Inference for the Thurstonian delta in the same-different protocol via the well known Wald statistic is shown to be inappropriate in a wide range of situations. We introduce the likelihood root statistic as an alternative to the Wald statistic to produce CIs and p-values for assessing difference as well as similarity. We show that the likelihood root statistic is equivalent to the well known G(2) likelihood ratio statistic for tests of no difference. As an additional practical tool, we introduce the profile likelihood curve to provide a convenient graphical summary of the information in the data about delta. On the basis of simulations, we show that the coverage probability of the Wald-based 95% CI for delta is often far from 95%, whereas the coverage probability of the profile likelihood-based Cl is close to the desired 95%. We also show how the likelihood framework can be used to combine information from independent experiments possibly using different discrimination protocols to obtain inference for a common delta. Finally, we provide a free R package with an implementation of the likelihood methodology presented in this paper.