Instrumented indentation tests provide an attractive means for obtaining data to characterize the plastic response of engineering materials. One difficulty in doing this is that the relation between the measured indentation force versus indentation depth response and the plastic stress-strain response is not unique. Materials with very different uniaxial stress-strain curves can give essentially identical curves of indentation force versus indentation depth. Zhang et al. (2019, “Identification of Plastic Properties From Conical Indentation Using a Bayesian-Type Statistical Approach,” ASME J. Appl. Mech., 86, p. 011002) numerically generated “experimental” conical indentation data and showed that using surface profile data and indentation force versus indentation depth data together with a Bayesian-type statistical analysis permitted the uniaxial plastic stress-strain response to be identified even for materials with indistinguishable indentation force versus indentation depth curves. The same form of hardening relation was used in the identification process as was used to generate the “experimental” data. Generally, a variety of power law expressions have been used to characterize the uniaxial plastic stress-strain response of engineering materials, and, of course, the form that gives the best fit for a material is not known a priori. Here, we use the same Bayesian statistics-based analysis but consider four characterizations of the plastic uniaxial stress-strain response and show that the identification of the hardening relation parameters and the associated uniaxial stress-strain response is not very sensitive to the form of the power law strain hardening relation chosen even with data that have significant noise.