Algorithms and diagnostics for the analysis of ranking data with the Extended Plackett-Luce model

The Plackett-Luce distribution (PL) is one of the most successful parametric options within the class of multistage ranking models to learn preferences on a given set of items from a sample of ranked sequences. It postulates that the ranking process is carried out by sequentially assigning the positions according to the forward order, that is, from the top (most-liked) to the bottom (least-liked) alternative. This assumption has been recently relaxed with the Extended Plackett-Luce model (EPL) with the introduction of the discrete reference order parameter, describing the rank attribution path. By starting from the formal proof of a special property of the EPL, related to the inverse ordering of the item probabilities at the first and last stage of the ranking process, we derive a diagnostic tool for the EPL distribution whose inferential utility is motivated from a double perspective. First, the novel statistic is proposed to test the appropriateness of the EPL assumption. In this regard, the new diagnostic contributes to fill the deficiency of goodness-of-fit methods for the family of multistage models. Besides model adequacy evaluation, we also show how the statistic can be exploited to construct a heuristic method that surrogates the likelihood approach for inferring the underlying reference order parameter. The usefulness of the proposals is illustrated with a simulation study and applications to real ranking data.