Remarks on optimal designs in a paired comparison experiment using the Bradley-Terry model

A number of methods for estimation of the parameters in a paired comparison experiment are considered. This is done by comparing the (asymptotic) variance-covariance matrices of the estimators. The methods considered are: Springall (1973), Beaver (1977a, 1977b) and El-Helbawy/Bradley (1978). It is shown that these three methods yield the same variance-covariance matrix. When the treatment parameters are equal, the variance-covariance matrix is proportional to the variance-covariance matrix, which is obtained when an ordinary least-squares method is used. In constructing optimal designs it is possible to restrict oneself to this variance-covariance matrix. Some errors by Springall (1973) are corrected. Keywords: Paired comparison; Bradley-Terry model; Optimal designs, Weighted least-squares; Factorials.