Midrank unification of rank tests for exact, tied, and censored data

Abstract

Ties, right censoring, left censoring, and interval censoring can all cause ambiguity about the rank of an observation. Therefore traditional rank tests for uncensored data without ties need to be adjusted in the presence of uncertainty about the rank assigned to an observation. A simple solution that can be used in any situation where the rank assigned to an observation is ambiguous is to replace the unknown rank of an observation with its midrank. We derive properties of the midranks and, in turn, generalizations of ranks tests for the two sample problem, matched pairs and competing risks in the presence of censoring. Several previously proposed rank tests for censored data are shown to be equivalent to midrank tests.