The fragility of standard inferential approaches in principal stratification models relative to direct likelihood approaches

Many empirical settings involve the specification of models leading to complicated likelihood functions, for example, finite mixture models that arise in causal inference when using Principal Stratification (PS). Traditional asymptotic results cannot be trusted for the associated likelihood functions, whose logarithms are not close to being quadratic and may be multimodal even with large sample sizes. We first investigate the shape of the likelihood function with models based on PS by providing diagnostic tools for evaluating ellipsoidal approximations based on the second derivatives of the log-likelihood at a mode. In these settings, inference based on standard approximations is inappropriate, and other forms of inference are required. We explore the use of a direct likelihood approach for parsimonious model selection and, specifically, propose comparing values of scaled maximized likelihood functions under competitive models to select preferred models. An extensive simulation study provides guidelines, for calibrating the use of scaled log-likelihood ratio statistics, as functions of the complexity of the models being compared.