The quantification of human perception through the study of psychometric functions $Ψ$ is one of the pillars of experimental psychophysics. In particular, the evaluation of the threshold is at the heart of many neuroscience and cognitive psychology studies, and a wide range of adaptive procedures has been developed to improve its estimation. However, these procedures are often implicitly based on different mathematical assumptions on the psychometric function, and unfortunately, these assumptions cannot always be validated prior to data collection. This raises questions about the accuracy of the estimator produced using the different procedures. In the study we examine in this letter, we compare five adaptive procedures commonly used in psychophysics to estimate the threshold: Dichotomous Optimistic Search (DOS), Staircase, PsiMethod, Gaussian Processes, and QuestPlus. These procedures range from model-based methods, such as the PsiMethod, which relies on strong assumptions regarding the shape of $Ψ$, to model-free methods, such as DOS, for which assumptions are minimal. The comparisons are performed using simulations of multiple experiments, with psychometric functions of various complexity. The results show that while model-based methods perform well when $Ψ$ is an ideal psychometric function, model-free methods rapidly outshine them when $Ψ$ deviates from this model, as, for instance, when $Ψ$ is a beta cumulative distribution function. Our results highlight the importance of carefully choosing the most appropriate method depending on the context.