Spatial extent inference (SEI) is widely used across neuroimaging modalities to study brain-phenotype associations that inform our understanding of disease. Recent studies have shown that Gaussian random field (GRF) based tools can have inflated family-wise error rates (FWERs). This has led to fervent discussion as to which preprocessing steps are necessary to control the FWER using GRF-based SEI. The failure of GRF-based methods is due to unrealistic assumptions about the covariance function of the imaging data. The permutation procedure is the most robust SEI tool because it estimates the covariance function from the imaging data. However, the permutation procedure can fail because its assumption of exchangeability is violated in many imaging modalities. Here, we propose the (semi-) parametric bootstrap joint (PBJ; sPBJ) testing procedures that are designed for SEI of multilevel imaging data. The sPBJ procedure uses a robust estimate of the covariance function, which yields consistent estimates of standard errors, even if the covariance model is misspecified. We use our methods to study the association between performance and executive functioning in a working fMRI study. The sPBJ procedure is robust to variance misspecification and maintains nominal FWER in small samples, in contrast to the GRF methods. The sPBJ also has equal or superior power to the PBJ and permutation procedures. We provide an R package https://github.com/simonvandekar/pbj to perform inference using the PBJ and sPBJ procedures
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