. The treatments are applied to n plots, arranged in b blocks with a common block size k. It is assumed that k< (v + 1) and that only one treatment is applied to each experimental unit. Associated with a design d is a (v + 1) × b incidence matrix N ∗ d = ((n dtj )) = [n d N ′ d ] ′ , where the non negative integer n dtj is the replication number of the t-th Paper received. June 1997; revised February 1998. AMS (1991) subject classification. Primary 62K05; secondary 62K10. Key words and phrases. Mixed effects model, A-optimality, BIB design, ABIB design, GD design, PBIB design, A-efficiency.