Elsevier

NeuroImage

Volume 22, Issue 2, June 2004, Pages 503-520
NeuroImage

Statistical parametric mapping for event-related potentials (II): a hierarchical temporal model

https://doi.org/10.1016/j.neuroimage.2004.02.013Get rights and content

Abstract

In this paper, we describe a temporal model for event-related potentials (ERP) in the context of statistical parametric mapping (SPM). In brief, we project channel data onto a two-dimensional scalp surface or into three-dimensional brain space using some appropriate inverse solution. We then treat the spatiotemporal data in a mass-univariate fashion. This implicitly factorises the model into spatial and temporal components. The key contribution of this paper is the use of observation models that afford an explicit distinction between observation error and variation in the expression of ERPs. This distinction is created by employing a two-level hierarchical model, in which the first level models the ERP effects within-subject and trial type, while the second models differences in ERP expression among trial types and subjects. By bringing the analysis of ERP data into a classical hierarchical (i.e., mixed effects) framework, many apparently disparate approaches (e.g., conventional P300 analyses and time-frequency analyses of stimulus-locked oscillations) can be reconciled within the same estimation and inference procedure. Inference proceeds in the normal way using t or F statistics to test for effects that are localised in peristimulus time or in some time–frequency window. The use of F statistics is an important generalisation of classical approaches, because it allows one to test for effects that lie in a multidimensional subspace (i.e., of unknown but constrained form). We describe the analysis procedures, the underlying theory and compare its performance to established techniques.

Introduction

Electroencephalography (EEG) measures voltage changes on the scalp induced by underlying neuronal activity. The EEG is thought to be caused by postsynaptic potential changes in cortical pyramidal neurons (Lopes da Silva and van Rotterdam, 1982). An important application of EEG is to assess differences among responses to different stimuli. Traditionally, one distinguishes between some stimulus-locked component and a residual (error) process. The stimulus-locked component is called the event-related potential (ERP). Often, because of a low signal-to-noise ratio, one measures several responses to estimate the average ERP, i.e., the common component of all ERPs with respect to a given stimulus type.

In cognitive neuroscience, the analysis of multisubject-averaged ERP data is well established. For example, guidelines for recording standards and publication criteria are described in Picton et al. (2000). In an ERP study, one analyses ERPs of several subjects under several trial types (conditions). The analysis proceeds, at each channel, by estimating a contrast or linear compound of the ERP for each subject. Typically, these linear combinations are averages over peristimulus time windows. The statistical test of differences among these subject-specific, trial type-specific contrasts is based on an analysis of variance (ANOVA) with appropriate corrections for nonsphericity. This or a similar procedure represents the conventional ERP analysis.

Another, more recent, analysis procedure is based on time–frequency decomposition. Here, the hypotheses relate to power in a specific frequency range within a peristimulus time window. This kind of analysis is often used on single trial data to characterise induced responses (non-time-locked), but can also be applied to averaged ERP data (Tallon-Baudry et al., 1998). The approach usually involves a continuous or discrete wavelet transform. The wavelet transform has also been used to reduce the ERP to one or a few wavelet parameters that capture, parsimoniously, the differences among trial types or groups of subjects. Some authors have used wavelet parameter estimates in a descriptive fashion (Thakor et al., 1993) or have derived statistics as the basis of inferences about ERP differences Basar et al., 1999, Trejo and Shensa, 1999.

Both the conventional and time–frequency analyses have been used to detect and make classical inferences about differences, between either trial types or groups of subjects. In this paper, we show that both analyses can proceed within a unified statistical framework using the same model. We propose a two-level hierarchical linear model as a general observation model for ERP data. In this model, conventional and time–frequency hypotheses can be tested using single- or multidimensional contrasts. The ensuing statistics are either t- or F-distributed, where the degrees of freedom are adjusted for nonsphericity. As an alternative to t- or F-statistics based on ordinary least squares, one can use the nonsphericity estimate to whiten the data and derive maximum-likelihood estimators. Inferences can be made at both levels of the model, resulting either in a fixed effects or random effects analysis. In addition to covering established characterisations, we will give examples of other useful contrasts that arise naturally from our framework, that do not conform to conventional or time–frequency analyses.

In the hierarchical model, the first level describes an observation model for multiple ERPs. The second level models the first-level parameters over subjects and trial types. These contain the differences or treatment effects one hopes to elicit by experimental design. Critically, to derive valid statistics at the second-level, one has to estimate the associated error covariance. It transpires that one can choose the observation model, at the first-level, to finesse nonsphericity estimation at the second. We will illustrate this using a discrete wavelet transform at the first level. The wavelet transform has two important features. The first is that it decomposes the ERP in the time–frequency domain, which gives a sparse representation of its salient features. The second is that the wavelet transform affords an efficient error covariance estimation at the second level. However, we stress that other useful (linear) transforms like the Fourier transform can also be used in the two-level approach. The approach described here pertains to the analysis of single voxel data. We assume that the error covariance matrix, at the second level, is known. In a subsequent communication, we will describe the estimation procedure in a mass-univariate setting (i.e., to spatially reconstructed ERP data).

This paper comprises three sections. In the first, we describe the mathematical basis of hierarchical models, paying special attention to the formulation of conventional analyses within this framework. In the second section, we provide worked examples using simulated and real data to demonstrate the use and flexibility of our approach. We conclude with a discussion of how this procedure relates to other analyses in the literature.

Section snippets

Hierarchical models

This section establishes the temporal model that is used, in various forms, in the next section. This is a two-level hierarchical model (where we estimate model parameters in a two-stage procedure, see below).

The model isy=X(1)β(1)(1)β(1)=X(2)β(2)(2)where y is the data vector, X(1) and X(2) are design matrices, β(1) and β(2) are parameter vectors, and ϵ(1) and ϵ(2) are normally distributed error vectors. The data y comprises stacked ERPs yij, where i = 1,…,Nsubjects, j = 1, …,Ntypes.N

Illustrative analyses

In this section, we apply the hierarchical and conventional methods to synthetic and real data. The synthetic data were designed to show that the hierarchical approach gives valid tests and retains sensitivity. Furthermore, we will demonstrate contrasts that can only be used in a hierarchical context. Analyses of real data are provided to illustrate the operational details, particularly contrast specification.

Summary and discussion

We have described a temporal model adopted by SPM for ERP data. The model pertains to voxel/channel data. To analyse time series of source-reconstructed ERP images, one needs a spatial model, which will be described in a future communication. The methods described here are implemented in Matlab software compatible with the SPM2 distribution and will be an integral part of future SPM releases.

Comparison to other methods

Here we focus on three examples of recently proposed procedures. One of these is used to analyse differences in power between groups. The other two rest on analyzing the time-dependent changes in power or phase following a stimulus. As mentioned above, this involves replacing the average ERP in the response or data-vector y with the average power or phase. This enables inferences about induced oscillations, as opposed to stimulus-locked oscillations that would be tested for using time–frequency

Conclusion

We have described a hierarchical observation model and associated inference procedures for the analysis of ERP data. This model is a generalisation of existing analysis techniques that rests upon standard estimation and classical inference methods. The most important aspect of this generalisation is that all the parameters pertaining to an ERP enter the observation model at the between-subject or second level. This is in contrast to conventional approaches where a single aspect (contrast of

Acknowledgements

The Wellcome Trust funded this work. We would like to thank Marcia Bennett for help in preparing this manuscript, and Rik Henson and Will Penny for helpful discussions.

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