Application of change-point theory to modeling state-related activity in fMRI
Martin Lindquist1* and Tor D. Wager2
1. Department of Statistics, Columbia University, New York, NY, 10027
2. Department of Psychology, Columbia University, New York, NY, 10027
* Corresponding author:
Department of Statistics
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Tor D. Wager
Department of Psychology
1190 Amsterdam Ave.
New York, NY 10027
Phone: (212) 854-5318
Fax: (212) 854-3609
Neuroimaging, particularly functional magnetic resonance imaging (fMRI), is a useful and increasingly popular tool for studying the brain activity dynamics underlying human psychology. fMRI data consists of brain images--approximately 100,000 brain 'voxels' (cubic volumes that span the 3-D space of the brain)--measured repeatedly every 2 s or so, with images nested within task conditions and conditions nested within participants. Analysis of these complex data sets is the focus of intensive research and development in the scientific community.
Most previous approaches towards analysis assume that the nature, timing and duration of the psychological processes are known (Wager, Hernandez, Jonides, & Lindquist, in press; Worsley & Friston, 1995). The analysis thus tests whether activity in a brain region (i.e., a voxel or set of voxels) is systematically related to the known input function. In these model-based approaches, the general linear model (GLM) is used to test for differences in activity among psychological conditions or groups of participants. For example, a task might require participants to inhibit an automatic tendency to make a motor response in favor of a less automatic one. The investigators might look for transient increases in brain activity when inhibition-demanding stimuli are presented, and test whether activity is greater on trials that require inhibition compared with those that do not.
However, in many areas of psychological inquiry, the precise timing and duration of psychological events is hard to specify a priori. For example, a participant watching a funny film might experience amusement that builds gradually over time. The time course of the psychological process is likely to vary idiosyncratically from participant to participant, and without moment-by-moment reports, the progression of amusement over time will be difficult to specify. The need for obtaining a valid and accurate measure of psychological activity to use as a predictor for brain activity is a limitation, as valid psychological indices are not always available. Self-reported emotion, for example, may not give an accurate picture of emotional experience in many situations. In addition, reporting emotions might change the underlying experience in important ways (Taylor, Phan, Decker, & Liberzon, 2003). Thus, our ability to localize emotional activity in the brain is only as good as our ability to accurately tell what someone is feeling through other means. Brain activity may provide a better measure of some aspects of emotional processing, but this information is difficult to uncover in the linear modeling system, because to the degree that brain activity departs from a priori predictions based on behavioral measures it will tend to go undetected.
Much of the literature on emotion has attempted to partially circumvent the problem raised by the uncertain time course corresponding to emotion by presenting brief emotionally evocative events of different types and looking for activity increases that follow a specified canonical model (Hariri, Mattay, Tessitore, Fera, & Weinberger, 2003; Lang et al., 1998; Ochsner et al., 2004; Wager, Phan, Liberzon, & Taylor, 2003). The cost of this approach is that these studies cannot investigate strong, naturalistic emotions. Many other processes are similarly difficult to assess psychologically, including anticipatory processes, shifts in strategy or performance, and changes in states of arousal.
A related problem is that even in tasks whose psychological profile is relatively easy to specify, the specification relies on logical task analysis and may not map precisely onto brain function. For example, an attention switching task might require participants to switch attention between two objects on some trials but not others (Wager, Jonides, Smith, & Nichols, 2005; Yantis et al., 2002). The knowledge of which trials are "switch trials" is used to build the GLM model; but the model cannot capture spontaneous shifts in attention that are not proscribed by the task instructions. Thus, there is a need to develop analyses appropriate for modeling activity whose psychological time course is uncertain.
An alternative analysis strategy that may be appropriate in such situations is the data-driven approach. Rather than taking psychological activity as given and testing for brain changes that fit the psychological model, data-driven approaches attempt to characterize reliable patterns in the data, and relate those patterns to psychological activity post hoc. One particularly popular data-driven approach in the fMRI community is independent components analysis (ICA), a variant on a family of analyses that also includes principal components and factor analysis (Beckmann & Smith, 2004; Calhoun, Adali, & Pekar, 2004; McKeown & Sejnowski, 1998). Recent extensions of these methods can identify brain activity patterns (components) that are systematic across participants (Beckmann & Smith, 2005; Calhoun, Adali, & Pekar, 2004), and can identify state-related changes in activity that can subsequently be related to psychological processes. However, these methods do not provide statistics for inferences about whether a component varies over time and when changes occur in the time series. In addition, because they do not contain any model information, they capture regularities whatever the source; thus, they are highly susceptible to noise, and components can be dominated by artifacts.
The development of hybrid data and model-driven approaches has the potential to mitigate some of the worst problems of each approach (Calhoun, Adali, Stevens, Kiehl, & Pekar, 2005; McIntosh, Chau, & Protzner, 2004). In this paper, we develop a hybrid approach for identifying changes in fMRI time series in individual and group data that allows for valid population inference. The approach uses ideas from statistical control theory and change point theory to model slowly varying processes with uncertain onset times and durations of underlying psychological activity. One of the main benefits of the methods presented here are that they are semi-model-free methods of detecting activation and are therefore insensitive to the phase lag of the hemodynamic response. In this sense they share some of the attractive features of data-driven analysis methods. However, on the other hand, they retain the inferential nature of the more rigid modeling approach.
The change-point analysis that we develop is a multi-subject extension of the exponentially weighted moving average (EWMA) change-point analysis (also called statistical quality control charts; (Neubauer, 1997; Roberts, 1959; Shehab & Schlegel, 2000)). Activity during a baseline period is used to estimate noise characteristics in the fMRI signal response. This activity is used to make inferences on whether, when, and for how long subsequent activity deviates from the baseline level. We extend existing EWMA models for individual subjects (a single time series) to include AR(p) and ARMA(1,1) noise processes, then develop a group analysis using a hierarchical model, which we term HEWMA (Hierarchical EWMA). Group-level changes are of primary interest in the neurosciences because group analyses can be used to make inferences about how a population of participants perform a task (often referred to as 2nd level or 'random effects' analyses in the fMRI literature; (Wager, Hernandez, Jonides, & Lindquist, in press)). Finally, we apply the method to the detection of differences between groups (e.g., patients vs. controls) or conditions within a group.
Once a systematic deviation from baseline has been detected in the group, we estimate the time of change and recovery time (if any). Variations across the brain in the onset and number of change points (CPs), and in the duration of a shift away from baseline activity, can be used to constrain inferences about the functions of brain regions underlying changes in psychological state. CPs are of potential interest for a number of reasons. First, they may provide a basis for discriminating anticipatory activity from responses to a challenge (e.g., activity that begins in anticipation of pain from that elicited by painful stimuli; (Koyama, McHaffie, Laurienti, & Coghill, 2005; Wager, 2005)). Second, CPs may be used to identify brain regions that become active at different times during a challenge (e.g., the early, mid, or late phases of a tonic painful stimulus). Third, CPs may provide meaningful characterizations of differences among individuals. In clinical studies, the onset time of brain responses to anxiety may provide clinically relevant markers of anxiety disorders. In studies of emotion, the speed of recovery from adverse events is thought to be an important predictor of emotional resilience (Fredrickson, Tugade, Waugh, & Larkin, 2003; Tugade & Fredrickson, 2004), and CPs could provide direct brain measures of recovery time. In cognitive psychology, brain CPs in problem solving and insight tasks may provide a direct neural correlate of traditional time-to-solution measures in cognitive studies (Cheng & Holyoak, 1985; Christoff et al., 2001).
The HEWMA method could be used to analyze fMRI data voxel-wise throughout the brain, data from regions of interest, or temporal components extracted using ICA or similar methods. Here, we provide power and false-positive rate analyses based on simulations, and we apply HEWMA to voxel-wise analysis of an anxiety-producing speech preparation task. We demonstrate how the method detects deviations from a pre-task-instruction baseline and can be used to characterize differences between groups in both evoked fMRI activity and CPs.
There are a large variety of change-point detection problems that present themselves in the analysis of time series and dynamical systems. In this paper, we apply ideas from statistical control theory and change point theory to model slowly varying brain processes (i.e., periods of enhanced neural activity detectable with fMRI) with uncertain onset times and durations. We first develop the method for detecting change points in a single time series using exponentially weighted moving averages (EWMA), and then develop a hierarchical extension (HEWMA) appropriate for multisubject fMRI studies.