On bias in the estimation of autocorrelations for fMRI voxel time-series analysis

For fMRI time-series analysis to be statistically valid, it is important to deal correctly with temporal autocorrelation in the noise. Most of the approaches in the literature adopt a two-stage approach in which the autocorrelation structure is estimated using the residuals of an initial model fit. This estimate is then used to "prewhiten" the data and the model before the model is refit to obtain final activation parameter estimates. An assumption implicit in this scheme is that the residuals from the initial model fit represent a realization of the "true" noise process. In general this assumption will not be correct as certain components of the noise will be removed by the model fit. In this paper we examine (i) the form of the bias induced by the initial model fit, (ii) methods of correcting for the bias, and (iii) the impact of bias correction on the model parameter estimates. We find that while bias correction does result in more accurate estimates of the correlation structure, this does not translate into improved estimates of the model parameters. In fact estimates of the model parameters and their standard errors are seen to be so accurate that we conclude that bias correction is unnecessary.