Doublethink: simultaneous Bayesian-frequentist model-averaged hypothesis testing

Helen Fryer, Nick Arning and I have posted our new preprint to arxiv. This is the first version of the paper that we have submitted for peer review. Doublethink addresses some long-standing questions in assessing evidence for the purposes of hypothesis testing.

Hypothesis testing is central to scientific enquiry, but conclusions can be heavily influenced by model specification, particularly which variables are included. Bayesian model-averaged hypothesis testing offers a solution, but the sensitivity of posterior odds and Bayesian false discovery rate (FDR) guarantees to prior assumptions limit the appeal. In hypothesis testing, we lack unifying results – like Bernstein-von-Mises’ Theorem – that predict convergence of Bayesian and frequentist results, even in large samples.

Our paper introduces new theory and a practical method, Doublethink, motivated by these issues:

• A key, and perhaps surprising, result is that Bayesian model-averaged hypothesis testing natively controls not only the Bayesian FDR, but also the frequentist strong-sense familywise error rate (FWER). This duality – which is general – seems to be unknown, or forgotten.
• For practical application, we derive large-sample asymptotic theory to quantify the rate at which the FWER is controlled. Specifically, we use a BIC-like model to characterize the tail probability of the model-averaged posterior odds via a chi-squared distribution.

• This result enables simultaneous control of Bayesian FDR and frequentist FWER at quantifiable levels and – equivalently – simultaneous reporting of posterior odds and asymptotic p-values.
• We explore the method’s benefits – like post-hoc variable selection – and limitations – like inflation – through a Mendelian Randomization study and detailed simulations, comparing Doublethink to Lasso, stepwise regression, the Benjamini-Hochberg procedure and e-values.

Besides the practical benefits of model-averaged hypothesis testing with frequentist guarantees, and the implications that entails for objective Bayesian hypothesis testing, these results offer fundamental insights likely to trigger renewed discussion of FDR, FWER and the reconcilability of p-values with evidence.

Doublethink is a novel addition to the emerging class of heavy-tailed combination tests. Since 2019, methods like the Cauchy combination test and harmonic mean p-value have surfaced as powerful tools for combining hypothesis tests despite inter-test dependence. Doublethink improves on these methods by allowing model uncertainty in the null hypothesis and by improving power.

We believe this paper will be of broad interest, addressing questions of importance to statistical methodology, big data analysis and scientific enquiry more generally.

Explanation of variables above

• The model-averaged p-value, adjusted for multiple testing, is p*.
• The model-averaged posterior odds, calculated from a Bayesian analysis, is PO.
• The number of variables in the analysis is ν.
• The prior odds of including each variable are μ.
• The sample size n is represented by ξ_n, which decreases as √n increases.

Doublethink methods paper

Today we release the first full draft of the Doublethink methods paper. This is an evolution of what was originally conceived as the supplement to the Doublethink COVID-19 paper. The wider significance of the results persuaded us to separate the two, which now focus on:

• Doublethink methods paper: Broad connections between Bayesian and classical hypothesis testing that we hope bring the best of both world by enabling scientists to simultaneously control the Bayesian false discovery rate and the classical familywise error rate, in big data settings.
• Doublethink COVID-19 paper: Identifying direct risk factors for COVID-19 hospitalization among 2000 candidate variables in 200,000 UK Biobank participants. Compares results to the literature and considers the limitations imposed by mediation and complex 'exposome-wide' association studies.

After soliciting colleagues for comments and another round of editing, we will move toward submission in later this year.

Introducing Doublethink: joint Bayesian-frequentist model-averaged hypothesis testing

This week Nick Arning, Helen Fryer and I released two related preprints describing a new method called Doublethink, and its application to identifying risk factors for COVID-19 hospitalization in UK Biobank:

• Doublethink: simultaneous Bayesian-frequentist model-averaged hypothesis testing. Fryer, Arning, Wilson (2023) arXiv doi: 10.48550/arXiv.2312.17566

• Identifying direct risk factors in UK Biobank with simultaneous Bayesian-frequentist model-averaged hypothesis testing using Doublethink. Arning, Fryer, Wilson (2024) medRxiv doi: 10.1101/2024.01.01.24300687

Doublethink: Bayesian-frequentist model-averaged hypothesis testing

Doublethink enables joint Bayesian and frequentist hypothesis testing when there is model uncertainty by interconverting Bayesian posterior odds and classical (frequentist) p-values. It has broad implications because (i) it reveals connections between the Bayesian approach to model averaging and the classical approach to multiple testing, and (ii) it brings the benefits of Bayesian model averaging to classical statistics.

Doublethink addresses two fundamental problems in hypothesis testing:

1. In classical tests, the statistical evidence that one variable directly affects an outcome generally depends on which other variables are assumed to directly affect it.
2. In Bayesian tests, the statistical evidence that one variable directly affects an outcome depends on the prior assumptions.

These issues are addressed by computing p-values from Bayesian model-averaged posterior odds, which (1) account for model uncertainty and (2) are theoretically invariant to prior assumptions, assuming large sample sizes.

Doublethink simultaneously controls the frequentist family-wise error rate (FWER) and the Bayesian false discovery rate (FDR). It builds on Johnson's Bayesian tests based on likelihood ratio statistics, and Karamata's theory of regular variation.

Identifying direct risk factors in UK Biobank with Doublethink

We applied Doublethink to identify direct risk factors for COVID-19 hospitalization in UK Biobank. This is a well-studied problem but we took an 'exposome-wide' approach in which we evaluated whether 1,900 variables measured in the UK Biobank each affected the outcome. This is still an under-utilized approach in epidemiology, which usually focuses on candidate risk factors.

Exposome-wide approaches have potential benefits over candidate risk factor approaches, including:

• The ability to discover unexpected results.
• Stringent control for multiple testing.
• Avoidance of bias in choosing candidate risk factors or deciding to publish.

However, we only studied the direct effects of variables on the outcome. This means we cannot make statements about the total (direct and indirect) effects of a variable, e.g. smoking, on the outcome, which are needed in applications like assessing potential interventions.

We identified individual variables and groups of variables that were 'exposome-wide significant' at 9% FDR and 0.05% FWER, after accounting for the direct effects of all other variables.

Comparing our results to over 100 published studies of COVID-19 in UK Biobank, we

• Recapitulated several commonly reported direct risk factors, e.g. age, sex, and obesity.
• Excluded others, e.g. diabetes, cardiovascular disease, and hypertension, which might be mediated through other variables that measure general comorbidity.
• Identified some infrequently reported direct risk factors, both individually, e.g. lung infection, and as groups, e.g. constipation/urinary tract infection, which might reflect underlying kidney disease.

The ability to test groups of variables, which increases sensitivity, was one of the benefits of Doublethink's model-averaging approach. It is particularly helpful in large biobanks that measure thousands of variables, because correlation between variables is pervasive, and can dilute the significance of individual variables that measure similar phenomena, like the numerous types of deprivation index. It serves as a flexible alternative to pre-analysis variable filtering algorithms, while controlling the risk of false positives by pre-defining significance thresholds for all possible tests.

To read more, please check out the preprints here and here.