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A Least Trimmed Square Regression Method for Second Level fMRI Effective Connectivity Analysis

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Abstract

We present a least trimmed square (LTS) robust regression method to combine different runs/subjects for second/high level effective connectivity analysis. The basic idea of this method is to treat the extreme nonlinear model variability as outliers if they exceed a certain threshold. A bootstrap method for the LTS estimation is employed to detect model outliers. We compared the LTS robust method with a non-robust method using simulated and real datasets. The difference between LTS and the non-robust method for second level effective connectivity analysis is significant, suggesting the conventional non-robust method is easily affected by the model variability from the first level analysis. In addition, after these outliers are detected and excluded for the high level analysis, the model coefficients of the second level are combined within the framework of a mixed model. The variance of the mixed model is estimated using the Newton–Raphson (NR) type Levenberg-Marquardt algorithm. Three sets of real data are adopted to compare conventional methods which do not include random effects in the analysis with a mixed model for second level effective connectivity analysis. The results show that the conventional method is significantly different from the mixed model when greater model variability exists, suggesting there is a strong random effect, and the mixed model should be employed for the second level effective connectivity analysis.

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Acknowledgments

This study is supported under the Centre of Excellence in Intelligent Systems (CoEIS) project, funded by InvestNI and the Integrated Development Fund, through ILEX. The data collection in this study was supported by a CIHR grant (# MOP53346) to Robert F. Hess) and Kathy T. Mullen (#MOP10819).

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The authors declare that we have no conflict of interest.

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Correspondence to Xingfeng Li.

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Appendices

Appendix 1: Levenberg-Marquardt Algorithm for the Mixed Effect Model

From Eq. 4, we can estimate the effect from the first level with \( E(y) = X\beta \), \( {\mathop{\text var}\nolimits} (y) = \varSigma \), where

$$ \varSigma = \sigma_{fixed}^2 + Q\sigma_{random}^2 $$
(A1)

\( \sigma_{fixed}^2 \) is the variance from the first level analysis, \( \sigma_{random}^2 \) is the random effects variance from the second level fMRI data analysis. In this study, we consider the simple case where the covariance components \( Q = I \) (Friston et al. 2005), where \( I \) is the \( n \times n \) identity matrix, and \( n \) is total number of runs/subjects to be used within the mixed effect model. From multivariate statistical analysis (Anderson 1984), we know that the probability density of the data \( y \) has the form of the multivariate normal distribution (Lynch and Bruce 1998), i.e.,

$$ f(y) = \left( {2\pi } \right){}^{{{- n \left/ {2} \right.}}}{\left| \varSigma \right|^{{{- 1 \left/ {2} \right.}}}}\exp \left[ { - \frac{1}{2}\left( {y - X\beta } \right){}^T{\varSigma^{- 1 }}\left( {y - X\beta } \right)} \right] $$
(A2)

Taking the natural logarithm of the expression on the right side of Eq. A2 yields the log-likelihood of \( \beta \) and \( \varSigma \) given the observed data \( (y,X) \) as (Anderson 1984; Harville 1977; Searle et al. 1992):

$$ \ell \left( {y,\beta } \right) = - \frac{n}{2}\ln \left( {2\pi } \right) - \frac{1}{2}\ln \left| \varSigma \right| - \frac{1}{2}\left( {y - X\beta } \right){}^T{\varSigma^{- 1 }}\left( {y - X\beta } \right) $$
(A3)

From matrix theory (Searle et al. 1992) and using:

$$ \frac{{\partial \ln \left| \varSigma \right|}}{{\partial \theta }} = tr\left( {{\varSigma^{- 1 }}\frac{{\partial \varSigma }}{{\partial \theta }}} \right) $$
(A4)
$$ \frac{{\partial \varSigma {}^{- 1 }}}{{\partial \theta }} = - {\varSigma^{- 1 }}\frac{{\partial \varSigma }}{{\partial \theta }}{\varSigma^{- 1 }} $$
(A5)

the score function \( S \) is calculated from Eq. A3 as:

$$ S = \frac{{\partial \ell }}{{\partial \theta }} = - \frac{1}{2}tr\left( {{\varSigma^{- 1 }}{\Sigma_{\theta }}} \right) + \frac{1}{2}{\left( {y - X\beta } \right)^T}\varSigma {}^{- 1 }{\Sigma_{\theta }}{\varSigma^{- 1 }}\left( {y - X\beta } \right) $$
(A6)

Since we are interested in the random effect in the second level analysis, we set \( \theta = \sigma_{random}^2 \), then we have \( \frac{{\partial \varSigma }}{{\partial \theta }} = \frac{{\partial \varSigma }}{{\partial \sigma_{random}^2}} = {\Sigma_{\theta }} = I \). Therefore, the score function \( S \) in Eq. A6 becomes:

$$ S = \frac{1}{2}\left[ { - tr\left( {{\varSigma^{- 1 }}} \right) + {{\left( {y - X\beta } \right)}^T}{\varSigma^{- 1 }}{\varSigma^{- 1 }}\left( {y - X\beta } \right)} \right] $$
(A7)

and the corresponding Hessian matrix is:

$$ \begin{array}{*{20}{c}} {H=\frac{{{{\partial }^{2}}\ell }}{{{{\partial }^{2}}\theta }}=\frac{1}{2}\left[ {tr\left( {{{\Sigma }^{{-1}}}{{\Sigma }_{i}}{{\Sigma }^{{-1}}}} \right)-{{{\left( {y-X\beta } \right)}}^{T}}{{\Sigma }^{{-1}}}{{\Sigma }_{i}}{{\Sigma }^{{-1}}}{{\Sigma }^{{-1}}}\left( {y-X\beta } \right)-{{{\left( {y-X\beta } \right)}}^{T}}{{\Sigma }^{{-1}}}{{\Sigma }^{{-1}}}{{\Sigma }_{i}}{{\Sigma }^{{-1}}}\left( {y-X\beta } \right)} \right]} \\ {=\frac{1}{2}\left[ {tr\left( {{{{\left( {{{\Sigma }^{{-1}}}} \right)}}^{2}}} \right)-2{{{\left( {y-X\beta } \right)}}^{T}}{{{\left( {{{\Sigma }^{{-1}}}} \right)}}^{3}}\left( {y-X\beta } \right)} \right]} \\ \end{array} $$
(A8)

The Newton–Raphson (NR) algorithm has been used to solve the ML and REML problems (Harville 1977). The NR method obtains the REML estimate of the vector of parameters \( \theta = \sigma_{random}^2 \), by starting with some initial value, i.e., \( {\theta^{(0) }} = {{{y^T {R_I}y}} \left/ {v} \right.} \), \( {\lambda^{(0) }} = 1000 \) , where \( {R_I} = I - X\left( {X^T X} \right){}^{+ }{X^T} \) and then iterating to a final solution using Levenberg-Marquardt algorithm (Marquardt 1963):

$$ {\widehat{\theta}^{{\left( {k + 1} \right)}}} = {\widehat{\theta}^{(k) }} - \left[ {H\left( {{{\widehat{\theta}}^{(k) }}} \right) + \lambda I} \right]{}^{- 1 }S\left( {{{\widehat{\theta}}^{(k) }}} \right) $$
(A9)

where \( H \) is the Hessian matrix of all second-order partial derivatives of the log-likelihood function (Eq. 4) with respect to the variance components; \( \lambda \) is damper factor, we set initial value \( \lambda = 1000 \) and double the value in each iteration; \( I \) is identity matrix.

Appendix 2: fMRI Data Collection and Pre-Processing

  1. 1.

    The first dataset: retinotopic mapping experiment

The first dataset was acquired using a retinotopic mapping stimuli experiment with a phased-encoded design. This study was performed with the informed consent of the subjects and approved by the Montreal Neurological Institute Research Ethics Committee of McGill University (Montréal, Canada). A Siemens 1.5 T Magnetom scanner was used to collect both anatomical and functional images in the first experiment. Eleven normal subjects (age 32.55 ± 4.5 years) were used in the first experiment (Li et al. 2007a, b). Briefly, anatomical images were acquired using a rectangular (14.5" × 6.5") head coil (circularly polarized transmit and receive) and a T1 weighted sequence (repetition time (TR) = 22 ms; echo time (TE) = 10 ms; flip angle = 30°) giving 176 sagittal slices of 256 × 256 mm2 image voxels. Functional scans for each subject were collected using a surface coil (circularly polarized, receive only) positioned beneath the subject’s occiput. Each functional imaging session was preceded by a surface coil anatomical scan (identical to the head coil anatomical sequence, except that 80 × 256 × 256 sagittal images of slice thickness 2 mm were acquired) in order to later co-register the data with the more homogeneous head-coil images. Functional scans were multislice T2*-weighted, gradient-echo, planar images (GE-EPI, TR = 3.0 s, TE = 51 ms, flip angle = 90°). The image volume consisted of 30 slices orthogonal to the calcarine sulcus. The field of view was 256 × 256 mm, the matrix size was 64 × 64 with a thickness of 4 mm yielding voxel sizes of 4 × 4 × 4 mm.

For functional data collection, phase-encoded designs (Engel et al. 1997; Sereno et al. 1995; Warnking et al. 2002) were used and each visual retinotopic experiment (phase-encoded design, travelling square wave) consisted of four acquisition runs for each eye (two eccentricity runs, two polar angle runs, two clockwise order runs, and two count-clockwise runs) and 128 image volumes acquired at three second intervals for the left and right eye of normal participants. Runs were alternated between the eyes in each case while the subject was performing a task to keep awake in the scanner. The eye not being stimulated was occluded with a black patch that excluded all light from the eye. Subjects monocularly viewed a stimulus back-projected into the bore of the scanner and viewed through an angled mirror. In addition, the middle temporal (MT) cortex or V5 cortex localizer experiment was conducted for seven normal subjects. The experiment consisted of two to five acquisition runs for both eyes using checkerboard contrast stimulus (Dumoulin et al. 2000). During the MT localizer scanning sessions, subjects binocularly viewed a stimulus back-projected into the bore of the scanner and viewed through an angled mirror. In the data pre-processing, dynamic motion correction for functional image time series for each run and for different runs were realigned at the same time by using the fmr_preprocess function (provided in the MINC software package: (http://noodles.bic.mni.mcgill.ca/ServicesSoftware/HomePage) with default parameters of three-dimensional Gaussian low-pass filtering. The first eight scans of each functional run were discarded due to start-up magnetization transients in the data, so that only 120 image volumes were used in each run. We defined the common boundaries of different visual areas (from V1 to V4) by calculating the retinotopic visual field sign map (Sereno et al. 1995; Warnking et al. 2002) information of each subject.

Because the direct average method can blur hemodynamic delay, a random sample method was applied to select one fMRI response from each region for the effective connectivity study. We repeated this 20 times, i.e., we obtained 20 networks from this dataset for the data analysis.

  1. 2.

    The second dataset: different cone contrast experiment

The second and third experiments were conducted using a random block design or counter-balanced block design (Mullen et al. 2008). These experiments were conducted within the constraints of the ethical clearance from the Medical Research Ethics Committee of the University of Queensland for MRI experiments on humans at the Centre for Magnetic Resonance (Brisbane, Australia). All magnetic resonance images in the second and third experiments were acquired using a 4T Bruker MedSpec system. A transverse electromagnetic head coil was used for radiofrequency transmission and reception. Head movement was limited by foam padding within the head coil in all MRI scans. Eight healthy observers were used as subjects (four female, mean age 41 years, age range: 31–54 years), five of whom were naive to the purpose of the study (Mullen et al. 2008). The subjects were instructed to maintain fixation on the provided fixation-point and trained prior to the scanning sessions to familiarize them with the task. All observers had normal or corrected-to-normal visual acuity. No participant had a history of psychiatric or neurological disorder, head trauma, or substance abuse. Stimuli were radial sine wave gratings or Cartesian sine wave checkerboards (both 0.5 cycles/degree) whose contrast phase reversed at 2 Hz or 8 Hz. All stimuli were presented in a temporal Gaussian contrast envelope (sigma = 125 ms). Four different stimulus conditions were used: achromatic (Ach), red-green (RG), blue-yellow (BY) stimuli, and a luminance (blank) condition in which only the fixation stimulus appeared. There were three different types (RG, BY, and Ach) that isolated L/M cone opponent, the S cone opponent or the achromatic (luminance) postreceptoral mechanisms respectively. The cone contrasts were set to high suprathreshold levels of 11 % (Ach), 4 % (RG), and 30 % (BY). The circular stimulus was viewed as 16° (full width) by ~12° (full height), since stimulus height was limited from top to bottom by the subject’s placement in the bore of the magnet. In the fixation condition, a white ring surrounded the small black fixation spot. Stimuli were presented time-locked to the acquisition of fMRI time-frames, i.e. every 3 s. Each stimulus was presented within a 500-ms time-window in a temporal Gaussian contrast envelope (sigma = 125 ms), with an inter-stimulus interval of 500 ms. In the remaining 1.5 s the subjects’ responses were recorded using an MR-compatible computer mouse. During the mean luminance (blank) condition an identical contrast discrimination task was performed for the fixation stimulus. The four stimulus types were presented in a random/counter-balanced block design (six presentations per block, duration = 18 s). Each block was repeated ten times giving a total of 240 presentations per scan, i.e., 12 min/scan. All results are based on data from two scans per experiment (480 presentations, 24 min).

For the fMRI studies, 241 T2*-weighted gradient-echo echoplanar images depicting BOLD contrast were acquired in each of 36 planes with TE = 30 ms, TR = 3,000 ms, in-plane resolution 3.6 mm and slice thickness 3 mm (0.6 mm gap). Two fMRI scans were performed in each session. In the same fMRI session, a high-resolution 3D T1 image was acquired using an MP-RAGE sequence with TI = 1,500 ms, TR = 2,500 ms, TE = 3.83 ms, and a resolution of 0.9 mm3. In the fMRI data pre-processing, the first two image volumes were cut because of magnetic instability, therefore we used only 239 image volumes in each run. The slices were taken parallel to the calcarine sulcus, and covered the entire occipital and parietal lobes and large dorsal-posterior parts of the temporal and frontal lobes. For more details regarding fMRI data pre-processing see (Hess et al. 2009). Identification of the early visual cortical areas of subject including V1, was performed in separate sessions with identical parameters except for the number of time-frames (128), number of fMRI scans (1–4) and slice orientation (orthogonal to the calcarine for the retinotopic mapping experiments).

  1. 3.

    The third dataset : same cone contrast experiment

All eight healthy observers from the second experiment also participated in this experiment. The stimuli were the same as experiment 2 except that these three stimuli (Ach, RG and BY) were presented with approximately the same cone contrasts (5–6 %). The subject viewed the stimuli with both eyes. Two fMRI scans were performed in each session. The fMRI and pre-preprocessing were acquired analyzed using the same procedure and parameters as in the dataset 2.

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Li, X., Coyle, D., Maguire, L. et al. A Least Trimmed Square Regression Method for Second Level fMRI Effective Connectivity Analysis. Neuroinform 11, 105–118 (2013). https://doi.org/10.1007/s12021-012-9168-8

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