Update 2023: My group has published a paper on spatially varying prewhitening: Parlak et al. “Sources of residual autocorrelation in multiband task fMRI and strategies for effective mitigation.” Frontiers in Neuroscience 16 (2023): 1051424. https://doi.org/10.3389/fnins.2022.1051424. We also developed a computationally efficient spatially variable prewhitening technique, implemented in the BayesfMRI R package (CRAN, Github).

I recently got around to reading the high-profile paper “Cluster failure: Why fMRI inferences for spatial extent have inflated false-positive rates” (Eklund et al. 2016), which called attention to the importance of properly processing and analyzing fMRI data. This lead me to an earlier paper titled “Does parametric fMRI analysis with SPM yield valid results?” (Eklund et al. 2012), which identified mismodeling of temporal autocorrelation as a major factor driving high false positive rates. From a statistical perspective, accounting for temporal autocorrelation is vital to obtain correct standard errors on model coefficients, which are the basis for hypothesis tests used to identify areas of activation in the brain, a primary goal in many fMRI studies. (In fact many group analyses actually ignore first-level standard errors, but these should ideally be incorporated into the second-level model.)

The 2012 paper pointed out two primary reasons for mismodeling of temporal autocorrelation: (a) using too low of a model order, such as AR(1), especially for faster TRs, and (b) assuming the same AR coefficients across the brain. They also recommend smoothing the coefficients to improve the estimates.

As a statistician, I was excited to try out this “correct” approach to prewhitening. Increasing the model order was no big deal (I used an AR(6) model for TR=0.72 seconds), but doing the prewhitening at the voxel level was a bit of a hassle. After much trial and error and a few new gray hairs, here’s what I’ve learned. Some of the computational strategies are specific to R, but they can probably be easily adapted to MATLAB or Python.

1. AR coefficients definitely seem to vary across the brain.

2. Averaging across subjects greatly improves the smoothness of the coefficient maps, which is presumably indicative of better estimation.

3. The prewhitening matrix (the square root inverse covariance matrix) needs to be computed quickly, since you’re going to be doing it separately for every voxel in the brain.

I’ll go into each of these in turn below, but first, let’s review why and how prewhitening is actually performed. fMRI task activation studies typically rely on fitting a regression model at many different locations in the brain, and ordinary least squares regression assumes uncorrelated errors. This assumption is the basis of how we compute standard errors of coefficient estimates, which in turn are the basis of hypothesis testing, which allows us to identify regions of the brain that are active in response to different tasks. The validity of these hypothesis tests depends primarily on (1) how correction for multiple comparisons was performed (the focus of the 2016 paper mentioned above) and (2) the accuracy of the standard errors of the model coefficients. If we ignore temporal autocorrelation, we will underestimate the true standard errors and obtain more false positives.

So how do we fix this?

**… this post was originally written on my old blog. Read the full post here. **