A Bayesian approach to False Discovery Rate and Power in Multiple Testing

Introduction The practical interest that motivated this study concerns the construction of brain networks as synthesis of patterns of functional connectivity. Functional connectivity is quantified through non-parametric correlation’s measures between non-overlapping anatomical ROIs. The final brain network is defined as a map containing only significant signals among brain regions. The statistical challenge inherent this approach deals with the multiple testing problem. To this purpose, we developed an innovative methodology to control for multiple testing in a Bayesian framework (Efron, 2010). In particular, we proposed a method of threshold selection which provides the optimal balance between the rate of false discoveries (FDR) and the power. Methods The Neuroradiology Department of the Niguarda Ca’ Granda Hospital acquired, with a 1.5 T General Electrics Scanner, MRI scans of two healthy women of 33 and 67 years old, both with a normal cognitive profile. Having defined 116 non-overlapping anatomical ROIs according to the automated anatomical labelling (AAL) atlas, we extracted 116 time-series representing the mean low-frequency fluctuations, being reflections of spontaneous neuronal activity. Separately for each subject, we computed a correlation matrix between the 116 time-series by means of the Spearman’s rank coefficient. With the aim of constructing brain maps, we defined the empirical estimates of Bayes FDR and power and we set up a methodology to control them simultaneously. In our paradigm, an estimate is said to be empirical when the a priori probability of a null hypothesis to be true, π_0, must be empirically estimated. Having set a threshold γ, which is the significant level of a single comparison (Sala, 2014), the proposed methodology identifies the balanced γ, namely the threshold which guarantees an FDR less than a maximum acceptable error, and a maximal power. Results We demonstrated that the Bayesian estimates of FDR and power show good theoretical properties, such as conservativeness and unbiasedness. Furthermore, we verified the robustness of our empirical estimates through some simulation studies under diverse patterns of dependency among p-values. In the light of these promising results, we applied our new methodology of threshold selection and brain network construction to fMRI data. Having selected the optimal threshold, separately for each participant, we constructed two brain networks, both with a rate of false discoveries less than 0.009 and a power greater than 0.5. At last, we compared the aforementioned networks through some network measures (Kolaczyk, 2010). Discussion Many well-established methodologies of threshold selection completely ignore the power thus leading to possibly misleading results. Conversely, our methodology satisfies both the need for a systematic and reproducible approach of threshold selection and the need for a simultaneous control of FDR and power of the tests. Furthermore, thanks to the Bayesian approach to estimate, any a priori information about π_0 can be easily integrated. With respect to the application on real data, we proved the feasibility of our new methodology. At last, the analysis of subject-specific brain networks, which evidences a less dense network for the young participant, leads us to ask new questions about the effects of the aging process on the activation of undifferentiated neural activity.