Generalized Spearman estimators of relative dose

In a biological assay the expected response may be transformed to a variable bounded between 0 and 1. If the transformed response is regarded as analogous to the tolerance distribution function, the mean of that distribution may be estimated for the standard and test preparations, and a simple estimator of the relative potency obtained. The special case where the identity transformation is used for a quantal response corresponds to Spearman's estimator, and our generalization has similar unbiasedness properties to that estimator. Asymptotic results are derived when the intervals between dose levels decrease and the sample of each dose level simultaneously increases. These results are evaluated for the case with equal sample sizes at regularly spaced values of the dose metameter. An approximate test for similarity is proposed. If the tolerance distribution is known up to a scale parameter, then the transformation may be chosen so that the estimator is asymptotically fully efficient. An application to the thermal disinfestation of wheat is given.