Hierarchical Bayes Analysis of Longitudinal Poisson Count Data

2 002 , V ol. 13, N o.2 p p . 22 7～234

Hie rarc h ic al B ay e s A n aly s i s o f L on g itu din al P oi s s on Coun t D at a

D al H o K im^{1 )} Im H e e S hin^{2 )} In S u n Ch oi^{3 )}

A b s tra c t

In th is paper , w e con sider hierarchical Bay e s g en er alized lin ear m odels for t h e an aly sis of lon g it u din al coun t dat a . S p ecifically w e int odu ce t h e hier ar chical Bay es r an dom effect s m odels . W e dis cu s s im plem ent at ion of th e B ay es pr ocedur es v ia M ark ov ch ain M on t e Carlo (M CM C ) int eg ration t echn iqu e s . T h e hierarchical Bay e m eth od is illu str at ed w it h a r eal dat a set an d is com pared w ith ot h er st at ist ical m et h ods .

K e y w ord s : R an dom effect s m odels , hier ar chical B ay es , lon g it u din al cou nt dat a , Gibb s s am plin g .

1 . In tro du c ti on

T h er e is con sider ably r ecent st atist ical in t ere st in t h e an aly sis of lon g it u din al dat a . Un lik e cr os s - sect ion al st u die s , w h er e sin gle ou t com e is m ea sur ed for ea ch in div idu al, lon git u din al st u dies in v olv e repeat ed m ea sur em en t s of in div idu als or subj ect s th rou g h t im e. T h is int r odu ce s a n at ur al corr elation am on g th e m ea sur em en t s w it hin a su bj ect w hich m u st b e t ak en int o accou nt for any st atist ical an aly sis . A lso, lon g it u din al st u die s can det ect ch an g e s ov er t im e w ith in in div idu als w h ich cros s - s ect ion al st u die s can n ot .

Dig g le, Lian g an d Zeg er (1994 ) prov ided a com pr eh en siv e fr equ en t ist an aly sis of lon git u din al dat a . T h ey ex t en ded t h e g en eralized lin ear m odel (GLM ) con cept s t o t h e an aly sis of lon gitu din al dat a . In part icu lar t h ey h av e con sider ed m ar gin al,

1. A s s ociat e Pr ofes s or , Departm ent of St at istics , Kyungpook National Univ er sit y , T aegu 702- 701, Kor ea

E - m ail : dalkim @knu .ac.kr

2. A s s ist ant Pr ofes s or , Divis ion of M edical St atis tics , Cathooolic Univ er sity of T aegu , T aegu 705- 718, Kor ea.

3. Lect ur er , Departm ent of St atistics , Kyungpook National Univer sity , T egu 702- 701, Kor ea.

r an dom effect s an d tr an sition m odels , an d h av e pr ov ided a v ariety of fr equ ent ist an aly ses for all t h es e m odels .

F or t h e r an dom effect s m odels , W aclaw iw an d Lian g (1994 ) con sider ed em pirical B ay es (EB ) appr oach an d Gh osh , Kim an d M ait i (1997 ) h av e con sider ed h ier ar ch ical B ay es (H B ) appr oach t o t h e an aly sis of lon git u din al bin ary dat a b a sed on g en er alized lin ear m ix ed m odels (GLM M ). Our obj ect iv e is t o in tr odu ce HB GLM M for t h e an aly sis of lon g it u din al cou nt dat a .

T h e r an dom effect s m odel in tr odu ces corr elation s w it hin a subj ect ov er tim e.

A s sign in g distribut ion s t o th e prior h y perparam et er s a s don e in a H B an aly sis also b uild s dep en den ce b et w een subj ect s acros s different t im e period s . T his en ables on e t o "b orrow st r en g th " fr om ot h er subj ect s a s w ell a s acros s t im e .

T h e out lin e of t h e r em ain in g sect ion s is a s follow . S ection 2 int r odu ces a g en er al HB r an dom effect s m odel. S ect ion 3 prov ides t h e H B an aly sis of a r eal dat a set an d com par es w it h th e ot h er an aly s es u sin g oth er com parable m et h ods .

2 . B ay e s i an Ge n e raliz e d Lin e ar M ix e d M o del s

S u ppose th at Y_ij den ot es th e r esp on se of t h e ith su bj ect (or in div idu al) at t h e j t h t im e ( j = 1 , , n_i ; i = 1 , , m ) . W e con sider t h e follow in g H B m odels :

(I) f ( y_ij| _ij) = ex p [ {y_{ij ij}- ( _ij)}/ _ij]h ( y_ij; _ij)

w h er e _ij( >0) ar e a s su m ed t o b e kn ow n , _ij= log _ij, _ij= x _ij^T + d _ij^T u_i+ log t_ij, , an d d_ij is su b set of x_ij an d u_i= ( u_i1, u_i2, , u_iq)^T. (II) ui

iid N ( 0 , G ) .

(III) an d G are m ar g in ally in depen den t w it h U n iform ( R^p) an d G In v ers e Wish art ( S , k ) .

N ot e t h at an offset , log tij, w a s int r odu ced t o t ak e accou nt of differ en t int erv al len gt h s in th e log - lin ear m odel. H ere t h e In v er se W ish art dist ribu tion h a s t h e pr ob ab ilit y den sity fun ct ion (pdf) of th e form

f ( G ) |G |- ( q + k ) / 2

ex p(- t ra ce ( SG^{- 1}) / 2).

T h e GLM M w a s als o con sider ed in Br eslow an d Clay t on (1993 ) a s w ell a s Zeg er an d K arim (1991) in a r elat ed bu t differ ent cont ex t .

T h e HB m et h od is im plem ent ed v ia t h e M CM C int eg ration t ech niqu e. (cf.

Gelfan d an d S m ith (1990)). T h is requ ir es g en er at ion of s am ples fr om t h e follow in g fu ll con dition als .

( i) [ | , G , y ] N(^{i j ( ( dij}ij^{) - log tij)}

T G d_ij xij,

i j

xij x ij T

d _ij^T G d_ij

- 1

( ii) p ( _ij| _{k l( k i , l j )}, G , , y ) ex p

{

}

( iii) p ( G | , , y ) _{i j}( d _ij^T G d_ij) ^{- 1/ 2}ex p

{

}

| G |- ( q + k ) / 2

ex p{- t ra ce ( S G^{- 1}) / 2}

It is ea sy t o g en er at e sam ples fr om t h e full con dition als giv en in (i ). H ow ev er , (ii ) an d (iii) ar e n ot a st an dar d den sit y fr om w hich on e can g en er at e sam ples ea sily . T h is difficu lt y is ov er com e by em ploy in g t h e M et r op olis - H a stin g s alg orit hm . A n alt ern at iv e appr oach t o g en er at e sam ple form (ii) w ou ld b e t o u s e t h e a daptiv e r ej ection sam plin g (A RS ) of Gilk s an d W ild (1992) sin ce ( _ij| ) ar e all log - con cav e . But t h e latt er is n ot pu r su ed h ere .

3 . D at a A n aly s i s

W e pr ov ide in th is section a H B an aly sis of a r eal dat a set giv en T h all an d V ail (1990) an d by Br eslow an d Clay t on (1993 ). F or each p at ien t , t h e n um b er of epilept ic s eizu r es w a s r ecor ded du rin g a b a s elin e period of eig ht w eek s . P atien t s w er e t h en r an dom ized t o t r eat m ent w it h t h e ant i- epilept ic dru g pr og ab ide, or t o placeb o in a ddit ion t o st an dar d ch em ot h er apy . T h e nu m b er of seizur es w a s t h en r ecor ded in four con s ecu tiv e t w o - w eek int erv als . Let Y_ij den ot e t h e P ois son cou nt s r espon s e fr om 0 t o infin it e corr espon din g t o occur th e epilept ic s eizu r es . T h e obj ectiv e is w h et h er or n ot t h e pr og abide r edu ces th e r at e of epilept ic seizur es . T h e su cces siv e s eizu r e cou nt s for 59 pat ient s . Cov ariat es ar e tr eatm en t (0, 1), 8 - w eek b a selin e s eizu r e cou nt s , an d ag e in y ear s .

W e fir st con sider th e follow in g m odels . M odel 1 is a log - lin ear m odel w it h a r an dom in t ercept a s follow s .

( I) log E ( Y_ij| u_i) = ₀+ ₁x_{ij 1}+ ₂x_ij2+ ₃x_{ij 1}x_{ij 2}+ u_i+ log ( t_ij) i = 1 , , 59 , j = 0 , , 4

w h er e x_{ij 1}=

{

{

0 if j = 0 ( II) ui

iid N ( 0 , ru^{- 1})

( III) = ( ₀, ₁, ₂, ₃)^T an d r_u ar e m ar gin ally in depen dent w it h U n iform ( R⁴) an d r_u Gam m a ( a / 2 , b / 2) .

T h en t h e j oin t p df of , , an d r_u g iv en y_ij is g iv en by f ( , , ru |yij)

i j [ ex p {yij ij- ( ij)}]

i j r_u^{1/ 2}ex p [ - r_u

2 ( _ij- log t_ij- x _ij^T )²]

i j r_u^{b/ 2}ex p { - ( r_u+ a) / 2}.

T o im plem ent th e M CM C in t egr at ion t echn iqu es in M odel 1, w e n eed t o g en er at e s am ples from th e fu ll con dit ion al dist ribu tion s lik e b elow :

( i) [ ru| , , y ] Gam m a ( {a + (

i j ij-

i j log tij-

i j x ij

T )²}/ 2 , { u_i+ b }/ 2)

( ii) [ | , y , r_u] N(^ru i j ( _ij- log t_ij) x _ij^T, r_u^{- 1}

i j ( x_ij x _ij^T)^{- 1})

( iii) p ( _ij| _{k l}( ( k , l) ( i , j ) ) , , r_u, y ) ex p {y_{ij ij}- ( _ij)

- r_u ( _ij- log t_ij- x_ij^T ) / 2}

T o com plet e th e hier ar chical M odel 1, w e a s sig n a Uniform ( R⁴) prior for

= ( ₀, ₁, ₂, ₃)^T, an d Gam m a ( 0 . 0005 / 2 , 0 . 0005 / 2) prior for r_u.

In M odel 2, w e add a secon d r an dom effect for t h e pr e/ post - tr eatm en t in dicat or ( x₂) a s follow s .

log E ( Y_ij|u_i) = ₀+ ₁x_{ij 1}+ ₂x_{ij 2}+ ₃x_{ij 1}x_ij2+ u_i1+ x_ij2u_i2+ log ( t_ij)

w h er e ui= ( ui1, ui2)^T is a s su m ed t o follow a biv ariat e n orm al dist rib ut ion w it h m ean 0 an d v arian ce m at rix G w it h elem ent s (^GG2 1^{11 G}G₂₂¹²). M or eov er , t h e h y perprior s ar e a s su m ed th at = ( ₀, ₁, ₂, ₃)^T an d G ar e m arg in ally in dep en den t w it h U n iform ( R⁴) an d G In v ers Wish art ( S , k) . Den ot e dij= ( 1 , xij2)^T. T h e in clu sion of ui2 allow s u s t o addr es s t h e con cern t h at t h er e m igh t b e h et erog en eity am on g subj ect s in th e r at io of th e ex pect ed seizu r e coun t s b efor e an d aft er th e r an dom izat ion . T h e degr ee of h et er og en eit y can b e m ea su r ed b y t h e m ag nit u de of G₂₂, t h e v arian ce of u_i2.

T o im plem ent t h e M CM C int eg ration t ech niqu es in M odel 2, w e g en er at e s am ples from th e fu ll con dit ion al dist ribu tion s a s follow s :

( i) [ | , G , y ] N

i j

( ij) - log tij

d _ij^T G d_ij x_ij,

i j

xij x ij T

d _ij^TG d_ij

- 1

( ii) p ( _ij| _{k l( k i , l j )}, G , , y ) ex p {y_{ij ij}- ( _ij)

- ( ( _ij) - x _ij^T - log t_ij)²/ {2 d _ij^TG d_ij}}

( iii) p ( G₁₁| G₁₂, G₂₂, , , y )

i j ( G₁₁+ 2x_ij2G₁₂+ x²_{ij 2}G₂₂) ^{- 1/ 2} ex p{^{- ( (2 ( G11ij) -+ 2xxij 2ijGT12+ x- log t2ij2G22ij))2} }

|G11G22- G²12| - ( q + k ) / 2

ex p{^{- 21 GG1111G22+ G- G22212} }

( iv) p ( G₁₂| G₁₁, G₂₂, , , y )

i j ( G₁₁+ 2x_ij2G₁₂+ x²_ij2G₂₂) ^{- 1/ 2} ex p{^{- ( (2 ( Gij11) -+ 2xxij 2ijGT12+ x- log t2ij 2G22ij))2} }

|G₁₁G₂₂- G²₁₂|- ( q + k) / 2

ex p{^{- 21 GG}11G¹¹₂₂^{+ G}- G²²²₁₂ }

( v) p ( G22| G12, G22, , , y )

i j ( G11+ 2xij2G12+ x²ij2G22) ^{- 1/ 2} ex p{^{- ( (2 ( Gij11) -+ 2xxij2ijGT12+ x- log t2ij 2G22ij))2} }

| G₁₁G₂₂- G²₁₂| - ( q + k ) / 2

ex p{^{- 21 GG}11G¹¹22^{+ G}- G²²²12 }

T o com plet e th e hier ar chical M odel 2, w e a s sig n a Uniform ( R⁴) prior for

= ( ₀, ₁, ₂, ₃)^T an d In v er s W ish art ((^{0 . 0005}0 0 . 0005⁰ ), 7)^{prior for G .}

In ou r set t in g G_ij, i = 1 , 2 ; j = 1 , 2 is a p osit iv e real n um b er , b ut M - H alg orit hm h a s all r eal nu m b er spa ce, i.e. t h e r an g e for M - H alg orith m is

( - , ) . H en ce w e n eed t o m odify t h e M - H alg orith m lik e b ellow .

T able 1: E st im ate and st and ard er r or s (in parent heses ) for the pr og abide d at a w ith 49t h ob serv ation .

M odel 1 M odel 2

V ariable A M L H B_M H HB_BU GS A M L H B_M H HB_BU GS

Inter cept ( 0) ^1.0

(0.15)

1.3012 (0.3302)

1.5012 (0.1302)

1.1 (0.14)

1.0344 (0.2322)

1.0182 (0.1134) T reat ment ( 1) ^{- 0.023}

(0.20)

- 0.0225 (0.1502)

- 0.0325 (0.1120)

0.050 (0.18)

0.0574 (0.1134)

0.0636 (0.1526)

T ime ( 2) ^0.11

(0.047)

0.1102 (- 0.0417)

0.1623 (0.0547)

0.002 (0.11)

0.0030 (0.0838)

0.00189 (0.0875) T rt by time ( 3) ^{- 0.10}

(0.065)

- 0.0986 (0.0584)

- 0.1092 (0.0658)

- 0.31 (0.015)

- 0.2938 (0.01175)

- 0.2961 (0.2152) G11

0.62 (0.12)

0.6145 (0.1364)

0.6614 (0.2364)

0.51 (0.10)

0.5117 (0.1499)

0.5242 (0.1089)

G₁₂ - - - 0.054

(0.056)

0.0667 (0.0462)

0.0623 (0.0489)

G₂₂ - - - 0.24

(0.062

0.2189 (0.0761)

0.2395 (0.1034)

M odified M et ropolis - H a st in g A lg orit hm : S in ce G_ij ( i = 1 , 2 : j = 1 , 2) is a v ariable w it h r an g e in p osit iv e r eal lin e, w e can u se a tr an sform at ion su ch a s G'_ij= log G_ij , t o m ap ( 0 , ) in t o ( - , ) , t h en u s e t h e tr an sition k ern el an d apply in g of t h e M - H alg orit hm t o t h e den sit y of G'_ij. A ft er on e t r an sit ion of th e M etr opolis - H a stin g alg orith m is don e th en w e t r an sform G'ij b ack t o t h e origin al s cale m ean s of G_ij = ex p G'_ij.

F or b ot h m odels , a bu rn - in of 7500 it er ation s w a s follow ed by a furt h er 15000 it er ation . T ab le 1 is b a sed on th e com plet e dat a an d T able 3 is b a s ed on t h e dat a w it h ou t 49t h ob s erv at ion . T h ese r esu lt s ar e b a sed on Gibb a s am plin g w it h 5 ch ain an d 15000 replicat ion follow in g Gelm an an d Rubin (1992).

T h e BU GS pr ov ide s a declar ativ e lan g u a g e for str aigh tfor w ar d sp ecificat ion of st atist ical m odels b a s ed on th e a s su m ed gr aphical st ru ct ur e, alt h ou g h th er e are s om e r est riction s on th e cla s s of m odels t h at can b e an aly sed curr en tly . A com piler t h en proces ses t h e m odel an d dat a an d set s up t h e sam plin g dist ribu tion s r equ ir ed for t h e Gibb s sam plin g . F in ally , appr opriat e s am plin g alg orit hm s ar e im plem ent ed t o sim ulat e v alu e s of t h un kn ow n qu ant ities in th e m odel. T o im plem ent BU GS for b ot h M odel 1 an d M odel 2, w e a s sig n a N ( 0 , 10⁴I ) prior in st ead of a Uniform ( R⁴) prior for = ( 0, 1, 2, 3)^T. A lso b ot h m odels w ere fitt ed u sin g t h e appr ox im ay e m ax im u m lik elih ood (A M L ) alg orith m . S ee Dig g le , Lian g an d Zeg er (1994) for det ailed calcu lat ion s .

T able 2 : E st im at e and st andard err or s (in parent hes es ) for t he pr og abide dat a w ith out 49t h ob serv ation .

M odel 1 M odel 2

V ariable A M L HB_M H HB_BU GS A M L H B_M H HB_BU GS

Inter cept ( 0) ^1.0

(0.14)

1.0012 (0.1302)

1.2022 (0.1252)

1.1 (0.13)

1.1222 (0.1532)

1.0843 (0.1341) T reat ment ( 1) ^{- 0.009}

(0.19)

- 0.0121 (0.1975)

- 0.0134 (0.0455)

- 0.029 (0.19)

- 0.0194 (0.1248)

- 0.0342 (0.0109)

T ime ( 2) ^0.11

(0.047)

0.1100 (0.0724)

0.1346 (0.2746)

0.10 (0.11)

0.0135 (0.0854)

0.0531 (0.0567) T rt by time ( 3) ^{- 0.30}

(0.070)

- 0.3181 (0.06243)

- 0.2897 (0.07423)

- 0.34 (0.15)

- 0.3326 (0.1141)

- 0.3453 (0.2133)

G₁₁ 0.53

(0.10)

0.5497 (0.2186)

0.6030 (0.2314)

0.46 (0.10)

0.5512 (0.1614)

0.5762 (0.0945) G12

0.014 (0.053)

0.0657 (0.0437)

0.0532 (0.0185) G22

0.22 (0.059)

0.2349 (0.0782)

0.2384 (0.0228)

In t h e r esult fr om T ab le 1 an d T able 2, t hr ee m eth od s are quit e com p ar able.

F or M odel 1, t h er e is m odest ev iden ce th at pr og ab ide is m or e effect iv e t h an th e placeb o in r edu cin g t h e occu rren ce of seizu r es ( ₃= - 0 . 0982 0 .0584 ). W ith p os sib le out lier pat ient nu m b er 49 delet ed , str on g er tr eatm en t effect is su g g est ed ( ₃= - 0 . 3 18 1 0 . 06242 ). F ocu sin g on th e r esu lt s of M odel 2 fit t ed t o th e com plet e dat a , subj ect s in t h e placeb o gr oup h av e ex pect ed seizur e r at e aft er t r eat m ent w h ich ar e estim at ed t o b e r ou g hly th e sam e a s b efor e t reatm en t ( ex p ( ₂) = ex p (0 . 003) = 1 . 003 ). F or th e prog abide gr ou p , th e s eizu re r at es ar e r edu ced aft er t r eat m ent by ab out 25.5 per cen t ( 1 - ex p (0 . 003 - 0 .293) = 0 . 255 ).

H en ce, th e t r eat m ent seem s t o h av e a m ode st effect . T h e estim at ed effect is

3= - 0 . 293 w it h a st an dar d error of.117. F in ally , if w e set a side of t h e m om ent p at ien t nu m b er 49, w h o h ad un u su ally h ig h s eizu re r at e s , ( ₃= - 0 . 3326 0 . 114 1).

T h e an aly sis w it h out p at ien t 49 is only ex plorat ory , an d is carried ou t in or der t o u n der st an d t his p atien t s ' s influ en ce on th seizur e coun t s an d perh ap s h a s special m edical problem .

R e f e re n c e s

1. Br eslow , N .E . an d Clay t on , D .G. (1993 ) A pprox im at e in g en eralized lin ear m ix ed m odel. J ournal of th e A m e rican S ta tis tical A s s ocia tion , 88, 9 - 25.

2. Gh osh , M ., Kim , D .H ., an d M ait i T . (1997 ). H ierarchical Bay esian an aly sis of lon gitu din al dat a . S ank hy a, S er. B , 59, 326 - 334.

3. Dig gle, P .J ., Lian g , K .Y . an d Zeg er , S .L . (1994 ). A naly s is of L ong it ud inal D a ta . Ox for d Univ er sity P r es s , Ox for d.

4. Gelm an , A . an d Rub in , D .B . (1992). Infer en ce fr om it er at iv e sim u lation u sin g m u lt iple s equ en ces (w it h dis cu s sion ). S ta t is tical S cience , 7, 457 - 511.

5. Gilk s , W .R . an d W ild , P . (1992). A dapt iv e rej ect ion sam plin g for Gibb s s am plin g . A pplied S t at istics , 41, 337 - 348.

6. Gelfan , A .E . an d S m it h , A .F .M . (1990). S am plin g - b a s ed appr oach es t o calcu lat in g m ar g in al den sit ies . J ournal of the A m er ican S ta t is t ical A s s ocia tion , 85, 398 - 409.

7. T h all, P .F . an d V ail, S .C. (1990). S om e cov arian ce m odels for lon gitu din al coun t dat a w it h ov er disp er sion . B iom e tr ics . 46, 657 - 671.

8. W alaw iw , M .A . an d Lian g , K .Y . (1994 ). Em pirical Bay es est im ation an d infer en ce for t h e r an dom effect s m odel w ith bin ary r esp on se. S ta t is tics in M ed icin e , 13, 542- 551.

9. Zeg er , S .L. an d K arim , M .R (1991). Gen er alized lin ear m odels w it h r an dom effect s : A Gibb s sam plin g appr oa ch . J ournal of the A m erican S ta tis t ical A s s ocia tion , 86, 79 - 86.

[ 2002년 3월 접수, 2002년 10월 채택 ]