Multiple hypothesis mapping of functional MRI data in orthogonal and complex wavelet domains

We are interested in methods for multiple hypothesis testing that optimize power to refute the null hypothesis while controlling the false discovery rate (FDR). The wavelet transform of a spatial map of brain activation statistics can be tested in two stages to achieve this objective: First, a set of possible wavelet coefficients to test is reduced, and second, each hypothesis in the remaining subset is formally tested. We show that a Bayesian bivariate shrinkage operator (BaybiShrink) for the first step provides a powerful and expedient alternative to a subband adaptive chi-squared test or an enhanced FDR algorithm based on the generalized degrees of freedom. We also investigate the dual-tree complex wavelet transform (CWT) as an alternative basis to the orthogonal discrete wavelet transform (DWT). We design and validate a test for activation based on the magnitude of the complex wavelet coefficients and show that this confers improved specificity for mapping spatial signals. The methods are applied to simulated and experimental data, including a pharmacological magnetic resonance imaging (MRI) study. We conclude that using BaybiShrink to define a reduced set of complex wavelet coefficients, and testing the magnitude of each complex pair to control the FDR, represents a competitive solution for multiple hypothesis mapping in fMRI.