A likelihood approach is considered as an alternative to the classical ridge analysis when the estimated stationary point of a quadratic response surface is a saddle point or it is not located inside the experimental region. We compute the path of maximum profile likelihood for the optimum point on the contours of hyperspheres centered at the design center. Plausibility regions around this path can provide information about the nature of the system inside the experimental region and about suitable regions where to conduct further experiments.