Testable Hypotheses for Unbalanced Neuroimaging Data

Unbalanced group-level models are common in neuroimaging. Typically, data for these models come from factorial experiments. As such, analyses typically take the form of an analysis of variance (ANOVA) within the framework of the general linear model (GLM). Although ANOVA theory is well established for the balanced case, in unbalanced designs there are multiple ways of decomposing the sums-of-squares of the data. This leads to several methods of forming test statistics when the model contains multiple factors and interactions. Although the Type I-III sums of squares have a long history of debate in the statistical literature, there has seemingly been no consideration of this aspect of the GLM in neuroimaging. In this paper we present an exposition of these different forms of hypotheses for the neuroimaging researcher, discussing their derivation as estimable functions of ANOVA models, and discussing the relative merits of each. Finally, we demonstrate how the different hypothesis tests can be implemented using contrasts in analysis software, presenting examples in SPM and FSL.