Valid conjunction inference with the minimum statistic

In logic a conjunction is defined as an AND between truth statements. In neuroimaging, investigators may look for brain areas activated by task A AND by task B, or a conjunction of tasks (Price, C.J., Friston, K.J., 1997. Cognitive conjunction: a new approach to brain activation experiments. NeuroImage 5, 261-270). Friston et al. (Friston, K., Holmes, A., Price, C., Büchel, C., Worsley, K., 1999. Multisubject fMRI studies and conjunction analyses. NeuroImage 10, 85-396) introduced a minimum statistic test for conjunction. We refer to this method as the minimum statistic compared to the global null (MS/GN). The MS/GN is implemented in SPM2 and SPM99 software, and has been widely used as a test of conjunction. However, we assert that it does not have the correct null hypothesis for a test of logical AND, and further, this has led to confusion in the neuroimaging community. In this paper, we define a conjunction and explain the problem with the MS/GN test as a conjunction method. We present a survey of recent practice in neuroimaging which reveals that the MS/GN test is very often misinterpreted as evidence of a logical AND. We show that a correct test for a logical AND requires that all the comparisons in the conjunction are individually significant. This result holds even if the comparisons are not independent. We suggest that the revised test proposed here is the appropriate means for conjunction inference in neuroimaging.