Robust and unbiased variance of GLM coefficients for misspecified autocorrelation and hemodynamic response models in fMRI

An antidote to the imager's fallacy, or how to identify brain areas that are in limbo

Traditionally, fMRI data are analyzed using statistical parametric mapping approaches. Regardless of the precise thresholding procedure, these approaches ultimately divide the brain in regions that do or do not differ significantly across experimental conditions. This binary classification scheme fosters the so-called imager's fallacy, where researchers prematurely conclude that region A is selectively involved in a certain cognitive task because activity in that region reaches statistical significance and activity in region B does not. For such a conclusion to be statistically valid, however, a test on the differences in activation across these two regions is required. Here we propose a simple GLM-based method that defines an "in-between" category of brain regions that are neither significantly active nor inactive, but rather "in limbo". For regions that are in limbo, the activation pattern is inconclusive: it does not differ significantly from baseline, but neither does it differ significantly from regions that do show significant changes from baseline. This pattern indicates that measurement was insufficiently precise. By directly testing differences in activation, our procedure helps reduce the impact of the imager's fallacy. The method is illustrated using concrete examples.

Fast and accurate modelling of longitudinal and repeated measures neuroimaging data

Despite the growing importance of longitudinal data in neuroimaging, the standard analysis methods make restrictive or unrealistic assumptions (e.g., assumption of Compound Symmetry--the state of all equal variances and equal correlations--or spatially homogeneous longitudinal correlations). While some new methods have been proposed to more accurately account for such data, these methods are based on iterative algorithms that are slow and failure-prone. In this article, we propose the use of the Sandwich Estimator method which first estimates the parameters of interest with a simple Ordinary Least Square model and second estimates variances/covariances with the "so-called" Sandwich Estimator (SwE) which accounts for the within-subject correlation existing in longitudinal data. Here, we introduce the SwE method in its classic form, and we review and propose several adjustments to improve its behaviour, specifically in small samples. We use intensive Monte Carlo simulations to compare all considered adjustments and isolate the best combination for neuroimaging data. We also compare the SwE method to other popular methods and demonstrate its strengths and weaknesses. Finally, we analyse a highly unbalanced longitudinal dataset from the Alzheimer's Disease Neuroimaging Initiative and demonstrate the flexibility of the SwE method to fit within- and between-subject effects in a single model. Software implementing this SwE method has been made freely available at http://warwick.ac.uk/tenichols/SwE.

Development of the Complex General Linear Model in the Fourier Domain: Application to fMRI Multiple Input-Output Evoked Responses for Single Subjects

A linear time-invariant model based on statistical time series analysis in the Fourier domain for single subjects is further developed and applied to functional MRI (fMRI) blood-oxygen level-dependent (BOLD) multivariate data. This methodology was originally developed to analyze multiple stimulus input evoked response BOLD data. However, to analyze clinical data generated using a repeated measures experimental design, the model has been extended to handle multivariate time series data and demonstrated on control and alcoholic subjects taken from data previously analyzed in the temporal domain. Analysis of BOLD data is typically carried out in the time domain where the data has a high temporal correlation. These analyses generally employ parametric models of the hemodynamic response function (HRF) where prewhitening of the data is attempted using autoregressive (AR) models for the noise. However, this data can be analyzed in the Fourier domain. Here, assumptions made on the noise structure are less restrictive, and hypothesis tests can be constructed based on voxel-specific nonparametric estimates of the hemodynamic transfer function (HRF in the Fourier domain). This is especially important for experimental designs involving multiple states (either stimulus or drug induced) that may alter the form of the response function.

Estimating and testing variance components in a multi-level GLM

Most analysis of multi-subject fMRI data is concerned with determining whether there exists a significant population-wide 'activation' in a comparison between two or more conditions. This is typically assessed by testing the average value of a contrast of parameter estimates (COPE) against zero in a general linear model (GLM) analysis. However, important information can also be obtained by testing whether there exist significant individual differences in effect magnitude between subjects, i.e. whether the variance of a COPE is significantly different from zero. Intuitively, such a test amounts to testing whether inter-individual differences are larger than would be expected given the within-subject error variance. We compare several methods for estimating variance components, including a) a naïve estimate using ordinary least squares (OLS); b) linear mixed effects in R (LMER); c) a novel Matlab implementation of iterative generalized least squares (IGLS) and its restricted maximum likelihood variant (RIGLS). All methods produced reasonable estimates of within- and between-subject variance components, with IGLS providing an attractive balance between sensitivity and appropriate control of false positives. Finally, we use the IGLS method to estimate inter-subject variance in a perfusion fMRI study (N=18) of social evaluative threat, and show evidence for significant inter-individual differences in ventromedial prefrontal cortex (VMPFC), amygdala, hippocampus and medial temporal lobes, insula, and brainstem, with predicted inverse coupling between VMPFC and the midbrain periaqueductal gray only when high inter-individual variance was used to define the seed for functional connectivity analyses. In sum, tests of variance provides a way of selecting regions that show significant inter-individual variability for subsequent analyses that attempt to explain those individual differences.

Statistical Analysis of fMRI Time-Series: A Critical Review of the GLM Approach

Functional magnetic resonance imaging (fMRI) is one of the most widely used tools to study the neural underpinnings of human cognition. Standard analysis of fMRI data relies on a general linear model (GLM) approach to separate stimulus induced signals from noise. Crucially, this approach relies on a number of assumptions about the data which, for inferences to be valid, must be met. The current paper reviews the GLM approach to analysis of fMRI time-series, focusing in particular on the degree to which such data abides by the assumptions of the GLM framework, and on the methods that have been developed to correct for any violation of those assumptions. Rather than biasing estimates of effect size, the major consequence of non-conformity to the assumptions is to introduce bias into estimates of the variance, thus affecting test statistics, power, and false positive rates. Furthermore, this bias can have pervasive effects on both individual subject and group-level statistics, potentially yielding qualitatively different results across replications, especially after the thresholding procedures commonly used for inference-making.

Functional connectivity analysis of fMRI data using parameterized regions-of-interest

Connectivity analysis of fMRI data requires correct specification of regions-of-interest (ROIs). Selection of ROIs based on outcomes of a GLM analysis may be hindered by conservativeness of the multiple comparison correction, while selection based on brain anatomy may be biased due to inconsistent structure-to-function mapping. To alleviate these problems we propose a method to define functional ROIs without the need for a stringent multiple comparison correction. We extend a flexible framework for fMRI analysis (Activated Region Fitting, Weeda et al. 2009) to connectivity analysis of fMRI data. This method describes an entire fMRI data volume by regions of activation defined by a limited number of parameters. Therefore a less stringent multiple comparison procedure is required. The regions of activation from this analysis can be directly used to estimate functional connectivity. Simulations show that Activated Region Fitting can recover the connectivity of brain regions. An application to real data of a Go/No-Go experiment highlights the advantages of the method.

Systems analysis of seed filling in Arabidopsis: using general linear modeling to assess concordance of transcript and protein expression

Previous systems analyses in plants have focused on a single developmental stage or time point, although it is often important to additionally consider time-index changes. During seed development a cascade of events occurs within a relatively brief time scale. We have collected protein and transcript expression data from five sequential stages of Arabidopsis (Arabidopsis thaliana) seed development encompassing the period of reserve polymer accumulation. Protein expression profiling employed two-dimensional gel electrophoresis coupled with tandem mass spectrometry, while transcript profiling used oligonucleotide microarrays. Analyses in biological triplicate yielded robust expression information for 523 proteins and 22,746 genes across the five developmental stages, and established 319 protein/transcript pairs for subsequent pattern analysis. General linear modeling was used to evaluate the protein/transcript expression patterns. Overall, application of this statistical assessment technique showed concurrence for a slight majority (56%) of expression pairs. Many specific examples of discordant protein/transcript expression patterns were detected, suggesting that this approach will be useful in revealing examples of posttranscriptional regulation.