MODELING TRIVARIATE CORRELATED BINARY DATA

Transcription

1 Al Azhar Bult of S Vol.6 No. Dmbr - 5. MODELING TRIVARIATE CORRELATED BINAR DATA Ahmd Mohamd Mohamd El-Sad Dartmt of Hgh Isttut for Sf Studs Maagmt Iformato Sstms Nazlt Al-Batra Gza Egt. ABSTRACT Ths ar rovds th stmato ad th tst rodurs for th masurs of assoato th orrlatd bar data assoatd wth ovarats. Th gralzd lar modl GLM usg th sral dd ad th dvlod altratv quadrat otal form AQEF rodurs ar mlod for th trvarat bar orrlatd outom varabls. Th odds ratos as masurs of assoato ar stmatd ad th arorat tsts ar suggstd. For omarso btw th two rodurs w usd th smulato stud ad a alato to a o roblm whh volvs th stmato of th masurs of assoato ad thr tsts. Th ovr-dsrso rtra s vstgatd for ths rodurs. Fall th dva ad sald dva ar usd as tsts for th goodss of ft of th modl to dtrm th bst rodur. Kwords: Trvarat Broull Dstrbuto Markov modl Gralzd lar modl Dva Lklhood rato tst Mamum lklhood stmators Altratv quadrat otal form. INTRODUCTION Th dd btw th rsos ad th laator varabls s of trst rasgl th rt studs sall wth orrlatd outom varabls assoatd wth ovarats. A quas-mamum lklhood stmat also kow as a sudo-lklhood stmat or a omost lklhood stmat s a stmat of a aramtr a statstal modl that s formd b mamzg a futo that s rlatd to th arthm of th lklhood futo but s ot qual to t. I otrast th mamum lklhood stmat mamzs th atual -lklhood futo. I lklhood aalss w must sf th atual form of th dstrbuto. I quas-lklhood w sf ol th rlatoshs btw th ma of outom ad ovarats ad btw th ma ad th vara futos. B adotg a quas-lklhood aroah ad sfg ol th ma-vara strutur w a dvlo mthods that ar alabl to svral ts of outoms varabls. To us th rgrsso modl usg th quas-lklhood mthod w must us th lk futo as a trasform btw th atural aramtrs ad rgrsso aramtrs. Ths lk futos ar dffrt from as to as aordg to th dstrbuto of orrlatd outom varabls. I ths stud our fous s o th trvarat as for th orrlatd bar data baus fw authors ar dvotd wth th trvarat as. Islam t al. [5 dvlod a w sml rodur to tak aout of th bvarat bar modl wth ovarat dd. Ths modl s basd o th tgrato of odtoal ad margal modls. Qaqsh [7 rstd a faml of multvarat bar dstrbutos for smulatg orrlatd bar varabls wth sfd margal mas ad orrlatos. Zhao ad Prt [ dsussd th sudo-mamum lklhood for aalzg orrlatd bar rsoss. Thr aramtrzato s basd o a sml arws modl whh th assoato btw rsoss s modld trms of orrlatos. Also Hagrt ad Zgr [ Hagrt [ rstd th odtoal -odds trrtato ad dvlod a gral aramtr lass of th sral dd modls that rmts th lklhood basd margal rgrsso aalss of bar rsos data. El-Sad t al. [ trodud a altratv quadrat otal form AQEF th bvart as to mak th quadrat otal form whh s rstd b Zhao ad Prt [ mor ralst trms of dfg th udrlg sudo-lklhood futo b modfg th ormalzg rodur th bvarat as. I ths ar th maor work s modlg th GLM ad th AQEF rodurs assoatd wth o ovarat. Ths rodur a b tdd for mor tha o ovarat wthout a loss of gralt MCullagh ad Nldr [6. Th gralzato of th assoato aramtrs a b

2 do wth sfd lk futos for th trvarat orrlatd bar rsoss varabls. H th bvarat AQEF wll b tdd to th trvarat as sml form also b modfg th ormalzg ross. Also to omar wth th AQEF rodur for th -odds ratos as masurs of assoato ad th rgrsso aramtrs w wll us th GLM rodur usg th sral dd ad th frst-ordr Markov modl. Sto rsts th trvarat Broull dstrbuto aml th ot robablts ad th -odds. Stos rsts th trvarat AQEF rodur ad sto rsts th trvarat GLM rodur usg sral dd rtra. Eah sto otas th stmato of atural aramtrs th stmato of rgrsso aramtrs th tstg hothss th goodss of ft of th modl ad ovr-dsrso rort. Fall Sto 5 dslas th umral amls usg R rogram for th smulato stud ad a alato to a o roblm usg th Huua Rags Data. Trvarat Broull Dstrbuto I ths sto w wll rst th ot robablt futo ad th -lklhood futo for thr orrlatd bar varabls havg th Broull dstrbuto. I ths as w a td for th trvarat Broull dstrbuto. If ad hav a Broull margals ah of whh taks th valu of thr or th t must b that has ol ght ossbl valus ad. For th trvarat bar data wth orrlatd bar outoms th ot mass futo s f Ahmd Mohamd Mohamd El-Sad... whr s Pr s th ot robablts.. ar Th orrsodg -lklhood futo of th ot mass futo for obsrvatos s.... Lt us df th followg aramtrs usg th rlatoshs btw th taos ad both of th margals th ot robablts ad th ovaras σ as: & E σ s E E E E σ E K E[ E q σ q q. Not that: f σ th th varabls ad ar ddt also f σ th th varabls ad ar ddt ad fall f σ th th varabls ad ar ddt. For aml usg th tao rort th bvartat as w hav: E [ σ E E E Th Tugls [8 usd ths rort to rst th ot robablts for th thr orrlatd bar varabls as: q q q q σ q σ q σ K q q q q q q q q q σ q σ σ q σ σ q σ σ q σ σ σ q σ q σ σ σ q σ q σ q σ σ σ σ σ K K K K K K K Th t stos la th aramtrs stmato ad tst rodurs for th AQEF ad th GLM rodurs th trvarat as as followg: Trvarat AQEF Produr I ths sto w wll td th bvarat Altratv Quadrat Eotal form AQEF whh s roosd b El-Sad t al. [ to th trvarat as. Ths futo rformulats th ot mass futo sml form. So th ot mass futo for th thr orrlatd bar varabls ad a b wrtt th altratv quadrat otal form as:

3 MODELING TRIVARIATE CORRELATED BINAR DATA f [ 5 [ [ whr ar th atural aramtrs. Th assoatd aramtrs th futo 5 a b wrtt as: P P P P P P P P P P P P P P P P P P Pr Pr Pr [. Pr Pr Pr If Pr Pr ths mas that ad ar odtoall ddt [ for gv. Ths a b wrtt as ρ. ρ ρ whr σ k ρ k k q q k k [ [ [ [ [. 8 As show from th rvous quatos 8 th ot robablts a b obtad asl rathr tha Tugls [8 th th quatos. Th t substo rsts th aramtrs stmato of th AQEF rodurs as follows:. Natural Paramtrs Estmato Usg th ot mass futo 5 th lklhood futo for obsrvatos a b wrtt as 9 [ ar th orrlato offts [8. To obta th ormalzg trm th ot futo 5 w a us th robablt ostrat: f. 6 I ths as th ormalzg trm a b obtad as Whr th ormalzg trm s dfd as show 7. Th frst drvatvs for th -lklhood futo 9 wth rst to ad ad ut t qual to zro ar 7 Smlfg th rsults of Tugls [8 for th ot robablts w a us th ot futo 5 to obta th ot robablts as:

4 Ahmd Mohamd Mohamd El-Sad. Solvg th stmatg quatos umrall w hav th stmats of atural aramtrs ad. Th w a us ths stmats to obta th stmats of th ot robablts th quatos 8. Th umbr of atural aramtrs ths rodur AQEF ar aramtrs.. Rgrsso Paramtrs Estmato W a us th t lk futos whh ar usd to trasform th atural aramtrs to th rgrsso aramtrs to sf th rgrsso modl as a futo of rgrsso aramtrs as followg:. Th th ot futo 5 usg th rgrsso aramtrs s bom: [ f

5 MODELING TRIVARIATE CORRELATED BINAR DATA 5 whr. Cosqutl th -lklhood futo for obsrvatos a b rssd as: [. Th frst drvatv for th -lklhood futo wth rst to ad ad uttg ths qual to zro rstvl w hav stmato quatos: Solvg th quatos umrall w gt th vtors of stmats ad. Th usg th quatos w hav th stmats of atural aramtrs ad. Th umbr of rgrsso aramtrs ths rodur AQEF ar aramtrs.. Tstg Hothss for Rgrsso Paramtrs Th Lklhood rato tst LRT a b usd to tst th rgrsso aramtrs. Aromatl LRT followg Ch-squar dstrbuto wth o dgr of frdom. W wll us th LRT to tst th ull hothss : H agast th altratv hothss : H. Th LRT a b wrtt as:

6 Ahmd Mohamd Mohamd El-Sad 6 [ χ : LRT 5 Also w a us th LRT to tst th ull hothss : H agast th altratv hothss : H. Th LRT a b wrtt as: [ χ : LRT 6 Smlarl w a us th LRT to tst th ull hothss : H agast th altratv hothss : H. Th LRT a b wrtt as: [ χ : LRT 7 Fall w a us th LRT to tst th ull hothss : H agast th altratv hothss : H. Th LRT a b wrtt as: [ χ : LRT 8 Th stmats of udr H a b obtad b solvg th quatos umrall.. Goodss of Ft of Modl Th dva a b usd to dtrm th goodss of ft of th modl. W a df th dva futo as: [ D χ : 9 whr s th umbr of stmatd rgrsso aramtrs [ s th -lklhood futo as a futo of th atural aramtrs s th ormalzg trm as a futto of atural aramtrs [ s th lklhood futo as a futo of. ad s th ormalzg trm as a futto of bar data..5 Ovr-Dsrso Crtra Th ovr-dsrso s had wh Var > Var µ. So th ovr-dsrso aramtr φ a b obtad from th rlato µ φvar Var. Th stmato of dsrso aramtr φ a b usd as a good masur for th ovr-dsrso rtra. So lt us df: Σ Var Cov Cov Cov Var Cov Cov Cov Var

7 MODELING TRIVARIATE CORRELATED BINAR DATA 7 Th quatt Σ follows th o-tral χ dstrbuto. Udr dd th stmator of dsrso aramtr φ s φ Var whr s th umbr of stmatd rgrsso aramtrs ad ar th stmatd margals. Th valu of φ s los to for th Broull data ma rflt abs of ovr-dsrso. Also w a us th sald dva futo D Sald D. φ as a masur of th goodss of ft of th modl. Th lowr valu s good ad surl t s bttr tha th dva futo 9 ad both of thm quals wh φ. Trvarat GLM Produr I ths sto w wll us th sral dd rort usg th frst-ordr Markov modl. Aordg to th odtoal s-odds trrtato of Hagrt ad Zgr [ Hagrt [. Th odtoal robablt of gv that s: Pr [ [ Also th odtoal robablt of gv that s: Pr [ [ Usg th quatos ad th followg sral dd rlatosh: f P P P w a obta th ot mass futo for th orrlatd bar varabls ad th otal faml form as: f [ [ 6 [ [ [ [ [. Natural Paramtrs Estmato I ths sto w wll rst th stmato of aramtrs of th trvarat Broull modl. For obsrvatos w a gt th -lklhood futo as: [ [ [ [ [ [ [ Takg th frst ordr drvatvs for 6 wth rst to ad ad uttg qual to zro w hav th stmatg quatos: 5

8 Ahmd Mohamd Mohamd El-Sad 8. 7 Solvg th stmatg quatos 7 umrall w hav th stmats ad. Th umbr of atural aramtrs ths rodur GLM ar aramtrs.. Rgrsso Paramtrs Estmato W d to stud th fft of ovarats o th ot futo 5 whh a b rssd as [ [ [ [ [ [ [ f 8 whr th atural aramtrs ad ad th rgrsso aramtrs ad ar dfd as show th quatos. For obsrvatos w a gt th -lklhood futo as [ [ [ [ [ [ [ 9

9 MODELING TRIVARIATE CORRELATED BINAR DATA 9 Takg th frst drvatvs for 9 wth rst to ad ad uttg ths qual to zro w hav Solvg th quatos umrall w gt th stmats ad. Th umbr of rgrsso aramtrs ths rodur GLM ar aramtrs.. Tstg Hothss for Rgrsso Paramtrs W a us th LRT to tst th ull hothss : H agast th altratv hothss : H. Th LRT tst a b wrtt as: [ χ : LRT Also w a us th LRT to tst th ull hothss : H agast th altratv hothss. : H Th LRT tst a b wrtt as: [ χ : LRT Th stmats udr H a b obtad b solvg th quatos umrall.. Goodss of Ft of Modl Th dva a b usd to dtrm th goodss of ft of th modl. So w a df th dva futo as [ D χ : whr [ [ [ [ [ [ [ ad

10 Ahmd Mohamd Mohamd El-Sad [. [ [ Also th sald dva as a goodss of ft of th modl a b usd usg th quato..5 Ovr-Dsrso Crtra Th stmator of dsrso aramtr φ a b usd as a good masur of th ovr-dsrso as show. I th t sto w wll us umral amls to la th dffrs btw th AQEF ad GLM rodurs usg th R rogram for th smulato ad alato studs. 5 Numral Eamls I ths sto w hav two substos th frst o las th smulato stud usg th grato of multvarat bar data ad th sod o dmostrats th alato stud usg th Huua Rags Data o a oal fld [. 5. Smulato Stud Tabl. Rsults of th AQEF ad GLM rodurs [ [ [ [ I th smulato stud w us th barsmclf akag of th R rogram to grat th multvarat bar data wth hagabl orrlato matr wth aramtr valu ρ.5 ad th margals..... Th frst thr olums from th gratd data ar sfd to th orrlatd bar rsoss ad. Th fourth olums s sfd to th laator varabl X. I ths stud w wll us larg saml sz 5. Th stmats Tabl ar obtad for th GLM ad AQEF rodurs usg th BBakag of R rogram [9. Tabl las th rsults for th AQEF ad GLM rodurs as follows: H th LRTs wll b omard wth χ So w a summarz th rsults from Tabl as followg: Estmat AQEF GLM Estmat AQEF GLM φ Sald Dva Log lklhood Valu LRT H : LRT H : LRT H : LRT H :.

11 MODELING TRIVARIATE CORRELATED BINAR DATA For th AQEF rodur: Smlarl for th GLM rodur w hav th rgrsso modl Th LRTs dmostrat sgfat assoato btw th orrlatd bar arws varabls assoatd wth laator varabl X. Howvr thr s o sgfat assoato btw all th orrlatd bar varabls. Ths dats that thr s a ar-ws frst-ordr dd btw th ars of bar varabls but th odds ratos for thr Broull varabls dmostrats that thr s o sod ordr assoato amog th thr Broull outom varabls. Th stmat of dsrso aramtr rflts th ovr-dsrso as φ.5 >. Th sald dva rflts th goodss of ft of th modl [Sald dva 8.76 < χ For th GLM rodur: Th LRTs dmostrat sgfat assoato btw th bar varabls ad ad sgfat assoato btw th bar varabls ad both ar assoatd wth laator X. Th stmat of dsrso aramtr rflts th ovr-dsrso as φ.9 >. Th sald dva also rflts th goodss of ft of th modl [ Sald dva.5 < χ Th rgrsso modls ar show blow as follows: For th AQEF rodur w hav th rgrsso modl ø ø 5. Alato Stud Sour: R-rogram Dr Nl Mthll Uvrst of Auklad ad [. Ths data wr olltd from th Huua Rags a small forst southr Auklad Nw Zalad. At 9 sts th forst th rs/abs of 7 lat ss was rordd as wll as th alttud. Eah st was of ara sz m. Th Huua Rags Data fram has 9 rows ad 8 olums. Th Alttud rrsts th th otuous ddt olum ad th ada btaw ad k olums ar orrlatd bar rsoss rs abs for th 7 lat ss. For ths data w us th olums ada btaw ad k as th ddt orrlatd bar varabls ad rstvl. O th othr had w wll us th olum alttud mtrs abov sa lvl as th otuous laator varabl X. Th stmats of th rgrsso aramtrs both th rodurs as lad Tabl a b obtad b solvg th stmatg quatos usg th BB-akag R rogram [9. Tabl las th rsults for th AQEF ad GLM rodurs as follows: H th LRT s wll b omard wth χ So as w obsrv from Tabl w hav th stmats of rgrsso aramtrs ad th tsts whh ar basd o th Huua Rags Data o oal obsrvatos. For th AQEF rodur:

12 Ahmd Mohamd Mohamd El-Sad Tabl. Rsults of th AQEF ad GLM rodurs Estmat AQEF GLM Estmat AQEF GLM φ.8.8 Th LRT s dmostrat sgfat assoato btw th bar varabls ad also o sgfat assoato btw th bar varabls ad both ar assoatd wth laator varabl X. It s also vdt that thr s sgfat assoato btw th bar varabls ad also sgfat assoato btw th bar varabls ad assoatd wth laator varabl X. Th stmat of dsrso aramtr rflts th ovr-dsrso as φ.8 >. Th sald dva rflts th goodss of ft of th modl [ Sald dva < χ. For th GLM rodur: Sald Dva Log lklhood Valu LRT H : LRT H : LRT H : LRT H :.99 - Th LRT s dmostrats sgfat assoato btw th orrlatd bar varabls ad ad sgfat assoato btw th orrlatd bar varabls ad both ar assoatd wth laator varabl X. Th stmat of dsrso aramtr rflts th ovr-dsrso as φ.8 >. Th sald dva also rflts th goodss of ft of th modl [ Sald dva < χ Th rgrsso modls a b show as follows: For th AQEF rodur w hav th rgrsso modl: Smlarl for th GLM rodur w hav th rgrsso modl:

13 7 MODELING TRIVARIATE CORRELATED BINAR DATA data. Bomtrs [5 Islam M. A. Chowdhur R. I. ad Brollas L.. A bvarat bar modl for tstg dd outoms. Bullt of th Malasa Mathmatal Ss Sot [6 MCullagh P. ad Nldr J. A Gralzd.6. lar modls sod dto. Chama ad Hall Lodo..9.9 Akowldgmt.6.7 Ma thaks for all M rofssors. REFERENCES [ Agrst A.. Catgoral data aalss sod dto. Wl Nw ork. [ El-Sad A. M. Islam M. A. ad Alzad A. A.. Estmato ad tst of masurs of assoato for orrlatd bar data. Bullt of th Malasa Mathmatal Ss Sot [ Hagrt P. J. ad Zgr S. L.. Margalzd mult-lvl modls ad lklhood fr wth dsusso. Statstal S 5-6. [ Hagrt P. J.. Margalzd trasto modls ad lklhood fr for tudal atgoral [7 Qaqsh B. F.. A faml of multvarat bar dstrbutos for smulatg orrlatd bar varabls wth sfd margal mas ad orrlatos. Bomtrka [8 Tugls J. L. 99. Som rrstatos of th multvarat Broull ad bomal dstrbutos. Joural of Multvarat Aalss [9 Varadha R. ad Glbrt P. D. 9. BB: A R akag for solvg a larg sstm of olar quatos ad for otmzg a hgh-dmsoal olar obtv futo. Joural of Statstal Softwar -6. [ T. W. 8. Th VGAM akag. R Nws [ T. W.. Th VGAM akag for atgoral data aalss. Joural of Statstal Softwar -. [ Zhao L. P. ad Prt R. L. 99. Corrlatd bar rgrsso usg a gralzd quadrat modl. Bomtrka