Analysis of fMRI time-series revisited--again

Friston et al. (1995, NeuroImage 2:45-53) presented a method for detecting activations in fMRI time-series based on the general linear model and a heuristic analysis of the effective degrees of freedom. In this communication we present corrected results that replace those of the previous paper and solve the same problem without recourse to heuristic arguments. Specifically we introduce a proper and unbiased estimator for the error terms and provide a more generally correct expression for the effective degrees of freedom. The previous estimates of error variance were biased and, in some instances, could have led to a 10-20% overestimate of Z values. Although the previous results are almost correct for the random regressors chosen for validation, the present theoretical results are exact for any covariate or waveform. We comment on some aspects of experimental design and data analysis, in the light of the theoretical framework discussed here.