Balanced 2(n) factorial designs when observations are spatially correlated

In this article we focus on the optimal factorial and fractional-factorial designs when observations within blocks are correlated. The topic was motivated by a problem when the pharmaceutical experimenter needed to develop a controlled release, once-daily tablet formulation. Typically, in order to compare different formulations, trials are conducted in healthy human volunteers where each formulation is administered and bioavailability is estimated. Since each subject is administered more than one formulation, the observations within subjects are correlated. Balanced designs for 2(n) factorial experiments when observations within blocks are spatially correlated, AR(1) with positive correlation (rho > 0), are characterized. An explicit construction and analytical proof of balanced designs for both 2(n) full and 2(n-1) fractional factorial experiments is provided. In order to illustrate the construction, two examples using a complete 2(3) factorial and a half replicate of 2(4) factorial experiment are provided. We consider the optimal or near-optimal designs provided by Cheng and Steinberg (1991), Martin et al. (1998c), and Elliott et al. (1999) as the starting point to obtain balanced designs. We compare the relative efficiencies of our balanced designs with these designs.