An Exact Confidence Interval for the Relative Potency Estimated from a Multivariate Bioassay

Abstract

In their paper on the combination of results from independent univariate bioassays, Armitage, Bennett, and Finney (1976) show how likelihood ratio (LR) tests can be used to test for agreement between assays and to derive a confidence interval for the common relative potency. Both Finney (1978, ?13.3) and V0lund (1980) adapt the strategy of Armitage et al. (1976) to the very similar problem of combining the results from the correlated variables of a single multivariate assay, a problem which had previously been well explored by Rao (1954). More recently, the analysis of a single multivariate assay has been reconsidered by Carter and Hubert (1985) and Laska, Kushner, and Meisner (1985). The main purpose of this paper is to describe a new, distributionally exact, confidence interval for the relative potency in a single multivariate assay. The intervals already described in the literature were derived from the asymptotic distributions of LR statistics or maximum likelihood (ML) estimators. The new exact interval can be derived from a likelihood score (LS) statistic, though it is fundamentally identical to the exact interval for the relative potency from several independent univariate assays derived by the author (Williams, 1978) using an approach suggested by E. J. Williams (1973). Section 2 of this paper establishes some notation and describes the preliminary statistical analysis. Section 3 summarizes the established confidence intervals that derive from asymptotic distributions of LR statistics or ML estimators. An additional LR interval, employing a Bartlett correction factor, is also mentioned. The new exact LS interval is presented in Section 4, and a numerical example is given in Section 5. Finally, Section 6 compares the confidence intervals obtained by these different methods.