Profile Likelihood and Incomplete Data

According to the law of likelihood, statistical evidence is represented by likelihood functions and its strength measured by likelihood ratios. This point of view has led to a likelihood paradigm for interpreting statistical evidence, which carefully distinguishes evidence about a parameter from error probabilities and personal belief. Like other paradigms of statistics, the likelihood paradigm faces challenges when data are observed incompletely, due to non-response or censoring, for instance. Standard methods to generate likelihood functions in such circumstances generally require assumptions about the mechanism that governs the incomplete observation of data, assumptions that usually rely on external information and cannot be validated with the observed data. Without reliable external information, the use of untestable assumptions driven by convenience could potentially compromise the interpretability of the resulting likelihood as an objective representation of the observed evidence. This paper proposes a profile likelihood approach for representing and interpreting statistical evidence with incomplete data without imposing untestable assumptions. The proposed approach is based on partial identification and is illustrated with several statistical problems involving missing data or censored data. Numerical examples based on real data are presented to demonstrate the feasibility of the approach.