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NeuroImage 10, 282–303 (1999)Article ID nimg.1999.0472, available online at http://www.idealibrary.com on

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Plurality and Resemblance in fMRI Data Analysis

Nicholas Lange,*,1 Stephen C. Strother,† Jon R. Anderson,† Finn Å. Nielsen,‡ Andrew P. Holmes,§Thomas Kolenda,‡ Robert Savoy,¶ and Lars Kai Hansen‡

*Statistical Neuroimaging Laboratory and Laboratory for Molecular Pharmacology, Mailman Research Center, McLean Hospital andonsolidated Department of Psychiatry, Faculty of Medicine, Harvard University, 115 Mill Street, Belmont, Massachusetts 02478-9106; ‡PET

Imaging Service, VA Medical Center and University of Minnesota; ‡Institute of Mathematical Modeling, Danish Technical University;§Wellcome Department of Cognitive Neurology, Functional Imaging Laboratory and Robertson Centre for Biostatistics, Department of

Statistics, University of Glasgow; ¶Rowland Institute and MGH-NMR Center

Received December 30, 1998

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We apply nine analytic methods employed currentlyn imaging neuroscience to simulated and actual BOLDMRI signals and compare their performances underach signal type. Starting with baseline time seriesenerated by a resting subject during a null hypothesistudy, we compare method performance with embed-ed focal activity in these series of three differentypes whose magnitudes and time courses are simple,onvolved with spatially varying hemodynamic re-ponses, and highly spatially interactive. We then ap-ly these same nine methods to BOLD fMRI time seriesrom contralateral primary motor cortex and ipsilat-ral cerebellum collected during a sequential fingerpposition study. Paired comparisons of results acrossethods include a voxel-specific concordance correla-

ion coefficient for reproducibility and a resemblanceeasure that accommodates spatial autocorrelation of

ifferences in activity surfaces. Receiver-operatingharacteristic curves show considerable model differ-nces in ranges less than 10% significance level (falseositives) and greater than 80% power (true positives).oncordance and resemblance measures reveal signifi-ant differences between activity surfaces in both dataets. These measures can assist researchers by identify-ng groups of models producing similar and dissimilaresults, and thereby help to validate, consolidate, andimplify reports of statistical findings. A pluralistictrategy for fMRI data analysis can uncover invariantnd highly interactive relationships between localctivity foci and serve as a basis for further discoveryf organizational principles of the brain. Results alsouggest that a pluralistic empirical strategy coupledormally with substantive prior knowledge can help toncover new brain–behavior relationships that mayemain hidden if only a single method is employed.1999 Academic Press

1 To whom correspondence and reprint requests should be ad-
oressed. Fax: (617) 855-3479. E-mail: [email protected].
282053-8119/99 $30.00opyright r 1999 by Academic Pressll rights of reproduction in any form reserved.

INTRODUCTION

Many statistical procedures have been proposed inecent years for the analysis of human blood oxygenevel dependent (BOLD) fMRI data. These methodsnclude cross-correlation (Bandettini et al., 1993), lin-ar models for smoothed and unsmoothed data (Fristont al., 1995b; Boynton et al., 1996; Aguirre et al., 1997;ohen, 1997; Zarahn et al., 1997); classical and modernultivariate methods, including varieties of singular

alue decomposition and eigenanalysis (Sychra et al.,994; Friston et al., 1995a; Bullmore et al., 1996b;trother et al., 1996; Worsley et al., 1997; McKeown etl., 1998a, 1998b, 1997; Hansen et al., 1999); timeeries and other frequency-domain methods accommo-ating hemodynamic delays (Friston et al., 1994a,995b; Forman et al., 1995; Worsley and Friston, 1995;ullmore et al., 1996a; Lange and Zeger, 1997; Mitrand Pesaran, 1997; Mitra et al., 1997; Tagaris et al.,997; Rajapaske et al., 1998); thresholds of randomelds (Friston et al., 1994b; Worsley, 1994; Poline et al.,997); nonlinear models (Friston, 1998; Vazquez andoll, 1998); artificial neural networks (Mørch et al.,997); test–retest reliability (Genovese et al., 1997;egeler et al., 1999), clustering (Baumgartner et al.,997; Moser et al., 1997; Goutte et al., 1999); physiologi-al models (Hu et al., 1995; Lee et al., 1995; Le and Hu,996; Buxton and Frank, 1997; Kim et al., 1997; Kimnd Ugurbil, 1997); event-related, single-trial studiesBuckner et al., 1996; Dale and Buckner, 1997; Buck-er, 1998; Friston et al., 1998a, 1998b). For overviewsnd critiques of statistical procedures for human BOLDMRI, see Constable et al. (1995), Lange (1996), Xiong1996), Lange (1997), Rabe-Hesketh et al. (1997), Wors-ey (1997), Aguirre et al. (1998a), Aguirre et al. (1998b),

cKeown et al. (1998), Petersson et al. (1998), Skudlar-ki et al. (1999), and Lange (1999). In this present work,MRI will always refer to human BOLD fMRI unless
therwise stated.

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283COMPARATIVE fMRI DATA ANALYSIS

In this paper, we compare several classes of statisti-al approaches for fMRI through development of aluralistic, empirical, and pragmatic analysis strategy.

pluralistic strategy supports and encourages mul-iple statistical perspectives by denying that any singlenalytic procedure can fully ‘‘explain’’ fMRI data. Anmpirical strategy regards all supposed facts aboutMRI time series as hypotheses to be tested in anvolving research environment. A pragmatic strategyudges the effectiveness of a statistical procedure forMRI according to its consequences in application.

hile many neuroscientists give tacit assent to statisti-al pluralism, empiricism, and pragmatism, few adoptpluralistic approach in their analyses of fMRI data,

pting instead to employ one method as determined inart by availability, expedience, or convenience. It isecessary to publish statistical results in fMRI re-earch due to obvious ethical limitations. Yet statisticalignificance as reported in the literature is both biasedBegg and Berlin, 1988) and insufficient for acceptancef neuroscientific interpretations and implications de-ived from such efforts. Statistical results must ‘‘makeense’’ in order to be accepted as bona fide contributionso updates of existing knowledge about the brain.lthough it has been demonstrated that some statisti-al procedures outperform others in certain functionaleuroimaging contexts, there does not exist a single,lobally optimal statistical procedure for the analysis ofny particular fMRI study. Model optimality mustnclude subject–matter components, such as explicitognitive neuroscientific assumptions and models ofrain function, in order to have requisite scientifictility in the field and not only optimality in a statisti-al sense. At issue is a tension between data and theory,resent in scientific investigations in general (see fornstance Duhem, 1954; Arbib and Hesse, 1986) andvident in the current plurality of models employed inunctional neuroimaging. On the one hand, fMRI datare theory-laden representations expressed in the lan-uage of the field. Yet, on the other hand, neuroscien-ific theory is empirically undetermined by these data,ince rival models may fit equally well or yield conflict-ng results. One resolution of the model identificationroblem is to incorporate a priori information in thenalysis process. However, there may be inconsisten-ies in prior knowledge and assumptions accumulatedn the neuroscientific literature that require the re-earcher to choose between competing versions of

‘ground truth.’’ An alternate strategy is to apply aariety of methods, without assessment of underlyingnconsistent assumptions and to report concordance oriscordance of results across the methods considered;ee for instance Karni et al. (1995) for use of severalimple statistical methods in an fMRI motor learningtudy. The language of statistical analysis is rich
nough to express new, data-based findings that chal- r
enge current neuroscientific theory. Yet must suchndings be articulated in simple statistical language inrder to be accepted, escaping the criticism of being

‘model dependent’’? Does hypothesized activity need toe detectable by a t test in order to be deemed ‘‘signifi-ant’’ by the imaging neuroscience community? Howuch understanding of fMRI statistics is required of

he neuroscientist in order for such data–theory feed-ack to be deemed both valid and important? What role,f any, is there for sophisticated data analyses in fMRIesearch including nonlinear models and general func-ion estimation methods such as artificial neural net-orks? How can one do more than detect and describe

patiotemporal structure in fMRI data and also ad-ance neuroscientific knowledge by testing explicitypotheses and assumptions about the transform fromeural activity to fMRI signal change? Questions suchs these motivated the present study and are revisitedn the Discussion.

Demonstration of the utility of a pluralistic, empiri-al, and pragmatic approach to fMRI data analysis isrovided by way of two examples. In the first hybridimulation example, we embed simple and complexignals with known features in baseline fMRI timeeries generated by a resting subject during a nullypothesis baseline study. We investigate three differ-nt types of fMRI time course magnitudes: 1) simple, 2)elayed and dispersed representing spatially variableemodynamic responses, and 3) highly interactive inwo distinct and highly correlated spatial clusters. Theecond example involves detection of unknown signaleatures in fMRI data generated during a simple motortudy by a different subject alternating between restnd self-paced sequential finger opposition. Statisticalnalyses of these two data sets employ nine differentrocedures covering a wide range of parametric andonparametric choices, including simple t tests, artifi-ial neural network models, and eigenimage analysessing a variety of basis functions. The first goal of this

nvestigation is to assess performance of this pluralityf methods in the presence of fMRI noise when ‘‘groundruth’’ is both known and unknown. The second goal iso develop and apply pairwise resemblance measuresor spatial activity patterns produced by the methods.

MATERIALS AND METHODS

Baseline Study

Whole-brain images consisting of 14 3.125 3 3.125 3.0-mm slices were collected using a GE Signa 1.5Tcanner at the MGH-NMR Center (Charlestown, MA),ith an integrated Advanced NMR Instascan echopla-ar imaging (EPI) system running an asymmetricpin-echo pulse sequence (TR 5 2.5 s, TE 5 70 ms,t 5 25 ms). A healthy volunteer was asked simply to
elax and remain still while 72 whole-brain baseline

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284 LANGE ET AL.

cans were collected. The fMRI time series were cor-ected for subject motion through application of a-parameter affine transformation (Woods et al., 1998a,998b). A 12 3 24-voxel patch containing primaryotor cortex was selected for analysis through com-

ined use of the functional and structural MR imagesollected concurrently. Longer time series were con-tructed from these null hypothesis data to enableodeling strategies to show possible advantages of

heir various forms of temporal averaging, as well as topply these results to the motor data; see next section.eries were resampled temporally in blocks of variableize while preserving temporal autocorrelation, anmportant determinant of significance levels for statis-ical signal detection (Purdon and Weisskoff, 1998).erial autocorrelation was respected separately at eachpatial location by using an information criterion (AIC,kaike, 1974) to estimate the order, q, of each temporalutoregressive process, ranging from q 5 0 (indepen-ence) to q 5 10 (highly autocorrelated). As anticipated,he most temporally correlated series were found inrey matter, where there is the most blood; the leastorrelated series arose from cerebrospinal fluid.

ybrid Simulation

Baseline fMRI space-time series are arranged in aatrix Y 5 [ ytv] of dimension T 3 V, where T 5 384

imepoints and V 5 288 voxels in a 12 3 24 voxel patch.ime series length corresponds to eight runs of 48imepoints each; effects of a simple on/off referenceunction operating in each run are defined in theollowing descriptions of the three experiments. Eacholumn of Y is a T 3 1 time series yv 5 ( y1v, . . . , yTv)T atrain location v 5 1, . . . , V ; each row of Y is a 1 3 Vpatial patch yt 5 ( yt1, . . . , ytV) of activity at time t 5, . . . , T. Each time series yv was linearly detrended byemoval of a least-squares regression line, i.e., fittingntercept and slope separately for each run within eachime series and replacing the series with residuals fromhis fit.

We embed signals of known characteristics in theaseline time series, disguising the signal to avoidubjective detection by researchers at different sitesVA Medical Center, Minneapolis, MN; Technical Uni-ersity of Denmark, Copenhagen; and Functional Imag-ng Laboratory, London, UK) and thus blinding the

FIG. 1. ‘‘Ground truth’’ images of active re

nvestigation with respect to prior information on fMRIignal structure. We then apply statistical methods ofnterest to these ‘‘signal plus noise’’ data and comparestimated features to known features. We aim to detecteak signals in lengthy series rather than strong

ignals in short series. Most if not all of our methodsould likely detect strong signals with acceptable

ensitivity and specificity, as has been demonstrated ineveral previous yet entirely simulated studies involv-ng many fewer methods (Constable et al., 1995; Xiongt al., 1996).Three experiments are performed in the baseline

tudy. In each experiment, two regions of activity areefined and signals with features specific to that experi-ent are added to the fMRI time series in these

egions. These regions of ‘‘ground truth’’ activity arehown in Fig. 1. In all three experiments, magnitudesf added signals at voxel locations in active regions,enoted by bv, are specified as some positive fraction,, of series standard deviation at each location:

bv 5 msyv (1)

here syv2 5 St51

T ( ytv 2 yv )2/(T 2 1) and yv 5 St51T ytv /T.

raction m is zero for inactive voxels outside ‘‘groundruth’’ regions. Linking magnitude and variance byquation (1) is reasonable since there is evidenceuggesting that, in the brain, variability tends to scaleith signal strength (Olshen and Shadlen, 1997), violat-

ng assumptions of ordinary least-squares regression inhich magnitude and variance are independent. Specif-

cs of parameters bv differ across the three experimentss follows:1. Simple. Spatial extents of embedded signals vary

n two irregularly shaped yet connected regions of 49nd 16 active locations as shown in the first panel ofig. 1. Active voxel locations turn on and off according

o a simple binary reference function x 5 (x1, . . . , xT)T

or which xt 5 0 when off and xt 5 1 when on, for t 5, . . . , T. Binary reference function x is periodic (12 off,4 on, 12 off) in eight runs of length 48 timepoints each.he collection of signal magnitudes in the two clusterst time t is the collection of scalars 55bvxt66. The fraction oftandard deviation used for signal magnitudes in thisrst experiment is set at m 5 0.15.

gions for the three hybrid simulations.

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2. Convolution with hemodynamic response functionHRF). Active voxel locations turn on and off accord-ng to a smoothed version of the reference function.his new reference function is defined as the convolu-ion of x with an HRF hv 5 (h1v, . . . , hTv). Each HRFas a known temporal delay and dispersion that variesith spatial location. Delays increase in radial fashionutward from the centers of two irregularly shaped yetonnected regions of 25 and 36 active locations ashown in the second panel of Fig. 1. Signal magnitudesv are again defined with m 5 0.15. The collection ofagnitudes at imaged brain locations v at time t is thus

he set 55ztvbv66, where

zv 5 [z1v . . . , zTv] 5 x ^ hv, (2)

ith ^ denoting convolution summation. The HRFs areodeled parametrically as gamma probability densi-

ies (Friston et al., 1994a; Leopold and Logothetis,996), with two parameters that vary spatially (Langend Zeger, 1997). This approach is one way to addresshe variability of hemodynamic responses in fMRI, anmportant aspect of design and analysis of such studiesAguirre et al., 1998c); see Appendix A.1 for more detail.

3. Exclusive-OR (XOR). Active clusters turn on andff in a coordinated fashion according to values of aoolean exclusive OR function taking the reference

unction and an unobserved binary random variable asrguments. A plausible example of an XOR functionith neurobiological meaning can be found in results ofisual perception and face matching experiments (Logo-hetis et al., 1994; Courtney et al., 1997; Ungerleider etl., 1998). These studies demonstrate regional activityifferences with respect to signal categories (faces andheir angles of presentation) and features (aspects ofaces shared by visual stimuli that are not faces). Somerain regions show increased activity when stimuli areither faces or nonfaces at various angles of presenta-ion, whereas other regions only show increased activ-ty when presented with faces (Logothetis et al., 1994).t is thus reasonable to imagine two unobserved binaryunctions, zL and zR, of alternating face stimuli (xt 5 0r xt 5 1) that contribute to changes in activity in twoegions. The first function, zL, could be sensitive to oneet of features independent of faces, being a binaryandom variable representing a form of transient task-elated (TTR) activity (McKeown et al., 1998a). Theecond function, zR, also a binary random variable,ould be sensitive to either faces or features indepen-ent of faces but not both simultaneously. That is, whent is off (features, not faces) the left and right spatialegions are either both on or both off together, yet whent is on (faces), one region is off and the other is on; seeig. 2 for an example of such logical patterns of activity.t is not assumed that values for these binary variables
re available; for purposes of this example, the unob- h
erved binary variables can be thought of merely asokens for cognitive processes involved in perception forhich little a priori information is available. Formally,

or the collection of activated spatial locations in theeft-hand cluster, generate a vector zL 5 (zL1, . . . , zLT)T

f 0’s and 1’s in blocks of duration equal to fourimepoints (set arbitrarily at 10 s in the presentxperiment), so that six such blocks span a single run inhe baseline study. Then, for the collection of activatedpatial locations in the right-hand cluster, define aorresponding vector, zR, which is the result of anlement-wise application of an XOR to zL and referenceunction x:

zRt 5 5zLt when xt 5 0

QzLt when xt 5 1

or t 5 1, . . . , T, where denotes complement. Theollection of signal magnitudes at time t is the set ofcalars 55bvzLt66 in the left-hand cluster and 55bvzRt66 in theight-hand cluster. This third experiment set m 5 1.00f temporal standard deviations, roughly six times theignal magnitudes of the preceding two experiments inrder that all methods may be given a full chance toemonstrate their ability (or inability) to detect thisignal. If values for the binary random variable arenown and available, then the XOR example could beubsumed under a general linear model frameworkhrough inclusion of interaction terms between theeference function and the binary random variable in aarge design matrix. Although there are many in-tances in the literature of interactions being tested inhe context of multiple linear regression, note that XORs a logical activation pattern that is not linearlyeparable (see for instance Duda and Hart, 1973) andies in a two-dimensional subspace. XOR is also anxample of a signal for which methods that averagever task blocks may not perform well (cf. Strother etl., 1995b for PET; Strother et al., 1996), whose tempo-al structure is nonstationary, and is not known a priorihen values of the binary variable are not available, as

FIG. 2. A highly interactive exclusive-OR (XOR) signal in twopatial clusters (see text).

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286 LANGE ET AL.

Motor Study

Using an identical pulse sequence as in the baselinetudy, a second healthy volunteer performed self-paced,eft-handed finger-to-thumb opposition alternating withest for three one-minute epochs (off, on, off) duringhich 24 whole-brain images per epoch were collected.hese three epochs constituted a run of 72 scans. Sevenuch runs were performed, with a 2- to 3-min gapetween runs. Regions containing primary motor cor-ex (a 12 3 24 voxel patch) and cerebellum (also 12 3 24oxels) were selected through combined use of theunctional MR and structural MR images collectedoncurrently. Data sets for both the baseline and motortudies are available through anonymous FTP from ourebsite (http://pet.med.va.gov:8080/plurality).

Data Analysis Methods

Data analysis methods employed in the baseline andotor studies include the following ‘‘single-voxel’’ proce-

ures: Student’s two-sample t (T), ‘‘standard’’ statisticalarametric mapping (SPM), Kolmogorov-Smirnov maxi-um deviation (K-S), a finite impulse response model

FIR), and a parametric Fourier technique (PFT). Eachrocedure is applied voxel-by-voxel, ignoring spatialutocorrelation. T, SPM, FIR, and PFT represent in-tances of the general linear model approach; K-S is aonparametric analogue of T. Linear detrending andur simple reference function obviate, to some degree,oncerns regarding adjustments for nuisance covari-tes and potential confounds such as global effectsStrother et al., 1995a; Andersson, 1997; Aguirre et al.,998b; Tegeler et al., 1999), an issue not addressed fullyerein beyond simple and commonly used detrendingtrategies. Nonparametric convolution modeling to esti-ate variable hemodynamic response functions is rep-

esented by FIR and a parametric analogue by PFT;hese functions are treated as known in design matri-es for SPM. Classical time series and system identifica-ion methods applied to fMRI data by Bullmore et al.1996a) to adjust for AR(1) errors by prewhiteningCochrane and Orcutt, 1949), and by Tagaris et al.1997) using Box-Jenkins (1976) procedures for autore-ressive integrated moving average (ARIMA) modelnalysis were not employed, yet could be the subject ofurther model comparisons. Use of the frequency do-ain PFT approach of Lange and Zeger (1997) ad-

resses the duality between trend and serial depen-ence in fMRI time series (see for instance Diggle,990; Lange, 1999) by including an explicit componentor hemodynamic responses in the trend term of theodel as well as allows for higher order autoregressive

rocesses. The PFT procedure was intended to repre-ent, in part, time series approaches to fMRI datanalysis. Note that many additional covariates would
eed to be added in order to subsume FIR and PFT in b
he general linear model framework, indicator vari-bles for each lag in the former and many gammarobability densities and their derivatives for the lat-er. ‘‘Multivoxel’’ procedures employing singular value,rincipal component and related decompositions in-lude a pruned feed-forward artificial neural networkANN) with one hidden layer, canonical variates analy-is (CVA) and independent component analysis (ICA)ith basis vector subset selection, and a representativef the newly emerging set of functional data analysisFDA) techniques.

Another way to view our taxonomy and thus avoid aotentially frightening ‘‘Babel’’ of statistical proceduress first to consider procedures that employ explicitly theeference function and perhaps additional explanatoryariables (T, SPM, K-S, FIR, PFT, ANN) versus thosehat do not (CVA, ICA, FDA); then to consider single-oxel (univariate) procedures (T, SPM, K-S, FIR, PFT)ersus multivoxel (multivariate) procedures (ANN, CVA,CA, FDA); and last to consider models that allow foronlinear mixing (ANN) versus those that do not (allthers). Appendix A.1 provides descriptions and cur-ent literature citations for each of these nine methods.

Performance and Resemblance

Method performance in the baseline and motor stud-es is measured by receiver-operating characteristicROC) curves, a pairwise concordance correlation coeffi-ient for reproducibility, and a pairwise resemblanceeasure for assessing residual structure in activity

urface differences between methods.

OC Curves

For the hybrid simulation data, we have groundruth and can thus learn properties of fMRI models bysing ROC curves (see for instance Metz, 1978) withouteduction of continuous signals to a single set of‘significant/not significant’’ binary indicators. For eachbserved value of an activity surface, define the falseositive ratio FPR (or, equivalently, Type I error prob-bility, or empirical significance level) as the number ofoxel locations declared falsely to be activated dividedy the total number of voxels considered. Define therue positive ratio TPR (or, equivalently, one minusype II error probability, statistical power, or sensitiv-

ty) as the number of voxel locations correctly declaredo be activated based on statistical functions of ob-erved fMRI signals at these locations divided by theotal number of voxels. Each distinct pixel grey level istself a threshold, generating a new value of TPRersus FPR. One ROC curve may thus be constructedor each model-signal combination, enabling compari-on of statistical procedures where it matters most inractice, at low FPR (empirical significance level be-ween 0 and 0.10) and high TPR (empirical power
etween 0.80 and 1.00).

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oncordance Correlation Coefficient forReproducibility

Lin (1989) introduced the following concordance cor-elation coefficient to evaluate reproducibility betweenandom variables S1 and S2:

rc 52rs1s2

s12 1 s2

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n Eq. (3), (µ1, µ2) and (s12, s2

2) are the means andariances of S1 and S2, respectively, and r is theirearson product–moment correlation. Lange et al.

1996) applied rc in fMRI model comparisons andtrother et al. (1997) applied r to measure reproducibil-

ty in PET. In the present case, there is one rc for each ofhe 36 distinct pairs of summary images for each of thehree experiments. The reproducibility statistic rc,lways less than r, can detect location and scale shiftshat r does not. Standardization of summary images bycaling them separately to have mean 0 and variance 1,istogram equalization, or isotonic regression, can miti-ate these differences, making rc and r equal or nearlyqual in such cases.

esemblance of Activity Surfaces

Neither ROC nor concordance correlation analysesake account of spatial autocorrelation in activity sur-aces, being invariant to paired voxel location permuta-ions. Since all activity surfaces considered exhibitignificant spatial autocorrelation, as indicated by bothoran (1948) and Geary (1954) statistics (not shown),

ny measure of activity surface resemblance mustccommodate spatial autocorrelation. One such resem-lance measure for comparing two activity surfaces iseveloped by fitting the voxelwise activity differencesetween the surfaces by a parametric model accountingor spatial autocorrelation and then testing for signifi-ant structure in this residual surface greater thanhat can be expected by chance alone. Two surfaces are

hen said to resemble each other if the test fails to rejecthe null hypothesis of resemblance, or to be signifi-antly dissimilar if the null hypothesis is rejected (inavor of an unspecified alternative). One class of modelsor activity surfaces that accounts for spatial autocorre-ation are linear models containing cubic polynomialunctions of spatial coordinates (see for instance Cressie,993), together with a specification for coordinate neigh-orhoods; see Appendix A.2.

RESULTS

Baseline Study

Figures 3–7 show results of the nine statistical
ethods applied to the three embedded signal types in r
he baseline study. Each summary image is scaledeparately by its own minimum (black) and maximumwhite). No post-hoc smoothing has been applied to anyf the results. Figure 3 contains results for simple focalctivity, Fig. 4 for focal activity convolved with spatiallyariable hemodynamic delays, and Fig. 5 for highlynteractive focal activity as represented by XOR sig-als. In these figures, visual impressions from thereyscale summary images in the top panel are madeore precise by their corresponding ROC curves in the

ottom left panel. The bottom right panel containsinary versions of the greyscale summaries in the topanel that have been thresholded at the 90th quantilef their empirical distributions and give examples ofheir shapes and spatial autocorrelations. No P valuesre derived for these examples of descriptive summa-ies, although such is certainly possible through appli-ation of random field theory.Examination of Fig. 3 indicates that all methods

emonstrate good performance for the simple focalctivity signal. Model sensitivity drops off slightlyhen model complexity increases, as expected for the

imple signal, for which T, SPM, and K-S outperform allthers. By this criterion, T is slightly better than SPMnd K-S at low false-positive rates even when modelingssumptions are purposely incorrect, concurring toome degree with other recent findings (Aguirret al., 1998a). For the convolution signal, Fig. 4 revealshat K-S, PFT, and FDA are the best detectors accordingo the empirical ROC curve comparison, with T andPM not far behind. Yet no method demonstratesdequate power at the chosen signal magnitude; notehat the vertical scale for true positive rate ends at.50. This behavior is again as expected for thesententionally weak signals. As seen in Fig. 5, modelesults show the greatest divergence for the XORignal. Each of the five single-voxel methods (T, SPM,-S, FIR, and PFT) produces a congeries of voxel valuesnd very low power, indicating that single-voxel ap-roaches can fail to detect highly interactive signals.or the XOR signal, multivoxel methods (ANN, ICA,nd FDA) clearly outperform the previous five,ven when the reference function is not used explicitlyCVA, ICA, and FDA). Both left- and right-hand clustersre detected approximately equally by the multivoxelethods, these clusters being active at the sameagnitude, spatial extent, and temporal duration on

verage. All multivoxel methods demonstrate very goodensitivity and specificity, with ANN besting the otherslightly.Table 1 displays pairwise concordance correlation

oefficients between summary images for convolutionnd XOR signals across all methods. For simple signalscorrelations not shown), all methods demonstrate con-iderable and roughly equivalent correlations in the
ange of 0.49 (FIR and ICA) to 0.93 (T, SPM, and FDA).

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288 LANGE ET AL.

or convolution signals, methods exhibit lower concor-ance, ranging from 0.08 (T and FIR) to 0.91 (T andPM), with low values among single-voxel (T, K-S, FIR)nd multivoxel (CVA, ICA, FDA) methods. It is ofnterest for the convolution example that the highestorrelations appear between single- and multivoxelethods (T or SPM and FDA; PFT and CVA). The

reatest range of correlations across methods is seen inhe XOR example, from a low of 20.05 (T and FDA) to aigh of 0.97 (CVA and ICA). In this case, measures ofssociation such as voxelwise correlations, being invari-nt to paired voxel location permutations, appear able

FIG. 3. Summary images for the fMRI data analysis methods:arametric mapping; K-S, Kolmogorov–Smirnov; FIR, finite impulsetworks; CVA, canonical variates analysis; ICA, independent compo

mages; bottom left, receiver–operating characteristic (ROC) curveuantile.

o carry much of the important differences between b

ingle- and multivoxel methods. Yet these pairwiseorrelations ignore any and all spatial autocorrelation.Table 2 gives resemblance indicators for activity

urfaces over all distinct pairwise comparisons, dis-layed separately for convolution and XOR signals.hese indicators highlight surface differences whileccounting for their spatial autocorrelations, as de-cribed in Appendix A2. A value of 0 indicates noesemblance (nominal P , 0.01) and a value of 1 indi-ates resemblance. Considerable heterogeneity in sur-ace shapes is seen across signals and methods. Notelso that there is no strict transitivity between resem-

ple signal. T, Student’s two-sample t; SPM, ‘‘standard’’ statisticalsponse; PFT, parametric fourier transform; ANN, artificial neural

nt analysis; FDA, functional data analysis. Top, greyscale summaryottom right, thresholds of greyscale summary images at the 90th

Sime rene

s; b

lance indicators, i.e., if surface A resembles surface B,

ararmloasobcb

fd

sEelva

pniq


nd B resembles C, it does not follow necessarily that Aesembles C. For the convolution signal, in general,ctivity surfaces for the single-voxel methods do notesemble each other and neither do the multivoxelethods resemble each other. There also appears to be

ittle resemblance between single- and multivoxel meth-ds (4 of 15), in particular for ICA. The most strikingnd discriminating results are again seen for the XORignal. There is resemblance among single-voxel meth-ds and among multivoxel methods yet no resemblanceetween them (0 of 15). These results strengthen theorrelation findings. Pairwise correlations and resem-

FIG. 4. Summary images for the fMRI data analysis methods: Coarametric mapping; K-S, Kolmogorov–Smirnov; FIR, finite impulsetworks; CVA, canonical variates analysis; ICA, independent compo

mages; bottom left, receiver–operating characteristic (ROC) curveuantile.

lance measures confirm visual, subjective impressions f

ormed while inspecting results from this plurality ofata analytic methods.

Motor Study

Figure 6 displays summary images from the motortudy in nine panels, one panel for each procedure.ach panel contains a summary image from contralat-ral primary motor (PM, top) and ipsilateral cerebel-um (CB, bottom) regions. There are some obviousisual similarities and differences between these im-ges. For PM, all methods appear to detect two activity

lution signal. T, Student’s two-sample t; SPM, ‘‘standard’’ statisticalsponse; PFT, parametric fourier transform; ANN, artificial neural


nvoe rene

s; b

oci, with maxima in the upper left and lower right

rwtaeapfI0PTta

twftrpiddb1sr

pniq

290 LANGE ET AL.

egions of the images. For CB, activity is more diffuse,ith a broad region extending from lower left upward

owards the center of the image and a small secondaryrea of activity slightly right of center. ANN imagesxhibit highest contrast and sharpest foci for both PMnd CB. As shown in Table 3, these data exhibit strongairwise correlations between methods, for PM rangingrom 0.44 (ANN and FDA) to 0.95 (T and SPM, CVA andCA) and for CB ranging from 0.40 (ANN and FDA) to.92 and 0.93 (again T and SPM, CVA and ICA); FIR andFT also show strong correlation for both regions.hese patterns of correlation suggest, in this example,hat there may be less of the type of highly interactive

FIG. 5. Summary images for the fMRI data analysis methodsarametric mapping; K-S, Kolmogorov–Smirnov; FIR, finite impulsetworks; CVA, canonical variates analysis; ICA, independent compo

mages; bottom left, Receiver–operating characteristic (ROC) curveuantile.

ctivity represented in the XOR example. The rela- a

ively lower correlation seen in comparisons of FDAith the others is likely due to absence of reference

unction and a possibly suboptimal choice of regulariza-ion parameter in this exploratory method; note theather diffuse nature of its activity surfaces as com-ared with the others. Table 4 displays resemblancendicators for the two regions. For PM, there is a highegree of resemblance among activity surfaces pro-uced by the single-voxel methods. Lack of resem-lance between single- and multivoxel methods (6 of5) may be indicative of spatial interactions missed byingle-voxel methods. FIR and PFT surfaces do notesemble any surface produced by multivoxel methods,

OR signal. T, Student’s two-sample t; SPM, ‘‘standard’’ statisticalsponse; PFT, parametric fourier transform; ANN, artificial neural


: Xe renes; b

nd the ANN surface does not resemble those of other

mtoPsta

tbapIs

a


ultivoxel methods. Patterns of resemblance indica-ors for ipsilateral cerebellum surfaces differ from thosef contralateral primary motor cortex. In contrast toM, there is a higher degree of resemblance betweeningle- and multivoxel methods for CB (11 of 15), wherehe behavior of FIR and PFT compared to the others

FIG. 6. Summary images produced by the nine fMRI data analysind ipsilateral cerebellum (CB, bottom in each panel).

ccounts for most of this difference. Figure 7 displays m

hree examples of activity surface differences producedy pairs of methods. While many of these differencesre attributable to random noise, there do appear to beatches of coherent, focal activity detection differences.n conclusion, the high degree of resemblance amongingle-voxel methods for the contralateral primary

ethods for contralateral primary motor cortex (PM, top in each panel)
s m
otor region suggests use of a signal magnitude map

faPiatoowa

soifsp

y act

TSKFPACI

PF

TSKFPACI

Kv

292 LANGE ET AL.

rom a relatively simple method such as T, SPM, or K-S,ugmented by hemodynamic delays as estimated byFT or FIR. ANN and CVA may provide some additional

nformation on multivoxel interactions. In this ex-mple, there is less evidence of such spatial interac-ions in ipsilateral cerebellum, due in part to evidencef activity that is more spatially diffuse than thatbserved in the contralateral primary motor region,hich exhibits general resemblance between single-

FIG. 7. (A) Focal activity not detected by T, yet detected by PFT aet detected by ANN for contralateral primary motor cortex. (C) Focal

TAB

Voxelwise Concordance Correlation Coefficients across

Convolution

SPM K-S FIR PFT ANN CVA ICA FDA

0.91 0.10 0.08 0.47 0.29 0.34 0.22 0.79PM 0.57 0.26 0.43 0.30 0.21 0.13 0.77-S 0.34 0.23 0.25 0.17 0.13 0.12IR 0.28 0.50 0.26 0.11 0.16FT 0.28 0.68 0.44 0.59NN 0.28 0.19 0.31VA 0.10 0.44

CA 0.18

Note. T, Student’s two-sample t; SPM, ‘‘standard’’ statistical paramFT, parametric fourier transform; ANN, artificial neural networks; CDA, functional data analysis.

TAB

Resemblance Indicators across Convolutio

Convolution

SPM K-S FIR PFT ANN CVA ICA F

1 1 0 0 1 1 0PM 1 0 0 1 0 0-S 0 0 1 1 0IR 0 1 0 0FT 1 1 0NN 0 1VA 0

CA

Note. Indicators are either 0, if P , 0.01, or 1. T, Student’s tolmogorov–Smirnov; FIR, finite impulse response; PFT, parametrariates analysis; ICA, independent component analysis; FDA, functi

nd multivoxel methods for this subject. t

DISCUSSION

Through development of a pluralistic data analysistrategy, this study has demonstrated the importancef model choice and its effects on empirical significancen summary images of focal activity as measured byMRI. Such empirical significance in turn affects andubserves tests of specific cognitive neuroscientific hy-othesis concerning human brain structure and func-

DA for the convolution signal. (B) Focal activity not detected by K-S,ivity not detected by T, yet detected by PFT for ipsilateral cerebellum.

1

nvolution and Exclusive-OR Signals and All Methods

Exclusive-OR


0.47 0.28 20.03 0.21 20.04 20.04 20.03 20.050.05 0.05 0.06 0.15 20.03 0.10 0.35

0.11 0.16 20.02 20.04 20.03 20.010.37 0.09 0.13 0.12 0.02

0.24 0.13 0.09 0.120.58 0.54 0.49

0.97 0.610.58

c mapping; K-S, Kolmogorov–Smirnov; FIR, finite impulse response;, canonical variates analysis; ICA, independent component analysis;

2

nd Exclusive-OR Signals and All Methods

Exclusive-OR


1 1 1 1 0 0 0 01 1 1 0 0 0 0

1 0 0 0 0 01 0 0 0 0

0 0 0 01 1 1

1 11

-sample t; SPM, ‘‘standard’’ statistical parametric mapping; K-S,ourier transform; ANN, artificial neural networks; CVA, canonicall data analysis.

nd F

LE

Co

etriVA

LE

n a

DA

00001010

woic fona

ion. Parametric and nonparametric collections of single-

abkccscfpfimocscalAf

mtd‘KiaethosFdfirheL

TSKFPACI

PF

TSKFPACI

Kv


nd multivoxel procedures were first considered in aaseline null hypothesis study under a variety ofnown signal complexities. Signal complexities in-luded time courses whose magnitudes were simple,onvolved with spatially variable hemodynamic re-ponses, and highly interactive in two spatially distinctlusters as represented logically by exclusive-OR (XOR)unctions. The same parametric and nonparametricrocedures were then applied in a simple sequentialnger opposition study involving contralateral primaryotor cortex and ipsilateral cerebellum. We have devel-

ped and applied several empirical methods for identifi-ation of model subgroups producing similar and dis-imilar results. These methods included a voxelwiseoncordance correlation coefficient for reproducibilitynd a resemblance measure for spatially autocorre-ated differences between pairs of activity surfaces.pplication of this pluralistic strategy to the baseline

MRI study has refined intuition and quantified perfor-

TAB

Voxelwise Concordance Correlation Coefficients for ContralaAll M

Contralateral Primary Motor (PM)


0.95 0.80 0.78 0.74 0.63 0.81 0.77 0PM 0.77 0.81 0.78 0.66 0.88 0.84 0-S 0.82 0.77 0.72 0.67 0.67 0IR 0.89 0.78 0.76 0.74 0FT 0.72 0.73 0.71 0NN 0.63 0.63 0VA 0.95 0

CA 0

Note. T, Student’s two-sample t; SPM, ‘‘standard’’ statistical paramFT, parametric fourier transform; ANN, artificial neural networks; CDA, functional data analysis.

TAB

Resemblance Indicators for Contralateral Primary Mot

Contralateral primary motor (PM)


1 1 1 0 1 0 1PM 1 1 0 1 0 0-S 1 1 1 1 1IR 1 0 0 0FT 0 0 0NN 0 0VA 1

CA

Note. Indicators are either 0, if P , 0.01, or 1. T, Student’s tolmogorov–Smirnov; FIR, finite impulse response; PFT, parametr
ariates analysis; ICA, independent component analysis; FDA, functiona
ance differences between models of varying complexi-ies for signals of varying complexities. Simple proce-ures such as single-voxel Student’s two-sample t, a

‘standard’’ statistical parametric mapping model, andolmogorov–Smirnov statistic failed to detect some

mportant spatial activity when the signal is defined asconvolution of hemodynamic responses and the refer-

nce function. Simple statistical procedures also failedo detect interactive processes in spatially distinct yetighly correlated regions of activity. Lack of sensitivityf single-voxel models against complex signals has beenhown in some cases to be quite severe, as was seen inig. 5, where XOR signals in the two activity foci wereetected only by multivoxel statistical procedures. Thisnding deserves further comment. The XOR signalesides in a two-dimensional linear subspace of theigh-dimensional space spanned by the observed data;ach eigenimage carries one orthogonal dimension.inear methods, whether single- or multivoxel, cannot

3

al Primary Motor Cortex and Ipsilateral Cerebellum acrossods

Ipsilateral Cerebellum (CB)


0.92 0.85 0.72 0.67 0.65 0.76 0.74 0.540.80 0.71 0.67 0.67 0.82 0.78 0.63

0.71 0.71 0.68 0.73 0.67 0.450.78 0.67 0.61 0.62 0.45

0.59 0.56 0.55 0.450.61 0.60 0.40

0.93 0.410.41

c mapping; K-S, Kolmogorov–Smirnov; FIR, finite impulse response;, canonical variates analysis; ICA, independent component analysis;

4

Cortex and Ipsilateral Cerebellum across All Methods

Ipsilateral cerebellum (CB)


1 1 0 0 1 1 1 11 0 0 1 1 1 1

0 0 1 1 1 01 0 1 0 0

1 1 1 01 1 0

1 00

-sample t; SPM, ‘‘standard’’ statistical parametric mapping; K-S,ourier transform; ANN, artificial neural networks; CVA, canonical

LE

tereth

DA

.59

.63

.46

.48

.47

.44

.59

.54

etriVA

LE

or

DA

11100011

woic f

l data analysis.

psemepvFehpspletaotfibmasastt

Osrsapcmdesneiiicactpsttsnc

uvopfwbWmtpsisms

emsmttpscleistcoLLptwtgpuopscdEscliMm

294 LANGE ET AL.

roduce a sole eigenimage that separates the two XORtates and instead map these states into two distinctigenimages. Multiple linear regression with a designatrix that contains an interaction between the refer-

nce function and the binary random variable are notossible to fit since the values of the binary randomariable are assumed to be unavailable in this example.or multi voxel methods CVA, ICA, and FDA, the twoigenimages associated with the largest variances ex-ibited spatial patterns demonstrated in Fig. 5, topanel; only the eigenimage with the largest variance ishown. CVA, ICA, and FDA detect these patterns in theresent case because the signal magnitudes are mucharger than those of the other two experiments, beingqual to 100% of voxelwise standard deviations. Hadhe magnitudes in the XOR experiment been as smalls they were in the other experiments, being only 15%f voxelwise standard deviations, then one can expecthat only ANN would detect activity. Carrying thesendings to the motor study, a lack of resemblanceetween activity surfaces produced by single- andultivoxel methods may indicate possible spatial inter-

ctions not detected by single-voxel methods. We findome evidence of this lack in the present example. Welso note that models estimating hemodynamic re-ponses (FIR and PFT) can produce activity surfaceshat contain information on focal activity not present inhose produced by other methods.

The original fMRI work of Kwong et al. (1992) andgawa et al. (1992) did not depend on sophisticated

tatistical procedures; a simple exponential model ofesponse to a simple visual stimulus and a t test forignificance of change in signal magnitude sufficed. Itppears that simple statistical analysis (T and K-S) andost hoc smoothing continues to be sufficient for someomparative fMRI analyses, such as in the recentultiinstitutional assessment of reliability and repro-

ucibility of activity in spatial working memory (Caseyt al., 1998). Yet fMRI research has evolved greatlyince the critical early studies to include many inge-ious and increasingly complex approaches to design ofxperiments and data analysis. Postprocessing (see fornstance Aguirre et al., 1998b) has also emerged as anmportant consideration. Whole brain data acquisitions becoming commonplace, increasing the extents andomplexities of observable spatiotemporal interactionsnd thus enabling tests of more elaborate and specificognitive neuroscientific hypotheses. While it is a statis-ical truism that fitting models improves estimationrecision (Altham, 1984), at issue in fMRI data analy-is is neither statistical estimation precision nor deriva-ion of a ‘‘significant’’ P value alone. Through exhaus-ive search, selective data exclusion, and heavymoothing, an artificially low P value can be attached toearly any desired result, potentially draining this
lassical measure of significance of ultimate meaning if s
sed in this manner. The fallacy of chasing small Palues is demonstrated, for instance, when activationsf very small spatial extent are deemed significant, inarticular when reporting extreme values of lengthyMRI time series. More important is the application ofell established statistical principles in service to theroader goal of increasing neuroscientific knowledge.hen there is no a priori reason to favor one empiricalodel over another, additional time and effort required

o apply several statistical procedures is minimal com-ared with that expended in experimental design,ubject recruitment, and data collection. Such a plural-stic data analysis strategy can help to simplify andtrengthen findings when there is concordance betweenethods, as well as to reveal new functional relation-

hips when there is disagreement between them.Additional perspective on the role of statistical mod-

ling in fMRI research is gained by considering theaturation of data analysis and interpretation in other

cientific fields of study. For instance, no predominantode of analysis was adopted during the early stages of

he now highly developed and regulated field of con-rolled clinical trials. Yet from the advent of the Coxroportional hazards model (Cox, 1972) onward, oneees a progressively more uniform and effective statisti-al treatment of these data. Although the currentandscape of statistical procedures for fMRI is notntirely consistent, and neither is any particular unify-ng form of analysis clear at present, one can expect toee similar increased consensus and effectiveness overime. One may also consider the role of statistics inognitive neurobiology, as for example in recent studiesf cell recordings in monkey visual cortex (Leopold andogothetis, 1996; Logothetis et al., 1996; Sheinberg andogothetis, 1997). These authors have shown thatarametric probability densities, such as gamma func-ions, can serve as benchmark processes for relativelyell known neural systems. Potential bias incurred by

aking a low-dimensional parametric approach is miti-ated by the degree to which available and reliablerior knowledge about the system under study wassed to build the model. In contrast, data-driven meth-ds suppress a priori dependence on low-dimensionalarametric forms to explore structure in lesser knownystems, as seen for instance in a recent functionallustering method (Tononi et al., 1998) and to accommo-ate nonstationary noise structures (Benali et al., 1997;verson et al., 1997). When considering two differenttatistical procedures, Sheinberg and Logothetis (1997)omment that their reported high percentages of modu-ating neurons did not depend on their specific multivar-ate analysis method (principle component analysis,

ahalanobis distance and Hotelling T 2) and that aore traditional yet somewhat arbitrary analysis

trengthened their findings.

tdWmoaaadwcsatadnrtwauslresmpbcotpndioasmtiia

ntscskevt

eaaositbmsiigtalwdstesscsemomht

A

t

tobrvzstlbwto

TiS


Yet what statistical procedure would be most effec-ive for accepting or rejecting existence of a new andistinct cortical processing area given fMRI evidence?ould linear models suffice or would more complexodels be required? The general linear model is used so

ften because it works so well in a wide range of appliedreas. For instance, until it was shown that certainctivity in human V1 as measured by fMRI behaved aslinear system under very limited conditions, as was

emonstrated by Boynton et al. (1996), linear modelsere employed primarily as a matter of convenience or

onvention. Profound nonlinearities have been ob-erved in fMRI time series (Vazquez and Noll, 1998)nd when nonlinear components are tested explicitlyhey are extremely prevalent and significant (Friston etl., 1998b). Suppose for the moment that a new andistinct cortical processing area does in fact exist, yeto statistical procedure familiar to the functional neu-oimaging community is effective in testing its exis-ence. If spatiotemporal features of the candidate areaere at all complex, as would likely be the case if therea had not yet been discovered, then a relativelynfamiliar and complex model would likely yield aignificant finding, whereas a simple voxelwise t test,inear model or Kolmogorov–Smirnov statistic may notegister significance. The neuroscientist could thenither deny the existence of the candidate area givenuch evidence built on unfamiliar grounds, or learnore about the evidentiary data and method before

assing scientific judgement. (There exist further possi-ilities: trust the opinion of a more statistically mindedolleague, ignore recent data altogether, and/or conductne’s own experiment.) For fMRI of lesser-known sys-ems, therefore, application of a plurality of statisticalrocedures, parametric and data-driven, linear andonlinear, would be most useful, regardless of concor-ance or discordance of results. Linear models mayndeed outperform nonlinear models when the amountsf available data involve only a handful of subjects andfew imaged brain regions. Yet as data available for

pecific studies increase, so too may the effectiveness ofore complex models increase, since bias–variance

radeoffs suggest that flexible models can reduce biasncurred by structured linear methods without increas-ng uncertainty of results (Friedman, 1994; Mørch etl., 1997).There is an increasing coalescence of neurobiological,

euroanatomical, signal processing, and modern statis-ical theory and practice in current fMR imaging neuro-cience. The most effective data analyses are those thatombine existing neuroscientific theory with existingtatistical theory to advance understanding in specificnowledge domains. Bayesian compromises betweenmpirical and substantive modeling approaches pro-ide a well understood formalism for such investiga-
ions. Bayesian methods have demonstrated recent e
ffectiveness in computational neuroanatomy (Miller etl., 1993, 1997) and in image registration (Ashburner etl., 1997); Lange (1997) provided a very brief discussionf the Bayesian paradigm applied in fMRI. For in-tance, prior knowledge regarding varieties of activityn motor cortex under static and dynamic force condi-ions (Ashe, 1997) and in normal and dysfunctionalasal ganglia-thalamocortical motor circuitry (Wich-ann and DeLong, 1993) can combine with fMRI time

eries such as those studied here to yield furthernsights. As specific knowledge of the fMRI signalncreases, principled analyses (Friston, 1998) involvingenuinely informative priors derived from other func-ional imaging modalities and from modern neuro-natomy (see for instance Kennedy et al., 1998; Mesu-am, 1998; Van Essen et al., 1998; Caviness et al., 1999)ill combine subject–matter knowledge with empiricalata to yield new results. Use of a pluralistic statisticaltrategy can help to reveal complex patterns of activityhat may remain undetected if only a single method ismployed. In addition, statistical pluralism can confirmpatiotemporal features of the fMRI signal by demon-trating reproducibility across a variety of methods andan simplify statistical reporting by grouping activityurfaces that resemble each other. A pluralistic strat-gy for analyses of fMRI time series can thus help toove the focus forward from detection and description

f spatio-temporal structure by use of a single ‘‘correct’’odel toward testing explicit cognitive neuroscientific

ypotheses regarding human brain structure and func-ion.

APPENDIX

1. Statistical Methods

Nine different statistical procedures are applied tohe previously described fMRI time series, as follows:

1. Student’s two-sample t statistic (T). Student’swo-sample t statistic is summarized by the collectionf simple linear regression models yv 5 xbv 1 ev atrain locations v 5 1, . . . , V, where bv is a scalaregression coefficient to be estimated and ev is a T-ector of independent and identically distributed, meanero Gaussian errors with assumed common variancev2. As pointed out by Lange (1996) for analyses of fMRIime series, Student’s two-sample t statistic is equiva-ent to Pearson product–moment cross-correlation rvetween yv and x up to a scale change since, rv 5 cvbv,here cv 5 sx

2/syv2 is the ratio of reference function and

ime series variances. Display least-squares estimatesf the bvs as a T-field summary image.2. ‘‘Standard’’ statistical parametric mapping (SPM).

his procedure, for the hybrid simulation data consist-ng of eight runs with no interrun gap, is a ‘‘standard’’PM analysis employing the general linear model for
ach time series, that use a smoothed reference func-

taesenab(dtbrhmg

ftNcMtmp

wtstowbatWamcpaeod1h

abtam

d6vcea

fst1qD

frSb(t

faaGhF6mcodF

dptvdt

fteas

296 LANGE ET AL.

ion, its temporal derivative, and a high-pass filter fordjustments of low frequency nuisance effects withinach run. Specifically, the design matrix for the hybridimulation data contains 11 covariates, being the refer-nce function convolved with an approximate hemody-amic response function, its temporal derivative tollow for small differences in hemodynamic delays, alock term, and an eight-component high pass filterfirst 8 discrete cosine basis functions). For the motorata, consisting of seven runs separated by time gaps,he design matrix contains 28 covariates (4 per run),eing 7 run-specific smoothed reference functions, 7un-specific temporal derivatives, 7 block terms, and 7alf-cosine drift basis functions. Denoting the designatrix for either data set by X, the SPM instance of the

eneral linear model for each fMRI time series is

yv 5 Xbv 1 ev (A.1)

or v 5 1, . . . , V voxels. Residual errors ev are assumedo have slight short term autocorrelation, so that ev ,(O, S) for unknown and nondiagonal variance–

ovariance matrix S. Model (A.1) is smoothed, as inayhew et al. (1998), with a known filter K, being in

his case a Toeplitz matrix that incorporates an approxi-ate hemodynamic response function for each time

oint, to yield smoothed model

Kyv 5 KXbv 1 Kev,

ith residual errors having variance–covariance ma-rix KSKT. Prewhitening (Cochrane and Orcutt, 1949;ee for instance Bullmore et al., 1996a for an applica-ion to fMRI) through setting K 5 S21/2 would beptimal if a reliable estimate of S were available;ithout such an estimate results are potentially veryiased. The SPM package therefore chooses a low biaspproach, potentially yielding high-variance predic-ions, by approximately KSKT by KKT. Using results oforsley and Friston (1995) and a Satterthwaite (1946)

pproximation, SPM obtains unbiased variance esti-ates and computes effective degrees of freedom for

ontrasts of model parameters. For all data sets in theresent SPM analysis, we consider a contrast thatssesses the average amplitude of the convolved refer-nce function across runs. We display SPM-field imagesf standardized estimates of that contrast which areistributed according to Student’s t distribution with41.76 and 117.11 effective degrees of freedom for theybrid simulation and motor data sets, respectively.3. Kolmogorov–Smirnov (K-S). This procedure yieldsstatistic for each voxel that is the maximum deviationetween empirical cumulative distributions for fMRIime series in on and off states. Producing a K-S-field isnonparametric alternative to the parametric T-field,
aking no assumptions about the shapes of on and off (
istributions (see for instance Press et al., 1992, pp.23–628). Specifically, rank from smallest to largest thealues in yv for which xt is off to obtain the empiricalumulative distribution Gv

0(t). Similarly, obtain thempirical cumulative distribution Gv

1(t) when xt is onnd compute the maximum difference

Dv 5 maxt

[Gv1(t) 2 Gv

0(t)]

or v 5 1, . . . , V. Display the results as a K-S-fieldummary image. Use of the K-S procedure with fMRIime series has been criticized recently (Aguirre et al.,998a) because all moments of the distributions inuestion contribute to the maximum difference statisticv, not only the first moment.4. Finite impulse response (FIR). This method is a

orm of moving average or, equivalently, finite impulseesponse model (see for example Oppenheimer andhafer, 1989) applied voxel-by-voxel; this method haseen adapted to the analysis of fMRI time seriesNielsen et al., 1997). Specifically, consider a simpleime series convolution model of form

ytv 5 o1#s,t2s#T

hv(t 2 s)x(s) 1 etv (A.2)

or all timepoints t and voxels v. In Eq. (A.2), hv(t 2 s) isn unknown hemodynamic response function (HRF) ofrbitrary shape, and, as in the T-field method, etv isaussian white noise. The number of components ofv(t 2 s) that enter into the model is a free parameter.or the present application, hv(t 2 s) 5 0 for all t 2 s ., so that the largest lag entering the model is 6. Fitodel (A.2) by ordinary least squares to obtain a

ollection of estimated signals, 55yv66, and compute input/utput ratios s(yv)/s(x), where s(·) is the standardeviation of its argument. Display the results as aIR-field summary image.5. Parametric fourier transform (PFT). This proce-

ure is a version of the iterative nonlinear methodroposed by Lange and Zeger (1997), being a generaliza-ion of Friston et al. (1994a) to allow for spatiallyarying hemodynamic delays. One way to accommo-ate hemodynamic effects in the time domain is throughhe following convolution model:

ytv 5 ztvbv 1 etv,

or t 5 1, . . . , T and v 5 1, . . . , V, where the designerm ztv is from Eq. (2) in the body text of the paper andtv is a random error that is temporally and spatiallyutocorrelated with other errors nearby in time andpace. Each unobservable hemodynamic response hv 5
h1v, . . . , hTv) is modeled as a two-parameter gamma

d

SidbHtdrtcidi

etroobsad(l(tSddhcwit

woosoYsaEa55c

nsmi

l‘sneavtvsaactma

c1ttsee1ldda1aesmtBfsstesirfit(cpf


ensity equal to

htv 5 gv(gvt)av21 exp (2gv)/G(av).

hape and scale parameters, av and gv respectively,dentify the members of this Pearson family of x2

istributions; parameters bv, av, gv are unknown and toe estimated, as distinct from SPM for which theseRFs are assumed to be known a priori. The expecta-

ion of each gamma density is av/gv, a quantity havingirect interpretation as local signal delay. This paramet-ic model is fit in the frequency domain after applica-ion of a discrete Fourier transform. The estimatedollection of 55bv66 is displayed as a PFT-field summarymage. The estimated collection of 55av/gv66 may also beisplayed as a hemodynamic delay field summarymage.

6. Artificial neural networks (ANN). This methodmploys a feed-forward artificial neural network that israined to classify images labeled as on or off by theeference function, x. Input to the network is theriginal image matrix Y decomposed in terms of anrthonormal basis of eigenimages and an orthogonalasis of time trends that together span the originalpace, i.e., an unsupervised (no x) principal componentnalysis (PCA) of multivariate time series. PCA is useduring the initial stage of ANN and the next methodcanonical variates analysis), and is conveniently formu-ated in terms of the singular value decompositionSVD). Trefethen and Bau (1997) give an insightfulreatment of the numerical linear algebra behind theVD; see also Lange (1999) and the following briefescription. When using the SVD it is customary toeal with the transpose of the data matrix as definedere, namely YT 5 [ yvt], whose rows yv 5 ( yv1, . . . , yvT)onsist of fMRI time series at voxels v 5 1, . . . , V, andhose columns yt 5 ( y1t, . . . , yVt)T consist of fMR

mages collected at times t 5 1, . . . , T. Application ofhe SVD to YT yields the unique decomposition

YT 5 ADBT; yvt 5 op51

P

avpdppbtp, (A.3)

here P 5 min (V, T ). The V 3 P matrix A consistsf orthonormal basis vectors, being the eigenvectorsf YTY; the P 3 P matrix D is a diagonal matrix ofingular values; the T 3 P matrix B also consists ofrthonormal basis vectors, being the eigenvectors ofYT. One can move easily between the vector spacespanned by A and B through the projection, scaling,nd orthogonal transformation operations afforded byq. (A.3) to examine orthogonal eigenimages 55a1, . . . , aP66nd their corresponding orthogonal time trends

b1, . . . , bP66. In terms of fMRI signal sources, the SVD
an identify a set of uncorrelated time sequences, c
amely the principal components dpbp enumerated byource index p. Thus, one can write the observed signalatrix YT as a weighted sum of fixed eigenimages ap, as

n Eq. (A.3).The artificial neural network contains one hidden

ayer with hyperbolic tangent activity functions and‘softmax’’ output normalization providing posterior clas-ification probabilities (Hintz-Madsen et al., 1996). Theetwork is pruned using a minimum generalizationrror criterion (Moody, 1991; Mørch, 1998; Hansen etl., 1999). Specifically, split the data into training andalidation sets of equal size and fit the network to theraining data. Using the fit as a guide to predict thealidation set, select network weights that minimizequared error loss between these predictions and thectual values in the validation set. ANN quantifiesctivity at each brain location by the network’s loss inlassification ability under a ‘‘leave-one-out’’ voxel dele-ion scheme; the resulting field is termed a ‘‘saliencyap’’ (Mørch et al., 1995). Display the saliency map as

n ANN-field summary image.7. Canonical variates analysis (CVA). We apply

anonical variates analysis (Rao, 1952; Mardia et al.,979; Gnanadesikan, 1997) to selected orthogonal timerends, defined at Eq. A.3, employing a design matrixhat includes indicators for the temporal ordering of thecans and that does not include the reference functionxplicitly. Statistically, CVA is a generalization of Fish-r’s linear discriminant analysis (Fisher, 1936, 1938,954) to handle many linear discriminants, or, equiva-ently, many canonical variates. This form of data-riven analysis in functional neuroimaging has beeneveloped for both PET (Clark et al., 1985; Moeller etl., 1987; Moeller and Strother, 1991; Strother et al.,995a, 1996) and fMRI (Sychra et al., 1994; Friston etl., 1995a; Strother et al., 1996; Worsley, 1997; Tegelert al., 1999). Prior to the SVD, we remove temporal andpatial drifts through subtraction of row and columneans from the data matrix, YT, so that elements sum

o zero separately in each row and column. Denote by* a T 3 U matrix of selected orthogonal time trends,

rom Eq. A.3, where U # P 5 min (V, T) is the size of aubset t , 551, . . . , T66. Columns of B* span a vectorpace of much smaller nominal dimension than that ofhe corresponding vector space spanned by orthogonaligenimages, A; we are working in a much smallerpace. Denote by Z a T 3 G design matrix of groupndicators, where G is the number of timepoints in eachun, with all of the first timepoints in each run in therst group, all of the second timepoints in each run inhe second group, and so forth. (By way of contrast,Friston et al., 1995a) chose a design matrix thatontained four Fourier basis functions for each of threereset experimental conditions in a reference function,or a total of twelve columns in their version of fMRI
anonical variates analysis.) Further, let M denote the

Gow

a

rtWg

Sm

ama1iSetptdismebrita1cttttsvrcer

CcawcutC

ps

sa

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298 LANGE ET AL.

3 U matrix of group means, so that ZM is the matrixf predictors given the group indicators, and computeithin- and between sample covariance matrices

SW 5 (B* 2 ZM)T(B* 2 ZM)/(T 2 G)

nd

SB 5 (ZM)T(ZM)/(G 2 1),

espectively; note that column means of B* are zero dueo our double-centering and hence do not appear in SB.e seek a canonical variate, e, that satisfies the

eneralized eigenvalue problem

SBe 5 lSWe. (A.4)

olving (A.4) for canonical variate e is equivalent toaximizing the objective function given by

Fe 5eTSBe

eTSWe,

ratio known as the generalized Raleigh quotient inathematical physics or as Fisher’s F statistic for

nalysis of variance in statistics (Duda and Hart,973). When SW is of full rank (nonsingular), Eq. (A.4)s converted to the standard eigenvalue problem,W21SBe 5 le, solved by application of the SVD (Mardiat al., 1979; Golub and Loan, 1996). Thus CVA may behought of as a ‘‘noise-informed’’ second SVD, where theooled within-group covariance matrix, SW, captureshe ‘‘noise’’ in the selected orthogonal time trendserived from the first SVD. A further remark concern-ng our subset selection is in order. Selection of theubset t of orthogonal time trends is not simply aatter of choosing those associated with the largest

igenvalues. Indices t need not be contiguous and maye found in practice to contain a mixture of componentsanging across their entire eigenspectrum. For physicalnterpretation of results, as here, it may well be thathe relations associated with the smallest eigenvaluesre those of greatest interest (Gnanadesikan, 1997, p.1). In their approach, Friston et al. (1995a) kept allomponents associated with eigenvalues larger thanhe average eigenvalue, which can lead to overfitting inhe present case. Instead, we start somewhat arbi-rarily with 50 orthogonal time trends associated withhe 50 largest eigenvalues from the first SVD. Theelected subset of orthogonal time trends is assessed byisual inspection, so one could perhaps argue that theeference function enters into the analysis implicitly, asould be said of the other methods that do not employ itxplicitly (ICA and FDA). We then sift these further by
egressing the first canonical variate from an initial u
VA on these 50 components and retain only thoseomponents that account for at least 2% of the varianceccommodated by the multiple linear regression. Last,e apply CVA a second time on the 11 to 18 remaining

omponents, depending on the particular data setnder study. Display the eigenimage associated withhe largest eigenvalue from this second CVA as aVA-field summary image.8. Independent component analysis (ICA). In inde-

endent component analysis we seek a linear decompo-ition of the data matrix given by

YT 5 SM; yvt 5 oq51

Q

svqmqt,

o that the spatial components, being the columns of S,re mutually independent:

1

V ov

svq(r) · Svq8

(r8) <1

V ov

svq(r) ·

1

V ov

svq8(r8)

or q Þ q8; i.e., the (r, r8) moments factor for indepen-ent signals. ICA decomposition can be obtained by anterative procedure if at most one of the spatial compo-ents is Gaussian (Comon, 1994; Bell and Sejnowski,995) and has been used extensively to model generalultivariate time series and in EEG analysis. Recently,

CA has been applied to fMRI by McKeown et al.1998b). In the present work, we have implemented the

cKeown et al. procedure using a modified Bell–ejnowski approach for identification of spatial andemporal patterns. The ICA summary image displayedere is an independent spatial source signal of maxi-um variance identified by inspection from among the

et of independent components. A major differenceetween CVA and ICA is the decomposition of the imageatrix employed to construct the set of basis vectors

sed in the analysis. While this decomposition is uniqueor CVA, since it is based on the SVD and a fixedeference function, it is only unique for ICA if theomponents are indeed independent. It is rarely if everhe case that, in the brain, there exist componentshich are completely independent.9. Functional data analysis (FDA). A newly emerg-

ng statistical research area is the field of functionalata analysis, or FDA (Ramsay, 1982; Silverman, 1985;amsay and Dalzell, 1991; Rice and Silverman, 1991;amsay and Silverman, 1997). FDA for fMRI treatsbserved time series as single entities rather than aserely sequences of individual observations. The term

‘functional’’ in this context refers to the intrinsic struc-ure of the data rather than their explicit form, al-hough for fMRI the term is quite apt. In practice thebserved data are discretely sampled versions of true
nderlying continuous functions. If these discrete val-

uoindimwt(dsdidadtacp‘

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tpaai

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es are assumed falsely to be errorless, then conversionf raw data points to true functional form involvesnterpolation. Otherwise, one employs smoothing tech-iques to remove observational errors that cause theata to be rougher than the true functions (see fornstance Hastie and Tibshirani, 1990; Green and Silver-

an, 1994). That is, one posits a model ytv 5 f (t, n) 1 etv,here t represents time (moment at which activity at

he brain location is sampled) and n represents spacelocation in the brain), and f (t, n) is a latent functionefined on the tensor product domain P 5 (t, n). Theampling rate or temporal resolution of the observedata, whose limits arise from constraints on the imag-ng device running continuous duty cycles, is a keyeterminant of what is possible in FDA above what islready available through existing statistical proce-ures. Another determinant of what FDA can do lies inhe local curvature of the functions themselves, defineds the second temporal derivative of f (t, n). Whereurvature is high, it is essential to have enough dataoints to estimate the function effectively, where

‘enough’’ depends on errors etv.The particular type of FDA used in the present

pplication is regularized PCA. Regularization, being aype of model-based smoothing that involves likelihoodnd penalty terms, is performed within the PCA itself;either the fMRI time series or the PCA results aremoothed separately. Specifically, for this application,he image matrix is expanded in cubic B-spline basisunctions B 5 55Bp(t), p 5 1, . . . , 4266, roughly one basisunction every nine timepoints, to obtain coefficientsatrix C. Defining J as the matrix of inner products

Bp, Bq8, K as the matrix of inner products 7D2Bp, D2Bq8,here D2 is the second derivative operator, and l as a

unable regularization parameter serving as a rough-ess penalty, find Cholesky decomposition LLT 5 J 1K. The roughness penalty was set at l 5 1026,mplying more closeness to an interpolated spline curvehan to a least-squares regression solution of theinimization problem. Having found L, solve the lin-

ar system LD 5 C for new coefficients D, and thenerform a classical principal component analysis on Do find eigenvectors U. Next, solve LTZ 5 U for Z andenormalize so that zp

TJzp 5 1 for each p. Last, trans-orm back to find eigenvectors z 5 ZTB and plot these asDA-field summary images. In the present case, sincehe summary image associated with the largest eigen-alue was a nearly constant image, we display theDA-field summary image associated with the second

argest eigenvalue.A.2. Resemblance indicators. The rationale behind

he resemblance measures developed for pairwise com-arisons of summary images given in Tables 2 and 4 iss follows. In the presence of considerable spatialutocorrelation within each summary image, and differ-
ng patterns of spatial autocorrelation between them,
oxelwise differences of any two such maps will alsoery likely be spatially autocorrelated. Therefore, anyormal assessment of how one map resembles anotherhat has the desirable property of being sensitive toaired voxel location permutations must employ someeighborhood structure between locations and takepatial autocorrelation into account. Similar consider-tions for single maps underlie much of the theory ofandom fields and their thresholds. Linear models forpatially autocorrelated maps (Ripley, 1981; Cressie,993) are well understood and are applied to theresent problem. We employ a 4-diagonal ‘‘bishop’’eighborhood structure to define neighborhood weightsatrix N; ‘‘rook’’ and ‘‘queen’’ neighborhoods yielded

imilar results. A conditional spatial autoregressionovariance structure of form S 5 (I 2 rN)21Ds2 wasmployed, where r and s2 are scale parameters esti-ated from the data and D is a diagonal matrix of

eighborhood weights taken simply as the identityatrix. A linear model with a mean function coeffi-

ients g, design matrix X consisting of cubic polynomi-ls of the spatial coordinates, and autocorrelated errorsith variance-covariance matrix S is fit to the surfacef voxelwise differences of each distinct pair of sum-ary images scaled to have mean 0 and variance 1. The

ariance–covariance matrix of mean function coeffi-ients is V21 5 XTSX. Under a null hypothesis ofesemblance, there is no structure in this residualurface and an omnibus statistic defined as d 5 gTV21gs equal to zero; the null hypothesis is rejected for largealues of d. The estimated value of this scalar isompared to a central x2 distribution with degrees ofreedom equal to the rank of V, in this case 8. Thessociated P value is reported as the resemblanceeasure for each summary image pair. Binary indica-

ors are given in Table 2, which are either 0 when P #.01 or 1 otherwise. As noted previously, several of theethods are instances of the general linear model (T,PM, FIR, PFT) and thus it would be inappropriate toive resemblance between these particular summarymages too much weight, as they fall within the samelass of statistical procedure. No strict adjustment of Palues for multiplicity has been performed, as thesendicators are intended to be descriptive rather thannferential.

ACKNOWLEDGMENTS

We thank the editors and two anonymous reviewers whose com-ents helped to improve the paper. This work was funded byational Institutes of Health Grants MH57180, as part of theuman Brain Project, and NS37483 to N.L.

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