Modelling Heterogeneous Dispersion in Marginal Models for Longitudinal Proportional Data

Transcription

1 Bmetrcal Jurnal 46 (24) 5, DOI:.2/bmj.252 Mdellng Hetergeneus Dspersn n Margnal Mdels fr Lngtudnal Prprtnal Data Peter X.-K. Sng*;, Zhengu Qu 2, and Mng Tan 3 Department f Mathematcs and Statstcs, Yrk Unversty, Trnt, ON, Canada M3J P3 2 B.C. Research Insttute fr Chldren s and Wmen s Health, Vancuver, BC, Canada V5Z 4H4 3 Greenebaum Cancer Center N9E7, Unversty f Maryland, Baltmre, MD 22, USA Receved 24 July 2, revsed 22 Octber 23, accepted 5 June 24 Summary Cntnuus prprtnal data s cmmn n bmedcal research, e.g., the pre-pst therapy percent change n certan physlgcal and mlecular varables such as glmerular fltratn rate, certan gene expressn level, r telmere length. As shwn n (Sng and Tan, 2) such data requres methds beynd the cmmn generalsed lnear mdels. Hwever, the rgnal margnal smplex mdel f (Sng and Tan, 2) fr such lngtudnal cntnuus prprtnal data assumes a cnstant dspersn parameter. Ths assumptn f dspersn hmgenety s mpsed manly fr mathematcal cnvenence and may be vlated n sme stuatns. Fr example, the dspersn may vary n terms f drug treatment chrts r fllw-up tmes. Ths paper extends ther rgnal mdel s that the hetergenety f the dspersn parameter can be assessed and accunted fr n rder t cnduct a prper statstcal nference fr the mdel parameters. A smulatn study s gven t demnstrate that statstcal nference can be serusly affected by mstakenly assumng a varyng dspersn parameter t be cnstant n the applcatn f the avalable GEEs methd. In addtn, resdual analyss s develped fr checkng varus assumptns made n the mdellng prcess, e.g., assumptns n errr dstrbutn. The methds are llustrated wth the same eye surgery data n (Sng and Tan, 2) fr ease f cmparsn. Key wrds: Cntnuus prprtns; Generalsed lnear mdels; GEEs; Lngtudnal data; Resdual analyss; Smplex dstrbutn; Varyng dspersn. Intrductn The cncept f dspersn parameter s a famlar ne n generalsed lnear mdels (GLMs). The dspersn parameter f a nrmal dstrbutn s smply ts varance; and the dspersn parameter f Pssn dstrbutns s always equal t, whch s the rat f varance t mean and where the verdspersn ccurs when such a rat s larger than. Dspersn mdels (Jørgensen, 997), as an extensn f the GLMs, nclude dspersn parameters descrbng the dstrbutnal shape, whch s beynd what the lcatn r mean parameter alne can descrbe. The smplex dstrbutn f Barndrff-Nelsen and Jørgensen (99) fr the errr term represents a specal dspersn mdel, and s useful fr mdellng cntnuus prprtnal data. Based n ths dstrbutn, Sng and Tan (2) develped a margnal mdel fr lngtudnal cntnuus prprtnal data, whch was used t analyse an eye surgery data. Smlar t Lang and Zeger s margnal mdels e.g. Dggle et al. (22), Sng and Tan (2) assumed a cnstant dspersn n ther mdel and ther fcus s n mdellng the trend cmpnent. A techncal advantage by settng a cnstant dspersn parameter s that, as shwn n Lang and Zeger s GEE (986) apprach, regressn ceffcents can be separately estmated frm the dspersn parameter. Ths s because the GEE can factrse a cnstant dspersn ut the estmatng equatn. * Crrespndng authr: e-mal: sng@mathstat.yrku.ca # 24 WILEY-VCH Verlag GmbH & C. KGaA, Wenhem

2 Bmetrcal Jurnal 46 (24) 5 54 Hwever, n practce the assumptn f hmgeneus dspersn may be questnable. Fr example, the magntude f dspersn may vary acrss drug treatment chrts due t dfferent rates f dsease prgressn r ver dfferent fllw-up tmes due t dfferent envrnmental expsures. It s clear that the margnal pattern f a ppulatn depends nt nly n ts averaged trend but als n ts dspersn characterstcs, as descrbed by the dspersn mdels. Therefre, ncrpratng varyng dspersn n the mdellng prcess allws us t assess the hetergenety f dspersn and t develp a smultaneus nference fr the entre margnal mdels cncernng bth trend and dspersn cmpnents. Such an access t the prfle f the dspersn parameter s mprtant, as shwn n ur smulatn studes n Sectn 4, mstakenly assumng a varyng dspersn t be cnstant n the applcatn f GEE methd culd cause sme serus prblems n statstcal nference. Fr example, the asympttc nrmalty thery fr the estmatrs may n lnger be vald, and ths thery s crucal t test fr statstcal sgnfcance fr the effects f sme cvarates f nterest. In addtn, a prper estmatn fr the dspersn parameter s appealng, fr example, n resdual analyss, where a standardsatn fr resduals s usually taken t stablse ther varances. The cmputatn f standardsed resduals always asks fr an apprprate estmate f the dspersn parameter. In ths paper, we prpse a new margnal mdel that cnssts f three cmpnents t be mdeled: the ppulatn-averaged effects, the dspersn pattern, and the crrelatn. In the cntext f lngtudnal data analyss, the frst versn f generalsed estmatng equatn apprach, knwn as f GEE, s prpsed by Lang and Zeger (986) and later extended by Prentce and Zha (99) t nclude a set f estmatng equatns n crrelatn parameters, referred t GEE2 n the lterature. As a matter f fact, estmatng a varyng dspersn parameter can be easly ncrprated wth the GEE2 usng the mean-varance relatnshp f the classcal GLMs r the expnental dspersn famly dstrbutns. Hwever, the mean-varance relatnshp s n lnger vald fr the smplex dstrbutn, because t s nt an expnental dspersn famly dstrbutn. Therefre, n ths paper, we suggest t add anther set f estmatng equatns t deal wth the dspersn cmpnent thrugh a certan mment prperty dfferent frm the mean-varance relatnshp. The resultng estmatng equatns extend the currently ppular GEE2, althugh t s stll called as GEE2 n the present paper. Mdellng dspersn parameter has been cnsdered by many authrs fr dfferent mdels n the lterature. Amng thers, Smyth (989) dscussed generalsed lnear mdels wth varyng dspersn fr crss-sectnal data, and Artes and Jørgensen (2) prpsed a mdel fr the ndex parameter f dspersn mdels, attemptng t attack ths prblem wth an underlyng applcatn clsely related t vn Mses dstrbutn fr lngtudnal crcular data. We fund ther methd dd nt wrk well fr the smplex dstrbutn. Pak (992) prpsed an estmatn prcedure that extends Lang and Zeger s GEEs by allwng bservatns frm dstrbutns wth dfferent dspersn parameters. Hwever, Pak s prcedure s nt applcable fr the smplex dstrbutn, because there s n clsed frm expressn fr the varance f the dstrbutn. By utlsng a certan mment prperty f the smplex dstrbutn, we cme up wth a dfferent slutn frm thse gven by Artes and Jørgensen (2) and Pak (992). In fact, because f dfferent perspectves and mdels, ur estmatng equatn fr the dspersn parameter f the smplex dstrbutn s smpler and numercally mre effcent than thers. The rest f the paper s rgansed as fllws. Sectn 2 presents dspersn margnal mdels wth varyng dspersn. An extended GEE2 s presented n Sectn 3, and Sectn 4 gves a smulatn study that demnstrates the mprtance f mdellng the dspersn parameter t cnduct a prper statstcal analyss n the presence f hetergeneus dspersn. Sectn 5 dscusses mdel dagnstcs thrugh resdual analyss. The prpsed methds are appled t re-analyse the eye surgery data n Sectn 6. Fnally we cnclude wth sme remarks. 2 Margnal Mdels T develp margnal smplex mdels fr lngtudnal cntnuus prprtnal data wth varyng dspersn, frst let y j, j ¼ ;...; n be the sequence f bserved repeated measurements n the th f m subjects, and t j, j ¼ ;...; n, be the sequence f crrespndng tmes n whch the measurements are taken n each subject. Asscated wth each y j are the values, x jk, k ¼ ;...; p, fp cvarates r expla- # 24 WILEY-VCH Verlag GmbH & C. KGaA, Wenhem

3 542 P. X.-K. Sng et al.: Mdellng dspersn natry varables. We assume that y j are realsatns f randm varables Y j whch fllw smplex dstrbutns Y j S ðm j ; s 2 j Þ, where m j 2ð; Þ are the mean parameters and s2 j > are the dspersn parameters, and bth may be specfed as functns f cvarates. The densty functn f the smplex dstrbutn s, suppressng ndces, gven by h =2 pðy; m; s 2 Þ¼ 2ps 2 fyð yþg 3 exp dðy; mþ ; y 2ð; Þ ; 2s2 where d s the unt devance, ðy mþ 2 dðy; mþ ¼ yð yþ m 2 ð mþ 2 ; and ts unt varance functn s vðmþ ¼m 3 ð mþ 3. See (Jørgensen, 997) fr mre detals. Let Y ¼ðY ;...; Y n Þ > ; x j ¼ð; x j ;...; x jp Þ > : We assume that Y ;...; Y m are ndependent. A margnal smplex mdel cnssts f three cmpnents gven as fllws. The frst cmpnent s a mdel t descrbe the ppulatn-averaged effects, where the mean parameter m j depends n the tmevaryng cvarates x j va a generalsed lnear mdel f the frm hðm j Þ¼x > j b ðþ where h s a knwn lnk functn and b ¼ðb ;...; b p Þ > s the regressn ceffcents t be estmated. The lnk functn s usually chsen t be the lgt lnk functn that maps the untary nterval t ð ; Þ. The secnd cmpnent s a mdel t descrbe the pattern f dspersn parameter s 2 j as a functn f cvarates z j (maybe a subset f x j ), gven by gðs 2 j Þ¼z> j g ð2þ where g s a knwn lnk functn and g ¼ðg ;...; g r Þ > wth g crrespndng t the ntercept term. T express the dspersn as f a multplcatve frm, the lgarthm lnk functn s used t btan a lg-lnear mdel and hence s 2 j ¼ exp Qr ðz> j gþ¼ ðe g k Þ z jk ¼ e g Q r ðe g k Þ z jk : k¼ k¼ The thrd cmpnent s fr mdellng crrelatn structure. The crrelatn between Y j and Y k s a functn f the lcatn parameters and perhaps f addtnal parameters, a ¼ða ;...; a q Þ >, namely, crr ðy j ; Y k Þ¼qðm j ; m k ; aþ ð3þ where qðþ s a knwn functn. Varus types f crrelatn structures may be used fr the q functn. Amngst thers, three cmmnly used n the analyss f lngtudnal data are the exchangeable, AR() and m-dependence crrelatns. It s nted that the justfcatn fr a chce f a crrelatn structure s n general a dffcult task due t lttle nfrmatn ver tme avalable. Hwever the Lang and Zeger s GEE apprach fr cnsstent parameter estmatn enjys the rbustness aganst msspecfcatn f crrelatn structure and hence has yelded ppularty n lngtudnal data analyss. 3 GEEs fr Parameter Estmatn Dente the mean vectr f subject by m ¼ðm ;...; m n Þ >. Let the scre vectr fr subject be u ¼ðu ;...; u n Þ > ; wth u j ¼ 2 d ðy j ; m j Þ; # 24 WILEY-VCH Verlag GmbH & C. KGaA, Wenhem

4 Bmetrcal Jurnal 46 (24) and under the regularty cndtns, Eðu j Þ¼ and therefre Eðu Þ¼. Frm Sng and Tan (2), the varance f u j s gven by var ðu j Þ¼ s2 j 2 Efd ðy j ; m j Þg ¼ 3s 4 j s2 j m j ð m j Þ þ vðm j Þ : Fllwng Sng and Tan (2), let w ¼ dag fvðm j Þg u be the wrkng vectr, and let RðaÞ be an n n wrkng crrelatn matrx wth a q vectr f crrelatn parameters a. S wrkng cvarance matrx fr w s V ¼ dag =2 var ðw j Þ RðaÞdag =2 var ðw j Þ : Therefre Lang and Zeger s GEE fr the smplex margn crrespnds t the estmatng equatn fr b gven by w ðb; g; aþ ¼ Pm D > A V w ¼ ; ð4þ where A ¼ dag fs 2 j vðm j Þ var ðu j Þg and D > > =@b. Fllwng Prentce and Zha (99), the GEE2 s frmed by addng an addtnal set f estmatng equatns fr the crrelatn parameters based n the standardsed scre resduals, defned by u j r j ¼ pffffffffffffffffffffffffffffff ¼ var ðu j Þ s j u j q ffffffffffffffffffffffffffffffffffffffffffffffffff : 2 Ed ðy j ; m j Þ It s easy t see that such scre resduals satsfy mment prpertes f Eðr j Þ¼, var ðr j Þ¼ and Eðr j r j Þ¼crr ðu j ; u j Þ¼crr ðw j ; w j Þ: The estmatng equatn fr the crrelatn parameter a then takes the frm w 3 ðb; g; aþ ¼ > H ðr Þ¼; ð5þ where r ¼ðr r 2 ; r r 3 ;...; r n r n Þ > ; H s a wrkng cvarance matrx and h ¼ Eðr Þ. The extended GEE2 cnssts f the equatns (4), (5), and an estmatng equatn fr the dspersn cmpnent gven as fllws, w 2 ðb; g; aþ ¼ S ðd s Þ¼ ; ð6þ where d ¼ðdðy ; m Þ;...; dðy n ; m n ÞÞ >, S s a wrkng cvarance matrx, and s ¼ Eðd Þ ¼ðs 2 ;...; s2 n Þ >. Nte that here we use the squared devance resduals, rather than the squared Pearsn resduals gven n Pak (992), t frm the thrd sets f estmatng equatns. The Crwder ptmal matrx fr S (Crwder, 987) s n fact the cv ðd Þ whch s n general nt easy t btan. A smple chce f S s the dentty matrx, leadng t the methd f mments estmatr fr g. Perhaps a better chce fr S s a dagnal matrx wth dagnal elements equal t the varances var fdðy j ; m j Þg ¼ 2ðs 2 j Þ2. See the appendx fr the prf f ths frmula n detal. Ths ndcates a gamma type f mean-varance relatn, that s, the unt varance functn s equal t the squared mean. Wth ths chce, the estmatng equatn wll effectvely prduce the quas-lkelhd estmatr f g as des the gamma regressn (Wedderburn, 974). Let q ¼ðb; g; aþ be the vectr f parameters t be estmated va the extended GEE2 fr whch the estmates are btaned by smultaneusly slvng the jnt equatns, 2 3 w ðb; g; aþ UðqÞ ¼Uðb; g; aþ ¼4 w 2 ðb; g; aþ 5 ¼ : (7) w 3 ðb; g; aþ # 24 WILEY-VCH Verlag GmbH & C. KGaA, Wenhem

5 544 P. X.-K. Sng et al.: Mdellng dspersn It s clear that the estmatng equatn UðqÞ ¼ s unbased, namely EUðqÞ ¼. Hence t fllws frm the standard thery f estmatng equatns that under sme mld regularty cndtns, the estmatr ^q ¼ð^b; ^g; ^aþ s cnsstent and m =2 ð^q qþ s asympttcally multvarate Gaussan wth zer mean and cvarance matrx f the frm lm mj ðqþ, where JðqÞ s the Gdambe nfrmatn matrx m gven by JðqÞ ¼S > R S: Detals f the senstvty matrx S ¼ Ef@UðqÞ=@q > g and f the varablty matrx R ¼ EfUðqÞU > ðqþg are gven n the appendx. Usng the Newtn-scrng algrthm, the slutn ^q fr the jnt equatn (7) can be btaned numercally by teratvely updatng the q values as fllws, q ðkþþ ¼ q ðkþ S U q ðkþ : 4 Smulatn Study T demnstrate the mprtance f prperly analysng the lngtudnal data n the presence f hetergeneus dspersn, we cnduct a smulatn study where the prprtnal data y S ðm ; s 2 Þ; ¼ ;...; 5, were generated ndependently accrdng t the fllwng margnal mdels: lgt ðm Þ¼ b þ b T þ b 2 S ; lg ðs 2 Þ¼g þ g T ; where cvarates T and S are varables f treatment grups ndcated by (,, ), and llness severty scre ranged n (,, 2, 3, 4, 5, 6) that s randmly assumed t each subject by a bnmal dstrbutn B (6,.5). Fr smplcty, we manly nvestgated hw the parameters b j s representng the ppulatnaveraged effects wuld be affected by the stuatn f the dspersn parameter. S, we cnsdered nly the ndependence crrelatn structure, fr whch we were able t smulate data. We tk three equally szed treatment grups, each wth 5 subjects. Usng the ntatn abve, we yeld x ¼ð; T ; S 3Þ > and z ¼ð; T Þ > n whch the severty cvarate was centralsed by the md-scre 3. Mrever, the true values were assgned as ðb ; b ; b 2 Þ¼ð:5; :5; :5Þ, ðg ; g Þ¼ð3; 2Þ. We ran the regressn ver 2 replcatns, and the crrespndng summares are lsted belw. Table reprts the summary statstcs f the parameter estmates frm the extended GEE2 apprach prpsed n the paper wth hetergeneus dspersn. These statstcs nclude mean pnt estmate, 2.5th and 97.5th percentles, emprcal standard devatn and mean standard errr fr each f the fve parameters. When the mdel wth the hmgeneus dspersn was used t ft the smulated data, the mean estmate f lg ðs 2 Þ was 9.2 wth the emprcal standard devatn equal t 8.6, cnsderably larger than the average standard errr.2 btaned frm the sandwch asympttc varance estmatr. Table Summary statstcs f the estmates, based n 2 replcatns generated frm the hetergenety mdel. True Value Hetergeneus Dspersn Hmgeneus Dspersn Mean (2.5%, 97.5%) Stdev* Stderr y Mean (2.5%, 97.5%) Stdev Stderr b (.5).4986 (.367,.668) (.275,.588) b (.5).546 (.6697,.3493) (.947,.9598) b 2 (.5).53 (.3973,.659) (.7345,.79) g ( 3.) ( , ) g ( 2.) 2.88 (.78, 2.284) * Emprcal standard devatn; y Mean standard errr # 24 WILEY-VCH Verlag GmbH & C. KGaA, Wenhem

6 Bmetrcal Jurnal 46 (24) Frm Table, we learned: () The pnt estmates ^b ; ¼ ; ; 2 n the pupulatn-averaged effects mdel () frm bth appraches are relatvely clse t each ther, althugh the mdel wth the hmgeneus dspersn prduces a lttle larger devatn frm the true values than the mdel wth the hetergeneus dspersn. () The 95% emprcal cnfdence ntervals frm the tw mdels have substantally dfferent cverage, zer beng ncluded n the ntervals gven by the hmgenety mdel as ppsed t zer beng excluded n thse gven by the hetergenety mdel, fr all three b parameters. Ths suggests that the hmgenety mdel lses ts pwer f dentfyng sme mprtant cvarates n the presence f hetergeneus dspersn. () The values f the emprcal standard devatn and the standard errr are very smlar n the hetergenety mdel, but clearly dfferent n the hmgenety mdel. Ths ndcates that the asympttc nrmalty thery fr the estmatrs frm the hmgenety mdel may be n lnger vald. T vsualse ths, we pltted the estmated denstes ver the 2 estmates fr each parameter n Fgure. Fgure Estmated denstes f the mdel parameters ver 2 replcatns usng data generated frm the hetergenety mdel. # 24 WILEY-VCH Verlag GmbH & C. KGaA, Wenhem

7 546 P. X.-K. Sng et al.: Mdellng dspersn Fgure ndcates that fr each parameter, the estmates frm the hetergenety mdel are evenly dstrbuted alng the parameter space and clearly frm a bell-shaped densty. In cntrast, the estmates frm the hmgenety mdel ccurs mre frequently n tal areas and clearly frm a heavy-taled densty. The densty fr the estmatr f g frm the hmgenety mdel has an extremely lng tal n the rght. In cnclusn, the asympttc nrmalty fr the estmatrs frm the hmgenety mdel s serusly n questn. Cnversely, we cnducted anther smulatn n that the true mdel had the hmgeneus dspersn. In partcular, data were generated smlarly as n the frst smulatn, except that nw the dspersn mdel s cnstant lg ðs 2 Þ¼g. The true values f b parameters are the same as abve, and set g ¼ 4, whch leads t a large dspersn arund 55. Table 2 gves the summary statstcs ver 2 replcatns. Table 2 Summary statstcs f the estmates, based n 2 replcatns generated frm the hmgenety mdel. True Value Hetergeneus Dspersn Hmgeneus Dspersn Mean (2.5%, 97.5%) Stdev* Stderr y Mean (2.5%, 97.5%) Stdev Stderr b (.5).57 (.2938,.788) (.2942,.75) b (.5).578 (.6943,.2885) (.6992,.294) b 2 (.5).59 (.3623,.688) (.369,.6857) g ( 4.) ( 3.726, 4.27) ( , 4.249) g (.).99 (.2874,.2889) Evdentally, Table 2 ndcates that the estmates frm the tw mdels are very clse, the null hypthess H : g ¼ cannt be rejected at the sgnfcance level :5 under the hetergenety mdel. As expected, the estmated denstes (nt shwn n the paper) f the parameters are very smlar between the tw mdels, and they are all very alke t nrmal densty curves. In summary, when a cnstant dspersn assumptn s n dubt, the hetergenety mdel seems t be necessary and advantageus t make prper statstcal nference. 5 Resdual Analyss * Emprcal standard devatn; y Mean standard errr We prpse t use tw types f resduals t frm dagnstcs fr the key mdel assumptns: () margnal dstrbutns, () lnk functns, and () the wrkng crrelatn structure. The frst ne s the standardsed scre resduals r j gven n (5), and the ther s the regular standardsed Pearsn pffffffffffffffffffffffffffffffff resduals, e j ¼ðy j m j Þ= var ðy j Þ, where var ðy j Þ has n clsed frm expressn as t nvlves the ncmplete gamma functn. See Jørgensen (997) fr the detals. The sample cunterpart f r j r e j s btaned by replacng parameters by ther crrespndng estmates, dented by ^r j r ^e j, accrdngly. Lke mst resdual analyses, ur resdual analyss belw s useful t detect strng sgnals asscated wth certan mdel assumptn vlatn. The smplex dstrbutn assumptn can be checked by the plt f ^e j aganst ^m j, whch ams t examne the mean-varance relatn. If ths assumptn s true, then var ðe j Þ¼, ndependent f mean m j. Therefre, pnts n the plt shuld randmly scatter arund the hrzntal lne at zer (the expectatn f resduals), wth apprxmately 95% pnts n the hrzntal band between 2 and 2. Any apparent departure frm ths wuld suggest ether a vlatn f the assumptn n dstrbutn r prbably a pr mdel ft. A seres f further nvestgatns are needed t dentfy whch factr s respnsble fr such departure. Ths apprach wuld becme mre relable as s 2 becmes large, because the mean-varance relatn becmes dmnated by mð mþ, a case smlar t that f a bnmal dstrbutn. # 24 WILEY-VCH Verlag GmbH & C. KGaA, Wenhem

8 Bmetrcal Jurnal 46 (24) Fllwng McCullagh and Nelder s (989), we use the plt f the adjusted dependent varable s j aganst the lnear predctr ^h j t check the chsen lnk functn. In ur settng, defne ( ) 3s 4 =2 j s j ¼ hðm j Þþ m j ð m j Þ þ s2 j uðy j; m vðm j Þ Þ; j ¼ ;...; n j ; ¼ ;...; m : Clearly, Eðs j Þ¼hðm j Þ snce Eðu j Þ¼, and var ðs j Þ¼Efs j hðm j Þg 2 ¼. If the lnk functn s apprprate, the plt f the estmates ^s j aganst ^h j ¼ x > ^b j shuld shw a straght lne wth apprxmately 95% pnts fallng nt a band wth the upper and lwer lmts f ^h j 2. As n generalsed lnear mdels, ths plt des nt suggest the best lnk functn fr the mdel, but rather nly gves an nfrmal check fr any strng vlatn f the used lnk. Althugh t s dffcult t mdel the true crrelatn structure f lngtudnal data, apprxmately crrect crrelatn structures allw regressn ceffcents t be estmated mre effcently. Thus, t s mprtant t assess the apprprateness f the wrkng crrelatn used n GEEs va resdual analyss. Nte that crr ðr j ; r j Þ¼crr ðw j ; w j Þ; mplyng that the true crrelatn f varable w j s equal t that f the standardsed scre resduals r j. Sme explratry prcedures presented n Sectn 3.4 f Dggle et al. (22) can be adpted fr w j s t examne the crrelatn f data. 6 An Example In ths sectn we re-analyse the phthalmlgcal data n the use f ntracular gas n retnal repar surgeres (Meyers et al., 992), wth a specal fcus n the hetergeneus dspersn. A prmary analyss f the data assumng the hmgeneus dspersn was dne by Sng and Tan (2). Brefly, the study was t nvestgate the decay curse f the ntracular gas n retnal repar surgeres prspectvely n 3 patents. The gas was njected nt the eye befre surgery and patents were fllwed three t eght (average f 5) tmes ver a three-mnth perd. The respnse varable y j was the percent f gas left n the eye recrded as prprtn (a percent). The questn was f the dsappearance f the gas s related t ther cvarates such as the cncentratn f the gas used. Sng and Tan (2) mdelled the gas vlume drectly usng a margnal mdel. Wth ur prpsed methd, we are able t test f the hmgeneus dspersn s true, and f nt s the mdel allws us t dentfy whch cvarates lead t hetergenety. T begn, the ppulatn-averaged effects mdel n Sng and Tan (2) s lgt ðm j Þ¼b þ b lg ðt j Þþb 2 lg 2 ðt j Þþb 3 x j ð8þ where t j s the tme cvarate f days after the gas njectn, and x j s the cvarate f gas cncentratn levels equal t, and, crrespndng t the cncentratn levels f 5%, 2% and 25%, respectvely. T ths mdel, the cmpnents f the estmatng functn w specfed by (4) are gven as fllws. D > ¼ X > dag fm j ð m j Þg ; D > A ¼ X > dag f3s 2 j vðm j Þþm j ð m j Þg; where X s f n 3 dmensn and ts jth rw s ð; lg ðt j Þ; lg 2 ðt j Þ; x j Þ, and var ðw j Þ¼s 2 j vðm j Þf3s2 j m2 j ð m j Þ2 þ g: Clearly the crrespndng senstvty matrx s S ¼ Pm D > A V A D. Als as ndcated n ther paper, AR() dependence seemed t ft the data the best, s ur analyss nly cncerns ths type f dependence, specfed as f the frst-rder ECM mdel, crr ðw j ; w j Þ ¼ exp ðajt j t j jþ, fr a <. When H s chsen t be the dentty matrx, the functn w 3 becmes FðaÞ ¼ Pm c > ðr h Þ ¼ # 24 WILEY-VCH Verlag GmbH & C. KGaA, Wenhem

9 548 P. X.-K. Sng et al.: Mdellng dspersn Table 3 Estmates, standard errrs and rbust z-statstcs frm the hetergeneus dspersn mdel fr the eye surgery data. Parameter b b b 2 b 3 g g g 2 a Estmate Stderr? z-statstc *Standard Errr where c ¼½jt t 2 j exp ð ajt t 2 jþ;...; jt n t n j exp ð ajt n t n jþš > and the crrespndng senstvty matrx S 33 ¼ Pm c > c. The mdel that addresses the hetergenety n tw cvarates f tme and gas cncentratn level takes the fllwng frm lg ¼ g þ g lg ðt j Þþg 2 x j : ð9þ s 2 j We ran the Newtn-scrng algrthm gven n Sectn 3 and fund estmates and standard errrs that are lsted n Table % 2% 25% Lg-dspersn Days Fgure 2 Ftted curves fr the pattern f hetergeneus dspersn ver tme acrss three treatment levels. # 24 WILEY-VCH Verlag GmbH & C. KGaA, Wenhem

10 Clearly, bth cvarates f tme and treatment are sgnfcant factrs attrbuted t the hetergeneus dspersn n mdel (9). Fgure 2 dsplays the ftted curves fr the pattern f dspersn prfle ver tme acrss three dfferent gas cncentratn levels. Based n the mdel wth the tme-varyng dspersn, ur fndngs fr ther parameters are very smlar t thse n Sng and Tan (2). Smlar t Sng and Tan (2), we fund that the quadratc tme term lg 2 ðt j Þ s sgnfcant, that the lnear tme lg ðt j Þ s nt sgnfcant, and that the gas cncentratn cvarate s margnally nsgnfcant, at the sgnfcance level.5. Als, The estmated lag- autcrrelatn ^q ¼ e^a ¼ :575ð:6Þ and ts z-statstc s , suggestng that q s sgnfcantly dfferent frm zer. In cntrast t the smulatn study, here we dd nt see dramatc dfferences between the results frm the hetergenety and hmgenety mdels. We gave the reasn as fllws. In the smulatn study we chse the ntercept and slpe parameters t be cmparable, respectvely 3 and 2, s that a change n the cvarate wuld greatly affect the sze f dspersn. Therefre, the results frm the hmgenety and hetergenety mdels were evdently dfferent. Hwever, n the data analyss the ntercept dmnates the cntrbutn t the dspersn ver the tw slpe ceffcents, mplyng that the verall dspersn remans mstly very large, and therefre n bg dfferences appeared n the results frm the tw types f mdels. Nw we cnsder the resdual analyss fr the abve mdel wth tme-varyng dspersn. Panel A f Fgure 3 shws the scatter-plt f the estmated standardsed Pearsn resduals ^e j s aganst the ftted mean values ^m j, t check the dstrbutn assumptn. The dashed lnes at 2 and 2 represent the asympttc 95% upper and lwer lmts, respectvely. The resduals seem t behave reasnably well as expected, nly three f them lyng utsde f the regn. The plt seems t be n agreement wth the smplex margnal dstrbutn. Panel B f Fgure 3 prvdes a rugh check f the lgt lnk functn used n the prpsed mdel, shwng the scatter-plt f the estmated adjusted dependent varables ^s j aganst the estmated lgt Bmetrcal Jurnal 46 (24) Pearsn resduals Adjusted dependent varable Ftted values Panel A: Checkng dstrbutn assumptn Lgt lnear predctr Panel B: Checkng lnk funktn Fgure 3 Dagnstc plts n the eye surgery data analyss. # 24 WILEY-VCH Verlag GmbH & C. KGaA, Wenhem

11 55 P. X.-K. Sng et al.: Mdellng dspersn lnear predctr ^h j. The tw sld lnes stand fr the asympttc 95% cnfdent bands wthn whch almst 96% pnts are cntaned. Ths clearly supprts the lgt lnk functn assumptn. Checkng the wrkng crrelatn seems t be nntrval, snce the data are measured at rregular tme pnts and the resduals avalable at a gven tme are sparse. S we feel that the prpsed methd fr checkng the wrkng crrelatn may nt be relable here. Alternatvely, Dggle s vargram plt (Dggle, 99) may be used here t reach an apprprate cnclusn. Hwever ths s nt the fcus f the paper, and hence the detals are mtted. 7 Cncludng remarks In ths paper we develped an apprach t mdellng the hetergeneus dspersn parameter, relaxng the usual assumptn f cnstant dspersn n Lang and Zeger s margnal mdels. An extended versn f GEEs was prpsed t estmate the parameters n the mdel fr dspersn. Thrugh the analyss f the eye surgery data, we fund that the dspersn can be a functn ver fllw-up tme as well as treatment arm, and that the shape f margnal dstrbutns s tme-varyng n addtn t the tme-varyng lcatns. Ths prpsed methd mprves the mdellng f lngtudnal data and prvdes a tl fr better understandng the margnal prfles f the lngtudnal cntnuus prprtnal data. The extended GEEs n ths paper was develped under the assumptn f n mssng values n data. Snce mssng values ften ccur n lngtudnal studes n practce, t wuld be f great nterest t further extend the prpsed GEEs t cnduct data analyss wth mssng values. It s knwn that GEEs prduce cnsstent estmatrs fr the mdel parameters when mssng values are cmpletely randm and gnred n the analyss. Hwever, when data cntan randm mssng values r nfrmatve mssng values, the cnsstency fr the GEEs estmatrs s generally n lnger vald. Reslvng ths ssue has been an actve research tpc n the lngtudnal data analyss. Fr example, Rbns et al. (995) suggested the nverse prbablty weghted GEEs that prduce cnsstent estmates f the drp-ut prcess s prperly mdelled. Anther apprach suggested by Pak (997) s t mpute the mssng values by the cndtnal expectatn gven the bserved data. Mre references can be fund n Dggle et al. (22), Verbeke and Mlenberghs (2), r Zegler et al. (998). Acknwledgements The authrs are grateful t the tw referees fr ther valuable suggestns and cmments that led t an mprvement f the paper. The authrs thank Dr. Sanfrd Meyers fr makng hs data avalable fr nclusn as an example. The frst tw authrs research was supprted by the Natnal Scence and Engneerng Research Cuncl f Canada. Appendx A Gdambe nfrmatn matrx Ths sectn gves the cmpnents f Gdambe nfrmatn matrx needed fr cmputng the estmated standard errrs fr estmates and hence fr cnstructng Wald test statstcs. The senstvty matrx s a 3 3 blck matrx, S ¼ S S 2 S > S 2 S 22 S 23 A; S 3 S 32 S 33 where clearly S 2 ¼, S 3 ¼, and S 23 ¼. Als the blck S 2 ¼ because Eu j ¼. S n general the S matrx takes the frm S S S 22 A; S 3 S 32 S 33 # 24 WILEY-VCH Verlag GmbH & C. KGaA, Wenhem

12 Bmetrcal Jurnal 46 (24) 5 55 and ts nverse matrx s S S S 22 A; S 33 S 3S S 33 S 32S 22 S 33 prvded that all dagnal blcks are nvertble. When the dstrbutn f r j r j s ndependent f the mean and dspersn parameters, bth S 3 and S 32 are. Therefre the matrx S becmes a blckdagnal matrx wth S ¼ Xm D > A V A D ; S 22 ¼ @g > and S 33 ¼ @a > : The varablty matrx R s als a 3 3 blck matrx, V V 2 V 3 V ¼ EfUðqÞU > ðqþg V 2 V 22 V 23 A: V 3 V 32 V 33 The nne blcks are detaled as fllws. V ¼ Efw w > g¼xm V 2 ¼ Efw w > 2 g¼xm V 3 ¼ Efw w > 3 g¼xm V 22 ¼ Efw 2 w > 2 g¼xm V 23 ¼ Efw 2 w > 3 g¼xm V 33 ¼ Efw 3 w > 3 g¼xm D > A V cv ðw ÞV A D ; D > A V cv ðw ; d > ; D > A V cv ðw ; r > > > S cv ðd Þ S S cv ðd ; r Þ H H cv ðr > : Because f symmetry, V 2 ¼ V > 2, V 3 ¼ V > 3, and V 32 ¼ V > 23. It s nted that cv ðw Þ¼dag fvðm j Þg cv ðu Þ dag fvðm j Þg, and an estmate f cv ðu Þ s btaned by pluggng the estmates ^m j and replacng cv ðu Þ by ^u ^u > n the expressn. The same apprach s appled t estmate the remanng blcks f V. # 24 WILEY-VCH Verlag GmbH & C. KGaA, Wenhem

13 552 P. X.-K. Sng et al.: Mdellng dspersn B Prf f mean-varance relatn Ths sectn presents the prf fr the frmula var fdðy j ; m j Þg ¼ 2ðs 2 j Þ2, Y j S ðm j ; s 2 jþ. Frm the appendx f Sng and Tan (2), suppressng crdnates, EfdðY; mþg ¼ s 2, and hence t s suffcent t shw that Efd 2 ðy; mþg ¼ 3ðs 2 Þ 2. A smple algebra leads t Efd 2 ðy; mþg ¼ Z d 2 ðy; mþ pðy; m; s 2 Þ dy rffffffffff l ð þ xþ 4 ¼ 2p x 4 Z fx 3 2 þð 4xÞ x 2 þ 2xð3x 2Þ x 2 þ 2x 2 ð3 2xÞ x 3 2 þ x 3 ðx 4Þ x 5 2 þ x 4 x 7 2 g f ðx; x; lþ dx; where l ¼ =s 2 and ( ) f ðx; x; lþ ¼exp l ð þ xþ 2 ðx xþ 2 2 x 2 : x Usng frmulas (5.4) (5.43) f (Jørgensen, 997), we btan and Z Z x 3 2 f ðx; x; lþ dx ¼ x 7 2 f ðx; x; lþ dx ¼ 2p 2 l 2 x 3 ð þ xþ 4 þ 3lx 4 ð þ xþ 2 þ 3x 5 l l 2 ð þ xþ 5 2p 2 l 2 ð þ xþ 4 þ 3lxð þ xþ 2 þ 3x 2 l l 2 x 2 ð þ xþ 5 : Pluggng these results and thse frm Sng and Tan (2), we get E d 2 ðy; mþ ¼ 3 s 2 ð Þ 2. References Artes, R. and Jørgensen, B. (2). Lngtudnal data estmatng equatns fr dspersn mdels. Scandnavan Jurnal f Statstcs 27, Barndrff-Nelsen, O. E. and Jørgensen, B. (99). Sme parametrc mdels n the smplex. Jurnal f Multvarate Analyss 39, 6 6. Crwder, M. (987). On lnear and quadratc estmatng functns. Bmetrka 74, Dggle, P. J. (99). Tme Seres: A Bstatstcal Intrductn. Oxfrd Unversty Press, Oxfrd. Dggle, P. J. Heagerty, P., Lang, K.-Y. and Zeger, S. L. (22). The Analyss f Lngtudnal Data, 2nd ed. Oxfrd Unversty Press, Oxfrd. Jørgensen, B. (997). The Thery f Dspersn Mdels. Chapman Hall, Lndn. Lang, K.-Y. and Zeger, S. L. (986). Lngtudnal data analyss usng generalzed lnear mdels. Bmetrka 73, McCullagh, P. and Nelder, J. A. (989). Generalsed Lnear Mdels, 2nd ed. Chapman and Hall, Lndn. Meyers, S. M., Ambler, J. S., Tan, M., Werner, J. C. and Huang, S. S. (992). Varatn f perflurprpane dsappearance after vtrectmy. Retna 2, Pak, M. C. (992). Parametrc varance functn estmatn fr nnnrmal repeated measurement data. Bmetrcs 48, 9 3. Pak, M. C. (997). The generalzed estmatng equatn apprach when data are nt mssng cmpletely at randm. Jurnal f Amercan Statstcal Asscatn 92, Prentce, R. L. and Zha, L. P. (99). Estmatng equatns fr parameters n means and cvarances f multvarate dscrete and cntnuus respnses. Bmetrcs 47, # 24 WILEY-VCH Verlag GmbH & C. KGaA, Wenhem

14 Bmetrcal Jurnal 46 (24) Rbns, J. M., Rtntzky, A. and Zha, L. P. (995). Analyss f semparametrc regressn mdels fr repeated utcmes n the presence f mssng data. Jurnal f the Amercan Statstcal Asscatn 4, Smyth, G. K. (989). Generalsed lnear mdels wth varyng dspersn. Jurnal f the Ryal Statstcal Scety, Seres B 5, Sng, P. X.-K. and Tan, M. (2). Margnal mdels fr lngtudnal cntnuus prprtnal data. Bmetrcs 56, Verbeke, G. and Mlenberghs, G. (2). Lnear Mxed Mdels fr Lngtudnal Data. Sprnger-Verlag, New Yrk. Wedderburn, R. W. M. (974). Quas-lkelhd functns, generalzed lnear mdels and the Gauss-Newtn methd. Bmetrka 6, Zegler, A., Kastner, C. and Blettner, M. (998). The generalsed estmatng equatns: an anntated bblgraphy. Bmetrcal Jurnal 4, # 24 WILEY-VCH Verlag GmbH & C. KGaA, Wenhem