A method for determining groups in nonparametric regression curves: application to prefrontal cortex neural activity analysis

h p://www.aimsp ess.com/jou nal/mbe MBE, 19(7): 6435–6454. DOI: 10.3934/mbe.2022302 Recei ed: 03 Ma ch 2022 Re ised: 21 Ma ch 2022 Accep ed: 28 Ma ch 2022 Published: 24 Ap il 2022 Resea ch a icle A me hod o de e mining g oups in nonpa ame ic eg ession cu es: Applica ion o p e on al co ex neu al ac i i y analysis Ja ie Roca-Pa di˜ nas1, Celes ino O d´ o˜ nez2,*and Lu´ ıs Mei a Machado3 1Depa men o S a is ics and Ope a ional Resea ch, Vigo Uni e si y, Vigo 36310, Spain 2Depa men o Mining Exploi a ion and P ospec ing, Geoma ics and Compu e G aphics Lab, O iedo Uni e si y, Mie es 33600, Spain 3Cen e o Ma hema ics, Minho Uni e si y, B aga 4704-553, Po ugal *Co espondence: Email: o donezceles ino@unio i.es; Tel: +34985458027. Abs ac : Gene alized addi i e models p o ide a lexible and easily-in e p e able me hod o unco - e ing a nonlinea ela ionship be ween esponse and co a ia es. In many si ua ions, he e ec o a con inuous co a ia e on he esponse a ies ac oss g oups de ined by he le els o a ca ego ical a i- able. When con on ed wi h a conside able numbe o g oups de ined by he le els o he ca ego ical a iable and a ac o -by-cu e in e ac ion is de ec ed in he model, i hen becomes impo an o com- pa e hese eg ession cu es. When he null hypo hesis o equali y o cu es is ejec ed, leading o he clea conclusion ha a leas one cu e is di e en , we may assume ha indi iduals can be g ouped in o a numbe o classes whose membe s all sha e he same eg ession unc ion. We p opose a me hod ha allows de e mining such g oups wi h an au oma ic selec ion o hei numbe by means o boo s ap- ping. The alidi y and beha io o he p oposed me hod we e e alua ed h ough simula ion s udies. The applicabili y o he p oposed me hod is illus a ed using eal da a om an expe imen al s udy in neu ology. Keywo ds: clus e ing o eg ession cu es; ac o -by-cu e in e ac ion; gene alized addi i e model; mul iple eg ession cu es; nonlinea eg ession; numbe o g oups 1. In oduc ion One o he main goals o s a is ical modeling is o unde s and he e ec o some explana o y a i- ables, Xi,i=1, ..., p, on a dependen a iable, Y, also known as he esponse. This ype o dependence is o en modeled using a gene alized linea eg ession model (GLM) [1] ha imposes a linea ela ion- ship be ween explana o y and dependen a iables wi hou speci ying in ad ance he unc ion ha links 6436 hem. Gene alized addi i e models (GAMs) [2, 3] o e an ex ension o he GLM h ough he inco - po a ion o nonlinea o ms o he explana o y a iables. GAM has been widely used as an e ec i e echnique o conduc ing nonlinea eg ession analysis in a ious ields such as economics, inance, he en i onmen , medicine, and biology. In a wide ange o hese applica ions, p oblems can be obse ed which i migh be use ul o compa e wo o mo e eg ession cu es. This o en occu s when i is nec- essa y o check i he e ec o a con inuous co a ia e on he esponse a ies ac oss g oups de ined by le els o a ca ego ical a iable. Tes ing he hypo hesis o equali y o he eg ession unc ions is a opic o s a is ical in e ence ha has been widely esea ched in he li e a u e. One impo an e iew on his opic may be ound in [4] (see hei Sec ion 7). Examples o me hods o clus e ing eg ession cu es using L2dis ance and o he me ics a e in [5–7]. Fo nonpa ame ic models, mos app oaches ely on smoo hing echniques, such as splines o Nada aya-Wa son [8], in he cons uc ion o he nonpa a- me ic es ima o s o eg ession unc ions. The gene al idea is o use hese nonpa ame ic es ima o s di ec ly, ei he by con as ing all indi idual es ima o s wi h he pooled es ima o , o by pe o ming pai wise compa isons. O he au ho s conside ed he use o empi ical p ocesses o a oid he necessi y o selec ing a smoo hing pa ame e equi ed in he cons uc ion o he nonpa ame ic es ima o s o eg ession unc ions [9–14]. The idea behind hese es s is o use he dis ibu ion o he esiduals and o compa e hem ia Kolmogo o -Smi no o C am´ e - on Mises ype s a is ics. ANOVA- ype es s a is ics ha e also been used o compa e eg ession unc ions (see o example [15] and [16]). Re - e ence [17] illus a es how he SiZe explo a o y ool is capable o compa ing mul iple cu es based on he esiduals. In [18] a ke nel-based nonpa ame ic app oach is conside ed while [19] p oposed a es ing p ocedu e o single index models. A second, hough ela ed ques ion is how o de e mine g oups among a se ies o eg ession cu es when he null hypo hesis o equali y o he eg ession unc ions is ejec ed. Though he a o emen ioned me hods can be used o compa e eg ession cu es, o he bes o ou knowledge, hose me hods canno be used o de e mine g oups among a se ies o eg ession cu es, o example, among g oups de ined by he le els o a ca ego ical a iable. Some o hem a e no ecommended when con on ed wi h a conside able numbe o cu es. I he null hypo hesis o equali y o cu es is ejec ed, hen his leads o he clea conclusion ha a leas one eg ession cu e is di e en . Howe e , hese me hods canno be used o asce ain whe he indi iduals can be g ouped in o a educed numbe o classes whose membe s all sha e he same eg ession unc ion o i all he eg ession cu es a e di e en om each o he . One na¨ ı e app oach would be o pe o m pai wise compa isons. Howe e , his app oach would lead o a la ge numbe o compa isons (e.g., 7 g oups would lead o 21 pai wise compa isons). This could be done bu wi hou he possibili y o de e mining g oups wi h simila eg ession cu es. A s a is ical p ocedu e o es ima e he unknown g oup s uc u e was p oposed in [20] and la e in [21]. The i s pape o hese au ho s was abou inancial ime se ies whe eas he second deals wi h en i onmen al s a is ics. Mo e ecen ly, he compa ison o su i al cu es was add essed [22]. Simila ly, in his pape , we p opose an app oach ha allows de e mining g oups ha sha e he same eg ession unc ion wi h an au oma ic selec ion o hei numbe . The p oposed me hod can be used, o ins ance, o es ablish g oups wi h simila mono onic ends in he eg ession cu es (e.g., o e he le els o ca ego ical a iables). The p oposed me hodology will be shown in he amewo k o he gene alized addi i e model wi h a bina y ou come and a logi link unc ion. The gene aliza ion o he p oposed me hods o o he link unc ions is possible. The emainde o he pape is o ganized as ollows: The ollowing sec ion p o ides he no a ion Ma hema ical Biosciences and Enginee ing Volume 19, Issue 7, 6435–6454. 6437 and he me hodological backg ound. Then, he pe o mance o he p oposed me hods is in es iga ed h ough simula ions, and hei usage is illus a ed h ough he analysis o a eal da a se om neu onal ac i i y in he p e on al co ex o monkeys du ing a disc imina ion ask. Finally, he las sec ion con ains a discussion and he main conclusions o he wo k. 2. Ma e ials and me hods 2.1. Ma hema ical model Le Ydeno e he bina y (0/1) a ge esponse and X=(X1,...,Xp) a se o pcon inuous co a ia es. In his eg ession con ex , he logis ic gene alized addi i e model exp esses he condi ional p obabili y P(X)=P(Y=1|X) as log P(X) 1−P(X)=α+ p X j=1 j(Xj) (2.1) whe e αis a cons an and each pa ial unc ion j ep esen s he e ec o he co a ia e Xj. These unknown smoo h pa ial unc ions a e modeled om he da a wi hou speci ying in ad ance any pa a- me ic s uc u e. The e o e, he GAM in (2.1) combines lexibili y wi h in e p e abili y, in which each o he addi i e componen s desc ibes he in luence o each co a ia e sepa a ely. In p ac ice, he e ec o a con inuous co a ia e Xjcan a y ac oss he le els 1,...,Ko a ca ego ical ac o F. Fo simplici y o no a ion, he e we conside ha only he e ec o he las co a ia e, Xp, depends on he le els o F, so he pu e GAM in (2.1) can inco po a e his ype o ac o -by-cu e ela ionship as ollows: logi (F,X)=log P(Y=1|F,X) 1−P(Y=1|F,X)=α+ p−1 X j=1 j(Xj)+                  g1(Xp) i F=1 g2(Xp) i F=2 . . .. . .. . . gK(Xp) i F=K (2.2) whe e g1,...,gK ep esen he speci ic e ec o Xp o each o he possible Kle els es ablished by he ac o F. 2.2. Es ima ing pa i ions The s udy o he pa ial unc ions g1,...,gKcan be use ul in he compa ison o wo o mo e g oups, which is an impo an p oblem associa ed wi h s a is ical in e ence. In e es cen e s on he null hypo h- esis H0:g1=. . . =gK, namely, ha he e ec o Xpdoes no depend on he le els o he ac o F. When he equali y o he Kcu es is ejec ed, leading o he clea conclusion ha a leas one cu e is di e en , i can be in e es ing o asce ain whe he g oups can be pe o med o , by con as , ha all hese cu es a e di e en om each o he . Mo e clea ly, i H0is ejec ed, i could be in e es ing o de- e mine: a) i any pai o cu es a e di e en om each o he , ha is gi,gj, o all i,j∈ {1,...,K}, Ma hema ical Biosciences and Enginee ing Volume 19, Issue 7, 6435–6454. 6438 o b) i he indices in {1,...,K}can be g ouped in a pa i ion o J(J<K) g oups I=(I1,...,IJ) wi h J<K, so ha gi=gj o each i,j∈Ik, o some k. No e ha Imus be a pa i ion o {1,...,K}and, he e o e, mus sa is y I1∪. . . ∪IJ={1,...,K}and Ii∩Ij=∅ o all i,j∈I. In some si ua ions, he possible pa i ion I=(I1,...,IJ), can be es ablished in ad ance, and he in e es is o alida e such pa i ion by es ing he null model logi (F,X)=α+ p−1 X j=1 j(Xj)+                 g1(Xp) i F∈I1 g2(Xp) i F∈I2 . . .. . .. . . gK(Xp) i F∈IJ (2.3) e sus he gene al model gi en in (2.2). Howe e , in gene al, he pa i ions a e unknown and will ha e o be es ima ed om he da a. Be o e explaining he p ocedu e o de e mining he g oups, le us in oduce some no a ion. Gi en a pa i ion I=(I1, . . . IJ), we de ine a unc ion GI:{1,...,K} → {1,...,J}so GI(k)=ji k∈Ij. Now, gi en a sample {Fi,Xi,Yi}n i=1and using he es ima es ˆgk(k=1,...,K) o he model in (2.2), he es ima ed pa i ion ˆ I=(ˆ I1, . . . ˆ IJ) can be ob ained by minimizing he ollowing C am´ e - on Mises ype dis ance min I=(I1,...,IJ) K X k=1Zxˆgk(x)−ˆ CGI(k)(x)2dx (2.4) Al e na i ely, he es ima ed pa i ion can be ob ained using he Kolmogo o -Smi no ype dis ance min I=(I1,...,IJ) K X k=1Zxˆgk(x)−ˆ CGI(k)(x)dx (2.5) In bo h cases he es ima ed cen oids o j=1,...,Ja e de ined as ˆ Cj(x)=PK k=1ˆgk(x)·I{GI(k)=j} PK k=1I{GI(k)=j}(2.6) The abo e minimiza ion p oblems can ha e a high compu a ional cos o la ge alues o K, because hey equi e he e alua ion o all he di e en combina ions o he Kcu es in o Jg oups. To sol e he quad a ic minimiza ion p oblem in (2.4) and L1-no m minimiza ion p oblem in (2.5) we p opose he use o he k-means o k-medians algo i hms. In bo h cases, he pa ial unc ions ha e o be es ima ed in a common g id o Xp,x• 1< . . . < x• N, o size N, leading o a ma ix             ˆg1(x• 1) . . . ˆgK(x• 1) . . .. . . ˆg1(x• N) . . . ˆgK(x• N)            N×K This ma ix will be he inpu o bo h heu is ic me hods, k-means and k-medians, and om hese, he es ima ed pa i ion ˆ I=(ˆ I1,...,ˆ IJ) is ob ained. Ma hema ical Biosciences and Enginee ing Volume 19, Issue 7, 6435–6454. 6439 2.3. De e mining he numbe o g oups The p ocedu e o ob ain he es ima ion pa i ion ˆ I=(ˆ I1,...,ˆ IJ) o a gi en J(J<K) has been explained in he p e ious sec ion. This sec ion ocuses on de e mining he numbe o g oups o he eg ession unc ions. Wi h his goal in mind, one possible app oach consis s in i ing he model in (2.3) o each alue o j=1,...,J, and selec ing he numbe o g oups K ha minimizes he Bayesian In o ma ion C i e ion (BIC). BIC is one o he mos widely used ools in s a is ical model selec ion. I s popula i y is de i ed om i s compu a ional simplici y and e ec i e pe o mance in many modeling app oaches. Ou simula ions on cu e clus e ing demons a e ha he BIC c i e ion can be conside ed a alid op ion, pe o ming qui e sa is ac o ily choosing he co ec numbe o clus e s in simula ion scena ios wi h a la ge numbe o cu es and complex da a se s. Ano he possibili y would be o es , o a gi en J, he null hypo hesis H0(J) ha a leas one pa i ion Io leng h Jexis s, so ha he model in (2.3) is ul illed, agains he al e na i e ha o any J-pa i ion a leas a g oup Ijexis s in which gm,gn o some m,n∈Ij. In o de o es H0(J), o a gi en sample {Fi,Xi,Yi}n i=1, we conside he ollowing wo s a is ics DCM = K X k=1 n X i=1ˆgk(Xip)−ˆg◦ Gˆ I(k)(Xip)2(2.7) DKS = K X k=1 n X i=1ˆgk(Xip)−ˆg◦ Gˆ I(k)(Xip)(2.8) whe e he es ima ed pa i ion ˆ I=(ˆ I1,...,ˆ IJ) is ob ained by sol ing he minimiza ion p oblems in (2.4) o (2.5), and ˆg◦ j unc ions a e ob ained om model (2.3) using he same pa i ion. Al e na i ely, we can use he de iance as an app op ia e measu e o disc epancy be ween obse ed and i ed alues and conside he ollowing likelihood a io es DLR =Pn i=1De i(Yi,ˆ Pi)−Pn i=1De i(Yi,ˆ P◦ i) Pn i=1De i(Yi,ˆ Pi)(2.9) which compa es he de iance o he null model (2.3) wi h he de iance o he gene al model in (2.2), whe e ˆ P◦ iand ˆ Pia e he es ima es alues o he ue p obabili ies Pi=P(Yi=1|Fi,Xi) unde he null and gene al model, espec i ely. The indi idual de iance De i(Yi,ˆ Pi) is de ined as De i(Yi,ˆ Pi)= −2Yilog ˆ Pi+(1 −Yi) log(1 −ˆ Pi). No e ha i H0(J) is e i ied, he alue o he es s a is ic Dshould be close o ze o. The decision ule based on each o he h ee s a is ics, D, consis s in ejec ing he null hypo hesis i Dis g ea e han (1 −α)-pe cen ile ob ained unde he null hypo hesis. Ne e heless, he heo y o de e mining such pe cen iles is no closed. To app oxima e he dis ibu ions o he es s a is ics, esampling me hods such as he boo s ap me hod can be applied [23–26]. The bina y boo s ap used in ou pape is a pa icula case o he boo s ap echniques sugges ed by [27] and [28] o in e ence in nonpa ame ic models wi h esponse belonging o he bina y amily. The bina y boo s ap in ol es he ollowing s eps: S ep 1. Using he o iginal sample da a {Fi,Xi,Yi}n i=1, compu e he es s a is ics Das explained abo e. Then, ob ain o i=1,...,n he es ima ed ˆ P◦ io he ue p obabili ies Pi=P(Yi=1|Fi,Xi) ob ained om he null model in (2.3). Ma hema ical Biosciences and Enginee ing Volume 19, Issue 7, 6435–6454. 6440 S ep 2. Fo b=1,...,B, gene a e he boo s ap sample {Fi,Xi,Y• i}n i=1, wi h Y• i∈Be noulli(ˆ P◦ i), and compu e he boo s ap s a is ics D•,b. No e ha he es ima ion o he pa i ion is equi ed in his S ep. Finally, he decision ule, based on each es s a is ic D, consis s in ejec ing he null hypo hesis i D>ˆ D1−α, whe e ˆ D1−αis he empi ical (1−α)-pe cen ile o he alues D•1,...,D•Bob ained p e iously. The boo s ap-based es in oduced he e can be e y use ul o au oma ically de e mine he numbe o g oups J. Speci ically, he ule p oposed is as ollows: he p ocedu e begins by es ing H0(1). I his hypo hesis is ue, we decide ha J=1 (all eg ession cu es a e equal). When he p e ious hypo hesis is ejec ed, i will be necessa y o es H0(2). I his new hypo hesis is no ejec ed, we decide ha J=2. When H0(2) is ejec ed, i will be equi ed o es H0(3) and so on un il a ce ain H0(J) is no ejec ed. Finally, no e ha we a e awa e ha his app oach could deal wi h he p oblem o mul iple hypo hesis es ing whe e a se o Jp- alues co esponding o he Jnull hypo heses, H0(1),H0(2),...,H0(J), a e gi en. E en hough se e al me hods ha e been p oposed o deal wi h his p oblem (see e.g. [29] o an in oduc ion o his a ea), he e is s ill a challenge because he e is no in o ma ion abou he minimum numbe o es s needed o apply hese echniques. Acco dingly, and conside ing ha ou ule inishes wi h a low numbe o hem, we ha e no conside ed his p oblem. 2.4. Simula ion s udy This sec ion epo s he esul s o a simula ion s udy conduc ed o e alua e he p ac ical pe o mance o he p oposed me hodology. The esponse Ywas gene a ed unde model (2.3) wi h log P(Y=1|F,X) 1−P(Y=1|F,X)=                      g1(X)=1.5Xi F∈I1={1,2,3} g2(X)=2X2−3 i F∈I2={4,5} g3(X)=2 sin(2X)−2 i F∈I3={6,7} g4(X)=2 cos(2X) i F∈I4={8,9} g5(X)=2 cos(2X)+aexp(0.5X) i F∈I5={10,11,12,13} g6(X)=4−2Xi F∈I6={14,15} (2.10) The ca ego ical co a ia e Xwas d awn om an uni o m dis ibu ion U[−2,2], and i s ac o s, F, we e gene a ed by aking a andom alue in {1,...,15}wi h associa ed p obabili ies p=(p1,...,p15)= n/Pnk,k=1,...,15, whe e: n=(n1,...,n15)=(1.0,2.0,1.0,1.5,1.0,2.0,1.5,2.0,2.0,1.0,1.0,1.0,1.5,1.5,1.0) In his way, and as can be seen in Table 1, we ha e conside ed unequal sample sizes o each (k= 1,...,15) cu e. We explo e he alidi y o he p oposed es s assuming ha we aim o es i he K=15 eg ession cu es can be g ouped in i e (J=5) g oups. To s udy he size and powe o he p oposed es s, di e en alues we e conside ed o a, anging om 0 o 2. I should be no ed ha g5(X)=g4(X)+ aexp(0.5X), so a alue o a=0 co esponds o a null hypo hesis o i e g oups, while a alue o a>0 co esponds o a null hypo hesis o six g oups. 2.5. Applica ion o neu al ac i i y in he p e on al co ex du ing a disc imina ion ask The da a analyzed in his sec ion come om labo a o y expe imen s o he ex a-cellula single uni ac i i y in he p e on al co ex o a monkey. These expe imen s we e conduc ed in he Labo a o y o Ma hema ical Biosciences and Enginee ing Volume 19, Issue 7, 6435–6454. 6441 Table 1. Sample size nk o each eg ession cu e k(k=1,...,15) o di e en o al sample sizes n=Pnk. n n1n2n3n4n5n6n7n8n9n10 n11 n12 n13 n14 n15 300 14 28 14 21 14 28 21 28 28 14 14 14 21 21 14 500 23 47 23 35 23 47 35 47 47 23 23 23 35 35 23 1000 47 95 47 71 47 95 71 95 95 47 47 47 71 71 47 2000 95 190 95 142 95 190 142 190 190 95 95 95 142 142 95 4000 190 380 190 285 190 380 285 380 380 190 190 190 285 285 190 Neu ophysiology o he Uni e si y o Vigo (Spain). The monkey was ained o disc imina e be ween di e en s imuli. The s imuli consis ed o s a iona y b igh line segmen s p esen ed on a moni o sc een in on o he monkey ha changed hei o ien a ion o e ime. A ial was ini ia ed when he monkey p essed a le e key wi h i s igh hand and hen wo s imuli ( e e ence and es ), each o 500 ms du a ion, we e p esen ed in sequence wi h a ixed in e -s imulus in e al (ISI: 1000 ms). A he end o he second s imulus, he subjec eleased he key in a 1200 ms ime window and p essed one o he wo swi ches (le o igh ), indica ing whe he he o ien a ion o he second s imulus was clockwise ( igh ) o coun e -clockwise (le ) o he e e ence s imulus. While he monkey wo ked on he ask, i s ex acellula uni ac i i y was eco ded. The monkey was ewa ded o co ec disc imina ion. To ga he enough da a and o accoun o he cell esponse a iabili y, he neu on was eco ded o e a numbe o T=80 ials. Fo each o he ials, we conside he angle ( ial speci ic), TA, co esponding o he es s imulus. The e e ence angle is 90◦. The ollowing TAs we e conside ed in he expe imen : •T A ∈ {78◦,102◦}: es s imuli mo e sepa a ed om he e e ence, and he e o e e y easy o disc imina e •T A ∈ {81◦, 99◦}: es s imuli is easy o disc imina e •T A ∈ {84◦, 96◦}: es s imuli is di icul o disc imina e •T A ∈ {87◦,93◦}: es s imuli closes o he e e ence, he e o e mo e di icul o disc imina e In his expe imen , he ou come o in e es is he neu onal ac i i y in he in e al ∈[ min, max]= [−500,3000] in ms. A each ins an and ial j=1,...,T, his ou come may hen be ep esen ed by a empo al bina y sequence, Yj , whe e Yj =1 i he e is a spike in [ , +1) ms and 0 o he wise. The explana o y a iables a e he ime when a spike was de ec ed and he in e al o ime ∆ since he las spike. The las one is an adjus men a iable ha akes in o accoun ha he neu on migh i e mo e easily i i has been i ed ecen ly. Acco dingly, o each ial, he da a se consis s o he ollowing in o ma ion {T A,( ,∆ ),Y } max = min We use gene alized addi i e modeling o assess whe he he associa ion be ween neu al ac i i y and decision-making depends on he di icul y o he disc imina ion ask ela ed o he angle o he es s imulus. We conside he ollowing GAM: Ma hema ical Biosciences and Enginee ing Volume 19, Issue 7, 6435–6454. 6442 log P(T A, ,∆ ) 1−P(T A, ,∆ )!=α+ (∆ )+                                      g1( ) i T A =78◦ g2( ) i T A =102◦ g3( ) i T A =81◦ g4( ) i T A =99◦ g5( ) i T A =84◦ g6( ) i T A =96◦ g7( ) i T A =87◦ g8( ) i T A =93◦ (2.11) whe e P(T A, ,∆ )=P(Y =1|F, ,∆ ), αis a ixed pa ame e , and gk o k=1,...,8 a e he speci ic ime unc ions associa ed o each o he es angles conside ed in he s udy. 3. Resul s 3.1. Resul s on simula ion da a se Type I e o a es and powe alues o he p oposed es s as a unc ion o aa e shown in Figu e 1. They we e calcula ed om 1000 simula ion uns using B=400 boo s ap samples a each epe i ion. We compa e he esul s o he h ee es s a is ics unde sample sizes n=1000 and 2000, a he signi icance le els o 0.05 and 0.10. As can be seen in Figu e 1, he h ee cu es show he expec ed beha io pa e n, wi h an inc ease in he powe as ainc eases, and an imp o emen in i as he sample size g ows. n=1000 a Powe (%) DCM DKS DLR 0.00 0.50 1.00 1.50 2.00 0 20 40 60 80 100 n=1000 a Powe (%) 0.00 0.50 1.00 1.50 2.00 0 20 40 60 80 100 n=2000 a Powe (%) 0.00 0.50 1.00 1.50 2.00 0 20 40 60 80 100 n=2000 a Powe (%) 0.00 0.50 1.00 1.50 2.00 0 20 40 60 80 100 Figu e 1. Rejec ion p obabili ies (powe o he hypo hesis es ) o he p oposed es s a is ics as a unc ion o a, o sample size n=1000 and n=2000 (uppe and lowe panels, espec- i ely) a a 0.05 and 0.10 (le and igh panels, espec i ely) signi ican le els ( ed line). Ma hema ical Biosciences and Enginee ing Volume 19, Issue 7, 6435–6454. 6443 The es s a is ic based on he likelihood a io es (labeled as DLR) is ha wi h he highes alues o powe o n=1000, bu he di e ences be ween he h ee es s a is ics a e minimal o n=2000. The C am´ e - on Mises– ype s a is ic es (labeled as DCM) e ealed a poo e beha io wi h lowe ejec ion p obabili ies o a sample size o 1000. Values o ype I e o , o he h ee es s a is ics and o di e en signi icance le els and di e en sample sizes (n=1000,2000, and 3000), a e epo ed in Table 2. The h ee es s a is ics wo k sa is ac o ily acco ding o he ype I e o , coming qui e close o he nominal le el ega dless o he sample size. Table 2. Es ima ed ype I e o (in %) and nominal le el pe cen ages (1, 5, 10, 15, and 20) o di e en sample sizes. le el DCM DKS DLR n=1000 1 1.4 1.3 1.0 5 5.5 5.3 5.6 10 9.7 9.8 11.5 15 14.5 15.3 16.5 20 18.7 20.0 21.1 n=2000 1 0.9 0.7 0.8 5 4.2 3.8 5.7 10 8.4 8.0 9.4 15 12.8 12.4 15.6 20 16.5 18.0 21.8 n=3000 1 0.9 0.7 1.2 5 4.2 4.0 6.5 10 8.8 7.6 11.1 15 13.0 12.1 16.2 20 16.2 17.0 22.3 In Table 3, we epo esul s ha can be used o e alua e he accu acy o he boo s ap-based al- go i hm in oduced in Sec ion 2.3. Again, di e en alues o awe e conside ed, anging om 0 o 2. Recall ha he alue a=0 co esponds o he null hypo hesis, which assumes ha he i een eg ession unc ions can be classi ied in o i e g oups, while, when a,0, he numbe o g oups is six. No e ha o selec he co ec numbe Jo g oups o eg ession unc ions, he boo s ap-based algo i hm mus i s ejec he i s null hypo hesis, H0(1), hen ejec he second hypo hesis, H0(2), and so on un il i accep s H0(5) i a=0, o un il H0(6) when a>0. Resul s shown in Table 3 o he h ee es s a is ics, using a nominal le el o 5%, display he numbe o imes ha he p ocedu e selec s he numbe o g oups J. Resul s in bold deno e he co ec classi ica ions acco ding o model (2.3), e ealing he high accu acy o he p oposed boo s ap-based algo i hm o a=0, showing he co ec numbe o g oups in pe cen ages qui e close o he nominal le el. As he alue o ainc eases, so does he pe cen age o cases in which he p oposed me hod sugges s six g oups. When compa ing he h ee es s a is ics, i can be obse ed ha all ha e a simila pe o mance wi h a small ad an age o he es s a is ic based on he likelihood a io es (labeled as DLR). Ma hema ical Biosciences and Enginee ing Volume 19, Issue 7, 6435–6454. 6450 han 90◦(clockwise o a ion) we e p e iously epo ed in a s udy [26]. Time(ms) Fi ing a e (spikes/s) −500 0 500 1000 1500 2000 2500 3000 0 5 10 15 20 25 30 Time (ms) Fi ing a e (spikes/s) Tes Re e ence ISI −500 0 500 1000 1500 2000 2500 3000 10 15 20 25 30 35 Figu e 5. Rep esen a ion o neu al ac i i y o e ime. Obse ed spikes pool coun s o spikes wi hin 10 ms in e als (le ) and he co esponding smoo hing e sion ( igh plo ). Time (ms) Fi ing a e (spikes/s) 1500 2000 2500 3000 10 20 30 40 50 60 78º 81º 84º 87º 93º 96º 99º 102º Figu e 6. Fi ed eg ession cu es o he eigh di e en expe imen al condi ions. Resul s o he boo s ap me hod using he h ee es s a is ics de ined in Sec ion 3 a e shown in Table 7. As can be seen, he h ee es s a is ics indica e ha he e a e wo g oups. DKD, and sligh ly less so DCM, a e close o ipping he balance in a o o h ee g oups. We ha e also applied he BIC c i e ion o de e mine he numbe o g oups. The esul s o his clus e solu ion also lead o he same conclusion, namely, J=2 g oups. These indings a e in ag eemen wi h hose epo ed in ou simula ions ha indica e ha BIC is mo e conse a i e, bu capable o de ec ing Ma hema ical Biosciences and Enginee ing Volume 19, Issue 7, 6435–6454. 6451 impo an di e ences. Table 7. P obabili y alues o es ing he null hypo hesis H0(J). Resul s based on he boo s ap me hod wi h di e en es s a is ics. JDCM DKS DLR 1<0.001 <0.001 <0.001 2<0.06 <0.05 <0.29 3<0.19 <0.13 <0.53 4<0.09 <0.76 <0.48 Clus e s o a ixed numbe o g oups be ween J=2 and J=5 a e shown in Figu e 7 o acili a e he comp ehension o he p oblem, al hough he esul s ob ained indica e ha he e a e only wo g oups. Fi ing a e (spikes/s) 12345 78º 81º 84º 87º 93º 96º 99º 102º 2 g oups Fi ing a e (spikes/s) 78º 81º 84º 87º 93º 96º 99º 102º 3 g oups Time (ms) Fi ing a e (spikes/s) 12345 1500 2000 2500 3000 78º 81º 84º 87º 93º 96º 99º 102º 4 g oups Time (ms) Fi ing a e (spikes/s) 1500 2000 2500 3000 78º 81º 84º 87º 93º 96º 99º 102º 5 g oups Figu e 7. Es ima ed eg ession cu es acco ding o he g oups o which hey belong. The cu es assigned in o wo (J =2), un il i e g oups (J =5) a e ep esen ed in each o he ou panels. The s a is ics used o es ima e he numbe o g oups ound only wo. 4. Conclusions In his wo k we p opose a boo s ap-based me hod o de e mine he numbe o g oups in gene alized addi i e models when he e ec o a con inuous co a ia e on he esponse a ies ac oss g oups de ined by le els o a ca ego ical a iable. The simula ion analysis con i ms he capaci y o he h ee p oposed s a is ical es s o ep oduce he heo e ical alues o ype I and ype II e o s, e en in he p esence o a Ma hema ical Biosciences and Enginee ing Volume 19, Issue 7, 6435–6454. 6452 la ge numbe o cu es. Howe e , he powe u ned ou o be highe o he es based on he likelihood es a io. As expec ed, he pe cen age o misclassi ica ions dec eases wi h he sample size, n, being almos null o n=4000. The simula ed da a we e also used o compa e ou me hod wi h BIC, a well-known app oach ha has been la gely used p e iously in clus e ing analysis o es ima e he numbe o clus e s in an a i icial da a se . The numbe o de ec ed g oups was he same o bo h me hods, bu he simula ion s udy showed ha BIC has a lowe s a is ical powe han he boo s ap me hod, al hough hey a e close o la ge sample sizes. Acco dingly, BIC could be used o ob ain an ini ial es ima ion o he numbe o clus e s ha could be uned using ou me hod. The applica ion o he p oposed me hod o an expe imen o de e mine he ac i i y o a neu on o a monkey subjec o isual s imuli led o he same conclusions when we compa ed ou app oach wi h BIC. In bo h cases he neu on only eac s o wo o he eigh s imuli, co esponding o he mos e iden di e ences wi h espec o a e e ence s a e. Rega ding he h ee s a is ics es ed, hey p o ide he same esul , al hough he e idence o wo clus e s is s onge o he s a is ic based on he likelihood a io es . Finally, i can be said ha al hough he p oposed me hod was designed o de ec g oups o eg es- sion cu es in gene alized addi i e models wi h a bina y esponse, i can be ex ended wi hou much di icul y o de e mine g oups in models wi h ano he kind o esponse. Acknowledgmen s This wo k was pa ially suppo ed by p ojec 2017/00001/006/001/097: Ayudas pa a el man- enimien o de ac i idades de in es igaci´on de ins i u os uni e si a ios de in es igaci´on y g upos de in es igaci´on de la Uni e sidad de O iedo pa a el eje cicio 2021. Lu´ ıs Mei a-Machado acknowledges inancial suppo om Po uguese Funds h ough FCT - ”Fundac¸˜ ao pa a a Ciˆ encia e a Tecnologia”, wi hin he p ojec s UIDB/00013/2020, UIDP/00013/2020. Ja ie Roca-Pa di˜ nas acknowledges inancial suppo om G an PID2020-118101GB-I00, Minis- e io de Ciencia e Inno aci´ on (MCIN/AEI /10.13039/501100011033). Con lic o in e es The au ho s decla e he e is no con lic o in e es . Re e ences 1. P. McCullagh, J. 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Van De Laan, Mul iple Tes ing P ocedu es wi h Applica ions o Genomics, Sp inge , Sp inge Se ies in S a is ics, New Yo k, 2007. ©2022 he Au ho (s), licensee AIMS P ess. This is an open access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion License (h p://c ea i ecommons.o g/licenses/by/4.0) Ma hema ical Biosciences and Enginee ing Volume 19, Issue 7, 6435–6454.