Identification of causal relations in neuroimaging data with latent confounders: An instrumental variable approach☆
Introduction
Inferring the causal structure of a cortical network is a central goal in neuroimaging (Smith et al., 2011). Various methods have been developed to infer causal relations from brain imaging data, including structural equation modeling (SEM) (Mcintosh and Gonzalez-Lima, 1994, Atlas et al., 2010), Granger causality (GC) (Granger, 1969, Kamiński et al., 2001, Gregoriou et al., 2009), dynamic causal modeling (DCM) (Friston et al., 2003, Daunizeau et al., 2011), and causal Bayesian networks (CBNs) (Ramsey et al., 2010, Grosse-Wentrup et al., 2011, Ramsey et al., 2011, Mumford and Ramsey, 2014, Weichwald et al., 2015). These methods commonly assume causal sufficiency; that is, they presume that all causally relevant variables have been observed. This assumption is often implausible, because various factors can confound a causal analysis. These factors include, but are not limited to, unmeasured brain regions in an fMRI analysis (Mcintosh and Gonzalez-Lima, 1994, Daunizeau et al., 2011, Friston et al., 2011), cardio-ballistic artifacts in ECoG recordings (Kern et al., 2013), and volume conduction of cortical and non-cortical current sources in EEG or MEG data (Grosse-Wentrup, 2009, Hipp and Siegel, 2013). Because it is not trivial to anticipate potential confounders, results obtained with methods based on causal sufficiency must be interpreted with caution.
Latent confounders can be addressed by the IC* (Pearl, 2000) and FCI algorithms (Spirtes et al., 2000, Zhang, 2008), which use the theory of ancestral graphs. Theoretically, both algorithms can distinguish genuine causal relations from spurious relations induced by latent confounders. In practice, the involved statistical tests are complex, which currently limits their application in neuroimaging to variables that are jointly Gaussian distributed (Waldorp et al., 2011). The assumption of jointly Gaussian distributed variables has been criticized as unreasonable for neuroimaging data (Hanson and Bly, 2001, Wink and Roerdink, 2006, Mumford and Ramsey, 2014).
We contribute to research on causal inference with latent confounders in two ways. First, we show that the statistical tests required to identify a genuine causal relation can be simplified when the experimental condition is randomized. Using the a priori knowledge that a randomized experimental condition cannot be caused by neural processes, we analytically prove that if two neural processes are modulated by an experimental condition, a single test of conditional independence is sufficient to establish a genuine causal relation between those processes. To emphasize the requirement that, in our approach, the experimental conditions must be randomized, we later refer to them as the stimuli presented to a subject. Second, by using linear regression, we reduce the required conditional independence test to a marginal independence test. This test is advantageous because asymptotically consistent statistical tests are readily available for marginal independence (Gretton et al., 2005, Gretton et al., 2008, Gretton and Györfi, 2010), but not for conditional independence (Fukumizu et al., 2008, Zhang et al., 2011). We prove that this linearized conditional independence test is sufficient but not necessary for conditional independence: while our test may fail to detect conditional independence if the assumption of linearity is not met, a positive test result implies that this assumption has been fulfilled. Taken together, our two contributions lead to a non-parametric version of the instrumental variable approach to causal inference (Angrist et al., 1996, Pearl, 2000). The resulting algorithm, which we term stimulus-based causal inference (SCI), can provide empirical evidence for a causal relation between two neural processes, even in the presence of latent confounders.
We demonstrate the performance of the SCI algorithm on simulated as well as on experimentally recorded EEG data. We first use a neural mass model for spectral responses in electrophysiology (Moran et al., 2007) to provide estimates of the power and of the false discovery rate (FDR) of the SCI algorithm for a variety of causal models. We then show how our method can be used to infer group-level causal relations on EEG data, which we recorded for a study on brain–computer interfacing (BCI) (Grosse-Wentrup and Schölkopf, 2014). In this study, subjects were trained via neurofeedback to self-regulate the amplitude of γ-oscillations (55–85 Hz) in the right superior parietal cortex (SPC), a primary node of the central executive network (CEN) (Bressler and Menon, 2010). Because transcranial magnetic stimulation (TMS) of the CEN has been found to modulate the medial prefrontal cortex (MPC) (Chen et al., 2013), we hypothesized that self-regulation of γ-power in the right SPC causes variations in γ-power in the MPC. Consistent with this hypothesis, the SCI algorithm determined the MPC to be modulated by the right SPC. We conclude the article with a discussion of the utility and of the limitations of causal inference to study the structure and the function of cortical networks.
We note that the SCI algorithm is applicable not only to EEG recordings but also to any neuroimaging data set that is based on randomized experimental conditions. We have condensed the SCI algorithm into one line of Matlab code, which is available at http://brain-computer-interfaces.net.
Section snippets
Methods
We begin this section by introducing the framework of causal Bayesian networks (CBNs), which our work is based on (cf. Ramsey et al., 2010, Grosse-Wentrup et al., 2011, Ramsey et al., 2011, Mumford and Ramsey, 2014, Weichwald et al., 2015 for applications of this framework in neuroimaging). We then present the sufficient conditions to establish causal influence of one cortical process on another in stimulus-based experiments (Section 2.2). In Section 2.3, we use linear regression to reduce the
Results
In this section, we study the performance of the SCI algorithm on simulated as well as on experimental data. We first investigate the power and the FDR of the SCI algorithm on simulated data that we generated with neural mass models for spectral responses in electrophysiology (Moran et al., 2007). We then demonstrate the application of the SCI algorithm to a group-level causal analysis on experimental EEG data that we recorded as part of a study on brain–computer interfacing (Grosse-Wentrup and
Discussion
We conclude the article with a discussion of the utility of causal inference in neuroimaging (Section 4.1), the strengths and (current) limitations of the SCI algorithm (Section 4.2), and the promise the SCI algorithm holds for the study of the neural basis of cognition (Section 4.3).
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M.S. was supported by the Centre for Integrative Neuroscience (Deutsche Forschungsgemeinschaft, EXC 307). The authors would like to thank Sebastian Weichwald for comments on an earlier version of this article.