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In a recent paper on complex-valued fMRI detection by Lee et al. (2007), a statistical model for magnitude and phase changes is presented (1). This follows a line of published research on the topic (2,3,4,5) motivated by the fact that fMRI phase data contains biological information regarding the vasculature contained within voxels (6,7). The Lee et al. (2007) model is elegant and computationally efficient but there are four items regarding it that need to be clarified in addition to its relationship to the Rowe (2005) model (5).
The Rowe (2005) model for detecting magnitude and phase changes in complex-valued data is
where at time t, t=1,…,n, yRt and yI are the observed real and imaginary observations. In addition, is the magnitude signal, is the tth row of a design matrix X describing temporal magnitude changes, β is a vector of magnitude regression coefficients, is the phase signal, is the tth row of a design matrix U describing temporal phase changes, γ is a vector of phase regression coefficients. Finally, ηRt and ηIt are the real and imaginary measurement error that are independent and identically distributed N(0,σ2) variables. Several hypothesis pairs are presented with suitable selection from Cβ=0, Cβ≠0, Dγ=0, and Dγ≠0.
The Lee et al. (2007) model is
where βR and βI are regression coefficients for the real and imaginary parts of the signal and all other variables are as previously defined. Lee et al. (2007) correctly describe that their model is to be used when the magnitude and phase design matrices are the same (U=X) in addition to the same contrast matrices (v=C=D).
The items that need to be clarified are that Lee et al. (2007) state that:
In spite of these inaccuracies, the Lee et al. (2007) model is elegant and is recommended when the magnitude and phase design matrices are identical with a column of ones for a constant baseline and a column with on/off (0/1 or −1/+1) elements for the reference waveform vector.
This work was supported in part by NIH R01 EB000215 and R01 EB007827.