An Introduction to Random Field Theory

This chapter is an introduction to the multiple comparison problem in functional imaging, and the way it can be solved using Random field theory (RFT). In a standard functional imaging analysis, we fit a statistical model to the data, to give us model parameters. We then use the model parameters to look for an effect we are interested in, such as the difference between a task and baseline. To do this, we usually calculate a statistic for each brain voxel that tests for the effect of interest in that voxel. The result is a large volume of statistic values. We now need to decide if this volume shows any evidence of the effect. To do this, we have to take into account that there are many thousands of voxels and therefore many thousands of statistic values. This is the multiple comparison problem in functional imaging. Random field theory is a recent branch of mathematics that can be used to solve this problem. To explain the use of RFT, we will first go back to the basics of hypothesis testing in statistics. We describe the multiple comparison problem and the usual solution, which is the Bonferroni correction. We explain why spatial correlation in imaging data causes problems for the Bonferroni correction and introduce RFT as a solution. Finally, we discuss the assumptions underlying RFT and the problems that arise when these assumptions do not hold. We hope this chapter will be accessible to those with no specific expertise in mathematics or statistics. Those more interested in mathematical details and recent developments are referred to Chapter 15.