Optimal saturated block designs when observations are correlated
Introduction
Consider comparing the relative effectiveness of treatments employing experimental units arranged into b blocks of size k. The standard linear model for the observation on unit u in block j when using design d isfor and . The components of (1) are an overall mean , the effect of treatment assigned to unit u in block j by design d, a block effect , and a random error with zero mean. Writing and for the vectors of treatment and block effects, and for the vector of yields arranged in lexicographic order, then (1) says the mean vector is for the unit/block incidence matrix, and the unit/treatment incidence matrix defined by the . Here and elsewhere, denotes a column of q ones (likewise a column of q zeros). Choice of design is choice of
It is well known that all treatment contrasts are estimable under d if and only if the information matrix for estimation of treatment effects, , is of rank ; equivalently, the rank of is . A design with this property is said to be connected. A necessary condition for d to be connected is that the number of rows of is at least its required rank, that is, .
Several papers have appeared in the literature which discuss optimality of connected block designs when the number of experimental units is minimal, . Let denote the class of all connected block designs having treatments, b blocks and constant block size satisfyingThen is the class of minimally connected designs. Alternatively, since estimation of block and treatment effects takes all degrees of freedom (there are no degrees of freedom remaining for error), this is the class of saturated designs. When observations are equivariable and uncorrelated, the A-, MV- and E-optimal designs in are known, and all connected designs are D-equal: see Bapat and Dey (1991), Mandal et al. (1991) and Dey et al. (1995). Here optimality of designs in is studied when observations are equivariable and correlated. For the D-optimality problem, an arbitrary correlation structure is considered. For other optimality problems this spatial correlation structure is assumed:whereIn addition it is everywhere required that the variance–covariance matrix for the entire observations vector y be positive definite, i.e.The common diagonal element of is denoted by . The earlier results mentioned above are for .
The paper proceeds as follows. Section 2 presents the basic properties of minimally connected designs in a fashion suited to the current endeavor. Section 3 identifies M-optimal (this includes A- and MV-optimal) designs for estimation of elementary treatment contrasts. That all connected designs are D-equal is established in Section 4. Section 5 studies the E-optimality problem, providing a sufficient condition and determining optimal designs in some special cases. Concluding remarks comprise Section 6.
Section snippets
Properties of minimally connected designs
This section presents a lemma from which several useful properties of minimally connected designs follow. Lemma 2.1 For each , there is only one unbiased estimator for any treatment contrast . Consequently, the ordinary least squares estimate (OLSE) and the general least squares estimate (GLSE) for are the same. Proof Suppose there are two unbiased estimators and for . Let and write . Then , which contradicts (2). □
The result
M-optimal designs for elementary treatment contrasts
The results established here will be for elementary treatment contrasts, but the concept of majorization optimality, also called M-optimality (Bagchi and Bagchi, 2001), can be more generally construed as follows (the reader is referred to Bhatia, 1997, for a complete discussion of majorization). Let be variances of a set of m contrasts of interest when estimated using design d, and let be the vector of the for . Let f be a monotonically increasing convex function.
D-optimal design
The D-value of a design d is the product of the positive eigenvalues of the Moore–Penrose inverse of the information matrix ; D-optimal designs minimize the D-value. The first step in attacking the D-optimality problem is to establish a useful expression for . Lemma 4.1 For any ,where is the variance–covariance matrix for any solution to the reduced normal equations for estimating treatment effects under design d, and is the all-ones
E-optimal design
A design is E-optimal if it minimizes (in d) the maximum eigenvalue of the Moore–Penrose inverse of the information matrix . For given d, this eigenvalue is the maximum over all normalized contrasts of var. The study of E-optimality to follow requires the Moore–Penrose inverse for the design displayed in (6). Corollary 5.1 for design is where V is the
Discussion
Spatial correlation of observations has been found to impose positional conditions on optimal designs. This is not surprising, for much stronger positional balancing is found in the optimality conditions determined for nonsaturated block designs with correlated errors in papers such as Kunert (1987), Morgan and Chakravarti (1988), Martin and Eccleston (1991), Bhaumik (1995) and Benchekroun and Chakravarti (1999). What may be surprising is that regardless of the strength of positive correlation,
Acknowledgments
This paper has evolved during the review process. We thank the referees for their patience and careful reading. J.P. Morgan was supported by National Science Foundation Grant DMS01-04195.
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