Effective degrees of freedom of the Pearson's correlation coefficient under autocorrelation
Authors: Stephen M Smith
Soroosh Afyouni
Thomas E Nichols
Journal: NeuroImage (2019)
Discipline: NEUROSCIENCE & BEHAVIOR
Keywords: Autocorrelation; Cross correlation; Functional connectivity; Graph theory; Pearson correlation coefficient
Affiliations: Oxford Big Data Institute, Li Ka Shing Centre for Health Information and Discovery, Nuffield Department of Population Health, University of Oxford, UK. Electronic address: ., Oxford Big Data Institute, Li Ka Shing Centre for Health Information and Discovery, Nuffield Department of Population Health, University of Oxford, UK; The Wellcome Centre for Integrative Neuroimaging, University of Oxford, UK. Electronic address: ., Oxford Big Data Institute, Li Ka Shing Centre for Health Information and Discovery, Nuffield Department of Population Health, University of Oxford, UK; The Wellcome Centre for Integrative Neuroimaging, University of Oxford, UK; Department of Statistics, University of Warwick, UK. Electronic address: .

Abstract:  The dependence between pairs of time series is commonly quantified by Pearson's correlation. However, if the time series are themselves dependent (i.e. exhibit temporal autocorrelation), the effective degrees of freedom (EDF) are reduced, the standard error of the sample correlation coefficient is biased, and Fisher's transformation fails to stabilise the variance. Since fMRI time series are notoriously autocorrelated, the issue of biased standard errors - before or after Fisher's transformation - becomes vital in individual-level analysis of resting-state functional connectivity (rsFC) and must be addressed anytime a standardised Z-score is computed. We find that the severity of autocorrelation is highly dependent on spatial characteristics of brain regions, such as the size of regions of interest and the spatial location of those regions. We further show that the available EDF estimators make restrictive assumptions that are not supported by the data, resulting in biased rsFC inferences that lead to distorted topological descriptions of the connectome on the individual level. We propose a practical "xDF" method that accounts not only for distinct autocorrelation in each time series, but instantaneous and lagged cross-correlation. We find the xDF correction varies substantially over node pairs, indicating the limitations of global EDF corrections used previously. In addition to extensive synthetic and real data validations, we investigate the impact of this correction on rsFC measures in data from the Young Adult Human Connectome Project, showing that accounting for autocorrelation dramatically changes fundamental graph theoretical measures relative to no correction. 

Retracted: No

PubPeer comments:
 It appears that the  Gordon panel in Figure 2A is either mislabeled or features the wrong parcellation. The Gordon parcellation does not include subcortical or cerebellar regions and is restricted to the cortical ribbon, but the image labeled  Gordon in Figure 2A includes both subcortical and cerebellar regions and extends considerably into the white matter. , 
 Thank you so much for catching this! There was in fact a copy and paste error, and that is the wrong image. Fortunately we re still at the proofing stage so we ll be able to fix the image before publication. The corrected figure is attached. ![file] ,