Relaxation Penalties and Priors for Plausible Modeling of Nonidentified Bias Sources

In designed experiments and surveys, known laws or design features provide checks on the most relevant aspects of a model and identify the target parameters. In contrast, in most observational studies in the health and social sciences, the primary study data do not identify and may not even bound target parameters. Discrepancies between target and analogous identified parameters (biases) are then of paramount concern, which forces a major shift in modeling strategies. Conventional approaches are based on conditional testing of equality constraints, which correspond to implausible point-mass priors. When these constraints are not identified by available data, however, no such testing is possible. In response, implausible constraints can be relaxed into penalty functions derived from plausible prior distributions. The resulting models can be fit within familiar full or partial likelihood frameworks. The absence of identification renders all analyses part of a sensitivity analysis. In this view, results from single models are merely examples of what might be plausibly inferred. Nonetheless, just one plausible inference may suffice to demonstrate inherent limitations of the data. Points are illustrated with misclassified data from a study of sudden infant death syndrome. Extensions to confounding, selection bias and more complex data structures are outlined.