Gaussian Copula Mixed Models with Non-Ignorable Missing Outcomes

Transcription

1 Avalable at Appl. Appl. Math. ISSN: Vol. 10 Issue 1 (June 015) pp Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) Gaussan Copula Mxed Models wth Non-Ignorable Mssng Outcomes N. Jafar E. Tabrz and E. Bahram Saman Department of Statstcs Faculty of Mathematcal Scence Shahd Behesht Unversty Tehran Iran Correspondence author e-mal : ehsan_bahram_saman@yahoo.com eceved: September 4 014; Accepted: November Abstract Ths paper s concerned wth the analyss of mxed data wth ordnal and contnuous outcomes wth the possblty of non - gnorable mssng outcomes. A copula-based regresson model s proposed that accounts for assocatons between ordnal and contnuous outcomes. Our approach entals specfyng underlyng latent varables for the mxed outcomes to ndcate the latent mechansms whch generate the ordnal and contnuous varables. Maxmum lkelhood estmaton of our model parameters s mplemented usng standard software such as functon nlmnb n. esults of smulatons concern the relatve bases of parameter estmates of jont and margnal models usng data wth non-gnorable outcomes. The proposed methodology s llustrated usng a medcal data obtaned from an observatonal study on women wth three correlated responses an ordnal response of osteopoross of the spne and two contnuous responses of body mass ndex and wastlne. The effect of the amount of total body calcum (Ca) job status (Job) type of dwellng (Ta) and age on all responses are nvestgated smultaneously. Keywords: Nongnorable mssng outcomes; Mxed outcomes; Latent Varables; Lkelhoodbased; Gaussan copula. AMS (010) No.: 6F03 81

2 8 Jafar et al. 1. Introducton Many statstcal applcatons nvolve jont analyss of multvarate data ncludng mxed ordnal and contnuous outcomes wth non-gnorable mssng values. For example n health studes pertanng to the maternal smokng effect on respratory llness n chldren we have a contnuous measure of the pulmonary functon and an ordnal measure of chronc symptoms n chldren. In medcal data set of osteopoross of the spne correlated outcomes are the ordnal outcome of osteopoross of the spne and contnuous outcomes of body mass ndex and wastlne and covarates that mght be due to ths type of job and dwellng. For the frst example separate analyss cannot assess the effect of maternal smokng on both outcomes. Also separate analyss gve based estmates for the parameters and we need to consder a method n whch these varables can be modelled jontly. So we need to model responses smultaneously. In the second example the smultaneous effect of the type of job and accommodaton on body mass ndex wast lne and osteopoross of the spne should be modelled jontly consderng mssng mechansms for each outcomes. Multvarate jont modellng of such mssng data often leads to complcatons n computaton due to a relatve lack of standard models. A number of jont modellng strateges for mxed outcomes have been studed n the lterature. The frst formulaton that has receved much attenton n mxed data lterature was ntroduced by Olkn and Tate's (1961) whch s called general locaton model. Ths model assumes multvarate normal dstrbuton for contnuous outcomes gven values of dscrete outcomes. The second formulaton ncludes the Cox and Wermuth (199) approach who suggest a logstc or probt condtonal dstrbuton for the bnary varable gven contnuous outcomes. The thrd formulaton was presented by Heckman (1978) n whch a general model for smultaneously analysng two mxed correlated responses s ntroduced and Catalano and yan (1997) extended and used the model for a cluster of dscrete and contnuous outcomes (vde also Ftzmaurce and Lard (1995) and Ftzmaurce and Lard (1997)). All the above references consder correlated nomnal and contnuous responses. Poon and Lee (1987) and Moustak and Knott (000) used a model for ordnal and contnuous responses wthout consderng any covarate effect. De Leon an Carrere (007) extended an approach smlar to that of Heckman (1978) and Sammel et al. (1997) for jontly modellng of a nomnal and a contnuous varable to jont modellng of bvarate ordnal and contnuous outcomes. All the above references dscuss dentfablty wth mposng some restrctons on the correlaton structure. Pnto and Normand (009) proposed a new parametrc constraned latent varable model to have dentfablty wthout restrctons on the correlaton structure. In such medcal studes often some of the subjects do not respond n some occasons whch cause for mssng outcomes. Much has been wrtten about statstcal methods for handlng ncomplete data. ubn (1976) and Lttle and ubn (00) defne the mssng mechansm as : (1) Mssng Completely At andom (MCA) : f mssngness s dependent nether on the observed responses nor on the mssng responses () Mssng At andom (MA) : f t s not dependent on the mssng responses ( gven the observed responses ) (3) Not Mssng At andom (NMA) : f t depends on the unobserved responses. MCA and MA are gnorable but NMA s nongnorable.

3 AAM: Intern. J. Vol. 10 Issue 1 (June 015) 83 A number of jont modellng strateges for mxed outcomes wth possblty of mssng values have been studed n the lterature. Lttle and Schluchter (1987) proposed the general locaton model wth the assumpton of gnorng the mssng data mechansm. Ganjal (003) used A model for mxed contnuous and dscrete bnary responses wth possblty of mssng responses. Bahram Saman et al. ( ) extended the model of Ganjal. Also Bahram Saman et al. (011) proposed a multvarate latent random effect model for mxed contnuous and ordnal longtudnal responses wth mssng responses. ang et al. (007) nvestgate an nferental method for mxed Posson and contnuous longtudnal data wth non-gnorable mssng values. The challenge s that models for jont dstrbutons of mxed outcomes wth nongnorable mssng values are uncommon. A recent alternatve strategy nvolves the use of copulas as dscussed n Sklar (1959) Song et al. (000) Newadomska-Bugaj and Kowalczyk (005) Zmmer and Trved (006) Kolev et al. (006) and Song et al. (009). A number of transton regresson models for non-gaussan responses have been proposed n lterature vde Benjamn et al. (003) for a revew. Several authors have recently adopted copulas to ndrectly construct mxed-outcome jont models. Copulas have been proved to be useful n practce when the jont dstrbuton of nterest s ether not avalable or dffcult to specfy but margnal dstrbutons can be specfed wth confdence lke n mxed-outcome settngs. Song et al. (000) nvestgate some copula-based regresson models for bvarate contnuous outcomes Zmmer and Trved (006) proposed trvarate copulas to model sample selecton and treatment effects. De Leon and Wu (011) proposed copula-based regresson models for bvarate mxed dscrete and contnuous outcomes. Our paper s concerned wth jont regresson models for correlated mxed ordnal and contnuous outcomes wth possblty of non - gnorable mssng outcomes constructed by usng copulas. We wll also extend De Leon and Wu' (011)'s approach and consder mssng data of the outcomes so our models are copula-based jont modellng of mxed data for bvarate and multvarate mxed ordnal and contnuous outcomes wth non-gnorable mssng outcomes. Ths paper s organzed as follows. We ntroduce a class of copula-based regresson models and the full lkelhood of the model for bvarate mxed outcomes wth non-gnorable outcomes n Secton. Smulaton results on the sample propertes of estmates are reported n Secton 3. Secton 4 llustrates the applcaton of the model to the medcal data. Fnally the paper concludes n Secton 5.. Model and Lkelhood.1. Bvarate Outcomes wth non-gnorable mssng values Let be an ordnal outcome wth D level and be a contnuous outcome. These outcomes are recorded for N ndvduals correlated and modeled smultaneously. Some outcome values may be mssng due to some reasons. Let and denote the underlyng latent varables for ordnal outcome the non-response mechansm for the ordnal varable and non-response

4 84 Jafar et al. mechansm for the contnuous varable respectvely. The ordnal varable of the th ndvdual wth D levels s defned as l1 f ( 0 1) lk 1 f [ k k 1) ld f [ D 1 D) where 1 D 1 are the cut-pont parameters wth 0 and response varables for respondng to and are defned respectvely as D. Also for the 0 ow. 1 0 and 0 ow. 1 0 and may be nterpreted as propensty of ndvdual as a latent varable to respond to and respectvely. The jont model s assumed to take the form : ( z ) ( z ) 3 ( z3 ) 3 4( z4 ) 4 (1) where E k 0. for k 134 and the covarance matrx of the vector of errors s

5 AAM: Intern. J. Vol. 10 Issue 1 (June 015) 85 where and are vectors of regresson coeffcents also ncludes an ntercept parameter but does not nclude any ntercept. Also z 1 z z 3 and z 4 are outcome specfc covarate vectors and 1 3 and 4 are lnk functons specfyng how the covarates are ncorporated n the margnal means. For example n the lnear models T T T T 1 z1 z 3 z 3 and 4 z4. Also the correlaton parameters jj for j j j 13 and j 34 should be estmated. If one of the correlaton parameters for j j j 1 and j 34 s found to be jj sgnfcant then we have a NMA mechansm and mssng mechansm cannot be gnored. On the other hand f for j j j 1 and j 34 are found to be 0s the mssng data s jj MCA and can be gnored. In ths model any multvarate dstrbuton can be assumed for the errors n the model. Here a multvarate Gaussan copula s assumed. We have to restrct at least one parameter to obtan an dentfable model. For dentfablty reasons we assume that Var Var Var. ( ) ( ) ( ) 1 To obtan the lkelhood functon we used the multvarate Gaussan copula. we can specfy the jont CDF of as F F F F and F F ( x y ) ( { F ( x )} { F ( y )} ) F ( x y r r ) ( { F ( x )} { F ( y )} { F ( r )} { F ( r )} ) y x 4 x y F ( x y r ) ( { F ( x )} { F ( y)} { F ( r )} ) 1 x 3 F ( { F ( x )} { F ( r )} { F ( r )} ) y x 134 y x F ( { F ( y )} { F ( r )} { F ( r )} ) y x 134 y x where s the standard normal dstrbuton functon multvarate normal dstrbuton wth covarance matrx y 13 3 s the cumulatve standard

6 86 Jafar et al. and 4 ; s the cumulatve standard multvarate normal dstrbuton wth matrx covarance where the margnal dstrbutons F F F and F are absolutely contnuous dstrbutons. To obtan jont the dstrbuton of four cases : 1 and and mssng mechansm we consder the followng Case 1 For the th ndvdual wth both mechansms s so we have and observed the jont dstrbuton of P( x y 1 1) and mssng F ( 1 y ) ( F 1 y 00) x l1 F ( 1 k y ) F ( 1 00) k y F ( ) ( 00) k y F k y x lk 1 F ( y ) ( 0 0 F y ) F ( ) ( 00) D y F D y x l D where F 1 ( y ) ( { F 1 ( )} { F ( y )} ) k k 1 F ( y 0) ( { F ( )} { F ( y )} { F (0)} ) k 3 k F ( k y 00) 3( { F ( k )} { F ( y )} { F (0) F (0)} 134).

7 AAM: Intern. J. Vol. 10 Issue 1 (June 015) 87 Case For the th ndvdual nether whose nor s observed the jont dstrbuton of and s P( 0 0) P( 0 0) F (00). In other words the jont dstrbuton of and wth possblty of mssng for both outcomes s specfed usng bvarate Gaussan copula as follow: Case 3 For the th ndvdual whose P( 0 0) ( { F (0)} { F (0)} ) s only observed the jont dstrbuton of and s So we have : P( x 0 1). F ( 10) F ( 100) x l1 x 1 l F ( 10) ( 0) k F k F (0) F ( D0) F ( 00) F ( k00) x lk 1 k1 F (00) F ( 0 0) x l D D where and F ( 0) ( { F ( )} { F (0)} ) y y F F F F ( 100) 3( { ( 1)} { (0)} { (0)} 134). y x Case 4 For the th ndvdual whose s observed the jont dstrbuton of and s

8 88 Jafar et al. P y P y ( 1 0) ( 0 0) P y P y ( 0) ( 0 0) F ( y 0) F ( y 00) n other words the jont dstrbuton of and mssng mechansms s specfed usng Gaussan copula as follow P y F y F 1 1 ( 1 0) ( { ( )} { (0)} ) ( { F ( y )} { F (0)} { F (0)} ) Also we have : f ( ) ( ). x y P x y y Lkelhood functon s the product of the jont dstrbuton of and and mssng mechansm for four cases and shows the smplfcaton obtaned by usng the assumpton of multvarate Gaussan copula for errors n the model... Multvarate Outcomes wth non-gnorable mssng values Suppose the vector of response for the th ndvdual s W ( ) 1 p ( p1) q where s for s 1... p are ordnal responses each wth D s levels and s for s p 1... q are contnuous responses. All responses are correlated. Let for s 1... p denote the underlyng random varable of the ordnal response for the th s ndvdual and s th outcomes wth D s levels. Defne l s 1s 0s s 1s j1 sj s s( j1) ld s( D 1) s sd 1... s l j D s s s where 0 s sd and s s ( s... 1 s( D s 1) ) s the vector of cut-ponts parameters for s1... p. Typcally when mssng data occur n an outcome we assume (... ) as the ndcator vector of respondng to and s. It s defned as 1 p

9 AAM: Intern. J. Vol. 10 Issue 1 (June 015) 89 s 1 0 s 0 otherwse and (... ) the ndcator vector for respondng to and ( p1) q s 1 0 s 0 otherwse s s defned as where s and s denote the underlyng latent varables of the non-response mechansm respectvely for the ordnal and contnuous varables. The jont model takes the form : ( z ) s 1... p (1) s 1 1 s s z s p q () s ( s ) s 1... ( z ) s 1... p (3) s 3 3 s s ( z ) s p 1... q (4) s 4 4 s s () where and d (1) () (3) (4) ( ) ~ MVN (0 ) (... ) for u 13 ( u) ( u) ( u) 1 p (... ) for u 4 ( u) ( u) ( u) ( p1) q where ( u) ( u) ( v) uu Var( ) for u 134 uv Cov( ) u v uv 134 and uv vu. Because of dentfablty problem we have to assume Var Var Var ( s ) ( y ) ( y ) 1 s s so ss j for j 134.

10 90 Jafar et al. s not zero then the mssng mechansm of response Note f one of the matrces s not at random. The vector s for s p 1... q ncludes an ntercept parameter but s and s for s 1... p and s for s p 1... q due to havng cut-pont parameters are assumed not to nclude any ntercept. Let Also let and J ( J J ) J obs obs obs Ms C J J { s : s observed } J ( J ) C obs obs s Ms obs and J { s : s observed } J ( J ) obs s { s J } obs s obs { s J } obs s obs C Ms obs. denote the set of contnuous and ordnal varables observed. Further let denote the set of underlyng random varable of the ordnal response of the th obs ndvdual defned as : { s J }. obs 1 s obs s Also the set of non-response mechansm for the contnuous and ordnal random varables whch s defned respectvely as: s s J s J { 1 obs} { 0 obs} obs s obs s s J s J { 1 obs} { 0 obs}. obs s obs j To obtan the lkelhood functon we used the multvarate Gaussan copula. The lkelhood of the model () s L f ( z... z ) where { Jobs } { Jobs } obs obs obs obs 1 4 P( C z... z ) obs obs obs obs Ms 1 4 ( ) { Jobs } C Ms { 0; s JMs 0; s JMs} s j

11 AAM: Intern. J. Vol. 10 Issue 1 (June 015) 91 and P( C z... z ) obs Ms obs 1 4 ( { F ( ); s J } { F ( y ); s J } 1 1 q ( j1) s obs s s s obs { F (0); s J } { F (0); s J }) 1 1 Ms s s ( { F ( ); s J } { F ( y ); s J } 1 1 q js obs s s s obs { F (0); s J } { F (0); s J }) 1 1 Ms s s Ms Ms P( C z... z ) obs Ms Ms obs 1 4 ( { F ( ); s J } { F ( y ); s J } 1 1 q ( j1) s obs s s s obs { F (0); s J } { F (0); s J } { F (0); s J }) 1 q ( { F ( js ); j Ms Ms Ms s s s 1 s J } { F ( y ); s J } obs s s obs { F (0); s J } { F (0); s J } { F (0); s J }) Ms Ms Ms s s s and P( C z... z ) obs Ms Ms obs 1 4 ( { F ( ); s J } { F ( y ); s J } 1 1 q ( j1) s obs j s s obs { F (0); s J } { F (0); s J } { F (0); s J }) Ms Ms Ms s s s F s J 1 q ( { ( js ); s 1 obs} { F ( y ); } s s s Jobs { F (0); s J } { F (0); s J } { F (0); s J }) 1 y 1 1 Ms Ms Ms s s s P( C z... z ) obs Ms obs 1 4 Ms Ms ( { F ( ); s J } { F ( y ); s J } 1 1 q ( j1) s obs s s obs j { F (0); s J } { F (0); s J } 1 1 Ms s s { F (0); s J } { F ( 0); sj }) j 1 1 Ms s s 1 1 q ( { F ( sj ); s Jobs} { F ( y ); } s s s Jobs { F (0); s J } { F (0); s J } 1 1 Ms s s { F (0); s J } { F (0); s J }). 1 1 Ms s s Ms Ms Ms Ms.3. Multvarate Outcomes wth gnorable mssng values We consder model () for fndng the condton for MA. Let

12 9 Jafar et al. W ( ) ( W W ) W ( ) ( W W ) and ( ) obs ms obs ms where (... ) ( p 1... ) (... ) 1 p q 1 p W obs s the vector of latent varables related to the observed part of W ( ) and W ms s the vector of latent varables related to the mssng part of W ( ). Accordng to our jont model the vector of responses along wth the mssng ndcators ( W ) ( Wobs Wms ) has a multvarate normal dstrbuton wth the followng covarance structure o o o m o m o m m m o m where o o cov( Wobs Wobs ) m m cov( Wms Wms ) o m cov( Wobs Wms ) cov( W ) o obs co ) v(. The jont densty functon of W and can also be parttoned as f W f W W f W ( ) ( ms obs ) ( obs ) where f W W and ( ms obs ) fw have respectvely a condtonal and a margnal normal ( obs ) dstrbuton. Accordng to the mssng mechansm defntons to have a MA mechansm the covarance matrx of the above mentoned condtonal normal dstrbuton cov( W W ) m o ms should satsfy the followng constrant obs mm m 1 m m o o o o m o o 1 1 m m m o o o m o m m o o o o 1 1 m o o o m o o o o o

13 AAM: Intern. J. Vol. 10 Issue 1 (June 015) 93 0 (3) 1. m o o o m So for obtanng the lkelhood functon we used the multvarate Gaussan copula wth constrant (3). o.4. Estmaton.4.1. Jont Estmaton Puttng l( ) as the log-lkelhood functon then let S( ) l( ) / be the score functon and ( ) l( ) / the Hessan matrx for obtanng the maxmum lkelhood estmate (MLE) ˆ we must solve the S( ) 0 ( jont estmaton). We know that the Fsher nformaton matrx s I( ) E{ ( )}= E{ S( ) S ( )}. It can be shown that ˆ s consstent and t has asymptotcally multvarate normal dstrbuton wth mean and covarance matrx gven by the nverse of the I( ). So the standard errors (SE) for ˆ are calculated from dagonals of ES( ˆ ) S ( ˆ ) 1. We used the functon pnorm for lkelhood evaluaton and the functon nlmnb whch do not requre the score functon for optmzaton n. One may choose dfferent startng values over multple runs of the teraton procedure and then examne the results to see whether the same soluton s obtaned repeatedly. When that happens one can conclude wth some confdence that a global maxmum has been found. For good ntal values for our applcaton we suggest the use of the results of separately analyzng contnuous and ordnal varables..4.. Margnal estmaton Often the maxmzaton of l( ) computatonally s not easy n practce so we use the method of nference functons for margns (IFM). Ths method frst estmates margnal parameters va margns then only uses the copula as a bass for estmatng the assocaton parameters. In other words the n IFM method the margnal models and the dependence between outcomes are specfed separately (Margnal estmaton). The IFM estmate has asymptotcally 1 1 multvarate normal dstrbuton wth mean and covarance matrx C J BJ where J s a block - dagonal matrx wth symmetrc dagonal blocks and B s a symmetrc block matrx. 1 1 Standard errors (SE) of are obtaned from the dagonals of C J BJ where J and Bare the respectve estmates of J and B obtaned from [va Joe and u (1996) and Harry and James (1998)]. 3. Smulaton Study In ths secton the frst consders a jont model for mxed ordnal and contnuous outcomes under the fve scenaros assembled from a margnal normal specfcaton for a margnal normal specfcaton for a margnal normal dstrbuton for the latent varable underlyng the ordnal outcome and the contnuous outcome the second a margnal model s based under NMA and MA mechansms for the fve scenaros. In both cases we adopt the Gaussan copula to

14 94 Jafar et al. construct the jont model. The results ndcate that the jont estmaton should be preferred to the margnal approach under NMA and MA mechansms however the two methods perform generally smlarly for mxed ordnal and contnuous responses wth non-gnorable mssng values. The relatve bases of the jont and margnal estmates are obtaned for the fve scenaros wth and wthout non-gnorable outcomes. The formula for the relatve bas of s as follows: ˆ elatve bas= 100% Ordnal - Normal Model wth non-gnorable mssng values Let be an ordnal outcome and be a contnuous outcome. These are obtaned for each of N subjects. Some of these values may be mssed. Contnuous varables and respectvely represent latent varables for ordnal outcome and latent varable related to mssng mechansm of. We defne ordnal varable for the th subject as follows : l1 1 f 1 l f 1 l3 3 f. The varables and and covarance matrx are generated by a multvarate normal dstrbuton wth zero mean We defne as 0. ow We assume the percentage of mssng values of to be 30 %. A total of M =1000 repeated samples ( ) of szes N=100 and N =00 were generated under fve scenaros where and =1 wth (A) (B) (C) (D) (E) We analyze the followng smple model

15 AAM: Intern. J. Vol. 10 Issue 1 (June 015) 95 E ( ) z E( ) z E 1 1 ( ) 1z (4) where the dstrbutons of and are respectvely N( 1+ z 1) N( 1z ) and N( 1z 1). We generate data by the same process as above and n estmatng the parameters we assume MA and NMA mechansms. 3.. Ordnal - Normal Model wth gnorable mssng values For our smulaton we have mssng values only for our contnuous varable and we may have Wms and Wobs. For the mssng mechansm we only need to defne as we do not have any mssng value for our ordnal response and we consder model (4) for fndng the condton for MA let 1 1. m m m o o m o o So the constrant (3) wll be reduced to m o 1 m o o o esults Table 1 presents results on the relatve bases of jont and margnal estmates obtaned under MA and NMA mechansms for the fve scenaros. The relatve bases for jont and margnal estmates of and under MA mechansm are generally larger than those for jont and margnal estmates under NMA mechansm.e. f data are not mssng at random such an assumpton on estmatng parameters leads to have based estmates of parameters. So f the mssng mechansm s NMA use of model () whch s assumed to be MA may lead to based estmates. A comparson of the relatve bases of jont and margnal estmates relatve bas of jont estmates suggest that the were generally smaller than those for margnal estmates. Fgures (1)-(3) show relatve Bases of jont and margnal estmates of and under NMA mechansm versus the values of 1 3 and 13. Sold and dashed plots correspond to relatve bases of jont estmates for N 100 and N 00 respectvely. Dotted and dashed-dotted plots correspond to those of margnal estmates for N 100 and N 00 respectvely. The parameter-specfc bases clearly ndcates that both full and margnal lkelhood approaches yeld reasonably unbased estmates wth NMA mechansm. Comparng parameter-specfc estmates show that relatve bases for margnal estmates were generally larger than those for jont estmates. So accordng to Fgures (1)-(3)

16 96 Jafar et al. relatve bases for margnal estmates of and versus the values of 1 3 and 13 are generally larger than those for jont estmates. 4. Applcaton 4.1. Osteopoross of the Spne Data The osteopoross of the spne data set s obtaned from an observatonal study on women n the Taleghan hosptal of Tehran Iran. These data record status of osteopoross of the spne as an ordnal outcome wth three levels for 581 patents. Albrand et al. (003) show some epdemologcal studes have dentfed clncal factors that predct the rsk of hp fractures n elderly women ndependently of the level of bone mneral densty (BMD) such as low body weght hstory of fractures and clncal rsk factors for falls. Also abdomnal obesty needs to be ncluded as a rsk factor for osteopoross and bone loss. Ther results showed that havng a lot of belly fat s more detrmental to bone health than havng more superfcal fat or fat around the hps. Excess fat around the belly may ncrease the rsk of women developng the brttle bone dsease osteopoross. So a bulgng wastlne puts women at rsk of osteopoross. We shall also try to fnd answers for some questons ncludng : (1) How does the type of dwellng affect the level of osteopoross wastlne and BMI of the patent? () How does the job status effect the level of osteopoross wastlne and BMI of the patent? (3) How do the amount of total body calcum and age affect the level of osteopoross wastlne and BMI of the patent? Also we consder the body mass ndex (BMI) and wastlne as contnuous outcomes. Covarates whch may affect the osteopoross of the spne and wastlne are amount of total body calcum (Ca) job status (Job) type of the dwellng (Ta) and age. Table : The varable of nterest and descrptve statstcs for them Dscreet Varables Type Levels Confdence nterval Osteopoross of the spne Ordnal None (6.8.)% Mld ( )% Severe ( )% Mssng ( )% Job status Bnary employee ( )% housekeeper ( )% Type of the dwellng Bnary house ( )% apartment ( )% Contnuous Varables Age Contnuous ( ) year Amount of total body calcum Contnuous ( ) mlgr wastlne BMI Contnuous Contnuous ( ) cm ( ) kgr/cm

17 AAM: Intern. J. Vol. 10 Issue 1 (June 015) 97 Table shows the lst type and descrptve statstcs of varables under study. Ths Table shows that the percentage of severe and mld osteopoross are more than that of none level. Also 67.4% of women lve n apartment and 58.7% of women are housekeeper. A frequency table for the osteopoross of the spne shows that 39% of values are mssng The Pearson correlaton between osteopoross of the spne and BMI responses osteopoross of the spne and wastlne responses and BMI and wastlne are r OSWastlne = 0.45 r OSBMI = 0.08 and r BMIWastlne = Based on the results our smulaton study we can expect to fnd a hgher value of correlaton by our model. These three varables osteopoross of the spne BMI and wastlne are endogenous correlated varables and they have to be modeled smultaneously. Takng nto account the correlaton leads us to obtan a more precse estmaton of standard errors of estmates and so a better nference. These three outcomes osteopoross of the spne wastlne and the ndcator varable for mssng mechansm of Osteopoross of the spne are endogenous correlated varables and they have to be modeled smultaneously. The jont model for these data s OS age Ca Ta Job Wast age Ca Ta Job age Ca Ta Job OS BMI age Ca Ta Job (5) The covarance matrx of the vector of errors ( ) for model (3) s Here ( ) ( ) ( ) ( ) and are parameters that should be estmated. A jont model to use the margnal and jont estmaton for model (3) s specfed by a multvarate Gaussan copula. For our applcaton we have mssng values only for our ordnal varable and we may have Wms OS and Wobs ( BMI Wast ). For mssng mechansm we only need to defne OS as we do not have any mssng value for our contnuous responses and

18 98 Jafar et al m m o o m 1 4 ( ) ( ) mo o so that the needed constrant wll be reduced to m m o o o o esults for Data esults of analyss the margnal and jont estmaton for model (5) wth mssng mechansm are gven n Table 3. For comparatve purposes four models are consdered. The frst model (model I) and the second model (model II) consder the jont estmaton wth NMA and MA mechansm for model (5). Also The thrd model (model III) and fourth model (model IV) uses the margnal estmaton wth NMA and MA mechansm for model (5). Model (I) shows a sgnfcant effect of Ca Ta and age on the value of osteopoross of the spne and sgnfcant effect of Ta on wastlne and shows a weak sgnfcant effect of age on BMI. From these effects we can nfer that the older the patent the lower the value of osteopoross of the spne ; people who lve n apartment have hgher low value of osteopoross of the spne than that of people who lve n a house and the more the amount of calcum of the body of the patent the hgher s the low value of osteopoross of the spne. None of the explanatory varables has any effect on the mssng ndcator for osteopoross of the spne. For model (I) correlaton parameters and 4 are strongly sgnfcant. They show a postve correlaton between wastlne and osteopoross of the spne ( 1 ˆ ) and t shows a postve correlaton between wastlne and BMI ( ˆ 4 ) and a postve correlaton between osteopoross and the mssng ndcator for the spne ( 13 ˆ ). Ths leads to have a NMA mechansm. Model (II) model (III) and model (IV) gve the same results as model (I). To compare model (I) and model (II) we have devance = (p-value < 0.001). So one may prefer model (I). For model (II) the estmated varance of wast and BMI ( ˆ1 and ˆ ) obtaned by model (I) are less than those of model (II). To compare model (I) and model (III) we have devance = (P-value < 0.001). Also for model (I) and model (IV) we have devance = (p-value < 0.001). So one may prefer model (I). Also comparng parameter-specfc estmates for model (I) model (III) model (II) and model (IV) show that loglkelhoods for margnal modellng are generally larger than those of jont modellng.

19 AAM: Intern. J. Vol. 10 Issue 1 (June 015) Concluson We have extended copula-based regresson models for mxed outcomes wth non-gnorable mssng values. For obtanng jont dstrbuton of dscrete and contnuous outcomes wth possblty of mssng values we consder four cases then usng bvarate and multvarate Gaussan copulas we mxed-outcome margnal regresson models. Two lkelhood estmaton strateges are proposed one method uses full lkelhood functon to estmate parameters smultaneously the other apples the IFM method to estmate parameters margnally and shared parameters jontly. A generalzaton of our model for longtudnal studes s stll an ongong research on our part. EFEENCES Albrand. G Munoz. F Sornay-endu. E DuBoeuf. F Delmas. P. D. (003). Independent predctors of all osteopoross-related fractures n healthy postmenopausal women : The OFEL Study. Bone 3(1): Bahram Saman E. Ganjal M. and Khodaddad A. (008). A Latent Varable Model for Mxed Contnuous and Ordnal esponses Journal of Statstcal Theory and Applcatons 7(3): Bahram Saman E. Ganjal M. and Eftekhar Mahabad S. (010). A Latent Varable Model for Mxed Contnuous and Ordnal esponses wth Nongnorable Mssng esponses Sankhya 7-B (1) : Bahram Saman E. Ganjal M. and Zahed H. (011). A Multvarate Latent andom Effect Model for Mxed Contnuous and Ordnal Longtudnal esponses wth Mssng esponses Journal of Statstcal Theory and Applcatons 10(3): Benjamn M. gby. Stasnopoulos D. (003). Generalzed autoregressve movng average models Journal of the Amercan Statstcal Assocaton Catalano P. J. (1997). Bvarate modellng of clustered contnuous and ordered categorcal outcomes Statstcs n Medcne Chb S. and Greenberg E. (1998). Analyss of multvarate probt model Bometrka Cox D. and Wermuth N. (199). esponse models for mxed bnary and quanttatve varables Bometrka 79 (3) De Leon A.. and Carrere K. C. (007). General mxed-data model : extenson of general locaton and grouped contnuous models. Canadan Journal of Statstcs De Leon A.. and Wu B. (011). Copula-based regresson models for a bvarate mxed dscrete and contnuous outcome. Statstcs n Medcne 30() Escarela G. Mena. and Castll-Morales A. (006). A flexble class of parametrc regresson models based on copulas : applcaton to polomyelts ncdence Statstcal Methods n Medcal esearch Ftzmaurce G. M. and Lard N. M. (1995). egresson models for a bvarate dscrete and contnuous outcome wth clusterng Journal of the Amercan Statstcal Assocaton

20 100 Jafar et al. Ftzmaurce G. M. and Lard N. M. (1997). egresson models for mxed dscrete and contnuous responses wth potentally mssng values Bometrcs Ganjal M. (003). A model for mxed contnuous and dscrete responses wth possblty of mssng responses Journal of Scences Islamc epublc of Iran 14 (1) Harry J. and James J. (1998). The Estmaton Method of Inference Functons for Margns for Multvarate Models Unversty of Brtsh Columba. Heckman J. J. D. (1978). Endogenous varable n a smultaneous Equaton system Econometrca 46 (6) Joe H. and u J. J. (1996). The estmaton method of nference functons for margns for multvarate models. Techncal eport Department of Statstcs Unversty of Brtsh Columba. Joe H. (1993). Parametrc famles of bvarate dstrbutons wth gven margnals Journal of Multvarate Analyss Kolev N. Anjos U. and Mendes B. (006). Copulas : a revew and recent developments. Statstcal Models Lttle. J. and ubn D. (00). Statstcal analyss wth mssng data Second edton New york Wley. Lttle. J. and Schluchter M. (1987). Maxmum lkelhood estmaton for mxed contnuous and categorcal data wth mssng values Bometrka Moustak I. Knott M. (000). Generalzed latent trat models. Psychometrka 65(3): Marshall A. (1996). Copulas margnals and jon dstrbutons. In : dstrbutons wth Fxed Margnals and elated Topcs (L. uschendorf B. Swetzer and M.D. Taylor (Eds.). Insttute of Mathematcal Statstcs : Hayward 13-. Newadomska-Bugaj M. and Kowalczyk T. (005). On grade transformaton and ts mplcatons for copulas Brazlan Journal of Probablty and Statstcs Olkn L. and Tate. F. (1961). Multvarate correlaton models wth mxed dscrete and contnuous varables Annals of Mathematcal Statstcs Oakes D. and tz J (000). egresson n a bvarate copula model Bometrka 87 () : Pnto A. T. and Normand S. L. T. (009). Correlated bvarate contnuous and bnary outcomes : Issues and applcatons Statstcs n Medcne Poon W.. and Lee S.. (1987). Maxmum lkelhood estmaton of multvarate polyseral and polychorc correlaton coefcents. Psychometrka 5 (3) : ubn D. B. (1976). Inference and mssng data Bometrca Sammel M. D. yan L. M. Legler J. M. (1997). Latent varable models for mxed dscrete and contnuous outcomes Journal of the oyal Statstcal Socety Seres B Methodologcal Song P. -K. L M. and uan. (009). Jont regresson analyss of correlated data usng Gaussan copula Bometrcs Sklar A. (1959). Fonctons de répartton à n dmensons et leurs marges Publcatons de l Insttut de Statstque de l Unversté de Pars Song P.. (000). Multvarate dsperson models generated from Gaussan copula Scandnavan Journal of Statstcs Tt E. and Kaark E. (1996). Generaton and nvestgaton of multvarate dstrbutons havng fxed dscrete margnals. In : dstrbutons wth Fxed Margnals and elated Topcs

21 AAM: Intern. J. Vol. 10 Issue 1 (June 015) 101 (L. uschendorf B. Swetzer and M.D. Taylor) Insttute of Mathematcal Statstcs : Hayward Van Ophem H. (1999). A general method to estmate correlated dscrete random vector. Econometrc Theory ang. Kang J. Mao K. and Zhang J. (007). egresson models for mxed Posson and contnuous longtudnal data Statstcs n Medcne Zmmer D. M. and Trved P. K. (006). Usng trvarate copulas to model sample selecton and treatment effects : Applcaton to famly health care demand Journal of Busness and Economc Statstcs Table 1: elatve bas of jont (J) and margnal (M) estmates of parameters wth NMA and MA mechansms under fve dfferent scenaros Scenaro Parameter (J) N α 1 β 1 β σ ρ 1 ρ 3 ρ 13 γ 1 θ 1 θ A NMA MA 00 0:01 0: Scenaro B NMA MA C NMA MA D NMA MA E NMA (M) MA N Parameter α 1 β 1 β σ ρ 1 ρ 3 ρ 13 γ 1 θ 1 θ

22 10 Jafar et al. Table 1 (Contnues) A NMA MA B NMA MA C NMA MA D NMA MA E NMA MA Table 3: The margnal and jont estmaton for model (3) wth NMA and MA mechansms. Parameter OS Age(α 1 ) Ca(α ) Ta (α 3 ) Job (α 4 ) Cut pont (θ 1 ) Cut pont(θ ) Jont Es.t S. E Es.t S. E Margnal Es.t S. E Es.t S. E Wast Constant(β 0 ) Age(β 1 ) Ca(β ) Ta(β 3) Job(β 4) σ 1 Jont Es.t S. E Es.t S. E

23 AAM: Intern. J. Vol. 10 Issue 1 (June 015) 103 Margnal Es.t S. E Es.t S. E BM I Constant(β0 ) Age(β 1 ) Ca(β ) Ta(β 3 ) Job(β 4 ) σ Jont Es.t S. E Es.t S. E Margnal Es.t S. E Es.t S. E OS Age(γ 1 ) Ca(γ ) Ta (γ 3 ) Job (γ4 ) Jont Es.t NMA S. E MA Es.t S. E Margnal NMA Es.t S. E MA Es.t S. E Correlaton ρ 1 ρ 3 ρ 14 ρ 3 ρ 4 ρ 34 Jont Es.t NMA S. E MA Es.t S. E Margnal Es. t Models NMA S. E Es.t S. E log-lkelhood Model I Jont NMA Model II Margnal Model III Jont MA Model IV Margnal

24 104 Jafar et al. Fgure 1: elatve Bases of jont and margnal estmates of and versus wth NMA mechansm ( ho1) 1 Fgure : elatve Bases of jont and margnal estmates of and versus ( ho13) 13 wth NMA mechansm

25 AAM: Intern. J. Vol. 10 Issue 1 (June 015) 105 Fgure 3: elatve Bases of jont and margnal estmates of and versus wth NMA mechansm ( ho3) 3