Logistic quantile regression for bounded outcomes

When research interest lies in continuous outcome variables that take on values within a known range (e.g. a visual analog scale for pain within 0 and 100 mm), the traditional statistical methods, such as least-squares regression, mixed-effects models, and even classic nonparametric methods such as the Wilcoxon's test, may prove inadequate. Frequency distributions of bounded outcomes are often unimodal, U-shaped, and J-shaped. To the best of our knowledge, in the biomedical and epidemiological literature bounded outcomes have seldom been analyzed by appropriate methods that, for one, correctly constrain inference to lie within the feasible range of values. In many respects, continuous bounded outcomes can be likened to probabilities or propensities. Yet, what has long been heeded when modeling the probability of binary outcomes with the widespread use of logistic and probit regression, so far appears to have been overlooked with continuous bounded outcomes with consequences at times disastrous. Logistic quantile regression constitutes an effective method to fill this gap.