Imprecision and Prior-Data Conflict in Generalized Bayesian Inference

Using External Information for More Precise Inferences in General Regression Models

Empirical research usually takes place in a space of available external information, like results from single studies, meta-analyses, official statistics or subjective (expert) knowledge. The available information ranges from simple means and proportions to known relations between a multitude of variables or estimated distributions. In psychological research, external information derived from the named sources may be used to build a theory and derive hypotheses. In addition, techniques do exist that use external information in the estimation process, for example prior distributions in Bayesian statistics. In this paper, we discuss the benefits of adopting generalized method of moments with external moments, as another example for such a technique. Analytical formulas for estimators and their variances in the multiple linear regression case are derived. An R function that implements these formulas is provided in the supplementary material for general applied use. The effects of various practically relevant moments are analyzed and tested in a simulation study. A new approach to robustify the estimators against misspecification of the external moments based on the concept of imprecise probabilities is introduced. Finally, the resulting externally informed model is applied to a dataset to investigate the predictability of the premorbid intelligence quotient based on lexical tasks, leading to a reduction of variances and thus to narrower confidence intervals.

Addressing potential prior-data conflict when using informative priors in proof-of-concept studies

Bayesian methods are increasingly used in proof-of-concept studies. An important benefit of these methods is the potential to use informative priors, thereby reducing sample size. This is particularly relevant for treatment arms where there is a substantial amount of historical information such as placebo and active comparators. One issue with using an informative prior is the possibility of a mismatch between the informative prior and the observed data, referred to as prior-data conflict. We focus on two methods for dealing with this: a testing approach and a mixture prior approach. The testing approach assesses prior-data conflict by comparing the observed data to the prior predictive distribution and resorting to a non-informative prior if prior-data conflict is declared. The mixture prior approach uses a prior with a precise and diffuse component. We assess these approaches for the normal case via simulation and show they have some attractive features as compared with the standard one-component informative prior. For example, when the discrepancy between the prior and the data is sufficiently marked, and intuitively, one feels less certain about the results, both the testing and mixture approaches typically yield wider posterior-credible intervals than when there is no discrepancy. In contrast, when there is no discrepancy, the results of these approaches are typically similar to the standard approach. Whilst for any specific study, the operating characteristics of any selected approach should be assessed and agreed at the design stage; we believe these two approaches are each worthy of consideration.