# DOE

## Seminar

### A-optimal block designs for the comparison of treatments with a control with autocorrelated errors

Seminar Room 1, Newton Institute

#### Abstract

There is an extensive literature on optimal and efficient designs for comparing /t/ test (or new) treatments with a control (or standard treatment) - see Majumdar (1996). However, almost all results assume the observations are uncorrelated. In many situations, it is more realistic to assume that observations in the same block are positively correlated, and there has been much interest in this case when all contrasts are of equal interest - see, for example, Martin (1996).Assuming that the estimation uses ordinary least-squares, Bhaumik (1990) found optimal within-block orderings under a first-order nearest-neighbour model NN(1) among some designs that would have been optimal test-control designs under independence. Cutler (1993) obtained some optimality results under a first-order autoregressive process AR(1) on the circle or the line, assuming generalised least-squares estimation for a known dependence. There are also some brief examples and discussion of the correlated case in Martin & Eccleston (1993, 2001). Here, we concentrate on generalised least-squares estimation for a known covariance. Results for independence, and Cutler's (1993) results for the AR(1), are for specific combinations of /t/, /b/, /k/, and use integer minimisation to ensure an optimal design exists. Here, we assume that the number of blocks /b/ is large enough for an optimal design to exist, and consider the form of that optimal design. This method may lead to exact optimal designs for some /b/, /t/, /k/, but usually will only indicate the structure of an efficient design for any particular /b/, /t/, /k/, and yield an efficiency bound, usually unattainable. The bound and the structure can then be used to investigate efficient finite designs.#### Video

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