# Handling Multiplicity in Neuroimaging Through Bayesian Lenses with Multilevel Modeling

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## Abstract

Here we address the current issues of inefficiency and over-penalization in the massively univariate approach followed by the correction for multiple testing, and propose a more efficient model that pools and shares information among brain regions. Using Bayesian multilevel (BML) modeling, we control two types of error that are more relevant than the conventional false positive rate (FPR): incorrect sign (type S) and incorrect magnitude (type M). BML also aims to achieve two goals: 1) improving modeling efficiency by having one integrative model and thereby dissolving the multiple testing issue, and 2) turning the focus of conventional null hypothesis significant testing (NHST) on FPR into quality control by calibrating type S errors while maintaining a reasonable level of inference efficiency. The performance and validity of this approach are demonstrated through an application at the region of interest (ROI) level, with all the regions on an equal footing: unlike the current approaches under NHST, small regions are not disadvantaged simply because of their physical size. In addition, compared to the massively univariate approach, BML may simultaneously achieve increased spatial specificity and inference efficiency, and promote results reporting in totality and transparency. The benefits of BML are illustrated in performance and quality checking using an experimental dataset. The methodology also avoids the current practice of sharp and arbitrary thresholding in the *p*-value funnel to which the multidimensional data are reduced. The BML approach with its auxiliary tools is available as part of the AFNI suite for general use.

## Keywords

Null Hypothesis Significance Testing (NHST) False Positive Rate (FPR) Type S and type M errors Regions of Interest (ROIs) General Linear Model (GLM) Linear Mixed-Effects (LME) modeling Bayesian Multilevel (BML) modeling Markov Chain Monte Carlo (MCMC) Stan Priors Leave-one-out (LOO) cross-validation## Notes

### Acknowledgments

The research and writing of the paper were supported (GC, PAT, and RWC) by the NIMH and NINDS Intramural Research Programs (ZICMH002888) of the NIH/HHS, USA, and by the NIH grant R01HD079518A to TR and ER. Much of the modeling work here was inspired from Andrew Gelman’s blog. We are indebted to Paul-Christian Bürkner and the Stan development team members Ben Goodrich, Daniel Simpson, Jonah Sol Gabry, Bob Carpenter, and Michael Betancourt for their help and technical support. The simulations were performed in the R language for statistical computing and the figures were generated with the R package ggplot2 (Wickham 2009).

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