Scoring rules in survival analysis

Scoring rules evaluate probabilistic predictions and (attempt to) measure the overall predictive ability of a model as a combination of calibration and discrimination [Gneiting and Raftery, 2007, Murphy, 1973]. Scoring rules have been gaining in popularity for the past couple of decades since probabilistic forecasts were recognised to be superior than deterministic predictions for capturing uncertainty in predictions [Dawid, 1984, 1986]. Formalisation and development of scoring rules has primarily been due to Dawid [Dawid, 1984, 1986, Dawid and Musio, 2014], Gneiting and Raftery [Gneiting and Raftery, 2007]; though the earliest measures promoting "rational" and "honest" decision making date back to the 1950s [Brier, 1950, Good, 1952]. In classification and (probabilistic) regression [Gressmann et al., 2018] settings there are established definitions for scoring rules and specific losses have been defined, most popular of which are the Brier score and Logloss. However the literature has been lacking for survival analysis, with no definition of a scoring rule being proposed until very recently, despite this losses are frequently utilised in the literature without justification or proofs about their properties. In this paper we will present a formal definition for a survival scoring rule as well as a second definition for an'approximate' survival scoring rule that can be utilised under specific conditions. We provide a brief review of losses in the literature and collate claims, proofs, and disproofs for properness of these loses.