In a multivariate bioassay the response variable is r-dimensional and the dose variable is one-dimensional. Thus, the graph of the dose response relation is a curve in the (r+1)-dimensional space. The similarity requirement for a test and a standard is divided into two parts: (i) a condition of marginal similarity corresponding to the usual similarity or parallelism of each response variable considered separately; (ii) a condition that all response variables give the same relative potency. These conditions correspond to two nested hypotheses for which a T2- and an asymptotic likelihood ratio test, respectively, are presented for the case of a multivariate normal response distribution and a parallel-line dose response model. Equations for direct calculation of the maximum likelihood estimate of the relative potency have been obtained. An asymptotic confidence set for the common relative potency is derived. These methods have been applied to a twin cross-over assay of insulin by the rabbit blood sugar method, in which the measurements of the blood sugar concentration at intervals after the insulin administration were regarded as the multivariate response. In comparison with the usual univariate analysis, multivariate methods allow additional tests of basic assumptions and they appear to provide a more efficient utilization of experimental data, leading to improved potency estimates.