Modelng sae-relaed fmri acvy usng change-pon heory Marn A. Lndqus 1*, Chrsan Waugh and Tor D. Wager 3 1. Deparmen of Sascs, Columba Unversy, New York, NY, 1007. Deparmen of Psychology, Unversy of Mchgan, Ann Arbor, MI, 48109 3. Deparmen of Psychology, Columba Unversy, New York, NY, 1007 ADDRESS: * Correspondng auhor: Marn Lndqus Deparmen of Sascs 155 Amserdam Ave, 10h Floor, MC 4409 New York, NY 1007 Phone: (1) 851-148 Fax: (1) 851-164 E-Mal: marn@sa.columba.edu
Absrac The general lnear model (GLM) approach has arguably become he domnan way o analyze funconal magnec resonance magng (fmri) daa. I ess wheher acvy n a bran regon s sysemacally relaed o some known npu funcon. However, becomes mpraccal when he precse mng and duraon of psychologcal evens canno be specfed a pror. In hs work, we nroduce a new analyss approach ha allows he predced sgnal o depend nonlnearly on he ranson me. I uses deas from sascal conrol heory and change-pon heory o model slowly varyng processes wh unceran onse mes and duraons of underlyng psychologcal acvy. Our approach s exploraory n naure, whle reanng he nferenal naure of he more rgd modelng approach. I s a mul-subjec exenson of he exponenally weghed movng average (EWMA) mehod used n change-pon analyss. We exend exsng EWMA models for ndvdual subjecs (a sngle me seres) so ha hey are applcable o fmri daa, and develop a group analyss usng a herarchcal model, whch we erm HEWMA (Herarchcal EWMA). The HEWMA mehod can be used o analyze fmri daa voxel-wse hroughou he bran, daa from regons of neres, or emporal componens exraced usng ICA or smlar mehods. We valdae he false-posve rae conrol of he mehod and provde power esmaes usng smulaons based on real fmri daa. We furher apply hs mehod o an fmri sudy (n = 4) of sae anxey. A oolbox mplemenng all funcons n Malab s freely avalable from he auhors. Keywords: fmri, Change-pon, EWMA
Inroducon The voxel-wse general lnear model (GLM) approach has arguably become he domnan way o analyze funconal magnec resonance magng (fmri) daa. Ths model s well-sued for esng wheher varably n a voxel s me course can be explaned by a se of a pror defned regressors ha model predced responses o psychologcal evens of neres. The GLM approach has proven parcularly powerful for dealng wh even-relaed desgns, as a sequence of sparse evens occurrng a random nervals affords a relavely specfc predced response, and a good f o he daa s ofen nerpreed n erms of sgnal evoked by a parcular psychologcal even (Wager, Hernandez, Jondes, & Lndqus, n press; Worsley & Frson, 1995). Mxed block/even-relaed desgns have been used o nvesgae sae-effecs n fmri (Vsscher, Mezn, Kelly, Buckner, Donaldson, McAvoy, Bhaloda, & Peersen, 003), bu nferences on saes are subjec o he lmaons n nerpreably of block desgns, and only saes ha are under relavely precse expermenal conrol (.e., hey can be urned on and off repeaedly by expermenal manpulaons) can be suded. New knds of sascal models are needed o capure sae-relaed changes n acvy, parcularly when psychologcal saes have unceran onse mes, emporal nensy profles, and duraons. As an example, consder an fmri sudy planned o assess he effecs n he bran of a new shor-acng drug (Fg. 1A) an even ha s no easly repeaed many mes n a block desgn. Researchers mgh be neresed n he knecs of he drug n dfferen bran areas. Some areas mgh respond o he nal admnsraon, and ohers mgh show susaned responses conssen wh plasma concenraons. Ye oher areas may aper off or ramp up over he perod of admnsraon, as he bran coordnaes an orchesraed response o dfferen componens of he experence. Sae-relaed acvy s crcal n many oher domans, ncludng drug pharmacology 3
(Breer e al., 1997; Wse, Wllams, & Tracey, 004), memory (Donaldson, Peersen, Ollnger, & Buckner, 001; Oen, Henson, & Rugg, 00), movaon and emoon (Gray, Braver, & Rachle, 00). For example, a parcpan wachng a funny flm mgh experence amusemen ha bulds gradually over me. Whou momen-by-momen repors, whch may aler he experence (Taylor, Phan, Decker, & Lberzon, 003), he progresson of amusemen over me wll be dffcul o specfy. Ths s rue n general for emoonal saes, whch are dffcul o urn on and off rapdly (requred for blocked or ER desgns) and may las longer han or no as long as he puave elcng even. The GLM's effecveness n modelng such sae-relaed acvy s lmed n several ways, and developng new approaches could lead o fruful alernaves. One ssue concerns he mach beween he desred nference and he nference provded by he model. For example, researchers may be neresed n makng nferences on he onse and duraon of bran acvy n a number of research sengs. Suppose a GLM s specfed ha ncorporaes a susaned (10-s long) response o a smulus, as s common wh epoch-relaed desgns. The GLM provdes nferences on he magnude of he hemodynamc responses. These nferences wll be approprae for esng hypoheses regardng he magnude, bu no for hypoheses concernng he duraon of acvaon. For example, suppose ha here s only a bref hemodyamc response o he smulus. The regressor wll f parally, and may well reach sascal sgnfcance as he GLM ess wheher he magnude of he regressor s non-zero. However, s clear ha nferrng a susaned response from hs f would be napproprae. Thus, gven a hypohess abou he duraon of acvy, would clearly be more nformave o perform nferences drecly on he duraon of acvaon. The GLM does no provde for hs ype of nference. Two oher relaed ssues wh he GLM nvolve he poenal for ms-modelng (Neer, 4
Kuner, Nachshem, & Wasserman, 1996), whch can reduce sensvy and lead o ncorrec nferences, and lmaons n reproducbly (Lou, Su, Lee, Ason, Tsa, & Cheng, 006; Genovese, Noll, & Eddy, 1997) for some paradgms. Frs, n he example above, even a paral f o he regressor may lead o a sgnfcan resul. Therefore, model sgnfcance alone s no enough o mply ha he specfed model s a beer choce han oher plausble alernaves. More flexble models reduce he rsk of ms-modelng and ncrease sensvy when he precse shape of he response canno be accuraely specfed a pror, and hs s an advanage of he flexble approach o modelng sae-relaed acvy we propose n hs paper. Second, suppose ha a bran regon reproducbly responds o a parcular emoon (e.g., anxey), hough he onse and me course vares across parcpans and sudes dependng on he parcular ask demands and ndvdual propenses. In hs case, here s no GLM model ha wll gve reproducble acvaon n he regon of neres. However, a more flexble model may capure conssences n acvaon magnude whle allowng for varaons n mng. For each of he suaons oulned above, model flexbly s crucal. Models ha perm more daa-drven esmaes of acvaon, and nferences on when and for how long acvaon occur, may be advanageous for dscoverng sae-relaed acvaons whose mng can only be specfed loosely a pror. Raher han reang psychologcal acvy as a zero-error fxed effec specfed by he analys (as n GLM analyss) and esng for bran changes ha f he specfed model, daa-drven approaches aemp o characerze relable paerns n he daa, and relae hose paerns o psychologcal acvy pos hoc. One parcularly popular daa-drven approach n he fmri communy s ndependen componens analyss (ICA), a varan on a famly of analyses ha also ncludes prncpal componens and facor analyss (Beckmann & Smh, 004; Calhoun, Adal, & Pekar, 004; McKeown & Sejnowsk, 1998). Recen exensons of hese 5
mehods can denfy bran acvy paerns (componens) ha are relable across parcpans, reang parcpan as a random effec (Beckmann & Smh, 005; Calhoun e al., 004), and can denfy sae-relaed changes n acvy ha can subsequenly be relaed o psychologcal processes. However, hese mehods do no provde sascs for nferences abou wheher a componen vares over me and when changes occur n he me seres. In addon, because hey do no conan any model nformaon, hey capure regulares whaever he source; hus, hey are hghly suscepble o nose, and componens can be domnaed by arfacs. Thus, he capacy for sascal nference s a srengh of he GLM, whereas he ably o dscover acvaon paerns ha are relaed o a relavely unconsraned psychologcal model (.e., he approxmae onse and offse of a menal sae under loose expermenal conrol) s a srengh of daa-drven mehods. In hs paper, we presen a model-drven approach for denfyng changes n fmri me seres n ndvdual and group daa ha allows for vald populaon nference. Wha ses hs work apar from he GLM approach s ha he predced sgnal depends non-lnearly on he model parameer (ranson me). In essence, he suggesed approach can be seen as an exenson of he GLM framework o allow for unknown ranson mes. The approach uses deas from sascal conrol heory and change-pon heory o model slowly varyng processes wh unceran onse mes and duraons of underlyng psychologcal acvy. Thus, lke daa-drven mehods, he model uses he daa o come up wh esmaes of wheher, when, and for how long acvaon occurred wh only mnmal specfcaon of a pror consrans. Concreely, n our model, one need only specfy he lengh of a no acvaon baselne perod and, for some applcaons, ha a bran regon s acvy level alernaes beween wo saes (e.g., smulaed and res). The esmaes can hen be compared wh psychologcal or physologcal parameers of neres such as subjecve rangs, behavor, or physologcal 6
responses o consran nerpreaon. Thus, a benef of he mehods are ha hey are semmodel-free mehods of deecng acvaon and are herefore nsensve o varaons n he phase lag and shape of he hemodynamc response across he bran. Though he mehods presened here share he aracve feaures of daa-drven analyss mehods, s sll n s core a model-drven approach and reans he nferenal naure of he more rgd modelng approach: populaon sascal nferences are made on acvaon paerns. The change-pon analyss ha we develop s a mul-subjec exenson of he exponenally weghed movng average (EWMA) change-pon analyss (also called sascal qualy conrol chars (Neubauer, 1997; Robers, 1959; Shehab & Schlegel, 000)). Acvy durng a baselne perod s used o esmae nose characerscs n he fmri sgnal response. Ths acvy s used o make nferences on wheher, when, and for how long subsequen acvy devaes from he baselne level. We exend exsng EWMA models for ndvdual subjecs (a sngle me seres) o nclude AR(p) and ARMA(1,1) nose processes, hen develop a group random effecs analyss usng a herarchcal model, whch we erm HEWMA (Herarchcal EWMA). The HEWMA mehod may be used o analyze fmri daa voxel-wse hroughou he bran, daa from regons of neres, or emporal componens exraced usng ICA or smlar mehods. Here, we provde power and false-posve rae analyses based on smulaons, and we apply HEWMA o voxel-wse analyss of an anxey-producng speech preparaon ask. We demonsrae how he mehod deecs devaons from a pre-ask-nsrucon baselne and can be used o characerze dfferences beween groups of ndvduals n boh evoked fmri acvy and changes n saes. The mehod also provdes nferences on he number and mng of changes n he sae of 7
acvy, denoed change-pons (CPs). Knowledge of he CPs may provde a bass for dscrmnang ancpaory acvy from responses o a challenge (e.g., acvy ha begns n ancpaon of pan from ha elced by panful smul; (Koyama, McHaffe, Lauren, & Coghll, 005; Wager, 005)). CP maps may also be used o denfy bran regons ha become acve a dfferen mes durng a challenge (e.g., he early, md, or lae phases of a onc panful smulus). These maps may provde meanngful characerzaons of dfferences among ndvduals: For example, he onse me of bran responses o anxey may provde clncally relevan markers of anxey dsorders. In sudes of emoon, he speed of recovery from adverse evens s hough o be an mporan predcor of emoonal reslence (Fredrckson, Tugade, Waugh, & Larkn, 003; Tugade & Fredrckson, 004), and CPs could provde drec bran measures of recovery me. In cognve psychology, bran CPs n problem solvng and nsgh asks may provde a drec neural correlae of radonal me-o-soluon measures n cognve sudes (Cheng & Holyoak, 1985; Chrsoff e al., 001). Inferences on he duraon of acvaon could have smlar advanages, ncludng ess on he duraon of pharmacologcal or emoonrelaed acvy and he duraon of cognve ask-se relaed acvy n cognve sae-em desgns. Mehods There are a large varey of change-pon deecon problems ha presen hemselves n he analyss of me seres and dynamcal sysems. In hs secon, we frs develop a mehod for deecng changes n acvaon paerns for a sngle me seres usng exponenally weghed movng averages (EWMA), and hen develop a herarchcal exenson (HEWMA) approprae for 8
mulsubjec fmri sudes. Furher, we nroduce mehods for esmang he exac mng and duraon of he deeced change. I. The exponenally weghed movng-average (EWMA) model r =,, 1 (e.g., an Gven a process ha produces a sequence of observaons x ( x x K ) T fmri me seres), we frs consder a wo-sae model where he daa s modeled as he combnaon of wo normal dsrbuons, one wh mean θ 0 and covarance marx Σ, and he second wh mean θ 1 and he same covarance Σ. Durng a baselne acquson perod, he process generaes a dsrbuon of daa wh mean θ 0, and whle n hs sae, he process s consdered o be n-conrol. The observaons follow hs dsrbuon up o some unknown me τ, he change-pon, when he process changes (.e., a new psychologcal sae resuls n ncreased or decreased neural acvy), resulng n he generaon of fmri observaons from he second dsrbuon wh mean θ 1 (see Fg. 1A). Whle n hs second sae, he process s deemed o be ou-of-conrol, or n he ou-of-conrol (OOC) sae. The sascal model for hs framework can be wren as follows: x = + ε for = 1, Kn (1) s x n where s denoes he sgnal and ε he nose a me. The sgnal s specfed by s θ 0 = θ1 for = 1, Kτ for = τ + 1, Kn () v ε =, follows a mean-zero normal dsrbuon wh covarance marx Σ,.e. and ( ε ε ) T,, 1 K ε n ( 0,Σ) r ε ~ N. (3) 9
The dagonals of he n x n marx Σ are he nose varance esmaes for each observaon, and he off-dagonals are he covarance among observaons nduced by auocorrelaon n he fmri me seres. A a laer sage we wll relax he consran of a sngle change-pon and allow s o move beween he wo saes. The problem of deecng and esmang change-pons has been suded exensvely n he change-pon and sascal qualy conrol leraure. In sascal conrol he exponenally weghed movng average (EWMA) conrol char has become a popular and flexble approach for deecng devaons from some baselne mean. I s based on he EWMA sasc, z. The sasc s a emporally smoohed verson of he daa, and s defned as follows: z λ for = 1, Kn (4) = x + ( 1 λ) z 1 or, equvalenly, z 1 = λ = j 0 j ( 1 λ) x j + (1 λ) θ 0 for = 1, Kn (5) where 0 < λ < 1 s a consan smoohng parameer chosen by he analys, and he sarng value z 0 s se equal o he baselne mean, θ 0. Thus, each value of z s a weghed average of he curren observaon x and he prevous value of he EWMA sasc. Noably, snce he EWMA sasc s a weghed average of he curren and all pas observaons, s relavely nsensve o volaons o he normaly assumpon. Smoohng he daa (.e., decreasng λ ) can ncrease power o deec devaons from he null model by regularzng he daa; however, he opmal choce of smoohng parameer depends on he naure of he devaons. A general rule of humb s o choose λ o be small (more smoohng) f one s neresed n deecng small bu susaned shfs n he process, and larger (less smoohng) f he shfs are expeced o be large bu bref. 10
The opmal choce of λ s dscussed n greaer deal n Lucas & Saccucc (Lucas & Saccucc, 1990). Whou loss of generaly we wll assume ha z = 0 0, and herefore we can rewre he sequence of EWMA sascs n marx noaon as r r z = Λx (6) r =,, 1 z n and where z ( z z K ) T 1 0 (1 λ) 1 (1 λ) 1 Λ=λ M O (1 λ) n 1 (1 λ) n 1 (1 λ) n K (1 λ) 1 (7) s a lower rangular smoohng marx. II. Sascal nference on he presence of acvaon n EWMA In general, we are neresed n makng nference on wo feaures of he model saed n Eqs. 1-3. Frs, we seek o develop sascal ess o deermne wheher a change n dsrbuon (.e., a deparure from he baselne sae of fmri acvy) has ndeed aken place,.e., wheher o rejec he null hypohessθ 0 = θ1. If a change s deeced, we would also lke o esmae when exacly he change ook place,.e. esmae he unknown parameer τ. For deecng acvaon, he null hypohess s ha here s a sngle, baselne sae where he mean (denoed µ) s consan. The alernave s ha here are wo or more saes wh dfferen mean acvy levels, hough here we consder only he smples wo-sae (baselne and acvaon) alernave. Thus, 11
H : μ = θ for = 1, Kn 0 0 H a : μ = θ 0 for = 1,Kτ and μ = θ1 for = τ +1, Kn (8) Our am s o assess he probably of observng he daa under he null hypohess ~ N( θ,σ 0 ) x r. In our analyss he parameers of he covarance marx are esmaed usng daa acqured durng a baselne perod n whch he subjec remans n a resng sae (Fg. 1A and 1B). For each EWMA sasc z followng he baselne perod, we compue a es sasc T : T = z θ 0 Var(z ). (9) where Var z ) s he varance of he EWMA sasc a me. The sasc T follows a - ( dsrbuon wh df degrees of freedom (dscussed below), provdng p-values accordng o classcal nference. Furher, conrol lms (analogous o confdence nervals) for deecng values of z ha vary sgnfcanly from baselne can be calculaed as follows: * θ ± Var( z ) (10) 0 where * s a crcal value from he -dsrbuon correspondng o he desred false posve-rae (see conrol lms n Fg. 1B). If z a any me exceeds he conrol lms, he process s deemed o have changed saes and he null hypohess s rejeced. The conrol lms ncorporae correcon for mulple ess across me, as dscussed below. EWMA sasc varance Under whe nose, can be shown (Mongomery, 000) ha ( 1 ( 1 ) ) λ Var( z ) = σ λ (11) λ For AR(1) nose, he mnmum auocorrelaon model approprae for fmri daa, Var z ) has prevously been derved (Schmd, 1997). Explc dervaons of Var z ) for a varey of ( ( 1
dfferen underlyng nose models (e.g. AR(p) and ARMA(1,1)) can be found n Lndqus and Wager (Lndqus & Wager, In press). To summarze, for an AR(p) process, he auocorrelaon funcon s gven by where p h γ ( k ) = A G m m (1) m= 1 G m are he roos of 1 φ z φ z p + K φ 0 (13) for m =1,K p and z 1 p = A m are consans (Brockwell & Davs, 00). In hs suaon, λγ (0) Var ( z ) = (1 (1 λ) ) + λ ( λ) p m m ( 1) (1 (1 λ) ) ( ) [ (1 λ) Gm ] G 1 λ G λ( λ) G m= 1 m A G (1 λ) m 1 λ m = 0. (14) For an ARMA(1,1) process, he auocorrelaon funcon can be calculaed (Brockwell & Davs, 00) as follows: 1 + θ + φθ σ γ ( h) = 1 φ ( 1 φθ )( φ θ ) σ φ 1 φ h-1 f h = 0 f h 1 Here, Var ( ) Z λγ = (1 (1 λ) ( λ) λ (1 λ) γ (1) ( 1) ( 1 ( 1 λ) ) ( λ)( 1 φ(1 λ) ) ( 0) λ φ ) + γ (1) ( φ(1 λ) ) ( ) 1 φ(1 λ) = 0 1 λ φ 13
In general, for a process wh fxed lengh and covarance marx Σ, he covarance marx of he me seres of EWMA sascs ( z r ), s gven by gven by he dagonal elemens of Σ *. T Σ * = ΛΣΛ Correcon for search across me and for spaal correlaon. Hence he varance of z r s n urn The p-values ha are calculaed a each me pon mus be conrolled for search over he me seres. Bonferron correcon s overly conservave because of posve dependence across me. A more sensve procedure, whch we adop, uses Mone Carlo negraon. Under he null hypohess, he sequence of es sascs T { T, T,, } r = follows a mulvarae - 1 K T n dsrbuon wh covarance marx Σ * and df degrees of freedom calculaed usng Saerhwae s approxmaon. Famlywse error rae (FWER) conrol across he me seres s provded by randomly generang vecors of n-lengh -values from he ( Σ *, df ) dsrbuon and usng her maxma o esmae a dsrbuon of maxmum null-hypohess -values (Nchols & Holmes, 00): T max = max (15) k { T } k { 1,Kn } We use he mulvarae T random number generaor provded n Malab (Mahworks, Nack, MA) wh 10,000 samples a each voxel, whch runs n less han 0.3s for n=00 on a personal compuer wh a.53 GHz Penum 4 processor and 1.00 GB of RAM. The 95 h percenle of he dsrbuon of T max provdes a crcal -value, * T, for wo-aled FWER conrol 1. If any T > T *, he voxel s sgnfcanly acvaed (or deacvaed) relave o he baselne perod. I should be noed ha he null dsrbuon of a lkelhood rao es for he hypoheses n Eq. 8 has been worked ou analycally n Hawkns (1977) and Worsley (1979) for he case when 1 The es s wo-aled because T max s defned based on he absolue value of T. 14
ε are ndependen and dencally dsrbued (d) normal random varables. However, he dsrbuons of he es sascs are raher complcaed and numercal mehods are used o sudy hem. Segmund (1985) provdes analyc approxmaons for hese dsrbuons. However, all he resuls are for he d normal case, whle we are dealng wh daa ha has sgnfcan auocorrelaon. Usng he permuaon mehods descrbed above allows us o expand he applcaon o comforably handle emporally auocorrelaed nose models. Fnally, we use false deecon rae (FDR) (Genovese, Lazar & Nchols (003)) o correc for correlaon over space. As FDR works on he p-values, s drecly applcable afer applyng EWMA. III. Esmaon and nference on change-pons (CPs) There are several mehods for esmang τ, he me pon a whch he sae shf akes place. In he zero-crossng mehod (Nshna, 199), he las me pon a whch he process crosses θ 0 before * T > T s he esmae of τ. More formally, le us assume ha me he EWMA sasc exceeds he conrol lms: A { T > T * } A s he frs = mn. (16) The change-pon esmaor of τ for an ncrease n he process mean s gven by he laes me pon pror o A n whch z les below ˆ θ 0, whch s he esmaed baselne mean. Tha s, we defne ˆ τ {, ˆ θ } A z max. (17) = 0 The change-pon esmaor of τ for a decrease s defned n an analogous manner. The mehod s man advanages are ha s concepually sraghforward and compuaonally effcen. 15
Acvaon duraon and mulple change-pons A lmaon of he zero-crossng mehod s ha only a sngle, nal CP s assessed, and here s no clear provson for assessng reurn o he baselne sae or he presence of mulple CPs. One possble approach would be o assume a reurn o baselne once he EWMA sasc crosses back across he conrol lm. Hence, he oal number of OOC pons can gve a rough ndcaon of he acvaon duraon, hough duraons are lkely o be based oward zero. As an alernave, we can use a Gaussan mxure model o classfy each observaon as eher belongng o he baselne (n conrol) or acvaed (OOC) dsrbuon. Ths procedure also allows us o esmae he lengh of me spen n he acvaed sae. Though, he mxure model s phlosophcally dfferen from change-pon esmaon usng he zero-crossng mehod ( s an esmaon raher han an nferenal echnque), we presen here as a flexble approach owards sudyng sae changes n an fmri me seres, whch can be exended o mulple acvaon saes. In he mxure model approach, we assume ha he daa has been pre-whened usng he covarance esmaes obaned from he EWMA sage of he analyss. We hen model he fmri me course as a mxure of wo normal dsrbuons, wh dfferen means, as follows: X ~ N( θ, ) for he baselne sae, and X ~ N( θ, ) for he acvaed sae (see Fg. 1C for 0 0 σ 1 an example). We can wre hs mxure as 1 1 σ X = ( 1 Δ) X + ΔX (18) 0 1 where he random varable Δ s equal o one wh probably p and equal o zero wh probably 1-p. The densy funcon of X can be wren f ( X x 0 X1 x) = (1 p) f ( x) pf ( ). (19) X + where X (x) s he normal probably densy funcon wh mean θ and varance σ. f 16
We can f hs model o he daa usng maxmum lkelhood mehods. The unknown parameers of he model are ( θ, θ, σ, σ, ) and he log-lkelhood can be wren: 0 1 1 p n 0, θ1, σ, σ, p ) = log X x ) = 1 ( f ( ) l( θ x (0) The parameers ha maxmze hs erm can be found usng he EM-algorhm (see Appendx A). Once we have deermned he maxmum lkelhood esmaes of he parameers n hs model, we need o classfy each daa pon accordng o whch sae hey belong o. Ths can be done usng Bayes formula. The probably ha a daa pon belongs n he acve (OOC) sae s gven by: pf X ( x ) 1 P ( acve x ) =. (1) f ( x ) X If P ( acve ) > 0. 5, hen he me pon s classfed as belongng o he acve sae, oherwse x s classfed as belongng o he orgnal baselne sae. The mxure model approach s aracve as does no requre one o specfy he number of change-pons presen n he me course a pror. I s mporan o noe ha n he EWMA framework he nose covarance was no assumed o vary beween he baselne and OOC saes. In hs secon we have relaxed hs assumpon o allow he covarance o vary (up o a scalng erm) beween he saes. Ths was done o ncrease he flexbly of he mxure model. However, we reurn o he assumpon of non-varyng nose covarance n he nex secon where he HEWMA framework s presened. IV. Herarchcal EWMA (HEWMA) for populaon nference The EWMA procedure oulned above s suable for sudyng a sngle me seres for an ndvdual subjec. In fmri analyss we are ypcally mos neresed n deecng wheher an 17
effec s presen over an enre group of subjecs. Group changes are of prmary neres n he neuroscences for makng populaon nferences. For hs reason, we developed a group analyss usng a herarchcal (.e., mxed effecs) exenson of EWMA. I should be noed ha here does exs a mulvarae exenson of he EWMA framework - he mulvarae exponenally weghed movng average (MEWMA) (Lowry, Woodall, Champ & Rgdon (199)). However, drec applcaon of MEWMA s no approprae for our purposes. For one, he MEWMA approach s only useful for performng a fxed-effecs analyss, whle we are prmarly neresed n mxed effecs analyss and he ably o make populaon wde nference. Secondly, he MEWMA approach does no dfferenae beween he drecon of he change n mean, whle we are neresed n sudyng acvaons and deacvaons separaely. To crcumven hese wo shorcomngs of he MEWMA approach, we nsead chose o develop he herarchcal exponenally weghed movng-average (HEWMA) model ha allows us o perform a mxed-effecs analyss on fmri group daa usng he same ype of analyss ha he EWMA mehod allows for sngle-subjec daa. We use he EWMA sasc and covarance marx defned prevously, ogeher wh a beween-subjecs covarance erm, o oban he HEWMA sasc, whch s a weghed populaon average, and s correspondng covarance marx. Thereafer, he Mone-Carlo procedure used n he EWMA framework s appled o ge p- values and es he hypohess of conssen acvaon whn he group, or alernavely dfferences n acvaon beween groups. The daa conss of a me course from one voxel, an ROI or a componen for each of m subjecs. Independen analyss s performed for each voxel/roi/componen me seres (.e., he massve unvarae approach). Le x denoe he daa for subjec a me, where = 1, Kn. Our herarchcal mxed-effecs model akes he followng form: = 1, Km and 18
x = s + ε () s = s + η (3) pop where η s he subjec-level nose erm a me and s, he underlyng sgnal for subjec, s consdered o be eher n a baselne or acvaed sae a each me pon. Fnally, he populaon sgnal s pop θ 0 = θ1 for = 1, Kτ for = τ + 1, Kn, (4) shows how he subjec mean sae-values are drawn from a larger populaon. In marx forma we can wre Eq. as x r r + r ε = s (5) r where x = ( x x Kx ) T, s = ( s s Ks ) T and ε ( ε ε Kε ) T 1,, n r 1,, n r =. Here ~ N( 0, Σ ) 1,, n ε r and he vecor s r can be consdered he equvalen of bea weghs, one wegh for each me pon. Smlarly, we can wre Eq. 3 as r r r s s + η (6) r s = pop pop pop pop where ( ) T pop = s s, Ks and he nose vecor r η s N ( 0, Σ B ). Here Σ B represens he 1, n nose covarance marx beween subjecs, and he off-dagonals are zero for ndependen subjecs; hus, r η s an n x 1 vecor of d (, ) subjec-level effec vares over me and may dffer among subjecs. N 0 σ B random varables. Ths mples ha he rue The mul-level model can be wren n sngle-level forma as: r x r s r r r s r = pop + η + ε = pop + ξ (7) 19
where ~ N( 0,Σ + Σ ) ξ r B ; ha s, he overall nose erm s dsrbued wh a varance equal o he sum of beween-subjecs and whn-subjecs varances. In he connuaon we wre V = Σ + Σ. Thus, n he HEWMA model he unknown parameers are θ,θ 0 1, τ, and he B varance componens Σ and Σ B. Noe ha f we have frs performed a sngle-level analyss (EWMA) on each subjec (see Eq. 4) we can use he EWMA sascs ( z for subjec a me ) n he second level. In hs case, we can assume ha he whn-subjec varance componens are known, and brough forward, from he frs level of analyss. Esmaon of he HEWMA sasc and s varance componens When calculang he oal varaon n subjec s EWMA sasc we wre, Σ V = Λ * V Λ T = ΛΣ Λ T + ΛΣ B Λ T = Σ +, (8) * * Σ B where * Σ s he varance brough forward from he sngle subjec analyss and * Σ B can be calculaed usng he fac ha Σ B = I σ. The only unknown erm ha needs o be esmaed n B whn he HEWMA framework s herefore he parameerσ. Hence, we can wre B Σ * B = αλ T Λ where Λ T Λ denoes he known poron and α he unknown poron of he covarance marx. In our approach we esmae he unknown varance componen usng resrced maxmum lkelhood (ReML). Our approach s equvalen o ha used n SPM (Frson e al., 00), whch uses an EM-algorhm o esmae he parameers of neres. To smplfy noaon we begn by rewrng he problem n marx form. Le, r z G r1 r r m [ z z L z ] T T T T = (9) 0
be he combned vecor of EWMA-sascs for all m subjecs (G denoes group). Recall, ha n he herarchcal model we can wre he covarance marx for each ndvdual subjec as V * * * = Σ + Σ B. Hence, follows ha he covarance marx for z r G can be wren: V * G V1 = 0 * V * O 0 * Vm (30) Furher, le G [ I I L I ] T = be an mn n n n n hs noaon we can defne he HEWMA-sasc, z r pop r z G z pop where he covarance marx pop pop generalzed leas squares regresson: pop marx where I n s he, as n n deny marx. Usng r = G + r ξ (31) r * ξ s N (, V ) 0 G T * 1 1 T * ( G VG G) G VG zg z r ˆ = 1r = m = 1 1 = 1 * 1. We can hen esmae he HEWMA sasc usng m * 1 r V V z. (3) Hence he esmae of z r pop wll be a weghed average of he ndvdual subjec s EWMA sascs. The covarance marx for he HEWMA sasc can be wren as: T * 1 ( G V G) 1 * 1 * 1 = = m V pop G V (33) = 1 In order o esmae he HEWMA sasc and s covarance marx, we frs need an esmae of * V. In our approach he whn-subjec componen he general form of * Σ s assumed known from he frs level and * Σ B s also assumed known up o a scalng erm. Hence we can wre he oal 1
varance for subjec as V * T * = α Λ Λ + Σ, where α s he unknown and Λ T Λ he known par of * Σ B. The problem of esmang he parameer α, s smlar o he varance componen esmaon procedure performed n SPM (Frson e al., 00). There hey fnd he so-called hyper-parameer usng an EM-algorhm and we wll follow he same general oulne here. Le, V * G = αq G + Σ G where T Q = Λ Λ and G I m Σ G * Σ1 = 0 Σ * O 0. (34) * Σ m The parameers α and Z pop are esmaed eravely usng he EM-algorhm n Appendx B. Correced p-values and populaon change-pon esmaes The fnal sep n he HEWMA framework s performng a Mone Carlo smulaon o ge correced p-values (searchng over me). Ths s done as descrbed n he EWMA secon, excep here we use Z pop and s covarance marx o calculae he relevan es-sascs and defne he mulvarae -dsrbuon. Fg. 6 shows an example of he whole procedure where he HEWMA group acvaon, Z pop, s esmaed from he me courses of 4 subjecs. Mone Carlo smulaons are used o fnd correced p-values. Fg. 6B shows he observed max T-sasc as a black lne overlad on he dsrbuon of he max T-sasc under he null hypohess. Fnally, FDR s used o correc for spaal correlaons. Change-pon esmaon can eher be performed drecly on he HEWMA sasc usng he mehods descrbed for he sngle subjec case, or performed on he ndvdual subjecs me courses. Though specfc nferenal echnques have no been developed for CP esmaon n he
group (HEWMA) analyss, he laer approach can be aken wh a few mnor aleraons of he sngle subjec framework. When usng he zero-crossng mehod, populaon nference s sraghforward usng a sgn permuaon es. Here he change-pon s defned o be he frs me pon, pror o he HEWMA sasc beng n he acve sae, n whch a sgnfcan number of he ndvdual me courses have crossed he baselne mean. Alernavely, s possble o esmae he change-pon for each ndvdual subjec and apply boosrap mehods o consruc ess and confdence nervals for he populaon change-pon. In addon, we have developed heory for usng MLE mehods for esmang he number of change-pons, as well as her mng. Ths maeral wll be publshed n a separae paper dealng solely wh change-pon esmaon. V. Smulaons In order o es he EWMA and HEWMA mehodology and he effcency of he changepon esmaon procedure we performed a se of hree smulaon sudes oulned n deal below. In Smulaon 1, we smulae a sngle-subjec daase wh four acve regons, each wh a dfferen acvaon onse me, and esmae boh he lkelhood of acvaon and CPs usng EWMA wh he zero-crossng mehod. In Smulaon, we assess he false posve rae (FPR) and power for he HEWMA mehod across values of he smoohng parameer λ. Smulaon 3 assessed power and FPR across varyng duraons of he baselne perod used for esmang varance componens. Noe ha n each of our smulaons he nose s consdered spaally ndependen. Ths was done o show he accuracy of he p-values n he case when hey are consdered uncorreced across space. In an applcaon o real daa, FDR (Genovese, Lazar & Nchols, 00) can be used 3
o correc hese p-values n he same manner ha hey are used o correc GLM-based p-values (Worsley & Frson, 1995). Smulaon 1. As shown n Fg., we consruced a 64 64 phanom mage conanng a square regon of sze 48 48 represenng a human bran. The mage nenses are assgned values of 1 or 0 for he pons nsde or ousde of he square, respecvely. Four smaller squares, wh dmensons 8 8, are placed nsde he larger square o smulae ROIs wh sac conras o he larger square. To smulae a dynamc mage seres, hs base mage s recreaed 50 mes accordng o a boxcar paradgm conssng of a prolonged perod of acvaon of lengh 50 me pons durng whch he sgnal whn he four squares ncreases o. The onse me of acvaon vares beween he four regons and akes values of 60, 80, 100 and 10 me pons. Hence, each regon has acvaon of smlar lengh and nensy, bu wh varyng onse mes. Nose smulaed usng an AR() model wh sandard devaon equal o 1 s added o each voxel s me course. We analyzed he smulaed daa se usng EWMA wh λ = 0. and an AR() nose model. We furher esmaed onse mes (change-pons) and p-values for all acve voxels. Smulaon. The nex smulaon sough o sudy he false posve rae (FPR) and perform power calculaons for he HEWMA mehod. Acual fmri nose was exraced from nonsgnfcan voxels of he bran (chosen because her HEWMA sascs gave rse o p-values above.95), obaned from he expermenal daa descrbed n he nex secon. In oal 50,350 nose me courses of lengh 15 me pons were ncluded n he sudy. The smulaon mmcked a group analyss conssng of 0 subjecs. For he FPR sudy, null hypohess daa wh no acvaon was creaed by randomly samplng me seres from he collecon of nose me courses. Ths was done for each of 0 subjecs and a random beween-subjec varaon 4
wh a sandard devaon of sze one hrd of he whn-subjec varaon was added o each subjec s me course. A sgnfcance level of α = 0. 05 was used o deermne acve voxels. For he power calculaons he same procedure was repeaed, wh he dfference ha an acve perod of lengh 50 me pons was added o he nose daa, wh nensy equvalen o a Cohen s d of 0.5 (Cohen, 1988). Ths concdes wh values observed n expermenal daa (Wager, Vazquez, Hernandez, & Noll, 005). The HEWMA mehod was performed on 5000 replcaons of each of hese wo daa ypes for λ values se equal o 0.1, 0.3, 0.5, 0.7 and 0.9. The analyss was furher performed usng nose models n he HEWMA framework correspondng o whe nose (WN), AR(1), AR() and ARMA(1,1) nose. For each smulaon he frs 60 me pons were used as a baselne perod. Smulaon 3. The procedure was dencal o ha for Smulaon, excep ha n hs smulaon, he baselne lengh was eher 0, 40, 60 or 80 me pons. The analyss was agan performed usng nose models n he HEWMA framework correspondng o whe nose (WN), AR(1), AR() and ARMA(1,1) nose. For each smulaon λ was se equal o 0. and a sgnfcance level of α = 0. 05 was used. VI. Expermenal fmri daa collecon and analyss Parcpans. We appled he HEWMA mehod o daa from 30 parcpans scanned wh BOLD fmri a 3T (GE, Mlwaukee, WI). The expermen was conduced n accordance wh he Declaraon of Helsnk and was approved by he Unversy of Mchgan nsuonal revew board. Sx parcpans were excluded because of moon or nonlnear normalzaon arfacs, leavng 4 parcpans. They were dscarded pror o analyss due o subsandard spaal 5
normalzaon and/or excessve head moon. I should be noed ha hese subjecs would also have been excluded from a sandard GLM analyss. Task desgn. The ask used was a varan of a well-suded laboraory paradgm for elcng anxey (Dckerson & Kemeny, 004; Gruenewald, Kemeny, Azz, & Fahey, 004; Roy, Krschbaum, & Sepoe, 001), shown n Fg. 5. The desgn was an off-on-off desgn, wh an anxey-provokng speech preparaon ask occurrng beween lower-anxey resng perods. Parcpans were nformed ha hey were o be gven wo mnues o prepare a seven-mnue speech, and ha he opc would be revealed o hem durng scannng. They were old ha afer he scannng sesson, hey would delver he speech o a panel of exper judges, hough here was a small chance ha hey would be randomly seleced no o gve he speech. Afer he sar of fmri acquson, parcpans vewed a fxaon cross for mn (resng baselne). A he end of hs perod, parcpans vewed an nsrucon slde for 15 s ha descrbed he speech opc, whch was o speak abou why you are a good frend. The slde nsruced parcpans o be sure o prepare enough for he enre 7 mn perod. Afer mn of slen preparaon, anoher nsrucon screen appeared (a relef nsrucon, 15 s duraon) ha nformed parcpans ha hey would no have o gve he speech. An addonal mn perod of resng baselne followed, whch compleed he funconal run. Hear rae was monored connuously, and hear rae ncreased afer he opc presenaon, remaned hgh durng preparaon, and decreased afer he relef nsrucon (daa wll be presened elsewhere). Because hs ask nvolves a sngle change n sae, as n some prevous fmri expermens (Breer & Rosen, 1999; Esenberger, Leberman, & Wllams, 003), and he precse onse me and me course of subjecve anxey are unknown, hs desgn s a good canddae for he HEWMA analyss. 6
Image acquson. A seres of 15 mages were acqured usng a T*-weghed, snglesho reverse spral acquson (graden echo, TR = 000, TE = 30, flp angle = 90) wh 40 sequenal axal slces (FOV = 0, 3.1 x 3.1 x 3 mm, skp 0, 64 x 64 marx). Ths sequence was desgned o enable good sgnal recovery n areas of hgh suscepbly arfac, e. g. orbofronal corex. Hgh-resoluon T1 spoled graden recall (SPGR) mages were acqured for anaomcal localzaon and warpng o sandard space. Image analyss. Offlne mage reconsrucon ncluded correcon for dsorons caused by magnec feld nhomogeney. Images were correced for slce acquson mng dfferences usng a cusom 4-pon sync nerpolaon and realgned (moon correced) o he frs mage usng Auomaed Image Regsraon (AIR; (Woods, Grafon, Holmes, Cherry, & Mazzoa, 1998)). SPGR mages were coregsered o he frs funconal mage usng a muual nformaon merc (SPM). When necessary, he sarng pon for he auomaed regsraon was manually adjused and re-run unl a sasfacory resul was obaned. The SPGR mages were normalzed o he Monreal Neurologcal Insue (MNI) sngle-subjec T1 emplae usng SPM (wh he defaul bass se). The warpng parameers were appled o funconal mages, whch were hen smoohed wh a 9 mm soropc Gaussan kernel. Indvdual-subjec daa were subjeced o lnear derendng across he enre sesson (15 mages) and analyzed wh EWMA. An AR() model was used o calculae he EWMA sasc ( z ) and s varance, and z r and he varance esmaes were carred forward o he group level HEWMA analyss. We used cusom sofware (see auhor noe for download nformaon and appendces) o calculae sascal maps hroughou he bran, ncludng HEWMA (group) and p-values for acvaons (ncreases from baselne) and deacvaons (decreases from baselne); ndvdual and group CPs (calculaed on he group HEWMA -me seres) usng he zero- 7
crossng mehod, acvaon duraon as esmaed by he number of OOC pons, and CP and runlengh esmaes usng he Gaussan mxure model descrbed above. Sgnfcan voxels were classfed no ses of voxels showng smlar behavor usng k- means cluserng. We consdered acvaons and deacvaons separaely, and used k-means cluserng on he group CP and longes acvaed run lengh (from he mxure model) o assgn voxels no classes wh smlar behavor. To do hs, we used he k-means algorhm mplemened n Malab 7.4 (Mahworks, Nack, MA) wh he v x marx of values for he v sgnfcan acvaed (or deacvaed) voxels as npu. Ths choce was arbrary and s prmarly for daa vsualzaon; oher cluserng algorhms may also be used effecvely. The number of classes was deermned by vsual nspecon of he jon hsogram of CP and duraon values. Twelve classes were used for he analyses repored here. Ses of conguous supra-hreshold voxels ( regons ) of he same class were he un of analyss for nerpreaon. Examnng he sysemac feaures n he me courses of regons of neres perms us o make nferences abou he role he regon plays n he speech preparaon ask. Raher han beng lmed o esng wheher a voxel s acvaed durng he preparaon nerval on average compared wh baselne perods, he HEWMA mehod can deec a number of dfferen ypes of neresng sysemac feaures, ncludng susaned acvaon durng he preparaon ask, ransen acvaon durng nsrucon presenaon, flucuaons n baselne acvy ha may be relaed o he sar of scannng, and ohers. Of parcular neres are voxels whose acvaon onse s near he me of ask onse (a 60 TRs or 10 s) and whose acvy s susaned hroughou he ask (a leas 60 TRs/10 s), whch may reflec susaned anxey. 8
Resuls I. Smulaon resuls Smulaon 1. We used EWMA o creae a sgnfcance map and a change-pon map ha accuraely depcs he dfference n onse me beween regons. Seng he smoohng parameer λ = 0. and he sgnfcance level α = 0. 05, we analyzed each voxel usng he EWMA procedure oulned n he Mehods secon. Fg. A depcs he heorecal sgnfcance map, wh equal amoun of acvaon presen whn each of he four acve regons. Fg. C depcs he acual sgnfcance map obaned usng EWMA. Ths ndcaes ha we were able o accuraely deec a large number of acve voxels whn he regon of acvy, wh a mnmal number of false posves ousde of he regon. Fg. B depcs he heorecal change-pon map, where he nensy vares dependng on he onse me. Fg. D depcs he acual change-pon map obaned by calculang he zero-crossng change-pon esmae for each voxel deemed acve n he pror analyss. The CP map provded accurae esmaes of he onse mes for he four regons, as ndcaed by he smlar nensy values for he rue values (Fg. C) and esmaes (Fg. D). Examnaon of he error n CP esmaon showed a dsrbuon ha was cenered a zero wh a slgh lef skew (mean = -.0, S. Dev. = 6.3, medan = 0 and IQR = 5). Smulaon. Fgs. 3A and 3B shows he FPR and power calculaons for each nose ype as a funcon of he smoohng parameer λ. Fg. 3A shows ha he number of false posves ncreases for each nose ype as a funcon of λ (.e., wh less smoohng). Ths s naural, as low values of λ enal a greaer amoun of smoohng, and mnmze he rsk of he null hypohess daa venurng oo far from he baselne mean. The nomnal alpha level of 0.05 s shown by he horzonal dashed lne. As λ ncreases, he amoun of smoohng decreases and he FPR exceeds α = 0. 05. The ARMA model performs worse han he oher models n FPR and 9
power, whle he oher models appear o behave n a smlar manner (hough are somewha conservave, wh FPRs near 0.01 wh low λ ). All models conrol he FPR appropraely wh λ.4. Sudyng Fg. 3B, appears ha hough he power ncreases slghly for each nose ype as a funcon of λ, does no vary n a sgnfcan manner. All nose models gave roughly equvalen resuls. However, we sugges he use of he AR() model as has he flexbly o model perodc nose oscllaons ha are ofen produced n fmri as a resul of physologcal changes. Fg. 5A shows an ROC curve for hs nose model correspondng o each of he fve smoohng parameers. In summary, our smulaon sudes ndcae ha a low value of λ wll gve a es wh reasonable power and low false posve raes. Increasng he value of λ wll lead o a slgh ncrease n power, bu a he cos of an ncrease n FPR. In he connuaon we use λ = 0.. Smulaon 3. Fg. 3C and 3D shows he FPR and power calculaons for each nose ype as a funcon of he baselne perod lengh. Sudyng Fg. 3C, s clear ha he number of false posves decreases for each nose ype as he baselne perod ncreases. Ths s naural as a long baselne perod allows for more daa o accuraely esmae he parameers of he model. The ARMA model performs sgnfcanly worse han he oher models, and gves rse o a dramacally nflaed FPR for baselne lenghs less han 60 me pons. Ths performance may be due o he fac ha he baselne perod s oo shor o ge an accurae esmae of he varance componens for hs model ype, as he ARMA parameer esmaon s more complex han for he oher models (.e. MLE vs. mehod of momens). Sudyng Fg. 3D, appears ha hough he power ncreases slghly for each nose ype as a funcon of baselne lengh, does no vary subsanally across baselne duraons excep for he ARMA model whch performed worse han 30
he oher models. We agan sugges he use of he AR() model, and Fg. 5B shows an ROC curve for hs nose model correspondng o each of he four baselne lenghs. In summary, our smulaon sudes ndcae ha an ncreased baselne leads o ncreased power as well as decreased FPR, whch s advanageous. However, s neresng o noe ha he WN, AR(1) and AR() models are all robus enough o handle a baselne perod as shor as 0 me pons. Naurally, hese values depend on he nose characerscs, so collecng more baselne daa (e.g., 60 me pons) s recommended. The ARMA model, on he oher hand, requres a baselne of a leas 60 me pons. II. FMRI resuls The HEWMA analyss on he expermenal daa revealed ask-relaed changes conssen wh prevous leraure on neuromagng of emoon (Phan, Wager, Taylor, & Lberzon, 004; T. D. Wager, K. L. Phan, I. Lberzon, & S. F. Taylor, 003) ncludng acvaons n dorsolaeral and rosral medal prefronal corces, mddle emporal gyrus, and occpal corex (Fgs. 6 and 7). Deacvaons were found n venral sraum and venral aneror nsula. The acvaon s conssen wh wha mgh be expeced n a cognvely complex ask, whch nvolved vsual cues a wo perods durng he ask, he menal effor and subvocal rehearsal requred o prepare a speech, and he anxey elced by he ask conex. In parcular, he rosral PFC has been srongly mplcaed n processng of self-relevan nformaon and he represenaon and regulaon of aversve emoonal saes (Ochsner e al., 004; Phan, Wager, Taylor, & Lberzon, 00; Qurk & Gehler, 003; Qurk, Russo, Barron, & Lebron, 000; Ray e al., 005; T. Wager, K. L. Phan, I. Lberzon, & S. F. Taylor, 003). Informaon abou he onse and duraon of acvaon provded by he HEWMA analyss can consran nerpreaon of he roles of hese 31
regons n ask performance. Esmaes were made of he me of onse and duraon of acvy for acvaed regons. The range of esmaed onse mes, from around 40 TRs (80 s) o around 180 TRs (360 s) from he sar of scannng ndcaes ha dfferen regons were acvaed a dfferen mes durng scannng. Mos sgnfcan voxels showed acvaon onses around he me of ask onse (60 TRs), when he vsual cue o begn speech preparaon was presened (Fg. 7). Lkewse, acvaon duraon esmaes from he Gaussan mxure model ranged from ransen ncreases (approx. 10 TRs or 0 s) o susaned ncreases (~80 TRs, 160 s). The -D hsogram of sgnfcan voxels by CP and esmaed duraon s shown n Fg. 7B. K-means classfcaon was used o provde a way o denfy classes of acvaed voxels wh smlar onse and duraon esmaes. Class membershp s ndcaed by color n he hsogram n Fg. 7B. The dversy of onse mes and duraons suggess ha a varey of dfferen GLM models would be requred o deec hese acvaons. Examnng he group me courses of voxels n hese regons corroboraes hs vew and provdes addonal evdence. Tme courses for wo paerns of responses are shown n Fg. 8, ncludng a medal prefronal regon showng susaned acvy hroughou he anxogenc ask and an occpal regon showng ransen responses o he nsrucon perods (when vsual smul were presened). The baselne perod n Fg. 8 s ndcaed by he shaded gray box n each panel, and he HEWMA-sasc me course s shown by he hck black lne (+/- one sandard error across parcpans, shown by gray shadng). Conrol lms are shown by dashed lnes; hus, he regon s sgnfcanly acvaed f he HEWMA-sasc exceeds he conrol lm a any me pon. Imporanly, he me course of acvaon n rmpfc parallels he srucure of he ask n ha acvaon begns around he onse of speech nsrucons and s susaned hroughou he 3
duraon of he preparaon nerval. Ths regon has been relaed n many neuromagng sudes o subjecve anxey and self-referenal processng (Breer & Rosen, 1999; Doughery e al., 004; Esenberger e al., 003; Ray e al., 005; Wang e al., 005), and s hus a neurophysologcally plausble canddae o show susaned acvaon relaed o he anxogenc ask. Dscusson Typcally sascal mehods n fmri can be caegorzed no wo broad caegores: hypohess and daa drven approaches. Hypohess-drven approaches es wheher acvy n a bran regon s sysemacally relaed o some known npu funcon. In hs approach, ypcally, he general lnear model (GLM) s used o es for dfferences n acvy among psychologcal condons or groups of parcpans. However, for many psychologcal processes, he precse mng and duraon of psychologcal acvy can be dffcul o specfy n advance. In hs suaon he GLM approach becomes mpraccal, as he psychologcal acvy canno be specfed a pror. Daa-drven mehods, such as ndependen componens analyss (ICA), gve an accoun of he daa usng few a pror assumpons. Insead, hey aemp o characerze relable paerns n he daa, and relae hose paerns o psychologcal acvy pos hoc. The man drawback of hese mehods s ha hey do no provde sascs for makng nferences abou wheher a componen vares over me and when changes occur n he me seres. The purpose of he HEWMA mehod s o allow for he deecon of sysemac changes n acvy wh a varey of onse mes and acvaon duraons. The dfferences n onse and duraon are lkely o reflec dfferences n funconal anaomy,.e., he way n whch each acvaed regon parcpaes n he ask. Boh he EWMA and HEWMA mehods for fmri daa analyss are desgned o deec regons of he bran where he sgnal devaes relably from a 33
baselne sae. The mehods make no a pror assumpon abou he behavor of hese changes, and he mehod wll deec acvaon and deacvaon, as well as regons wh boh shor and prolonged acvaon duraon. In hs sense, boh EWMA and HEWMA can be hough of as searches for acvy dfferences across me, correcng for he mulple comparsons esed and accounng for he correlaon among observaons. Once a sysemac devaon from baselne has been deeced, he second sep n he analyss enals esmang when exacly he change ook place, as well as he recovery me (f any). Ths esmaon procedure can be performed usng he zero-crossng mehod or a Gaussan mxure model. Oher mehods are under developmen as well (e.g. an MLE approach) and we presen he mehods above as a sarng pon for furher work. Once hese esmaes are obaned, we can cluser he acve voxels no groups whose esmaes behave n a smlar manner. Ths allows us o classfy regons and even dscard of regons for whch he acvaon was rggered by effecs such as drf or movemen. As a fnal sep, afer esmang he changepon for each acve voxel n he bran, we summarze he resuls n a Change-pon Map (CPM). A CPM s an mage of he bran wh a color-coded change-pon mask supermposed, whose nensy vares dependng on he esmaed onse me of acvaon. In hs paper we have assumed ha he change pons were fxed n me across groups of subjecs. We feel hs provdes a useful frs sep owards esmang he laency of processes n he bran. There may be suaons where he change-pon, correspondng o a ceran smul, dffers across subjecs. In hs case may prove benefcal o nsead allow he change-pons o vary randomly across subjecs. Ths opc s ofen referred o as mul-pah change-pon problems (Asgharan & Wolfson, 001), and wll be he focus of fuure work. 34
In addon, we have chosen o derend he fmri me courses pror o he EWMA/HEWMA analyss n order o remove nusance parameers (e.g. drf). However, would be relavely sraghforward o exend he EWMA framework o smulaneously esmae he rend regressors whle performng he change-pon deecon. For example, here are mehods are esng for change-pons n smple lnear regresson (Km & Segmund 1989), whch would drecly allow for he modelng of drf componens n he model. Whle we have no furher explored hese echnques a hs me, hs s an neresng drecon for fuure research. Recommended choces for analyss parameers The man decsons ha need o be made before applyng hese mehods o expermenal daa are he choce of he smoohng parameer λ and he lengh of he baselne perod. Accordng o our power and false-posve rae analyses, a relavely low value of λ gves hgh power, wh a srong conrol of he FPR. In our analyss of expermenal daa we ypcally choose a value of 0., as hs appears o gve rse o an adequae amoun of smoohng for he analyss of fmri daa. The opmal value of λ wll vary dependng on wheher bref or susaned changes are of greaer neres, wh lower values beng more approprae for more susaned acvy. The lengh of he baselne perod s anoher ssue. The daa whn hs perod s used o esmae he baselne mean, as well as he whn-subjec varaon, and our smulaons show ha, unsurprsngly, longer baselne perods produce more accurae esmaes. However, our analyss ndcaes ha he mehod s relavely sable for even very shor baselne perods for whe nose, AR(1) and AR() models. The ARMA model showed ncreased sensvy o shor baselne perods wh drascally ncreased FPR and decreased power. Though some models gve reasonable resuls wh relavely shor baselne perods, a longer baselne perod of a leas 60 35
me pons s recommended o ensure sable esmaes of he baselne acvaon varance. The AR() model may be mos approprae for fmri daa, as has he flexbly o model perodc nose oscllaons ha are ofen produced n fmri as a resul of physologcal changes (e.g., pulsale moon of he bran due o breahng and cardac acvy). Poenal applcaons HEWMA appears o be an approprae analyss for group fmri daa, parcularly when s no possble o replcae expermenal manpulaons whn subjecs (e.g., a sae anxey nducon ha canno be repeaed whou changng he psychologcal naure of he sae). Emoonal responses are one prme canddae for applcaons of he mehod. Bu here are a number of oher domans n whch may be useful as well, and he mehod apples o any longudnal daa wh enough observaons so ha repeaed measures ANOVA (for example) s mpraccal. HEWMA may be parcularly useful for areral spn labelng and perfuson MRI sudes, whch measure bran acvy over me whou he complcang facors of sgnal drf and hghly colored nose n fmri (Lu, Wong, Frank, & Buxon, 00; Wang e al., 005). Anoher poenal use s n denfyng voxels of neres and characerzng bran responses n ecologcally vald asks, such as free vewng of flms. In a recen paper, for example, parcpans wached a 60 mn segmen of an acon move (Hasson, Nr, Levy, Fuhrmann, & Malach, 004). The nvesgaors examned he me-course of acvy hroughou he bran and assessed wheher ncreases n parcular regons were sysemacally relaed o feaures of he flm (e.g., presenaon of scenes, hands, faces). HEWMA could be used n hs suaon o denfy voxels ha respond conssenly across parcpans durng vewng, whch would reduce false posves by provdng a reduced se of voxels of neres and provdng some quanave ools for characerzng he duraon and number of acvaed perods. 36
HEWMA could also be appled o changes n sae-relaed acvy evoked by learnng, e.g., onc ncreases bran acvy as a funcon of experse, or o sudes of onc ncreases followng soluons o nsgh problem-solvng asks. Experse resuls n funconal and srucural reorganzaon of corex (Klgard & Merzench, 1998; Kourz, Bes, Sarkhel, & Welchman, 005), and HEWMA could be used o more precsely characerze he me-course of boh ypes of changes (.e., do shfs occur gradually or suddenly?) In an nsgh ask, parcpans are presened wh a problem ha requres a novel combnaon of elemens (Bowden, Jung-Beeman, Fleck, & Kounos, 005; MacGregor, Ormerod, & Chroncle, 001). Once parcpans solve he ask, here s a qualave shf n her undersandng of how he elemens of he problem relae ha canno be reversed (he soluon s obvous once one knows ). Ths profound shf s poorly undersood, n par because approprae mehods have no been devsed o sudy s bran mechansms n healhy parcpans. Anoher poenal use s n longudnal sudes of bran funcon or srucure, and how hey change wh developmen or wh he progresson of a neurologcal or psycharc dsorder. For example, Mayberg and colleagues have conduced several longudnal sudes of resng FDG PET acvy n depressed paens over he course of reamen (Goldapple e al., 004; Mayberg e al., 00). The me course correspondng o bran acvy changes hroughou he reamen process s unknown, and HEWMA could be used o locae regons ha respond o reamen and denfy he me a whch hey do so. Conclusons In hs paper we developed a new approach, HEWMA, ha can be used o make nferences abou ndvdual or group fmri acvy, even when condons are no replcaed (e.g. 37
a sngle expermenal nducon of emoon). The HEWMA mehod s an exenson of EWMA, a me seres analyss mehod n sascal process-conrol heory and change-pon heory, o mulsubjec daa. I perms populaon nference, and can be used o analyze fmri daa voxelwse hroughou he bran, daa from regons of neres, or emporal componens exraced usng ICA or smlar mehods. Smulaons show ha he mehod has accepable false-posve rae conrol, and applcaon o an fmri sudy of anxey shows ha produces reasonable and novel resuls wh emprcal daa. A oolbox mplemenng all funcons n Malab s freely avalable from he auhors (see Auhor Noe). The HEWMA approach can complemen GLM-based and purely daa-drven mehods (such as ICA) by provdng nferences abou wheher, when, and for how long sysemac saerelaed acvaon occurs n a parcular bran regon. Alhough s developed here for fmri daa analyss, he mehod could be useful n deecon of devaon from a baselne sae n any ype of me seres daa, ncludng ASL, longudnal sudes of bran srucure or PET acvy, and ohers. 38
Auhor Noe We would lke o hank Doug Noll, Sephan F. Taylor, Barbara Fredrckson, and Lus Hernandez for provdng daa wh whch o develop he model, and o he developers of SPM sofware for her exremely useful ools. Sofware mplemenng he EWMA and HEWMA analyses s avalable for download from hp://www.columba.edu/cu/psychology/or/, or by conacng he auhors. 39
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Fgure Capons Fgure 1. A schemac overvew of: (A) he model of rue acvaon, (B) he EWMA sasc and s conrol bounds, and (C) he Gaussan mxure model (llusraed usng a separae smulaon). The parameer τ represens he rue change-pon, and τˆ s esmae. N 0 s he frs me pon afer he baselne perod, whle N s he oal number of me pons n he poenalacvaon perod. N ooc s he frs me pon ha s posvely denfed as acvaed, and s used o calculae τˆ n he zero-crossng mehod. Conrol lms are crcal values for he EWMA sasc, z, correcng for mulple dependen comparsons across me. In he mxure ˆ model, τ 1 4 are esmaes of when here s an acve-nonacve or nonacve-acve sae change ˆ andω1 are esmaes of he duraons of wo example acvaon perods. Fgure. Resuls from a smulaed sngle run expermen (Smulaon 1) usng an effec sze equal o 1 and an AR() nose model. (A-B) The rue sgnfcance and change-pon maps, (C) he sgnfcance map obaned from he EWMA analyss of he daa (usng λ = 0. and he AR() conrol bounds) and (D) he change-pon map esmaed usng he zero-crossng mehod. To he rgh are examples of boh acve and non-acve me courses ploed ogeher wh her conrol lms. The daa s represened by he lgh gray lne and he EWMA sasc by he dark black lne. Fgure 3. (A-B) Smulaon. Smulaed power and false posve raes for HEWMA wh varyng smoohness parameer λ, and a fxed baselne lengh of 60 TRs. (C-D) Smulaon 3. Same plos wh fxed λ = 0. and varyng baselne lengh. Fgure 4. (A) Smulaon. Recever operang characersc (ROC) curves show he fracon of rue posves (rue posve rae, TPR) vs. he fracon of false posves (false posve rae, FPR) 44
for HEWMA usng our recommended nose model (AR()) wh varyng smoohness parameer λ, and a fxed baselne lengh of 60 TRs. The opmal prmary hreshold for dscrmnang acve and nacve voxels s p <.08. In hese smulaons, hgher lambdas (less smoohng) produced more sensve resuls (bu wh false posve raes ha exceed he nomnal hreshold; see Fg. 3). Sensvy wh real daases wll vary dependng on he smoohness of he rue underlyng sgnal. (B) Smulaon 3. ROC plos wh fxed λ = 0. and varyng baselne lengh. Fgure 5. A schemac of he expermenal ask desgn for he fmri sudy. See ex for a dealed explanaon. Fgure 6. (A) HEWMA group acvaon, z r pop, n he medal fronal corex and he esmaed change-pon for onse of acvaon (CP, green lne). (B) Resuls of Mone Carlo smulaons for fndng correced p-values. Black lne: observed max T; dsrbuon: null hypohess max T. (C) Case weghs calculaed by akng he nverse of Eq. (8). Weghs are based on varably durng he baselne nerval. Hgher varance wll resul n a lower wegh for ha subjec. (D) The ndvdual me courses for he 4 subjecs. Fgure 7. (A) Lef: Regons wh sgnfcan acvaons n HEWMA (correced over me and false dscovery rae (FDR) correced a alpha =.05 over space; (Genovese, Lazar, & Nchols, 00)). Rgh: Regons color-coded accordng o K-means classfcaon (7 classes). (B) Hsogram of number of voxels by CP and acvaon duraon, color-coded by esmaed class. (C) Axal slces of regons shown n (A). 45
Fgure 8. The bran surface s shown n laeral oblque and axal vews. Sgnfcan voxels are colored accordng o sgnfcance. Increases are shown n red-yellow, and decreases are shown n lgh-dark blue. (A) Tme course from a regon showng susaned acvy n rosral medal PFC (sngle represenave voxel). The baselne perod s ndcaed by he shaded gray box, and he HEWMA-sasc s shown by he hck black lne (+/- one sandard error across parcpans, shown by gray shadng). The conrol lms are shown by dashed lnes. (B) A smlar plo showng ransen responses o presenaon of ask nsrucons n vsual corex. 46
Fgure1 47
Fgure 48
Fgure 3 49
Fgure 4 50
Fgure 5 51
Fgure 6 5
Fgure 7 53
Fgure8 54
Appendx A EM-algorhm for Gaussan Mxure Model The fmri me course s modeled as a mxure of wo normal dsrbuons, wh dfferen means and varances: X ~ N( θ, ) and X ~ N( θ, ). We can wre hs 0 0 σ 0 1 1 σ 1 as X = ( 1 Δ) X 0 + ΔX 1, where he random varable Δ s equal o one wh probably p and equal o zero wh probably 1-p. The densy funcon of X can be wren f X ( X + x 0 X1 x) = (1 p) f ( x) pf ( ). where X (x) s he normal probably densy funcon wh mean θ and varance σ. The f log-lkelhood can be wren: n 0, θ1, σ 0, σ 1, p ) = log X x ) = 1 l( θ x ( f ( ) The unknown parameers ( θ, θ, σ, σ, ) ha maxmze hs erm can be found usng he EMalgorhm. 0 1 0 1 p Perform he followng wo seps unl convergence: E-sep: pf ˆ ˆ ˆ X ( x, ) θ j Σ j ˆ γ = for = 1, KT (1 pˆ) f ( x ˆ θ, Σˆ ) + pf ˆ ( x ˆ θ, Σˆ ) X 1 j j X j j M-sep: T 1 pˆ = ˆ γ, T = 1 T ˆ γ x ˆ = 1 θ j = T, ˆ γ = 1 T T ˆ γ ( x ˆ θ j )( x ˆ θ j ) = 1 ˆ σ j = for j = 1, T ˆ γ = 1 55
Appendx B EM-algorhm for HEWMA an Le Λ be he lower rangular smoohng marx (defned n Eq. 7) and G = [ I I L I ] T mn n marx where I n s he n n deny marx. Furher, le z r G be he combned vecor of n n n EWMA-sascs for all m subjecs and V = α * G Q G + Σ G s covarance marx, where T Q = Λ Λ and G I m Σ G * Σ1 = 0 Σ * O 0. * Σ m In our mplemenaon he whn-subjec componen and * Σ s assumed known from he frs level, QG s also assumed known. The unknown parameer α s esmaed eravely usng he EM-algorhm below. Ths s equvalen o he varance componen esmaon performed n SPM (Frson e al., 00). Perform he followng wo seps unl convergence: E-sep: V * G = αq G + Σ G T ( * 1 G V ) 1 C = G G M-sep: P * 1 * 1 T * 1 = VG VG GCG VG 1 g = r 1 H = r 1 T r r T ( PQ ) + r( P Q Pz z ) G T ( P Q PQ ) G G G G G α = α + H 1 g 56