Combining voxel intensity and cluster extent with permutation test framework.

In a massively univariate analysis of brain image data, statistical inference is typically based on intensity or spatial extent of signals. Voxel intensity-based tests provide great sensitivity for high intensity signals, whereas cluster extent-based tests are sensitive to spatially extended signals. To benefit from the strength of both, the intensity and extent information needs to be combined. Various ways of combining voxel intensity and cluster extent are possible, and a few such combining methods have been proposed. Poline et al.'s [NeuroImage 16 (1997) 83] minimum P value approach is sensitive to signals whose either intensity or extent is significant. Bullmore et al.'s [IEEE Trans. Med. Imag. 18 (1999) 32] cluster mass method can detect signals whose intensity and extent are sufficiently large, even when they are not significant by intensity or extent alone. In this work, we study such combined inference methods using combining functions (Pesarin, F., 2001. Multivariate Permutation Tests. Wiley, New York) and permutation framework [Holmes et al., J. Cereb. Blood Flow Metab. 16 (1996) 7], which allow us to examine different ways of combining voxel intensity and cluster extent information without knowing their distribution. We also attempt to calibrate combined inference by using weighted combining functions, which adjust the test according to signals of interest. Furthermore, we propose meta-combining, a combining function of combining functions, which integrates strengths of multiple combining functions into a single statistic. We found that combined tests are able to detect signals that are not detected by voxel or cluster size test alone. We also found that the weighted combining functions can calibrate the combined test according to the signals of interest, emphasizing either intensity or extent as appropriate. Though not necessarily more sensitive than individual combining functions, the meta-combining function is sensitive to all types of signals and thus can be used as a single test summarizing all the combining functions.