Regular articleGeneral multilevel linear modeling for group analysis in FMRI
Introduction
Functional magnetic resonance imaging studies are typically used to address questions about activation effects in populations of subjects. This generally involves a multisubject and/or multisession approach where data are analyzed in such a way that allows for hypothesis tests at the group level Holmes, and Friston, 1998, Worsley et al., 2002, e.g., in order to assess whether the observed effects are common and stable across or between groups of interest.
Calculating the level and probability of brain activation for a single subject is typically achieved using a linear model of the signal together with a Gaussian noise model for the residuals. This model is commonly referred to as the General Linear Model (GLM) and much attention to date has been focused on ways of modeling and fitting the (time series) signal and residual noise at the individual single-session level Bullmore et al., 1996, Woolrich et al., 2001, Worsley and Friston, 1995.
In order to carry out higher-level analyses it is straightforward to formulate a complete single-level GLM that relates various parameters of interest at the group level to the full set of (time series) data available Cnaan et al., 1997, Matthews et al., 1990. In FMRI, where the human and computational costs involved in data analysis are relatively high, however, it is desirable to be able to make group-level inferences using the results of separate first-level analyses without the need to reanalyse any of the individual subject data; an approach commonly referred to as the “summary statistics” approach to FMRI analysis (Holmes and Friston, 1998). Within such a two-level approach, group parameters of interest can easily be refined as more data become available. The natural question to address is if and how a two-level approach to group analysis restricts the type of hypotheses we are able to infer at the group level.
In order to be able to generate results that extend to the population, we also need to account for the fact that the individual subjects themselves are sampled from the population and thus are random quantities with associated variances. It is exactly this step that marks the transition from a simple fixed-effects model to a mixed-effects model2 and it is imperative to formulate a model at the group level that allows for the explicit modeling of these additional variance terms Frison and Pocock, 1992, Holmes, and Friston, 1998.
In this article we revisit the single-level mixed-effect GLM for FMRI group analysis which relates the effect of interest at the group level (e.g., difference of patients and controls) to the individual FMRI time-series. This model is well known and encompasses both the familiar single-subject models and a general group model Cnaan et al., 1997, Everitt, 1995, Everitt, 1995, Matthews et al., 1990. By using the full GLM at the group level, one obtains a very general and flexible framework which allows examination of more complex relationships beyond the simple tests such as mean group activation. An example of a more general group analysis would be to test whether activation is correlated with some other variable such as drug dosage or disability score.
The main result shows that this single-level GLM can be decomposed into an equivalent two-level version so that group analyses can be performed using only the lower-level parameter estimates and their (co-)variances, from the individual subject analyses. The practical consequence of this model is that it is possible to perform valid group analyses in two stages: first the individual subject analyses, and second, a single group analysis performed on the output of the combined estimates from the individual subject analyses. Furthermore, there are very few restrictions placed upon the model and so it can be used in very general conditions, such as when the estimates at the individual subject level are obtained using different prewhitening for each subject or indeed different regressors. We finally give examples of how, in this general framework, group tests of interest, like paired or unpaired t tests, can easily be formulated. The model equivalence result applies not only to FMRI studies but to any mixed-effects GLM that can be split into two levels in the same way as presented here.
All models used here are instances of the univariate GLM. This means treating each single, registered voxel separately and assuming Gaussian noise. Parameter estimation is being performed (at all levels) using the prewhitening approach Bullmore et al., 1996, Woolrich et al., 2001, which is known to be the best linear unbiased estimator (BLUE) for known variances for these models (Searle et al., 1992).
In this article we make a distinction between the problems involved in modeling of FMRI data at the group level and estimation of the relevant (co-)variances. While estimation of variances is an important issue when implementing such models (and is discussed in Section III), we consider it best to treat this as a separate problem. An article on practical issues of estimation, using a novel fully Bayesian method, is currently being prepared and these techniques form the basis of what has been implemented in the latest software release of FSL.
Section snippets
II. models
To begin with we consider the familiar two-level univariate GLM for FMRI. That is, the model that in the first level deals with individual subjects, relating time series to activation, and in the second level deals with a group of subjects or sessions (or both), relating the combined individual activation estimates to some group parameter, such as mean activation level.
III. estimation of variance components
In the previous sections it was assumed that all variance terms are known a priori. In practice, these quantities are unknown and will need to be estimated as part of the model fitting. Variance component estimation is a challenging task in itself, having generated a variety of approaches. Any approach to variance estimation (or combination of approaches) can easily be combined with the multilevel GLM to provide a practical multilevel method; this section discusses some of the more popular
IV. examples
In this section we show how various group-level parameters of interest can easily be calculated within the GLM framework. This amounts to specifying a suitable group design matrix XG, a covariance structure VG, and possibly a contrast vector CG. Note that unlike the case of first-level designs, the mean parameter value is often of interest and hence the design matrix, XG, must always model the group mean activation; that is, the unit vector must always be included in the span of XG.
For several
Conclusion
In this article we have shown that it is possible to efficiently test, using only first-level parameter and (co-)variance estimates, general hypotheses for a mixed-effects3 group analysis model within the framework of the multilevel GLM under the BLUE for known variances. In particular, we have demonstrated the equivalence between a multilevel GLM and a single-level version if
Acknowledgements
The authors thank Mark Woolrich, Didier Leibovici, Timothy Behrens, and Joe Devlin for the contribution they made by taking part in many helpful discussions and thinking through examples and counterexamples which helped to greatly improve the manuscript. Funding from the UK MRC and EPSRC and GSK is gratefully acknowledged.
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These authors contributed equally to this work.