Residual analysis for detecting mis-modeling in fMRI

The voxel-wise general linear model (GLM) approach has arguably become the dominant way to analyze functional magnetic resonance imaging (fMRI) data. The approach relies on specifying predicted patterns of signal change a priori. In this work we develop methods for detecting mis-modeling in the GLM framework, and derive mathematical expressions for quantifying the effects this has on bias and power. We show that even a relatively small amount of mis-modeling can result in severe power loss, and can inflate the false positive rate beyond the nominal value. Due to the massive amount of data, examining the appropriateness of the model is challenging in fMRI. We propose a simple procedure involving the residuals that can be used to identify possible voxels or regions of the brain where model misfit may be present. The key idea is that if there is model misfit - such as a misspecification of onset, duration, or response shape - residuals will be systematically larger in mis-modeled segments of the time series. By looking at the weighted sum of consecutive residuals using a moving window, our method can pick out regions of a residual time series in which the residuals are consistently larger than expected by chance, while ignoring spurious large residuals that are expected based on the noise distribution. It may also be used more generally for identifying artifacts in fMRI time courses. We investigate the effectiveness of this method using a simulation study, and by applying it to an fMRI dataset. We develop a method and accompanying software for creating whole-brain maps showing power loss and bias due to mis-modeling. Such maps could be a valuable tool in assessing violations of statistical assumptions and informing about differences in the shape and timing of the hemodynamic response function (HRF) across the brain.