### Efficiency, optimality, and differential treatment interest

Morgan, JP *(Virginia Tech)*

Friday 15 August 2008, 17:00-17:30

Seminar Room 1, Newton Institute

#### Abstract

Standard optimality arguments for designed experiments rest on the assumption that all treatments are of equal interest. One exception is found in the "test treatment versus control" literature, where the control is allocated special status. Optimality work there has focused on all pairwise comparisons with the control, making no explicit account of how well test treatments are compared to one another. In many applications it would be preferable to choose a design depending on the relative importance placed on contrasts involving the control to those of test treatments only. This is an example of where a weighted optimality approach can better reflect experimenter goals.
When evaluating designs for comparing $v$ treatments, weights $w_1,\ldots,w_v$ ($\sum_iw_i=1$) can be assigned to account for differential treatment interest. These weights enter the evaluation through optimality measures, leading to, for example, weighted versions of the popular A, E, and MV measures of design efficacy. Families of weighted-optimal designs have been identified for both blocked and unblocked experiments.
The theory for weighted optimality leads quite naturally to the notion of weight-balanced designs. Weighted balance and partial balance incorporate the concepts of efficiency balance and its generalizations that have been built on the foundation laid by Jones (1959, JRSS-B 21, 172-179). These balance ideas are closely tied to the weighted E criterion.

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