Generalized least-squares method applied to fMRI time series with empirically determined correlation matrix.

Functional magnetic resonance imaging (fMRI) time series analysis and statistical inferences about the effect of a cognitive task on the regional cerebral blood flow (rCBF) are largely based on the linear model. However, this method requires that the error vector is a gaussian variable with an identity correlation matrix. When this assumption cannot be accepted, statistical inferences can be made using generalized least squares. In this case, knowledge of the covariance matrix of the error vector is needed. In the present report, we propose a method that needs stationarity of the autocorrelation function but is more flexible than autoregressive model of order p (AR(p)) models because it is not necessary to predefine a relation between coefficients of the correlation matrix. We tested this method on sets of simulated data (with presence of an effect of interest or not) representing a time series with a monotonically decreasing autocorrelation function. This time series mimicked an experiment using a random event-related design that does not create correlation between scans. The autocorrelation function is empirically determined and used to reconstitute the correlation matrix as the toeplitz matrix built from the autocorrelation function. When applied to simulated time series with no effect of interest, this method allows the determination of F values corresponding to the accurate false positive level. Moreover, when applied to time series with an effect of interest, this method gives a density function of F values which allows the rejection of the null hypothesis. This method provides a flexible but interpretable time domain noise model.