Impact of autocorrelation on functional connectivity

Abstract

Although the impact of serial correlation (autocorrelation) in residuals of general linear models for fMRI time-series has been studied extensively, the effect of autocorrelation on functional connectivity studies has been largely neglected until recently. Some recent studies based on results from economics have questioned the conventional estimation of functional connectivity and argue that not correcting for autocorrelation in fMRI time-series results in "spurious" correlation coefficients. In this paper, first we assess the effect of autocorrelation on Pearson correlation coefficient through theoretical approximation and simulation. Then we present this effect on real fMRI data. To our knowledge this is the first work comprehensively investigating the effect of autocorrelation on functional connectivity estimates. Our results show that although FC values are altered, even following correction for autocorrelation, results of hypothesis testing on FC values remain very similar to those before correction. In real data we show this is true for main effects and also for group difference testing between healthy controls and schizophrenia patients. We further discuss model order selection in the context of autoregressive processes, effects of frequency filtering and propose a preprocessing pipeline for connectivity studies.

Keywords: Autocorrelation; Autoregressive process; Functional connectivity; Independent component analysis; Resting-state fMRI.

Figure 1

A) Empirical bias and standard deviation of estimation of true ρ_w,z = +0.5 based on r_x,y and r_w,z for different combinations of AR(1) coefficients (α and β in Eq. (5) & (6)) for time-series x and y and different sample sizes (length of time-series x and y) of (64, 256, 1024) obtained from 10000 simulations. The empirical results are compared with theoretical bias and standard deviation of r_w,z and r_x,y derived in Eq. (3) & (4) and Eq. (13) & (16) respectively. The whiskers show standard deviation (square root of variance of the estimator). It is evident that theoretical and empirical results agree with each other. For equal coefficients, estimation of ρ_w,z based on r_x,y is unbiased. For different AR(1) coefficients, estimation is biased. The variance of the estimator increases as the product of AR(1) coefficients of x_t and y_t increases. B) Top row: Histogram of corrected and uncorrected empirical Pearson correlation coefficients (r_w,z and r_x,y) obtained from 10000 simulations based on Eq. (5) & (6) with sample size of 256 and true correlation of +0.5 for 3 different combination of α and β (Eq. 5 & 6). Bottom Row: Scatter plot of uncorrected correlation coefficients, r_x,y, against corrected correlation coefficients r_w,z. Correlation coefficient between r_w,z and r_x,y is provided in the bottom row scatter plots (r).

Figure 2

Spatial maps of selected 47 independent components grouped based on functionality into 7 categories: subcortical (5 components), auditory (2 components), visual (11 components), senorimotor (6 components), attention/cognitive control (13 components), default-mode network (8 components) and cerebellar (2 components).

Figure 3

Durbin-Watson statistics histogram for A: Uncorrected IC time-series for healthy controls B: Uncorrected IC time-series for schizophrenia patients C: Corrected IC time-series for healthy controls D: Corrected IC time-series for schizophrenia patients. Autocorrelation correction successfully concentrated DW statistics around 2 which is a sign for absence of autocorrelation.

Figure 4

A,B: Mean and standard deviation of FNC grouped by functionality of brain networks (Figure 2) for healthy controls before and after autocorrelation correction. C: -log(p-value)×sign of t-statics after subject-wise 1-sample t-test on each FNC pair before and after autocorrelation correction. Although the FNC values alter noticeably before and after autocorrelation correction, p-values remain very similar.

Figure 5

A,B: Mean and standard deviation of FNC grouped by functionality of brain networks (Figure 2) for schizophrenia patients before and after autocorrelation correction. C: -log(p-value)×sign of t-statics after subject-wise 1-sample t-test on each FNC pair before and after autocorrelation correction. Although the FNC values alter noticeably before and after autocorrelation correction, p-values remain very similar.

Figure 6

A: Difference in mean of FNC between healthy controls and schizophrenia patients (healthy-patients) grouped by functionality of brain networks (Figure 1) before and after autocorrelation correction. B: -log(p-value)×sign of t-statics after subject-wise 2-sample t-test between controls and patients before and after autocorrelation correction. Although the differences in FNC values between healthy controls and patients alter noticeably before and after autocorrelation correction, p-values of 2-sample t-test remain very similar.

Figure 7

A: Histogram of corrected and uncorrected FNC values (pooled all subjects and pairs) for healthy controls and schizophrenia patients. B: Scatter plot of uncorrected FNC values against corrected FNC values for healthy controls and schizophrenia patients. Correlation coefficient between corrected and uncorrected FNC values is high for both groups (r = +0.89). Compare these results with simulation results in Figure 1B (especially for α = β = 0.5). C: Scatter plot of –log(p_value) × sign (T_Statictics) before and after autocorrelation correction for healthy controls and schizophrenia patients (these are scatter plots of color-coded values in Figure 4C and 5C).

Figure 8

Histogram of AR coefficient for pooled IC time-series for all subject for healthy controls and schizophrenia patients if all time-series are corrected with AR(1).

Figure 9

Autocorrelation function with 95% confidence interval lines and amplitude of frequency spectra for two fMRI time-series, x(t), y(t) before autocorrelation correction (left column), after autocorrelation correction with AR(4) model (middle column) and after frequency filtering with a order 6 Butterworth passband filter with cutoff frequencies of 0.01 Hz and .10 Hz (right column). While autocorrelation correction improves the autocorrelation function (all values are inside 95% confidence interval), frequency filtering introduce back the autocorrelation in a more severe and complicated manner.

Figure 10

A: Canonical HRF function B: Best model order based on AIC for correcting autocorrelation of samples taken with different TR (repetition time) from the HRF function. Best model order increases exponentially as TR decreases.

Figure 11

A: Two correlated low frequency time-series (200 time-points) B: After adding high frequency noise to the original time-series in part A (SNR = 20db) C: Time-series in part B passed through AR(1) process with coefficients of +0.6. Autocorrelation acts as a low pass filter and enhances the correlation between two noisy signals in part B close to the original level in part A.