Estimating the effective degrees of freedom in univariate multiple regression analysis

The general linear model provides the most widely applied statistical framework for analyzing functional MRI (fMRI) data. With the increasing temporal resolution of recent scanning protocols, and more elaborate data preprocessing schemes, data independency is no longer a valid assumption. In this paper, we revise the statistical background of the general linear model in the presence of temporal autocorrelations. First, when detecting the activation signal, we explicitly account for the temporal autocorrelation structure, which yields a generalized F-test and the associated corrected (or effective) degrees of freedom (DOF). The proposed approach is data driven and thus independent of any specific preprocessing method. Then, for event-related protocols, we propose a new model for the temporal autocorrelations ("damped oscillator" model) and compare this model to another, previously used in the field (first-order autoregressive model, or AR(1) model). In the case of long fMRI time series, an efficient approximation for the number of effective DOF is provided for both models. Finally, the validity of our approach is assessed using simulated and real fMRI data and is compared with more conventional methods.