Seven transmembrane receptors were originally named and characterized based on their ability to couple to heterotrimeric G proteins. The assortment of coupling partners for G protein-coupled receptors has subsequently expanded to include other effectors (most notably the βarrestins). This diversity of partners available to the receptor has prompted the pursuit of ligands that selectively activate only a subset of the available partners. A biased or functionally selective ligand may be able to distinguish between different active states of the receptor, and this would result in the preferential activation of one signaling cascade more than another. Although application of the "standard" operational model for analyzing ligand bias is useful and suitable in most cases, there are limitations that arise when the biased agonist fails to induce a significant response in one of the assays being compared. In this article, we describe a quantitative method for measuring ligand bias that is particularly useful for such cases of extreme bias. Using simulations and experimental evidence from several κ opioid receptor agonists, we illustrate a "competitive" model for quantitating the degree and direction of bias. By comparing the results obtained from the competitive model with the standard model, we demonstrate that the competitive model expands the potential for evaluating the bias of very partial agonists. We conclude the competitive model provides a useful mechanism for analyzing the bias of partial agonists that exhibit extreme bias.