The average rate of change for continuous time models

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1 Behvior Reserch Mehods 9, (), doi:.78/brm...68 he verge re of chnge for coninuous ime models EN ELLE Universiy of Nore Dme, Nore Dme, Indin he verge re of chnge (ARC) is concep h hs been misundersood in he pplied longiudinl d nlysis lierure, where he slope from he srigh-line chnge model is ofen hough of s hough i were he ARC. he presen ricle clrifies he concep of ARC nd shows unequivoclly he mhemicl definiion nd mening of ARC when mesuremen is coninuous cross ime. I is shown h he slope from he srigh-line chnge model generlly is no equl o he ARC. Generl equions re presened for wo mesures of discrepncy when he slope from he srigh-line chnge model is used o esime he ARC in he cse of coninuous ime for ny model liner in is prmeers, nd for hree useful models nonliner in heir prmeers. he nlysis of chnge is imporn in mny fields for ssessing he effecs of he pssge of ime on some dependen vrible. -vrying nd ime-invrin covries cn be incorpored ino he nlysis in n effor o undersnd nd model inerindividul differences in chnge. Mny imes, nlysis of chnge procedures re imporn wih or wihou experimenl mnipulion. Modern concepulizions of he nlysis of chnge regrd inrindividul chnge o be he sring poin for longiudinl d nlysis (e.g., Collins, 996; Meh & Wes, ; Rudenbush, ; Rogos, Brnd, & Zimowsi, 98; Rogos & Wille, 98). hus, before ggreging over individuls in mulilevel model frmewor, prerequisie for modeling chnge prmeers s dependen vribles is h he chnge prmeers be hemselves meningful. he presen ricle focuses on single individul rjecory, since specifying he individul-level model is necessry bu no sufficien condiion of meningful model for collecion of individuls for phenomenon h is repeedly mesured. Some reserch quesions demnd resonbly lrge ime spn beween mesuremen occsions nd cn resonbly expec o obin only relively smll number of repeed observions. For exmple, reserchers sudying cdemic chievemen over school yer cnno expec o obin lrge number of mesuremens bsed on comprehensive exminions. his is due in pr o he logisics of collecing comprehensive mesuremens, s well s he relively slow chnge in chievemen. Reserchers sudying opics such s mril sisfcion, depression, employee sisfcion, employee moivion, nd so forh generlly fll ino similr siuions. However, oher reserch quesions cn be ddressed wih insrumens h mesure he vrible of ineres coninuously, or nerly so, or les wih relively lrge number of mesuremen occsions. For exmple, her re, elecricl civiy of he her, blood flow o vrious regions of he brin, eyegze posiion nd moun of movemen, body movemen, nd respirion cn be mesured lierlly or essenilly coninuously. Becuse behviorl nd biologicl sysems re inexricbly lined, more nd more reserch is cuing cross rdiionl behviorl/psychologicl nd medicl/ physiologicl reserch opics, nd growing lis of journl iles suggess h scienific progress cn be nd is being mde by bridging vrious specs of behvior/psychology nd medicine/physiology. As formerly dispre fields coninue o blend, wys of collecing d will coninue o evolve, some of which will consis of mesuremens h re en coninuously or nerly so. As such, new opporuniies will emerge for sudying he nure of behvior nd biologicl sysems, s well s, nd perhps mos impornly, he inercion of he wo. elley nd Mxwell (8) discussed he verge re of chnge (ARC) generlly nd derived mesures of discrepncy beween he ARC nd he regression coefficien from he srigh-line chnge model for discree number of ime poins. he ARC describes he verge or ypicl re of chnge over some ime inervl of ineres for priculr rjecory nd is hus prsimonious mesure h cn poenilly describe compliced process, regrdless of he funcionl form of chnge. Alhough he concep of he ARC in longiudinl conex is ppeling nd seems o be srighforwrd, he echnicl underpinnings hve no received much forml enion (cf. elley & Mxwell, 8; Seigel, 97). he regression coefficien from he srigh-line chnge model hs ofen been he wy in which such succinc descripion of chnge over ime hs been emped. Alhough using single. elley, elley@nd.edu 9 he Psychonomic Sociey, Inc. 68

2 RAE OF CHANGE FOR CONINUOUS IME MODELS 69 vlue s descripor of poenilly compliced process of chnge hs n inuiive ppel, he presen wor will demonsre h he regression coefficien from he srigh-line chnge model is generlly no equl o he ARC for given rjecory. Aggreging cross individuls in mulilevel modeling conex when he focus of ineres is he overll ARC is hus generlly problemic nd will end o led o bised esimes. he purpose of he presen ricle is o exend he wor of elley nd Mxwell o he cse of coninuous ime. In so doing, he limiing cse of coninuous ime models cn be developed nd exmined, so h he discrepncy beween he slope from he srigh-line chnge model nd he ARC cn be beer undersood. Mhemicl Form of he ARC elley nd Mxwell (8) deiled he mhemicl underpinnings of he ARC, which we summrize here. he re of chnge of nonvericl srigh line h psses hrough wo poins, (, ) nd (, ), is he slope of he line, where represens some bsis of ime (e.g., monoonic rescling) nd is coninuous funcion of ime, f( ), he h mesuremen occsion (, ). he slope of he line connecing wo poins is he chnge in divided by he chnge in ime: Slope f f, () where f( ) is he dependen vrible, is he chnge in he dependen vrible, nd is he chnge in ime. In he limi s pproches zero, Equion yields he insnneous re of chnge when evlued specific ime vlue: d d f f lim f ( ), () where d /d is he derivive of wih respec o, which will be represened s f (). he men vlue for coninuous funcion h is differenible over he inervl o is given s f c f d ( ), () where f c represens he men vlue of he funcion of ineres, which is f () for he ARC (Finney, Weir, & Giordno,, p. ; Sewr, 998, p. 7). Since Equion yields he men of coninuous differenible funcion, nd Equion is specil cse of coninuous funcion, combining he wo equions will yield he men of he derivives (i.e., insnneous res of chnge) of he funcion from o. When he limiing equion for derivive is combined wih he men of coninuous funcion, i cn be shown h he men of derivives, which is lierlly he ARC, cn be wrien s ARC f f f f d. (C) As cn be seen in Equion C, he mhemicl definiion of he ARC is he chnge in divided by he chnge in ime during some specified inervl. Equion C is well-nown in nlyic clculus (e.g., Finney e l.,, pp ; Sewr, 998, pp. 6 7 nd 8), where, regrdless of he funcion, he men of ll of he derivives evlued over specified coninuous inervl mus equl /. In he conex of longiudinl d nlysis, he mhemics underlying he ARC re no generlly well nown (cf. elley & Mxwell, 8; Seigel, 97), which hs led o some confusion in he pplied longiudinl d nlysis lierure. As single mesure describing overll chnge, he ARC holds promise. he problem, however, is h in n emp o convey n esime of he ARC, reserchers hve used he slope from he srigh-line chnge model. As elley nd Mxwell showed in he cse of discree ime, he slope from he srigh-line chnge model generlly is no equl o he ARC. As monioring insrumens incresingly llow for more mesuremen occsions o be obined in he sme ime inervl so much so h some re essenilly coninuous nd ohers re pproching coninuous discussion of he ARC in he conex of coninuous ime is pproprie nd is provided here. (A) (B) Discrepncy Beween he Regression Coefficien From he Srigh-Line Chnge Model nd he ARC he discrepncy beween he regression coefficien nd he ARC will be qunified by wo prmeers: he bis nd he discrepncy fcor. For fixed vlues of ime, he bis is operionlly defined by Equion (see below), where SLCM is he generl represenion of he slope from he srigh-line chnge model (i.e., n individul s ordinry les squres regression slope), is condiionl on he rue funcionl form of chnge, nd E[] represens he expeced vlue of he rndom vrible in brces. For fixed vlues of ime, he second prmeer h describes he discrepncy is he discrepncy fcor nd is operionlly defined s E E SLCM f f S LCM ARC, (6) B E SLCM E f f SLCM ARC ()

3 7 ELLE where, gin, is condiionl on he rue funcionl form of chnge. In siuions where B (implying ), inerpreing SLCM s if i were he ARC yields no inconsisency in reserch conclusions or inerpreion. However, when B (nd by implicion ), concepulizing SLCM s he ARC my be problemic nd cn poenilly led o misinformed conclusions regrding inrindividul chnge, inerindividul chnge, nd group differences in chnge. Alhough imes inerpreion of B my be more srighforwrd hn inerpreion of, i is lso poenilly rbirry due o he poenil rescling of ime nd/or he dependen vrible. We include boh so h, depending on he priculr siuion, eiher or boh my be used. Exmining he Bis in he ARC When Is Coninuous In he cse of coninuously mesured ime vlues, he ordinry slope from he srigh-line chnge model generlizes, wih he use of inegrion rher hn summion, o d d, (7) where SLCMC is he regression coefficien for he srighline chnge model when ime is coninuous. Equion 7 cn be rewrien s he inegrl of sum fer expnding he numeror nd he denominor: d d. (8) Becuse he inegrl of sum is he sum of he inegrls, Equion 8 cn be rewrien s Equion 9, shown below. Relizing h ( )/ nd h d in he siuion of coninuous ime, he ls wo componens in he numeror of Equion 9 re equl nd of opposie sign, leding o simplificion of he numeror becuse he wo componens cncel. Alernively, second perspecive for undersnding why he wo compo- nens cncel in he numeror of Equion 9 cn be seen by rewriing he ls wo componens s Becuse d. is he firs momen bou he men, his quniy mus lwys equl zero (Sur & Ord, 99, chp. ). In he following subsecions describing siuions where ime is coninuous, he reduced form of Equion 9, d d d d d will be pplied o liner nd hen o nonliner models., () When Cn Be Wrien As Liner Funcion of Any funcionl form cn be represened by power series, such h he sum of squred deviions beween he vlues of he rue funcion nd he vlues pproximed by he power series cn be mde o be infiniesimlly smll by dding enough polynomil powers nd coefficiens (Finney e l.,, chp. 8; Sewr, 998, secion 8.6). A power series in he longiudinl conex is limiing sum of coefficiens muliplied by posiive ineger powers of ime. Such power series is given s M f lim m m, M m () where m is he coefficien ( m ) for he mh power (m,..., M). Alhough power series is infinie by definiion, nown funcionl forms cn be represened by finie sums. In generl, he following finie sum cn be used o impose or pproxime some nown or unnown funcionl form of chnge nd is more generl hn he power series, since he powers of ime re no limied o nonnegive inegers (s is he definiion of polynomil chnge model), bu cn e on ny rel vlues: f, () d d d d d (9)

4 RAE OF CHANGE FOR CONINUOUS IME MODELS 7 where ( ) represens he h (,..., ; ) power. he inercep of priculr chnge curve is he sum of he s whose is zero. In he specil cse where [, ], he inercep is, which, sricly speing, is n indeermine form when. However, due o l Hôpil s rule, which uses derivives o evlue he converging limi of funcion h would oherwise be indeermine under sndrd lgebric rules, he quniy by sndrd convenions (Finney e l.,, secion 7.6; Sewr, 998, secion.). When evluing he equions given in his secion by compuer, cre should be en o ensure h he priculr progrm defines s (rher hn, e.g., reurning n error messge). Generl resuls emerge for B nd by relizing h funcionl forms of chnge cn generlly be represened by Equion. he following secion mes use of his fc when exmining B nd for ny model liner in is prmeers. will be replced by Equion so h he resuls will be in he mos generl form of models liner in heir prmeers. Replcing in Equion wih he finie sum of Equion yields d d. () d d Crrying ou he inegrion nd replcing wih is definiion [ ( )/] yields Equion, shown below. Afer simplifying boh he numeror nd he denominor, he generl equion for he regression coeffi- cien from he srigh-line chnge model when cn be wrien in he form of Equion nd when ime is coninuous is given by Equion, shown below. I is useful o noe h Equion does no consrin he vlues of or, he number of componens defining (i.e., ), or he vlues of nd. he ARC when is defined s sum of coefficiens muliplied by powers of ime cn be wrien s he following: ARC. (6) Becuse he slope (Equion ) nd he ARC (Equion 6) hve been defined when is expressed s specil cse of Equion, generl expressions emerge for B nd. he generl bis for he presen siuion is found by subsiuing Equions nd 6 ino Equion (see Equion 7, below). he generl discrepncy fcor is hen found by subsiuing Equions nd 6 ino Equion 6 (see Equion 8, below). I cn be shown h when f( ) is defined s liner or qudric chnge curve, SLCMC ARC nd B from Equion 7 is zero (nd hus ). hus, in he cse of coninuous ime, if he funcion governing chnge is srigh-line chnge model or qudric chnge model, no problems rise when SLCMC is inerpreed s he ARC. However, for he generl cse of ny liner model oher hn srigh-line or qudric model, B (nd hus ). hus, inerpreing SLCMC s if i were he ARC poenilly leds o misleding conclusions. Alhough. () 6. () B 6. (7) 6. (8)

5 7 ELLE forml proof hs no been provided for he mos generl cse, nlyic nd empiricl evluion of he equions for wide vriey of models liner in heir prmeers when ime is coninuous yields B for nonrivil prmeer combinions. However, wh is cler is h, in generl, SLCMC ARC unless he funcionl form is liner, qudric, or some combinion of liner nd qudric. hus, inerpreing he slope from he srigh-line chnge model s hough i is he ARC generlly leds o bised esimes of he ARC. Ofen in pplied longiudinl reserch he iniil vlue of ime is represened s zero ( ). his is especilly rue in experimenl sudies when represens bseline mesure of some ribue (prees) before remen begins. Anoher reson why mny imes equls zero is becuse ime is ofen scled such h he inercep represens he iniil (sring) vlue. In he specil cse where is replced by zero, Equions 7 nd 8 cn be simplified. he simplified slope when he iniil vlue of ime (or scled ime) is zero cn be wrien s 6. (9) he ARC for such series defined by Equion cn be wrien s ARC, () where is he inercep of he priculr chnge curve. Recll h he inercep is simply he sum of he coefficiens whose equls zero. If no equls zero when, hen iself equls zero nd he chnge curve goes hrough he origin. he generl expression for B when [, ] is obined by subrcing he righ-hnd side (RHS) of Equion from he RHS of Equion 9: B 6. () he generl expression for in his siuion is obined by dividing Equion 9 by Equion : 6. () Of course, since Equions nd re specil cses of Equions 7 nd 8, i holds rue h when he funcionl form of chnge is liner or qudric chnge curve, he regression coefficien for he srigh-line chnge model nd he ARC re equivlen. Agin, however, s he equions in his secion show, i is generlly he cse h SLCMC ARC. he exc vlues of B nd/or P cn be found wih he pproprie equion(s) from he presen secion when ime is coninuous. he nex secion dels wih he cse where f( ) equls ech of he nonliner models previously discussed. When Conforms o Cerin Nonliner Funcions of Fiing sisicl model liner in is prmeers o longiudinl d is generlly srighforwrd. As he phenomenon under sudy grows incresingly more complex, he order of he polynomil chnge model cn be incresed ccordingly, unil he prediced scores resonbly correspond wih he observed scores. Nonliner models of he sme complex phenomenon cn ofen be more inerpreble nd prsimonious, nd re generlly more vlid beyond he observed rnge of d, when compred wih liner models (Pinheiro & Bes, ). Furhermore, i is ofen he cse h he prmeers in nonliner models cn be esily inerpreed, wheres once polynomil model is beyond qudric, he mening of he higher order prmeers ypiclly offers lile meningful inerpreion. An exmple of such difference beween nonliner nd liner models reles o sympoes. In polynomil chnge models, sympoic vlues cnno generlly be modeled for he sympoe o hold beyond he rnge of he observed d. hus, reserchers who me use of polynomil rends mus ccep h heir model will necessrily fil some poin beyond he rnge of he d cully colleced. Such scenrios cn poenilly led o indeque models where impossible vlues re prediced. o demonsre problems h rise when d ruly follow nonliner funcionl forms ye re modeled by srighline chnge models, hree nonliner chnge models will be presened so h ler he bis nd discrepncy fcor cn be developed for ech. he seleced nonliner models re he sympoic regression chnge curve, he Gomperz chnge curve, nd he logisic chnge curve. Alhough wide vriey of nonliner models exis, hese models of chnge were chosen becuse hey re especilly helpful for pplied reserch. A brief inroducion o ech is given here bsed on he descripions found in elley nd Mxwell (8). he Asympoic Regression Chnge Curve he generl sympoic regression chnge curve ofen referred o s he negive exponenil chnge model describes fmily of poenil regression models where he dependen vrible pproches some limiing vlue s ime increses. A generl sympoic regression equion for single rjecory ws given by Sevens (9) s, () where is he sympoic vlue pproched s, is he chnge in from o (i.e., represens ol chnge in ), nd ( ) is sclr h defines he fcor by which he deviion beween nd is reduced for ech uni chnge of ime, hus reflecing he re which. Equion cn be equivlenly wrien s exp, ()

6 RAE OF CHANGE FOR CONINUOUS IME MODELS 7 where log() ( ) nd cn be hough of s scling prmeer (Sevens, 9). he Gomperz Chnge Curve he Gomperz chnge model is nonliner model h is ofen used in he biologicl sciences. he symmeric sigmoidl form of he Gomperz chnge curve offers n opion for hose who see o model cerin ypes of nonliner rends. he generl hree-prmeer Gomperz chnge model for single rjecory cn be wrien s exp exp, () where is he sympoe s. he prmeers nd define he poin of inflecion on he bsciss. he poin of inflecion on he ordine is / exp(), which is pproximely 7% of he sympoic chnge (Rowsy, 98, chp. nd pp. 6 67; Winsor, 9). he Logisic Chnge Curve he logisic chnge model is noher nonliner sigmoidl model h provides noher opion for modeling chnge over ime in he behviorl sciences. he generl hree-prmeer logisic chnge model for single rjecory cn be wrien s, (6) exp where is he sympoe s. he prmeers nd define he poin of inflecion on he bsciss /. he poin of inflecion on he ordine is /, % of he sympoic chnge (chp. nd pp of Rowsy, 98; Winsor, 9). Nonliner Models for he Anlysis of Chnge In his secion, nd ARC re derived for he sympoic chnge curve (Equion ), he Gomperz chnge curve (Equion ), nd he logisic chnge curve (Equion 6). Generl equions re presened for nd ARC for hese nonliner models, hus llowing one o compue B by subrcion nd/or by division, s needed. he derivions proceed in mnner nlogous o (lbei no s deiled s, for spce considerions) he wy hey did for he derivions presened in he previous secion for models liner in heir prmeers. he discrepncy in he sympoic regression chnge model. he regression coefficien for he srigh- line chnge model pplied o chnge h follows n sympoic regression (lso ermed negive exponenil) model in he cse of coninuous ime is given in Equion 7, shown he boom of his pge, where subscrips will be used AR in his cse for sympoic regression o idenify he priculr nonliner chnge model. he ARC for he sympoic regression model, obined by subsiuing Equion ino Equion C, is given s he following: ARC AR exp exp. (8) he vlue of B for he sympoic regression model is hus obined by subrcing he RHS of Equion 8 from he RHS of Equion 7, nd is obined by dividing he RHS of Equion 7 by he RHS of Equion 8. he discrepncy in he Gomperz chnge model. he regression coefficien for he srigh-line chnge model pplied o chnge h follows Gomperz chnge model in he cse of coninuous ime is obined by firs expressing G s in Equion 9 (see below), where Ei is he exponenil inegrl. he exponenil inegrl is defined s Ei( qx, ) g exp( xg) dg, q g () wih q being nonnegive ineger nd x some lgebric expression (Abrmowiz & Segun, 96). Given G, he slope for he Gomperz chnge model is equl o he following: SLCM G 6, () CGC where he subscrip GC denoes he Gomperz chnge model. he ARC for he Gomperz chnge model is given in Equion (below). he vlue of B for he Gomperz chnge model is hus obined by subrcing he RHS of Equion from he RHS of Equion, nd is obined by dividing he RHS of Equion by he RHS of Equion. he discrepncy in he logisic chnge model. he regression coefficien for he srigh-line chnge model pplied o chnge h follows logisic chnge model in he cse of coninuous ime is obined by defining L, L, L, nd L (see Equions 6, nex pge). In L he dilogrihm funcion is required. he funcion dilog (Lewin, 98) is defined s Equion 7: AR exp exp 6exp exp exp exp exp / (7) G exp exp Ei,exp Ei,exp d (9) exp ARC GC exp exp exp ()

7 7 ELLE dilog( x) x log( g). g dg g (7) he four logisic componens re hen combined wih he oher necessry prmeers in he following mnner: LC L L L L 6 /, (8) where LC denoes logisic chnge. he ARC for he logisic chnge model is given in Equion 9 (see below). he vlue of B for he logisic chnge model is hus obined by subrcing he RHS of Equion 9 from he RHS of Equion 8, nd is obined by dividing he RHS of Equion 8 by he RHS of Equion 9. Alhough i would be dvngeous o show generlly wheher i is possible for AR ARC AR, GC ARC GC, nd/or LC ARC LC, he presen ime no mhemiclly rcble soluion ws obinble due o he complicions h rise wih he nonliner funcionl forms used. Anlyic nd empiricl invesigions hve shown h for nonrivil cses, he regression coefficien from he srigh-line chnge model nd he ARC re no generlly equl. For ny specific siuion, given he equions provided, he exc vlue of B nd cn be deermined. Exmples of he Discrepncies Alhough generl equions re presened for he bis nd discrepncy fcors, i cn be difficul o discern wheher he bis nd discrepncy fcors moun o ny meningful deviions beween he ARC nd he slope from he srigh-line chnge model. Figures,, nd show plos of sympoic regression, Gomperz, nd logisic chnge models, respecively, for differen combinions in he cse of coninuous ime for nd vlues when [, ] nd is fixed. he purpose of he figures is o show he reder vriey of nonliner funcionl forms wih vriey of prmeer vlues, o illusre how he chnge models discussed in he presen wor generlize o vriey of rjecories h migh be useful in pplied reserch. In ddiion o illusring he rjecories hemselves, he priculr prmeer vlues governing he curves hve been included op he priculr plo. Wihin ech of he plos is he vlue of SLCM, ARC, B, nd. From he plos i cn be seen h B is someimes posiive (i.e., when SLCM ARC, implying h ) bu in oher siuions i is negive (i.e., when SLCM ARC, implying h ). I is imporn o noe h B nd for he differen scenrios exmined re specific o he seleced prmeers nd he chosen ime inervl. he exc vlues of B nd re rbirry o lrge exen, since modificion of he prmeers will chnge he B nd vlues. However, he priculr exmples of chnge curves provided in Figures,, nd re hough o consis of vriey of relisic chnge curves. he srigh line wihin ech plo represens he prediced scores given ime (i.e., he regression line) for he srigh-line chnge model, wheres he nonliner rend represens he rue chnge for he priculr siuion. Alhough i is difficul o sy wh lrge discrepncy would be, discrepncy fcor s smll s.76 (boom lef of Figure ) nd one s lrge s. (boom righ of Figure ) seem o be very problemic. Cerinly, commonly used sisics would be regrded s problemic if heir expeced vlues were.76 imes smller or. imes lrger hn heir corresponding populion vlues. Furhermore, he smlles nd lrges discrepncy fcors shown in he figures (i.e., he.76 nd he. noed bove) would hve been surpssed hd differen prmeer vlues been used. hus, he figures re men o supplemen he mhemicl derivions wih exmples showing vriey of chnge curves nd he corresponding bis nd discrepncy fcor of ech. Discussion Confusion exiss in he lierure regrding he definiion nd inerpreion of he ARC. Becuse mny monioring sysems re now cpble of recording informion coninuously or ner coninuously over ime, i is imporn o consider he effecs of esiming nd inerpreing he slope from srigh-line chnge model s he ARC. As is shown in he presen ricle, here is generlly bis when using he slope from he srigh-line chnge model s if i were he ARC. hree srighforwrd, sufficien condiions cn be described such h here is no discrepncy when using he srigh-line chnge model o esime he ARC when ime is coninuous: () L log exp log exp log exp log exp () L log exp log exp log exp log exp L dilog exp exp exp dilog exp exp exp () L log exp log exp (6) ARC LC exp exp (9) exp exp

8 RAE OF CHANGE FOR CONINUOUS IME MODELS 7,,,, 6,, 8,,,, SLCM.7 ARC.9 B.7.86 SLCM. ARC.99 B SLCM.688 ARC.999 B..6 SLCM. ARC B.6.8 SLCM. ARC B.7.7,,,, 6,, 8,,,, SLCM.6 ARC.97 B SLCM.68 ARC.99 B..67 SLCM. ARC.999 B.78.6 SLCM.9 ARC B.8.8 SLCM.667 ARC B..7,,,, 6,, 8,,,, SLCM.8 ARC B SLCM.96 ARC.98 B.9.86 SLCM. ARC.988 B SLCM.8 ARC.998 B.8.6 SLCM. ARC B.6.8 Figure. Illusrion of he srigh-line chnge model fi o vriey of sympoic chnge curves long wih SLCM, ARC, B, nd, given he prmeers h re specified.

9 SLCM. ARC. B.6.76,, SLCM.78 ARC.89 B.6.69,, SLCM.96 ARC.6 B SLCM.79 ARC.77 B.7.8,, SLCM 6.68 ARC.99 B.79.6,, SLCM ARC.966 B.7.,, SLCM 6.9 ARC.966 B.96.96,, SLCM 6.96 ARC.988 B.97.96,, SLCM 6. ARC.99 B.7.8,, SLCM 7. ARC.99 B.8.6,, SLCM 6.76 ARC.998 B.78.8,, SLCM.9 ARC.999 B.96.87,, Figure. Illusrion of he srigh-line chnge model fi o vriey of Gomperz chnge curves long wih ; SLCM, ARC, B, nd, given he prmeers h re specified SLCM.86 ARC.6 B.97.88,, SLCM.9 ARC.67 B.6.9,, SLCM 6.8 ARC.77 B..99,, ,, 76 ELLE

10 RAE OF CHANGE FOR CONINUOUS IME MODELS 77,,,, 6,, 8,,,, Figure. Illusrion of he srigh-line chnge model fi o vriey of logisic chnge curves long wih SLCM, ARC, B, nd, given he prmeers h re specified SLCM.6 ARC.8 B.7.6 SLCM.8 ARC.6 B.88.9 SLCM.79 ARC.79 B SLCM.9 ARC.78 B.7. SLCM.8 ARC.76 B..67,,,, 6,, 8,,,, SLCM.9 ARC. B.9.96 SLCM.899 ARC. B.8.6 SLCM 6.8 ARC.8 B.8.7 SLCM 6.87 ARC.898 B.89.8 SLCM 6. ARC.98 B ,,,, 6,, 8,,,, SLCM. ARC. B.6.88 SLCM.6 ARC.6 B..99 SLCM.79 ARC.79 B..6 SLCM 6.6 ARC.9 B.68. SLCM 6.68 ARC.96 B.666.6

11 78 ELLE. he rue funcionl form of chnge consiss of only liner componen.. he rue funcionl form of chnge consiss of only qudric componen.. he rue funcionl form of chnge consiss of only some combinion of liner nd qudric componens. Of course, Condiions nd re specil cses of Condiion when he qudric nd liner componens re zero. hus, s his ricle hs shown, he slope from he srigh-line chnge model nd he ARC re no generlly equl o one noher for n individul rjecory when ime is mesured coninuously. his is no o sy h no oher funcions cn hve n ARC h equls he slope from he srigh-line chnge model, bu generlly i is he cse. Cerinly, specil cses of oher funcions cn be mde so h he slope from he srigh-line chnge model nd he ARC re equl. However, such is generlly no he cse, nd in mos circumsnces here will be some degree of bis. his ricle hs shown h he bis beween he ARC nd he slope from he srigh-line chnge model cn be posiive or negive nd smll or lrge, poenilly yielding misleding conclusions regrding chnge over ime. I cn be shown (e.g., elley & Mxwell, 8) h when he bis is nonzero nd ll oher hings re equl, he lrger he number of ime poins, he lrger he discrepncy beween he slope from he srigh-line chnge model nd he ARC when he bis is nonzero. hus, i is especilly imporn o undersnd he relionship beween he slope from he srigh-line chnge model nd he ARC in he cse of coninuous or nerly coninuous ime, since in such siuions he discrepncy beween he slope from he srigh-line chnge models reches is mximum for ny given scenrio. AUHOR NOE I hn Sco E. Mxwell, Seven M. Boer, Dvid A. Smih, nd Joseph R. Rusch for helpful commens nd suggesions o previous versions of his ricle. Correspondence concerning his ricle should be ddressed o. elley, Deprmen of Mngemen, Universiy of Nore Dme, Nore Dme, IN 66 (e-mil: elley@nd.edu). REFERENCES Abrmowiz, M., & Segun, I. (96). Hndboo of mhemicl funcions. New or: Dover. Collins, L. M. (996). Mesuremen of chnge in reserch on ging: Old nd new issues from n individul growh perspecive. In J. E. Birren &. Schie (Eds.), Hndboo of he psychology of ging (h ed., pp. 8-8). Sn Diego: Acdemic Press. Finney, R. L., Weir, M. D., & Giordno, F. R. (). homs clculus (h ed.). New or: Addison Wesley. elley,., & Mxwell, S. E. (8). Delineing he verge re of chnge. Journl of Educionl & Behviorl Sisics,, 7-. Lewin, L. (98). Polylogrihms nd ssocied funcions. New or: Norh-Hollnd. Meh, P. D., & Wes, S. G. (). Puing he individul bc ino individul growh curves. Psychologicl Mehods,, -. Pinheiro, J., & Bes, D. (). Mixed-effecs models in S nd S-Plus. New or: Springer. Rowsy, D. A. (98). Nonliner regression modeling: A unified prcicl pproch. New or: Deer. Rudenbush, S. W. (). Compring personl rjecories nd drwing cusl inferences from longiudinl d. Annul Review of Psychology,, -. Rogos, D., Brnd, D., & Zimowsi, M. (98). A growh curve pproch o he mesuremen of chnge. Psychologicl Bullein, 9, Rogos, D., & Wille, J. B. (98). Undersnding correles of chnge by modeling individul differences in growh. Psychomeri,, -8. Seigel, D. G. (97). Severl pproches for mesuring verge res of chnge for second degree polynomil. Americn Sisicin, 9, 6-7. Sevens, W. (9). Asympoic regression. Biomerics, 7, Sewr, J. (998). Clculus: Conceps nd conexs. Cincinni, OH: Broos/Cole. Sur, A., & Ord, J.. (99). endll s dvnced heory of sisics: Disribuion heory (6h ed., Vol. ). New or: Wiley. Winsor, C. P. (9). he Gomperz curve s growh curve. Proceedings of he Nionl Acdemy of Sciences, 8, -8. NOE. Mulilevel models re lso equivlen or closely reled o rndom effecs models, hierrchicl (non)liner models, len chnge curves, nd mixed effecs models. hus, regrdless of he verbige given o such models, he issues discussed in he presen ricle re eqully pplicble. (Mnuscrip received Mrch, 8; revision cceped for publicion November, 8.)