Elsevier

NeuroImage

Volume 42, Issue 3, September 2008, Pages 1078-1093
NeuroImage

A unified framework for group independent component analysis for multi-subject fMRI data

https://doi.org/10.1016/j.neuroimage.2008.05.008Get rights and content

Abstract

Independent component analysis (ICA) is becoming increasingly popular for analyzing functional magnetic resonance imaging (fMRI) data. While ICA has been successfully applied to single-subject analysis, the extension of ICA to group inferences is not straightforward and remains an active topic of research. Current group ICA models, such as the GIFT [Calhoun, V.D., Adali, T., Pearlson, G.D., Pekar, J.J., 2001. A method for making group inferences from functional MRI data using independent component analysis. Hum. Brain Mapp. 14, 140–151.] and tensor PICA [Beckmann, C.F., Smith, S.M., 2005. Tensorial extensions of independent component analysis for multisubject FMRI analysis. Neuroimage 25, 294–311.], make different assumptions about the underlying structure of the group spatio-temporal processes and are thus estimated using algorithms tailored for the assumed structure, potentially leading to diverging results. To our knowledge, there are currently no methods for assessing the validity of different model structures in real fMRI data and selecting the most appropriate one among various choices. In this paper, we propose a unified framework for estimating and comparing group ICA models with varying spatio-temporal structures. We consider a class of group ICA models that can accommodate different group structures and include existing models, such as the GIFT and tensor PICA, as special cases. We propose a maximum likelihood (ML) approach with a modified Expectation–Maximization (EM) algorithm for the estimation of the proposed class of models. Likelihood ratio tests (LRT) are presented to compare between different group ICA models. The LRT can be used to perform model comparison and selection, to assess the goodness-of-fit of a model in a particular data set, and to test group differences in the fMRI signal time courses between subject subgroups. Simulation studies are conducted to evaluate the performance of the proposed method under varying structures of group spatio-temporal processes. We illustrate our group ICA method using data from an fMRI study that investigates changes in neural processing associated with the regular practice of Zen meditation.

Introduction

Independent component analysis is becoming increasingly popular for analyzing functional neuroimaging data. Compared to the conventional analysis tools such as general linear model (GLM), a key advantage of ICA is that it is a data-driven approach and does not rely on a priori model of brain activity. Therefore, ICA is applicable to cognitive paradigms where prior knowledge of the expected brain time course is not available. ICA could also be used as an exploratory tool to identify and distinguish various types of signals.

ICA has been successfully applied to single-subject fMRI analysis (Beckmann and Smith, 2004, McKeown et al., 1998, Petersen et al., 2000). However, the extension of ICA to group inferences is not as straightforward as in the case of GLM because ICA does not have a pre-specified design matrix, and both the time courses and the spatial maps need to be estimated for each subject (Calhoun and Adali, 2006). Several methods have been proposed to perform group ICA analysis on fMRI data aggregated across multiple subjects. Calhoun et al. developed the GIFT approach (Calhoun et al., 2001), which consists of an initial data-reduction through PCA for each subject, followed by the temporal concatenation of the reduced data across subjects, and a final ICA decomposition of the concatenated data. Back-construction and statistical comparison of individual maps is performed following the ICA estimation. More recently, Beckman and Smith (2005) proposed a tensor probabilistic ICA (PICA) that factors the multi-subject data as a trilinear combination of three outer products, representing the loadings in the temporal, spatial and subject domains, respectively. Tensor PICA is derived from parallel factor analysis (PARAFAC) (Harshman and Lundy, 1984) and it is a natural extension of the two-way product factoring of the single-subject ICA, which factors the data as a combination of two outer products of loadings in the temporal and spatial domains. Other group methods that have been proposed include the approach by Svensén and colleagues (Svensén et al. (2002)), which concatenates multi-subject fMRI data in the spatial domain and extracts independent components with subject-specific spatial maps associated with common time courses across subjects. Schmithorst and Holland also proposed a group ICA method, which performs PCA reduction and ICA decomposition on the data averaged across subjects (Schmithorst and Holland, 2002).

Among the existing methods, the GIFT and tensor PICA are the most frequently used for performing group ICA analysis of multi-subject fMRI data. These two methods share several similarities: both methods are spatial ICA approaches, i.e., they assume statistical independence of the spatial maps of the extracted components and both methods provide estimation of group spatial maps by performing ICA on the aggregated group data. On the other hand, the GIFT and tensor PICA have important distinctions. A major difference lies in the structure of the group spatio-temporal processes that is assumed in the ICA decomposition. The tensor PICA approach decomposes the multi-subject fMRI data as a trilinear combination of three outer products representing group spatial maps, group time courses and subject loadings. That is, subjects are associated with the same set of group spatial maps and time courses but differ in the magnitude of loading on the group spatio-temporal processes. The GIFT decomposition of the data, on the other hand, implies group spatial maps and subject-specific time courses. Furthermore, the estimation and statistical inference procedure for the GIFT and tensor ICA model are also distinctive and are tailored to their specific model structure. The GIFT approach belongs to the classical noise-free ICA framework which assumes that data are completely characterized by the estimated sources and the mixing matrix. Based on the noise-free assumption, the GIFT reconstructs a subject's spatial map from the group ICA estimation by multiplying the inverse of the block of the mixing matrix corresponding to the subject with the observed data from the subject. The statistical inference for spatial activation is then performed through a “random effects” inference on the individual maps. The tensor PICA approach is a probabilistic ICA approach which assumes the observed data is the combination of a set of statistically non-Gaussian sources aggregated through the mixing matrix and additive Gaussian noises. Voxel-wise Z-scores are calculated by dividing the estimated spatial maps by the noise standard deviation. The Z-scores are then modeled with the Gaussian/Gamma mixture where the Gaussian component represents the background noise and the Gamma distribution models brain activation. The statistical inference for spatial activation is performed through calculation of the posterior probability for activation based on the mixture model for the voxel-wise Z-scores.

Since the existing group ICA models assume different group structures in the ICA decomposition and their estimation and statistical inference procedures are based on the particular model structure, these methods may produce different results when applied to the same fMRI data. It is desirable to develop a more general framework for group ICA that could accommodate varying structures of group spatio-temporal processes. It is also important to develop a statistical method to select an appropriate group structure for a particular data set.

An important task in analyzing multi-subject fMRI data is to characterize and compare brain activity between subjects from different groups, such as subjects with/without certain psychiatric conditions or subjects assigned to various treatment arms. Current group ICA methods do not take into account subjects' group identification in the ICA decomposition. Group comparisons are typically performed as post-ICA-estimation analysis often by comparing independent components estimated separately in each group. There are several limitations associated with the existing group comparison approaches. First, when independent components are estimated separately in each group, it is necessary to first identify matching components in different groups. Because independent components are not ordered as in the case of principal components analysis, this is often done by identifying independent components in each group that are associated with a pre-specified spatial or temporal template, which requires some prior information on the spatial distribution or temporal dynamics of the underlying source signal (Calhoun et al., 2004). Furthermore, it is possible for spatial ICA to split a spatio-temporal structure into two or more temporally correlated components, which creates another difficulty in selecting matching components from different groups. Recently, Calhoun et al. proposed an approach that performs a group ICA on combined data from both subject groups and then reconstructs subject-specific maps and time courses for group comparisons (Calhoun et al., in press). This new approach avoids the need for matching components between groups. However, the group identification is still not incorporated directly in the ICA decomposition. The second issue with the existing group ICA comparison approaches lies in the interdependence of group comparisons on the temporal and spatial domains. Preferably, group comparison in one of the domains should be performed while controlling for the group difference in the other domain. For example, GLM estimates and compares group spatial maps by regressing each subject's data against the same temporal paradigm. Such approach does not naturally apply to ICA because both the time courses and spatial maps are estimated from data. Within common group ICA comparison approaches (but see Calhoun et al., in press), group independent components are estimated separately in each group and thus differ both in their spatial images and time courses. Hence, group comparison in either the temporal or the spatial domain is confounded by the group difference in the other domain. The above limitations of the existing approaches arise mainly because the group comparisons are performed indirectly as a second-stage analysis after estimating independent components separately in each group. Hence, it is desirable to develop a group ICA method that could directly incorporate subjects' group information in the ICA decomposition and therefore provide a formal statistical method for group comparisons.

In this paper, we propose a unified framework for fitting group ICA models that are based on varying structures of group spatio-temporal processes. We consider a class of group spatial ICA models, assuming independence in the spatial domain. The proposed models decompose multi-subject fMRI data into group spatial maps and a group mixing matrix that reflects the assumed structure of group spatio-temporal processes, such as the trilinear product structure of tensor PICA. This class of models incorporates existing methods such as the GIFT and tensor PICA as special cases. Furthermore, by specifying an appropriate structure for the group mixing matrix, the proposed model can directly incorporate subjects' group information in the ICA decomposition. For model estimation and statistical inference, we propose a maximum likelihood approach. Latent spatial source signals are modeled with Gaussian mixture distributions where the various Gaussian components model the probability density of background noise and BOLD effects respectively. We develop a modified EM algorithm to obtain the maximum likelihood estimates of the parameters in the group ICA models.

To help select an appropriate structure of group spatio-temporal processes, we present statistical tests for making model comparisons between group ICA models assuming different group structures. The statistical tests are based on the difference in the maximum log-likelihoods under two ICA model structures and are known as the likelihood ratio test (LRT). The LRT can be used to assess the validity of a group structure in an fMRI data set. Furthermore, a statistical test based on the LRT is developed to examine group differences in the spatial modes' associated time courses between subject groups. We also develop a local LRT for comparisons of different group structures for a subset of independent components. The local LRT is motivated by the fact that ICA components extracted from fMRI data often reflect different kinds of signals including task-related, transiently task-related, physiology-related and artifact-related ones (Calhoun and Adali, 2006, McKeown and Sejnowski, 1998). Given the varying characteristics of ICA components, it may be desirable to assume a specific group structure only upon components that are of interest or expected to have particular properties while not imposing this structure upon components whose properties are unknown or unlikely to conform to the chosen structure. Another motivation for the local LRT is that we may be interested only in a subset of the signals that are relevant to the study objectives. The local LRT provides more precise evaluation of an assumed group structure on the selected components.

The remainder of this paper is organized as follows. In the Methods section, we introduce a class of group ICA models and show that existing group ICA models such as the GIFT and tensor PICA can be viewed as special cases within this class. We then present a maximum likelihood (ML) approach with a modified Expectation–Maximization (EM) algorithm for the estimation of the proposed class of models. To compare between group ICA models, we introduce likelihood ratio tests and show how to use the LRT to assess the goodness-of-fit of a group structure in a data set and to test group differences between subject groups. The Results section evaluates the performance of the proposed method on simulated data, and also illustrates its application to real fMRI data from a study investigating changes in neural processing associated with the regular practice of Zen meditation. Finally, a concluding section summarizes and provides further discussion about the presented method and findings.

Section snippets

Methods

In this section, we first describe the class of group ICA models that is able to subsume varying structures of group spatio-temporal processes. We then present the maximum likelihood approach for model estimation and likelihood ratio tests for performing model comparisons.

Simulation studies

In the following, we compare the performance of different group ICA models under the various types of spatio-temporal structure embedded in the simulated datasets. We first consider the results from the single-group simulation study and compare the results from the Tensor Model and Full Model. The accuracy of the model estimates is measured by the spatial correlations between the estimated and true spatial maps (Fig. 2A) and the temporal correlations between the estimated and true time courses (

Discussion

We have presented a unified framework for group ICA analysis of multi-subject fMRI data, by introducing a class of group ICA models that is able to (1) accommodate varying types of structures of group spatio-temporal processes, including those assumed by existing group ICA methods such as the GIFT and tensor PICA; (2) provide a formal statistical framework for group comparisons by incorporating group identification in the ICA decomposition; and (3) allow component-specific group structures to

Acknowledgments

The authors thank the referees and editor for helpful comments. Y.G. was partially supported by the National Institute of Mental Health (NIH grant R01-MH079251). The imaging data were collected as part of a pilot study awarded to G.P. by the Emory Center for Research on Complementary and Alternative Medicine in Neurodegenerative Diseases (NIH grant P30-AT00609).

References (35)

  • BinderJ.R. et al.

    Neural correlates of lexical access during visual word recognition

    J. Cogn. Neurosci.

    (2003)
  • BiswalB.B. et al.

    Blind source separation of multiple signal sources of fMRI data sets using independent component analysis

    J. Comput. Assist. Tomogr.

    (1999)
  • BroR.

    Multi-Way Analysis in the Food Industry. Models, Algorithms, and Applications

    (1998)
  • CalhounV.D. et al.

    Unmixing fMRI with independent component analysis

    IEEE Eng. Med. Biol. Mag.

    (2006)
  • CalhounV.D. et al.

    A method for making group inferences from functional MRI data using independent component analysis

    Hum. Brain Mapp.

    (2001)
  • Calhoun, V.D., Pearlson, G.D., Maciejewski, P., Kiehl, a.K.A., in press. Temporal lobe and ‘default’ hemodynamic brain...
  • CorbettaM. et al.

    Control of goal-directed and stimulus-driven attention in the brain

    Nat. Rev., Neurosci.

    (2002)
  • Cited by (0)

    View full text