Journal of Mathematical Psychology

Joural of Mahemacal sychology 56 (22) 34 355 Coes lss avalable a ScVerse SceceDrec Joural of Mahemacal sychology oural homepage: www.elsever.com/locae/mp Sascal measures for workload capacy aalyss Joseph W. Houp, James T. Towsed Deparme of sychologcal ad Bra Sceces, Idaa Uversy, Bloomgo, IN 4745, USA a r c l e f o a b s r a c Arcle hsory: Receved 4 Augus 2 Receved revsed form 5 Aprl 22 Avalable ole 2 Jue 22 Keywords: Meal archecure Huma formao processg Capacy coeffce Noparamerc Race model A crcal compoe of how we udersad a meal process s gve by measurg he effec of varyg he workload. The capacy coeffce (Towsed & Nozawa, 995; Towsed & Weger, 24) s a measure o respose mes for quafyg chages performace due o workload. Despe s precse mahemacal foudao, ul ow rgorous sascal ess have bee lackg. I hs paper, we demosrae sascal properes of he compoes of he capacy measure ad propose a sgfcace es for comparg he capacy coeffce o a basele measure or wo capacy coeffces o each oher. 22 Elsever Ic. All rghs reserved. Measures of chages processg, for sace deerorao, assocaed wh creases workload have bee fudameal o may advaces cogve psychology. Due o her parcular sregh dsgushg he dyamc properes of sysems, respose me based measures such as he capacy coeffce (Towsed & Nozawa, 995; Towsed & Weger, 24) ad he relaed Race Model Iequaly (Mller, 982) have bee gag employme boh basc ad appled secors of cogve psychology. The purvew of applcao of hese measures cludes areas as dverse as memory search (Rckard & Bac, 24), vsual search (Krummeacher, Gruber, & Müller, 2; Weder & Muller, 2), vsual percepo (Edels, Towsed, & Algom, 2; Scharf, almer, & Moore, 2), audory percepo (Fedler, Schröer, & Ulrch, 2), flavor percepo (Veldhuze, Shepard, Wag, & Marks, 2), mul-sesory egrao (Hugeschmd, Hayasaka, effer, & Laure, 2; Rach, Dederch, & Colous, 2) ad hrea deeco (Rchards, Hadw, Beso, Weger, & Doelly, 2). Whle here have bee a umber of sascal ess proposed for he Race Model Iequaly (Goda, Rehl, & Bluro, 22; Mars & Mars, 23; Ulrch, Mller, & Schröer, 27; Va Zad, 22), here has bee a lack of aalycal work o ess for he capacy coeffce. A hs po, he oly quaave es avalable for he capacy coeffce s ha proposed by Edels, Dok, Brow, ad Heahcoe (2). Ths es s lmed o esg a crease workload from oe o wo sources of formao. Correspodg auhor. E-mal addresses: houp@daa.edu (J.W. Houp), owse@daa.edu (J.T. Towsed). Furhermore, reles o he assumpo ha he uderlyg chaels ca be modeled wh he Lear Ballsc Accumulaor (Brow & Heahcoe, 28). I hs paper, we develop a more comprehesve sascal es for he capacy coeffce ha s boh oparamerc ad ca be appled wh ay umber of sources of formao. The capacy coeffce, C(), was orgally veed o provde a precse measure of workload effecs for frs-ermag (.e., mmum me; referred o as OR or dsucve processg) processg effcecy (Towsed & Nozawa, 995). I complemes he survvor eraco coras (SIC), a ool whch assesses meal archecure ad decsoal soppg rule (Towsed & Nozawa, 995). The capacy coeffce has bee exeded o measure effcecy exhausve (AND) processg suaos (Towsed & Weger, 24). I fac, s possble o exed he capacy coeffce o ay Boolea decso rule (e.g., Blaha, 2; Towsed & Edels, 2). The logc of he capacy coeffce s relavely sraghforward. We demosrae hs logc usg a comparso bewee a sgle source of formao ad wo sources for clary; bu he reasog readly exeds o more sources ad he heory seco apples o he geeral case. Cosder he experme depced Fg., whch a parcpa respods yes f a do appears eher slghly above he md-le of he scree, slghly below, or boh ad respods o oherwse (a smple deeco ask). Opmally, he parcpa would respod as soo as she deecs ay do, wheher or o boh were prese. We refer o hs as frsermag processg, or smply OR processg. Aleravely, parcpas may wa o respod ul hey have deermed wheher or o a do s prese boh above ad below he mdle. Ths s exhausve or AND processg. I hs desg, AND processg s sub-opmal, bu oher desgs, such as whe he 22-2496/$ see fro maer 22 Elsever Ic. All rghs reserved. do:.6/.mp.22.5.4

342 J.W. Houp, J.T. Towsed / Joural of Mahemacal sychology 56 (22) 34 355 The capacy coeffce for AND processg s defed a aalogous maer, wh he cumulave reverse hazard fuco (cf. Chechle, 2; Towsed & Edels, 2; Towsed & Weger, 24) place of he cumulave hazard fuco. The cumulave reverse hazard fuco, deoed by K() s defed as, Fg.. A llusrao of he possble smul Edels e al. (submed for publcao). The acual smul had much lower coras. We refer o he codo wh boh he upper ad lower dos prese as (for prese prese). A dcaes oly he upper do s prese (for prese abse). A dcaes oly he lower do s prese ad AA dcaes eher do s prese. I he OR ask, parcpas respoded yes o, A or A. I he AND ask, parcpas respoded yes o oly ad o oherwse. parcpa mus oly respod yes whe boh dos are prese, AND processg s ecessary. If formao abou he presece of eher of he dos s processed depedely ad parallel, he he probably ha he OR process s o complee a me s he produc of he probables ha each do has o ye bee deeced. Le F be he cumulave dsrbuo fuco of he compleo me, ad he subscrp X Y dcae ha he fuco correspods o he compleo me of process X uder smulus codo Y. Usg hs oao, he predco of he depede, parallel, frsermag process s gve by, F AB AB () F A AB () F B AB (). () We make he furher assumpo ha here s o chage he speed of deecg a parcular do due o chages he umber of oher sources of formao (ulmed capacy). Thus, we ca drop he referece o he smulus codo, F A AB () F A A () F A (), F B AB () F B B () F B () F AB () ( F A ()) ( F B ()). (2) We refer o a model wh hese collecve assumpos ulmed capacy, depede ad parallel as a UCI model. Ths predco of equaly forms he bass of he capacy coeffce ad plays he role of he ull hypohess he sascal ess developed hs paper. To derve he capacy coeffce predco for he UCI model, we eed o pu Eq. (2) erms of cumulave hazard fucos (cf. Chechle, 23; Towsed & Ashby, 983, pp. 248 254). We do hs usg he relaoshp H() l( F()), where H() f /( F) ds s he cumulave hazard fuco. F AB AB () ( F A ()) ( F B ()) l F AB AB () l (( F A ()) ( F B ())) l ( F A ()) l ( F B ()) H AB () H A () H B (). (3) The capacy coeffce for OR processg s defed by he rao of he lef ad rgh had sde of Eq. (3), C OR () H AB () H A () H B (). (4) Ths defo gves a easy way o compare agas he basele UCI model performace. From Eqs. (3) ad (4), we see ha UCI performace mples C OR (). Usually, respose mes are assumed o clude some exra me ha are o smulus depede, such as he me volved pressg a respose key oce he parcpa has chose a respose. Ths s ofe referred o as base me. We do o rea he effecs of base me hs work. Towsed ad Hoey (27) show ha base me makes lle dfferece for he capacy coeffce whe he assumed varace of he base me s wh a reasoable rage. K() f ds l (F). (5) F If he parcpa has deeced boh dos whe boh are prese a AND ask, he he mus have already deeced each do dvdually. If he dos are processed depedely ad parallel, hs meas, F AB AB () F A AB A()F B AB (). (6) Wh he addoal assumpo of ulmed capacy, we have, F AB () F A ()F B (). (7) As above, we ake he aural logarhm of boh sdes o oba he predco erms of cumulave reverse hazard fucos. l (F AB ()) l (F A ()F B ()) l (F A ()) l (F B ()) K AB () K A () K B (). (8) Wh he cumulave reverse hazard fuco, relavely larger magude mples relavely worse performace. Thus, o maa he erpreao of C() > mplyg performace s beer ha UCI, he capacy coeffce for AND processg s defed by, C AND () K A() K B (). (9) K AB () Wh he defos of he OR ad AND capacy coeffces had, we ow ur o ssues of sascal esg. We beg by adapg he sascal properes of esmaes for he cumulave hazard ad reverse cumulave hazard fucos o esmaes of UCI performace. The, based o hose esmaes, we derve a ull-hypohess-sgfcace es for he capacy coeffces. We do o esablsh he sascal properes of he capacy coeffce self, bu raher he compoes. The sascal es s based o he samplg dsrbuo of he dfferece of he predced UCI performace ad he parcpa s performace whe all arges are prese. The dfferece form leads o more aalycally racable resuls ha a es based o he rao form. Noeheless, he logc of he es s he same, f he values are sascally sgfca (ozero for he dfferece; o oe for he rao) he he UCI model may be reeced.. Theory The capacy coeffces are based o cumulave hazard fucos ad reverse cumulave hazard fucos. Alhough would be heorecally possble o use he machery developed for he emprcal cumulave dsrbuo fuco o sudy he emprcal cumulave hazard fucos, usg he dees meoed above, H() log( F()), K() log(f()) (cf. Towsed & Weger, 24), we sead use a drec esmae of he cumulave hazard fuco, he Nelso Aale (NA) esmaor (e.g., Aale, Borga, & Gessg, 28). Ths approach grealy smplfes he mahemacs because may of he properes of he esmaes follow from he basc heory of margales. There s o parcular advaage of oe approach over he oher erms of her sascal quales (Aale e al., 28).

J.W. Houp, J.T. Towsed / Joural of Mahemacal sychology 56 (22) 34 355 343.. The cumulave hazard fuco Le Y() be he umber of resposes ha have o occurred as of mmedaely before ad le T be he h eleme of he ordered se of respose mes RT ad le be he umber of respose mes. The he NA esmaor of he cumulave hazard fuco s gve by, Ĥ() Y(T T ). () Iuvely, hs esmaor resuls from esmag f () by / wheever here s a respose ad zero oherwse; ad esmag F() by he umber of respose mes larger ha dvded by. Ths gves a esmae of h() f ()/( F) (/)/(Y()/) /Y() wheever here s a respose ad zero oherwse. Iegrag across s [, ] leads o he sum Eq. () o esmae h ds. Aale e al. (28) demosrae ha hs s a ubased esmaor of he rue cumulave hazard fuco ad derve he varace of he esmaor based o a mulplcave esy model. We oule he argume here, he exed he resuls o a esmaor for reduda arge UCI model performace usg daa from sgle arge rals. We beg by represeg he respose mes by a coug process. 2 A me, we defe he coug process N() as he umber of resposes ha have occurred by me. The esy process of N(), deoed λ(), s he probably ha a respose occurs a fesmal erval, codoed o he resposes before, dvded by he legh of ha erval. For oaoal coveece, we also roduce he fuco J(), f Y() > J() () f Y(). If we roduce he margale, M() N() λ ds, he we ca wre he cremes of he coug process as, dn() λ()d dm(). (2) Uder he mulplcave esy model, he esy process of N() ca be rewre by λ() h()y(), where Y() s a process represeg he oal umber of resposes ha have o occurred by me ad h() s he hazard fuco. Ths model arses from reag he respose o each ral as a dvdual process. Each respose has s ow hazard rae h () ad a dcaor process Y () whch s f he respose has o occurred by me ad oherwse. If he hazard rae s he same for each respose ha wll be cluded he esmae, he he coug process N() follows he mulplcave esy model wh h() h () ad Y() Y (). Rewrg Eq. (2) usg he mulplcave esy model gves, dn() h()y()d dm(). If we mulply by J()/Y(), ad le J()/Y() whe Y(), he, J() J() dn() J()h()d Y() Y() dm(). By egrag, we have, J Y dn Jhds J Y dm. The egral o he lef s he NA esmaor gve Eq. (), usg egral oao. The frs erm o he rgh s our fuco of eres, he cumulave hazard fuco (up ul max, whe all resposes have occurred) whch we deoe H (). The las erm s a egral wh respec o a zero mea margale, whch s ur a zero mea margale. Hece, E Ĥ() H (). Thus, he NA esmaor s a ubased esmae of he rue cumulave hazard fuco, ul max. For sascal ess volvg Ĥ(), we wll also eed o esmae s varace. To do so, we use he fac ha he varace of a zero mea margale s equal o he expeced value of s opoal varao process. 3 J Because he egral dm s wh respec Y o a zero mea margale, he opoal varao process s gve by, Ĥ() H () J dn. Thus, Var Ĥ() H J () E dn. Therefore, a esmaor of he varace of Ĥ() H () s gve by, ˆσ 2 () J H dn (T T ). The asympoc behavor of he NA esmaor s also well kow. Ĥ() s a uformly cosse esmaor of H() ad Ĥ() H() coverges dsrbuo o a zero mea Gaussa margale (Aale e al., 28; Aderse, Borga, & Kedg, 993)..2. The cumulave reverse hazard fuco To esmae he capacy coeffce for AND processg, we mus also adap he NA esmaor o he cumulave reverse hazard fuco. Based o he NA esmaor of he cumulave hazard fuco, we defe he followg esmaor, ˆK() G(T T ). (3) Here G s a esmae of he cumulave dsrbuo fuco gve by he umber of resposes ha have occurred up o ad cludg s. Iuvely, hs correspods o esmag f ()/F() d by seg f () / wheever here s a observed respose ad usg G() o esmae F(), dovealg cely wh he NA esmaor of he cumulave hazard fuco. Each of he properes meoed above for he NA esmaor of he cumulave hazard (ubasedess, cossecy ad a Gaussa lm dsrbuo) also hold for he esmaor of he cumulave reverse hazard. Smlar o J() for he cumulave hazard esmae, we eed o rack wheher or o he esmae of he deomaor of Eq. (3) s zero, Q () f G() > f G() (4) max K () Q kds. (5) 2 See Appedx A. for deals. 3 The opoal varao process s also kow as he quadrac varao process. We use he oao [M] o dcae he opoal varao process of a margale M.

344 J.W. Houp, J.T. Towsed / Joural of Mahemacal sychology 56 (22) 34 355 Theorem. ˆK() s a ubased esmaor of K (). Theorem 2. A ubased esmae of he varace of ˆK() s gve by, ˆσ 2 () K G 2 (T T ). (6) Theorem 3. ˆK() s a uformly cosse esmaor of K(). Theorem 4. ˆK() K() coverges dsrbuo o a zero mea Gaussa process as he umber of respose mes used he esmae creases..3. Esmag UCI performace Havg esablshed esmaors for he cumulave hazard fuco ad cumulave reverse hazard fuco, we ow eed a esmae of he performace of he UCI model based o he sgle arge respose mes. O a OR ask, he UCI model cumulave hazard fuco s smply he sum of he cumulave hazard fucos of each of he sgle arge codos (cf. Eq. (3)). I hs model, he probably ha a respose has o occurred by me, F UCI (), s he probably ha oe of he sub-processes have fshed by me, F (). Hece, k F UCI () ( F ()) k log ( F UCI ()) log ( F ()) H UCI () k log ( F ()) k H (). (7) Based o hs, a reasoable esmaor for he cumulave hazard fuco of he race model s he sum of NA esmaors of he cumulave hazard fuco for each sgle arge codo, k Ĥ UCI () Ĥ (). (8) To fd he mea ad varace of hs esmaor, we reur o he mulplcave esy model represeao of he sub-processes saed above. We use he oao as above, k H () UCI h J ds. (9) We wll also eed o dsgush amog he ordered respose mes for each sgle arge codo. We use T o dcae he h eleme he ordered se of respose mes from codo. Theorem 5. Ĥ UCI () s a ubased esmaor of H UCI (). Theorem 6. A ubased esmae of he varace of Ĥ UCI () s gve by, ˆσ 2 () k T (T ). (2) Theorem 7. Ĥ UCI () s a uformly cosse esmaor of H UCI (). Theorem 8. Le k where s he umber of respose mes used o esmae he cumulave hazard fuco of he compleo me of he h chael. The, ĤUCI H UCI coverges dsrbuo o a zero mea Gaussa process as he umber of respose mes used he esmae creases. Assumg a UCI process o a AND ask, he probably ha a respose has occurred by me, F UCI (), s he probably ha all of he sub-processes have fshed by me, F (). Hece, k F UCI () F () k log (F UCI ()) log F () K UCI () k log (F ()) k K (). (2) Thus, o esmae he cumulave reverse hazard fuco of he UCI model o a AND ask, we use he sum of NA esmaors of he cumulave reverse hazard fuco for each sgle arge codo, k ˆK UCI () ˆK (). (22) The esmaors of he UCI cumulave reverse hazard fucos rea he sascal properes of he NA esmaor of he dvdual cumulave reverse hazard fuco: Because we are usg a sum of esmaors, he cossecy, ubasedess ad Gaussa lm properes all hold for he UCI AND esmaor us as hey do for he UCI OR esmaor. Theorem 9. ˆKUCI () s a ubased esmaor of K UCI (). Theorem. A ubased esmae of he varace of gve by, ˆσ 2 () k T ˆKUCI () s G 2 (T ). (23) Theorem. ˆKUCI s a uformly cosse esmaor of K UCI. Theorem 2. Le k where s he umber of respose mes used o esmae he cumulave hazard fuco of he compleo me of he h chael. The, ˆKUCI K UCI coverges dsrbuo o a zero mea Gaussa process as he umber of respose mes used he esmae creases..3.. Hadlg es I heory he uderlyg respose me dsrbuos are couous ad hus here s o chace of wo exacly equal respose mes. I pracce, measureme devces ad rucao lm he possble observed respose mes o a dscree se. Ths meas repeaed values are possble. Aale e al. (28) sugges wo ways of dealg wh hese repeaed values. Oe mehod s o add a zero mea, small-varace radom-value o he ed respose mes. Aleravely, oe could use he umber of resposes a a parcular me he umeraor of Eqs. () ad (3). If we le d() be he umber of resposes ha occur exacly a me, hs leads o he esmaors, Ĥ() d(t ) (24) Y(T T ) ˆK() d(t ) G(T T ). (25).4. Hypohess esg Havg esablshed a esmaor of UCI performace, we ow ur o hypohess esg. Here, we focus o a es wh UCI

performace as he ull hypohess. I he followg, we wll use he subscrp r o refer o he codo whe all sources dcae a arge (.e., he reduda arge codo a OR ask or he arge codo o a AND ask). We beg wh he dfferece bewee he parcpa s performace whe all sources dcae a arge ad he esmaed UCI performace, Ĥ r () Ĥ UCI () T (r) ˆK r () ˆKUCI () T (r) Y r (T (r) ) G r (T (r) ) J.W. Houp, J.T. Towsed / Joural of Mahemacal sychology 56 (22) 34 355 345 k T k T Y (T ) (26) G (T ). (27) Followg he same le of reasog gve for he ubasedess of he cumulave hazard fuco esmaors, he expeced dfferece s zero uder he ull hypohess ha H r () H UCI (). Furhermore, he po-wse varace of he fuco s he sum of he po-wse varaces of each dvdual fuco, T (r) T (r) r (T (r) ) G 2 r (T (r) ) k T k T (T ) (28) G 2 (T ). (29) The dfferece fucos coverge dsrbuo o a Gaussa process. Hece, a ay, uder he ull hypohess, he dfferece wll be ormally dsrbued. We ca he dvde by he esmaed varace a ha me so he lm dsrbuo s a sadard ormal. I geeral, we may wa o wegh he dfferece bewee he wo fucos, e.g., use smaller weghs for mes where he esmaes are less relable. Le L() be a predcable wegh process ha s zero wheever ay of Y or Y r s zero (G or G r s zero for AND), he we defe our geeral es sascs as, Z OR T (r) Z AND T (r) L(T ) Y r (T (r) ) G r (T (r) ) k T k T L(T ) Y (T ) (3) G (T ). (3) Theorem 3. Uder he ull hypohess of UCI performace, E (Z OR ). Theorem 4. A ubased esmae of he varace of Z OR s gve by, Var (Z OR ) () L 2 (T ) k T (T ) L 2 (T ) (T ). T Theorem 5. Assume ha here exss sequeces of cosas {a } ad {c } such ha, lm a lm c L () c Y () l <, y <. ad Y () r y r <, Furher, assume ha here exss some fuco d such ha for all δ, lm f r (a /c ) 2 L () 2 k Y r Y d for all s T δ. The (a /c )Z OR () coverges dsrbuo o a zero mea Gaussa process as he umber of respose mes used he esmae creases. The Harrgo Flemg wegh process (Harrgo & Flemg, 982) s oe possble wegh fuco, L() S( ) ρ Y r() Y () Y r () Y (). (32) Here, S( ) s a emprcal survvor fuco esmaed from he pooled respose mes from all codos, S() N r N Y s r. (33) Y Fg. 2 depcs he relave values of he Harrgo Flemg weghg fuco across a rage of respose mes ad dffere values of ρ. Respose mes were sampled from a shfed Wald dsrbuo ad he esmaors were based o 5 samples from each of he sgle chael ad double arge models. Whe ρ, he es correspods o a log-rak es (Mael, 966) bewee he esmaed UCI performace ad he acual performace wh reduda arges (see Aderse e al., 993, Example V.2., for a dscusso of hs correspodece). As ρ creases, relavely more wegh s gve o earler respose mes ad less o laer respose mes. Varous oher wegh processes ha sasfy he requremes for L() have also bee proposed (see Aale e al., 28, Table 3.2, for a ls). Because he ull hypohess dsrbuo s uaffeced by he choce of he wegh process, s up o he researcher o choose he mos approprae for a gve applcao. I he smulao seco, we use ρ, esseally a log-rak (or Ma Whey f here are o cesored respose mes) es of he dfferece bewee he esmaed UCI performace ad he performace whe all arges are prese (Aale e al., 28; Mael, 966). We have chose o prese he Harrgo Flemg esmaor here because of s flexbly ad specfcally wh ρ for he smulao seco because reduces o ess ha are more lkely o be famlar o he reader. Ay wegh fuco ha sasfes he codos o L() could be used. A more horough vesgao of he approprae fucos for respose me daa wll be a mpora ex sep, bu s beyod he scope of hs paper. For a sascal es of C AND (), we smply swch Y() wh G() ad he bouds of egrao. Theorem 6. Uder he ull hypohess of UCI performace, E (Z AND ). Theorem 7. A ubased esmae of he varace of Z AND s gve by, Var (Z AND ) () T L 2 (T ) G 2 (T ) k T L 2 (T ) G 2 (T ). Theorem 8. Assume ha here exss sequeces of cosas {a } ad {c } such ha, lm a lm c L () c G () l <, g <. ad G () r g r <,

346 J.W. Houp, J.T. Towsed / Joural of Mahemacal sychology 56 (22) 34 355 Fg. 2. O he lef, he emprcal survvor fuco based o respose mes sampled from Wald dsrbuos s show. The rgh shows he relave values of he Harrgo Flemg (Harrgo & Flemg, 982) weghg fuco for he capacy es a varous values of ρ. Furher, assume ha here exss some fuco d such ha for all δ, lm f r (a /c ) 2 L () 2 k G r G d for all s T δ. The (a /c )Z AND () coverges dsrbuo o a zero mea Gaussa process as he umber of respose mes used he esmae creases. Ths leads o a reverse hazard verso of he Harrgo Flemg esmaor, L() F() ρ G r() G () G r () (34) G () F() N r N G s r. (35) G Therefore, oe ca use he sadard ormal dsrbuo for ull-hypohess-sgfcace ess for UCI-OR ad UCI-AND performace. Uder he ull hypohess, Z OR() U OR () Var (Z OR ()) Z AND() U AND () Var (Z AND ()) 2. Smulao d N (, ) (36) d N (, ). (37) I hs seco, we exame he properes of he NA esmaor of UCI performace as well as he ew es sasc usg smulaed daa. To esmae ype I error raes for dffere sample szes, we use hree dffere dsrbuos for he sgle arge compleo mes, expoeal, shfed Wald ad exgaussa. Frs, we smulaed resuls for he es sascs wh hrough 2 samples per dsrbuo. For each sample sze, he ype I error rae was esmaed from smulaos. I each smulao, he parameers for he dsrbuo were radomly sampled. For he expoeal rae parameer, we sampled from a gamma dsrbuo wh shape ad rae 5. For he shfed Wald model, we sampled he shf from a gamma dsrbuo wh shape ad rae.2. aramerzg he Wald dsrbuo as he frs passage me of a dffuso process, we sampled he drf rae from a gamma dsrbuo wh shape ad rae ad he hreshold from a gamma dsrbuo wh shape 3 ad rae.6. The rae of he expoeal he exgaussa dsrbuo was sampled from a gamma radom varable wh shape ad rae 25. The mea of he Gaussa compoe was sampled from a Gaussa dsrbuo wh mea 25 ad sadard devao ad he sadard devao was sampled from a gamma dsrbuo wh shape ad rae.2. I each smulao, respose mes were sampled from he correspodg dsrbuo wh he sampled parameers. Double arge respose mes were smulaed by akg wo depede samples from he sgle chael dsrbuo, he akg eher he mmum (for OR processg) or he maxmum (for AND processg) of hose samples. As show Fg. 3, he ype I error rae s que close o he chose α,.5 across all hree model ypes ad for he smulaed sample szes. Ths dcaes ha, eve wh small sample szes, usg he sadard ormal dsrbuo for he es sasc works que well. Nex, we exame he power as a fuco of he umber of samples per dsrbuo ad he amou of crease/decrease performace he reduda arge codo. To do so, we focus o models ha have a smple relaoshp bewee her parameers ad capacy. For OR processes, we use a expoeal dsrbuo for each dvdual chael, F A () F B e λ ad H A () H A () λ. The predco for a UCI or model hs case would be H AB () 2λ. Thus, chages wll drecly correspod o chages he rae parameer. To explore a rage of capacy, we used a rage of mulplers for he rae parameers of each chael for smulag he double arge respose mes, F A,ρ () F B,ρ () e ρλ so ha H AB,ρ () 2ρλ. Ths leads o he formula H AB,ρ /H AB ρ for he rue capacy. For AND processes, we use F A () F B () ( e λ ) 2/ρ as he dsrbuo for smulag respose mes whe boh arges are prese. Alhough ρλ o loger correspods drecly wh he rae parameer, hs approach maas he correspodece α C AND. I geeral, he power of hese ess may deped o he dsrbuo of he uderlyg chael processes. We lm our explorao

J.W. Houp, J.T. Towsed / Joural of Mahemacal sychology 56 (22) 34 355 347 Fg. 3. Type I error raes wh α.5 for hree dffere chael compleo mes. For each model, double arge respose mes were smulaed by akg he m or max of depedely sampled chael compleo mes for OR or AND models respecvely. The error raes are based o smulaos for each model wh hrough 2 rals per dsrbuo. Fg. 4. ower of he U OR ad U AND sascs wh α.5 as a fuco of capacy ad umber of rals per dsrbuo for capacy above. These resuls are based o wo expoeal chaels, wh eher a OR or AND soppg rule, depedg o he sasc. Usg hese models, he power of U OR ad U AND s he same. o hese models because of he aalyc correspodece bewee he rae parameer ad he value of he capacy coeffce. Wh oher dsrbuos, he relaoshp bewee he capacy coeffce ad he dffereces he parameers across workload ca be que complcaed. These parcular resuls are also depede o he weghg fuco used. Here, we focus o he log-rak ype wegh fuco, bu may oher opos are possble (e.g., Aale e al., 28, p. 7). These esmaes apply o cases whe he magude of he weghed dfferece hazard fucos bewee sgle ad double arge codos mach hose predced by he expoeal models. Boh he U OR ad U AND es sascs, appled o he correspodg OR or AND model, had he same power. Fgs. 4 ad 5 show he power of hese ess as he capacy creases ad as he umber of rals used o esmae each cumulave hazard fuco creases. For much of he space esed, he power s que hgh, wh celg performace for early half of he pos esed. Wh a rue capacy of., he power remas que low, eve wh up o 2 rals per dsrbuo. However, wh a hgher umber of rals, he crease power s que seep as a fuco of he rue capacy. Wh 2 rals per dsrbuo, he power umps from roughly.5 o.9 as he rue capacy chages from. o.2. O he low ed of rals per dsrbuo,.e., o 2, reasoable power s o acheved eve up o a capacy of 3.. Wh 3 rals per dsrbuo, he power creases roughly learly wh Fg. 5. ower of he U OR ad U AND sascs wh α.5 as a fuco of capacy ad umber of rals per dsrbuo for capacy less ha. These resuls are based o wo expoeal chaels, wh eher a OR or AND soppg rule, depedg o he sasc. Usg hese models, he power of U OR ad U AND s he same. he capacy, achevg reasoable power aroud C() 2.5. The es was o as powerful wh lmed capacy. The power sll creases wh more rals ad larger chages capacy, however he crease s much more gradual ha ha he super capacy plo. Dffereces he power as a fuco of capacy for super ad lmed are o be expeced gve ha he capacy coeffce s a rao, whle he es sascs are based o dffereces. However, hs does o expla he dffereces power see here. ar of he dfferece s due o a dfferece scale. The super capacy plo has sep szes of. whle he lmed capacy plo has sep szes of.4. We chose dffere sep szes so ha we could demosrae a wder rage of capacy values. Furhermore, he weghg fuco here may be more sesve o super capacy. The effec of he weghg fuco o hese ess, ad deermg he approprae weghs f oe s eresed oly oe of super or lmed capacy, s a mpora fuure dreco of hs work. 3. Applcao I hs seco we ur o daa from a rece experme usg a smple do deeco ask. I hs sudy, parcpas were show oe of four ypes of smul. I he sgle do codo, a whe.2 do was preseed drecly above or below a ceral fxao o a oherwse black backgroud. I he double do codo,

348 J.W. Houp, J.T. Towsed / Joural of Mahemacal sychology 56 (22) 34 355 Fg. 6. Capacy coeffce fucos of he e parcpas from Edels e al. (submed for publcao). The lef graph depcs performace o he OR ask, evaluaed wh he OR capacy coeffce. The rgh graph depcs performace o he AND ask, evaluaed wh he AND capacy coeffce. he dos were preseed boh above ad below fxao. Each ral bega wh a fxao cross, preseed for 5 ms, followed by eher a sgle do smulus, a double do smulus or a erely black scree preseed for ms or ul a respose was made. arcpas were gve up o 4 s o respod. Each ral was followed by a s er-ral erval. O each day, parcpas were show each of he four smul 4 mes radom order. arcpas compleed wo days worh of rals wh each of wo dffere sruco ses. I oe verso, he OR ask, parcpas were asked o respod Yes o he sgle do ad double do smul ad oly respod o whe he erely black scree was preseed. I he oher verso, he AND ask, parcpas were o respod Yes oly o he double do smulus ad No oherwse. See Fg. for example smul. For furher deals of he sudy, see Edels, Towsed, Hughes, ad erry (submed for publcao). The capacy fucos for each dvdual are show Fg. 6. Upo vsual speco, all parcpas seem o be lmed capacy he OR ask. O he AND ask, all of he parcpas had super capacy for some poro of me, wh oly hree parcpas showg C() less ha oe, ad hose oly for laer mes. Table shows he values of he sasc for each dvdual. The es shows sgfca volaos of he UCI model for every parcpa o boh asks. These daa are based o 8 rals per dsrbuo, so based o he power aalyss he las seco, s o surprse ha he es was sgfca for every parcpa. These daa dcae ha parcpas are dog worse ha he predced basele UCI processg o he OR ask, ad beer ha he UCI basele o he AND ask. O he OR ask, C OR < dcaes eher hbo bewee he do deeco chaels, lmed processg resources, or processg worse ha parallel (e.g., seral). O he AND ask, C AND > dcaes eher faclao bewee he do deeco chaels, creased processg resources wh creased load, or processg beer ha parallel (e.g. coacve). Gve he aure of he smul ad prevous aalyses (Houp & Towsed, 2), s ulkely ha he falure of he assumpo of parallel processg s he explaao for hese resuls. Lkewse, here s o reaso o beleve ha a chage he avalable processg resources would be dffere bewee he asks, alhough resources may be lmed for boh versos (cf. Towsed & Nozawa, 995). We beleve he mos lkely explaao of hese daa s a crease faclao bewee he do deeco processes he AND ask. Table Resuls of he es sasc appled o he resuls of he wo do experme of Edels e al. (submed for publcao). OR ask AND ask.2 ** 4.5 ** 2 4.4 ** 3.8 ** 3 8.22 ** 3.88 ** 4 6.85 ** 8.62 ** 5 9.2 ** 6.5 ** 6 2.72 * 22.27 ** 7 5.4 **.49 ** 8 5.9 **.59 ** 9 5.53 ** 5.75 ** * p <.. ** p <.. 4. Dscusso I hs paper, we developed a sascal es for use wh he capacy coeffce for boh mmum-me (OR) decso rules ad maxmum-me (AND) decso rules. We dd so by exedg he properes of he Nelso Aale esmaor of he cumulave hazard fuco (e.g., Aale e al., 28) o esmaes of ulmed capacy, depede parallel performace. Ths approach yelds a es sasc ha, uder he ull hypohess ad he lm as he umber of rals creases, has a sadard ormal dsrbuo. Ths allows vesgaors o use he sasc a varey of ess beyod us a comparso agas he UCI performace, such as comparg wo esmaed capacy coeffces. As par of developg he sasc, we demosraed wo oher mpora resuls. Frs, we esablshed he properes of he esmae of UCI performace o a OR (AND) ask gve by he sum of cumulave (reverse) hazard fucos esmaed from sgle arge rals. Ths cluded demosrag ha he esmaor s ubased, cosse ad coverges dsrbuo o a Gaussa process. Furhermore, developg he esmaor for UCI AND processg, we exeded he Nelso Aale esmaor of cumulave hazard fucos o cumulave reverse hazard fucos. Despe beg less commo ha he cumulave hazard fuco, he cumulave reverse hazard fuco s used a varey of coexs, recely cludg cogve psychology (see Chechle, 2; Edels, Towsed e al., 2; Towsed & Weger, 24). Noeheless,

J.W. Houp, J.T. Towsed / Joural of Mahemacal sychology 56 (22) 34 355 349 we were uable o fd exsg work developg a NA esmaor of he cumulave reverse hazard fuco. Ths esmaor ca also be used o esmae he reverse hazard fuco he same way he hazard fuco s esmaed by he muhaz package (Hess & Gelema, 2) for he sascal sofware R (R Developme Core Team, 2). I hs mehod, he cumulave hazard fuco s esmaed wh he Nelso Aale esmaor, he smoohed usg (by defaul) a Epaechkov kerel. The hazard fuco s he gve by he frs order dfferece of ha smoohed fuco. Alhough he sascal es s based o he dfferece bewee predced UCI performace ad rue performace whe all arges are prese, he es s vald for he capacy rao. If oe reecs he ull-hypohess ha he dfferece s zero, hs s equvale o reecg he hypohess ha he rao s oe. Noeheless, we have o developed he sascal properes of he capacy coeffces. Isead, we have demosraed he small-sample ad asympoc properes of he compoes of he capacy coeffces. I fuure work, we hope o develop Bayesa couerpars o he prese sascal ess. Oe advaage of hs approach would be he ably o cosder poseror dsrbuos over he capacy coeffce s rao form, raher ha beg resrced o he dfferece. There are addoal reasos o explore Bayesa aleraves as well. We wll o repea he varous argumes deph, bu here are boh praccal ad phlosophcal reasos why oe mgh prefer such a alerave (cf. Kruschke, 2). We are also eresed a more horough vesgao of he weghg fucos used hese ess. I could be ha some weghg fucos are more lkely o deec super-capacy, whle ohers are more lkely o deec lmed capacy. Furhermore, he effecs of he weghg fuco are lkely o vary depedg o wheher hey are used wh he cumulave hazard fuco or he cumulave reverse hazard fuco. Whle he capacy coeffce has bee appled a varey of areas wh cogve psychology, he lack of a sascal es has bee a barrer o s use. Ths work removes ha barrer by esablshg a geeral es for UCI performace. Ackowledgmes Ths work was suppored by NIH-NIMH MH 5777-7 ad AFOSR FA955-7--78. Appedx. roofs ad supporg heory A.. The mulplcave esy model The heory of he NA esmaor for cumulave hazard fucos s based o he mulplcave esy model for coug processes. Here, we gve a bref overvew of he model ad he correspodg model for coug processes wh reverse me. For a more horough reame, see he Aderse e al. (993) ad Aale e al. (28). Also, Aale e al. (29) gves a good roduco o he opc. Suppose we have a sochasc process N() ha racks he umber of eves ha have occurred up o ad cludg me, e.g., he umber of respose mes less ha or equal o. The esy of he coug process λ() descrbes he probably ha a eve occurs a fesmal erval, codoed o he formao abou eves ha have already occurred, dvded by he legh of ha erval. The process gve by he dfferece of he coug process ad he cumulave esy, M() N() λ ds urs ou o be a margale Aale e al. (28, p. 27). Uder he mulplcave esy process model, we rewre he esy process of N() by λ() h()y(), where Y() s a process for whch he value a me s deermed gve he formao avalable up o mmedaely before. Of parcular eres hs work s esmag he cumulave hazard fuco H() h() d. We wll also use he cocep of he coug process reverse me N. Hece, we wll sar a some max, possbly fy, ad work backward hrough me coug all of he eves ha happe a or afer me. The mulplcave model akes a smlar form, wh a process G() ha s deermed by all he formao avalable afer, place of Y(), λ() k()g(). We wll use hs model whe we are eresed esmag he cumulave reverse hazard fuco K() max k() d. Oe mpora cosequece of he mulplcave esy model s ha he assocaed margales are locally square egrable (cf. Aderse e al., 993, p. 78), a requreme for much of he heory below. A.2. Margale heory Before gvg he full proofs of he heorems hs paper, we wll frs summarze he heorecal coe ha wll be ecessary, parcularly some of he basc properes of margales. Iuvely, margales ca be hough of as descrbg he forue of a gambler hrough a seres of far gambles. Because each game s far, he expeced amou of moey he gambler has afer each game s he amou he had before he game. A ay po he sequece, he ere sequece of ws ad losses before ha game s kow, bu he oucome of he ex game, or ay fuure games s ukow. More formally, a dscree me margale s defed as follows: Defo. Suppose X, X 2,... are a sequece of radom varables o a probably space (Ω, F, ) ad F, F 2,... s a sequece of σ -felds F. The {(X, F ) :, 2,...} s a margale f: () F F, () X s measurable F, () E( X ) s fe ad (v) E(X F ) X almos surely. Assumg X correspods o he gambler s forue afer he h game ad F correspods o he formao avalable afer he h game, he frs wo codos correspod o he oo ha afer a parcular game, he resul of ha game ad all prevous games s kow. The hrd codo correspods a codo ha he expeced amou of moey a gambler has afer ay game s fe. The fal codo correspods o he faress of he gambles; he expeced amou of moey he gambler has afer oe more game s he moey he has already. Margales ca also be defed for couous me. I hs case, he dex se ca be, for example, he se of all mes >, R. The we requre ha for all s <, F s F sead of (). The secod wo codos are he same afer replacg he dscreely valued dex wh he couously valued. The fal requreme for couous me becomes E(X F s ) X s for all s. The expecao fuco of a margale s fxed by defo, bu he chage he varably of he process could dffer amog margales ad s ofe que mpora. I addo o he varace fuco, here are wo useful ways o rack he varao, he predcable ad opoal varao processes. For dscree margales he predcable varao process s based o he secod mome of he process a each sep codoed o he σ -feld from he prevous sep. Formally, X E (X X ) 2 F. (A.)

35 J.W. Houp, J.T. Towsed / Joural of Mahemacal sychology 56 (22) 34 355 The opoal (quadrac) varao process s smlar, bu he secod mome a each sep s ake whou codog, [X] (X X ) 2. (A.2) To geeralze hese processes o couous me margales, evely spl up evely spl up he erval [, ] o subervals, ad use he dscree defos, he ake he lm as he umber of sub-ervals goes o fy, X lm [X] lm 2 E X/ X ( )/ F ( )/ 2 X/ X ( )/. (A.3) (A.4) For reverse hazard fucos, he cocep of a reverse margale wll also be useful. These processes are margales, bu wh me reversed so ha he margale propery s based o codog o he fuure, o he pas. A sequece of radom varables..., X, X, X 2,... s a reverse margale f..., X 2, X, X,... s a margale. The defo of a reverse margale ca be geeralzed o couous me by he same procedure as a margale. Zero mea margales wll parcularly useful he proofs ha follow. Smlar o he equaly of he varace ad he secod mome for zero mea uvarae radom varables, he varace of a zero mea margale s equal o he opoal varao process ad he predcable varao process. Also, he sochasc egral of a predcable process wh respec o a zero mea margale s aga a zero mea margale. Iformally, we ca see hs propery by examg a dscree approxmao o he margale wh [, ] dvded o sub-ervals, I() The, k H k M k M (k ). (A.5) E(I() ( )/ F ) k (k ) E H k M M F k k (k ) E H k M M F k ( ) E H k M() M F ( ) H k E ((M()) F ) M. (A.6) Aale e al. (28) also gve he predcable ad opoal varao processes of he egral of a predcable process, here H(), wh respec o a coug process margale, M, H dm () H 2 λ ds (A.7) H dm () H 2 dn. (A.8) Furhermore, f M,..., M k are orhogoal margales (.e., for all, M, M ) he, H dm () H 2 λ ds (A.9) k k k H dm () H 2 dn. (A.) From he reverse-me relaoshp, he same properes hold for reverse margales, wh he lower boud of egrao se o > ad he upper boud se o or max. We wll make use of he margale ceral lm heorem for he proofs volvg lm dsrbuos of he esmaors. Ths heorem saes ha, uder cera assumpos, a sequece of margales coverges dsrbuo o a Gaussa margale. There are varous versos of he heorem ha requre varous codos. For our purposes, we wll use he codo ha here exs some o-egave fuco y such ha sup s [,] Y () / y coverges probably o, where Y () s as defed above (he umber of resposes ha have o occurred by me s). Wh he addoal assumpo ha he cumulave hazard fuco s fe o [, ], hs s suffce for he margale ceral lm heorem o hold. If he margales are zero-mea ad, assumg s s he smaller of s ad s 2, cov(s, s 2 ) s h(u)/y(u) du Aderse e al. (993, pp. 83 84). I our applcaos, y s he survvor fuco of he respose me radom varable. We wll also eed a aalogous heorem for reverse margales for he heory of cumulave reverse hazard fucos. Wh cumulave reverse hazard fucos, we ca o loger assume K() <, bu we ca assume lm K(). Thus, we wll eed a fuco g() such ha sup G () s [, ] / g place of he codos o y(). Furhermore, he covarace fuco of he lm dsrbuo, cov(s, s 2 ), wll deped o he larger of s ad s 2. I hs case, g s he cumulave dsrbuo fuco of he respose me radom varable. Thus, f we addoally assume ha whe >, K() <, he dsrbuo of he processes he lm as he umber of samples creases s aga a Gaussa margale (Loyes, 969). A.3. roofs of heorems he ex A.3.. The cumulave reverse hazard fuco As a remder, Q () I {G() > } ˆK() G(T T ) K () max Q kds. Theorem. ˆK() s a ubased esmaor of K (). (A.) roof. Ths ca be derved he same maer ha he Ĥ s show o be ubased Aale e al. (28, pp. 87 88), bu wh me reversed. Isead of usg he defo of he coug process N() Eq. (2), we use reversed coug processes N() whch s a some max, ad creases as decreases. We represe he esy of hs process as λ() k()g(), where k() s he reverse hazard fuco a me ad le M() be he reverse max margale M() N() λ ds. The, he rasos of hs process ca be wre formally as, d N() k()g()d d M(). Le Q ()/G(), he, Q () G() d N() Q ()k()d Q () G() d M().

J.W. Houp, J.T. Towsed / Joural of Mahemacal sychology 56 (22) 34 355 35 Iegrag from o max ad mulplyg hrough by, we have, max Q G d N max max Q kds Q G d M. The lef-had sde s ˆK, he frs erm o he rgh s K (), ad he fal erm s a zero mea reverse margale. Hece, ˆK() s a ubased esmaor of K (). Theorem 2. A ubased esmae of he varace of ˆK() s gve by, ˆσ 2 K () T G 2 (T ). (A.2) roof. From Theorem, ˆK() K () s a zero mea reverse margale. Thus, Var ˆK() K () E ˆK() K () max Q E G d M. max Q The egral G d M s wh respec o a zero mea margale wh me reversed, so he opoal varao process s, max Q max G d Q M G 2 d N. Hece, ˆσ 2 () max Q G 2 d N G 2 (T ). T Lemma. ˆK() K () max roof. From he proof of Theorem, ˆK() K () max Q G d M. Q G kds. (A.3) Thus, he predcable varao process s (cf. Aderse e al., 993, p. 7), max Q 2 max Q 2 d M Gk ds G G max Q k ds. G (A.4) Theorem 3. ˆK() s a uformly cosse esmaor of K(). roof. By Leglar s Iequaly (e.g., Aderse e al., 993, p. 86) ad Lemma, r sup ˆK () K () > η s [, max ] δ max η r Q () 2 G () kds > δ. For each s [, max ], here s some posve probably of observg a respose a or before s. Hece, as, here wll be fely may resposes observed a or before s, so, f s [, max ] G(). Because Q () ad k <, max Q () k G () ds. Ths mples ha for ay posve δ, max lm r Q () k G () ds > δ. Ad hus for ay posve ϵ, lm r sup ˆK () K () ϵ. s [, max ] Addoally, max lm K K Q () ds. s Therefore, sup s [,max ] ˆK () K. Theorem 4. ˆK() K() coverges dsrbuo o a zero mea Gaussa process as he umber of respose mes used he esmae creases. roof. I he lm as he umber of samples creases, for ay >, Q () oly whe k() so s suffce o demosrae he covergece of ˆK() K (). Frs, we mus sasfy he codo ha here mus exs some g() such ha G()/ g() for all [τ, max ]. Ths follows drecly from he Glveko Caell Theorem (e.g., Bllgsley, 995, p. 269) by og ha G()/ s he emprcal cumulave dsrbuo fuco. The we may apply he reverse margale ceral lm heorem ad he cocluso follows. A.3.2. Esmag UCI performace As a remder, J() I {Y() > } Ĥ() Y(T T ) H () Jh ds. Theorem 5. Ĥ UCI () s a ubased esmaor of H UCI (). roof. k E ĤUCI () E Ĥ () k E Ĥ () k H () H UCI (). (A.5) (A.6) (A.7) Theorem 6. A ubased esmae of he varace of Ĥ UCI () s gve by, ˆσ 2 () k T (T ). (A.8)

352 J.W. Houp, J.T. Towsed / Joural of Mahemacal sychology 56 (22) 34 355 roof. To deerme he varace, we use he opoal varao of he margale. Assumg ha he coug processes for each process are depede, we have he followg relao (e.g., Aale e al., 28, p. 56), k Y () dm () k () dn (). Because, uder he ull hypohess, Ĥ UCI () k H () s a zero mea margale, k k Var Ĥ UCI () H () E Ĥ UCI () H () k k T dn (T ). Theorem 7. Ĥ UCI () s a uformly cosse esmaor of H UCI (). roof. Suppose s he umber of respose mes used o esmae Ĥ (). For each, Ĥ () s a cosse esmaor of H (), so ) sup Ĥ( H. s [,] The, sup Ĥ() () UCI H UCI() s [,] k sup Ĥ ( ) s [,] k H k Ĥ( sup ) H s [,] k sup s [,] ) Ĥ( H. Theorem 8. Le k where s he umber of respose mes used o esmae he cumulave hazard fuco of he compleo me of he h chael. The, ĤUCI H UCI coverges dsrbuo o a zero mea Gaussa process as he umber of respose mes used he esmae creases. roof. For he dsrbuo o coverge, Y / mus coverge probably o some posve fuco y() for all. Ths happes as log as he proporos of / coverge o some fxed proporo p, ad y () p ( F ()). The, we have as he lm dsrbuo a Gaussa process wh mea ad covarace, Cov(s, ) k σ 2 (m(s, )) for s, >. Theorem 9. ˆKUCI () s a ubased esmaor of K UCI (). roof. k E ˆKUCI () E ˆK () k E ˆK () k K () UCI (). Theorem. A ubased esmae of he varace of gve by, ˆσ 2 () k T G 2 (T ). ˆKUCI () s (A.9) roof. By Theorem 9, ˆKUCI () k K () s a zero mea margale so, k k Var ˆK UCI () K () E ˆK UCI () K () k max k T G 2 () d N () G 2 (T ). Theorem. ˆKUCI s a uformly cosse esmaor of K UCI. roof. Suppose s he umber of respose mes used o esmae ˆK. For each, ˆK s a cosse esmaor of K, so sup ˆK ( ) K. s [, max ] The, () sup ˆK UCI K UCI s [, max ] k k sup ˆK ( ) K s [, max ] k ( sup ˆK ) K s [, max ] k sup ˆK ( ) K. s [, max ] Theorem 2. ˆKUCI K UCI coverges dsrbuo o a zero mea Gaussa process as he umber of respose mes used he esmae creases. roof. Le k where s he umber of respose mes used o esmae he cumulave hazard fuco of he compleo me of he h chael. For he dsrbuo o coverge, Y / mus coverge probably o some posve fuco y() for all. Ths happes as log as he proporos of / coverge o some fxed

J.W. Houp, J.T. Towsed / Joural of Mahemacal sychology 56 (22) 34 355 353 proporo p, ad y () p ( F ()). The, we have as he lm dsrbuo a Gaussa process wh mea ad covarace, Cov(s, ) k σ 2 (m(s, )) for s, >. (A.2) L 2 k dn r L 2 dn L 2 (T ) k T (T ) L 2 (T ) (T ). T A.3.3. Hypohess esg Lemma 2. Uder he ull hypohess of UCI performace, Z OR () roof. Z OR () Y r dm r Y r dn r Y r dm r k k k k Y dm Y Y h ds Y r dm r h r k Y dm. Y dn Y r Y rh r ds Y dm k h ds k Y r dm r Y dm k h r h ds Y r dm r k Y dm. Theorem 3. Uder he ull hypohess of UCI performace, E (Z OR ). roof. Accordg o Lemma 2, Z OR s he dfferece of wo egrals wh respec o zero mea margales. The expecao of a egral wh respec o a zero mea margale s zero. Hece, he cocluso follows from he leary of he expecao operaor. Theorem 4. A ubased esmae of he varace of Z OR s gve by, Var (Z OR ) () L 2 (T ) k T (T ) L 2 (T ) (T ). roof. T Var (Z OR ()) E [Z OR ()] k E Y dm r Y dm Theorem 5. Assume ha here exss sequeces of cosas {a } ad {c } such ha, lm a lm c L () c Y () l <, y <. ad Y () r y r <, Furher, assume ha here exss some fuco d such ha for all δ, lm f r (a /c ) 2 L () 2 k Y r Y d for all s T δ. The (a /c )Z OR () coverges dsrbuo o a zero mea Gaussa process as he umber of respose mes used he esmae creases. roof. Here we use he verso of he margale ceral lm heorem as saed Aderse e al. (993, Seco 2.23). Frs, we mus show ha he varace process coverges o a deermsc process. a () Z OR c a L () c Y () r dm() r k a L () c Y () dm() L () 2 c 2 Y () 2 Y () r α r ds r L () 2 k c 2 L () c Y () 2 Y () 2 a α r Y () r α ds k a α Y () ds. Hece, from he assumpos ad by he domaed covergece heorem, a k () Z OR l 2 y r α r y α ds <. c The secod codo holds because for all s [, ], a c Y r ad a c Y for all. Therefore, he cocluso follows from he margale ceral lm heorem.

354 J.W. Houp, J.T. Towsed / Joural of Mahemacal sychology 56 (22) 34 355 Lemma 3. Uder he ull hypohess of UCI performace, Z AND () roof. Z AND () max max max max k G r d Mr G r d Nr G r d Mr max k max k k G r G rh r ds max max max max max G d M max max G G h ds G r d Mr h r G r d Mr h r G r d Mr k max k h ds max Y d M. G d N G d M k G d M k h ds max k G d M. Theorem 6. Uder he ull hypohess of UCI performace, E (Z AND ). roof. Accordg o Lemma 3, Z AND s he dfferece of wo egrals wh respec o zero mea margales. The expecao of a egral wh respec o a zero mea reverse margale s zero. Hece, he cocluso follows from he leary of he expecao operaor. Theorem 7. A ubased esmae of he varace of Z AND s gve by, Var (Z AND ) () T roof. L 2 (T ) G 2 (T ) Var (Z AND ()) E [Z AND ()] max k E G d Mr max T L 2 G 2 d Nr L 2 (T ) G 2 (T ) k T k k T max max L 2 (T ) G 2 (T ). L 2 (T ) G 2 (T ). G d M L 2 G 2 d N Theorem 8. Assume ha here exss sequeces of cosas {a } ad {c } such ha, lm a lm c L () c G () l <, g <. ad G () r g r <, Furher, assume ha here exss some fuco d such ha for all δ, lm f r (a /c ) 2 L () 2 k G r G d for all s T δ. The (a /c )Z AND () coverges dsrbuo o a zero mea Gaussa process as he umber of respose mes used he esmae creases. roof. max a () a L Z AND c c k max c 2 () G () r d M() r L () c G () d M() L () 2 G () 2 G () r α r ds r L () 2 max a k max c 2 max L () c G () 2 G () 2 a α r G () r α ds k a α G () ds. Hece, from he assumpos ad by he domaed covergece heorem, max a k () Z AND l 2 g r α r g α ds c <. The secod codo holds because for all s [, max ], a c G r ad a c G for all. Therefore, he cocluso follows from he reverse margale ceral lm heorem. Refereces Aale, Aderse, e al. (29). Hsory of applcaos of margales survval aalyss. Elecroc Joural for Hsory of robably ad Sascs, 5(). Aale, O. O., Borga, Ø., & Gessg, H. K. (28). Survval ad eve hsory aalyss: a process po of vew. New York: Sprger. Aderse,. K., Borga, Ø., & Kedg, N. (993). Sascal models based o coug processes. New York: Sprger. Bllgsley,. (995). robably ad measure (3rd ed.). New York: Wley. Blaha, L. M. (2). A dyamc Hebba-syle model of cofgural learg. h.d. Thess. Idaa Uversy, Bloomgo, Idaa. Brow, S. D., & Heahcoe, A. (28). The smples complee model of choce respose me: lear ballsc accumulao. Cogve sychology, 57, 53 78.

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