Analysis of Mixed Correlated Bivariate Negative Binomial and Continuous Responses

Transcription

1 Avalable at Appl. Appl. Math. ISSN: Vol. 8, Issue 2 (Deember 2013), pp Applatons and Appled Mathemats: An Internatonal Journal (AAM) Analyss of Mxed Correlated Bvarate Negatve Bnomal and Contnuous Responses F. Raze, E. Bahram Saman and M. Ganjal Department of Statsts Shahd Behesht Unversty Tehran, Iran ehsan_bahram_saman@yahoo.om Reeved: Aprl 31, 2013; Aepted: November 26, 2013 Abstrat A general model for the mxed orrelated negatve bnomal and ontnuous responses s proposed. It s shown how to onstrut parameter of the models, usng the maxmzaton of the full lkelhood. Influene of a small perturbaton of orrelaton parameter of the model on the lkelhood dsplaement s also studed. The model s appled to a medal data, obtaned from an observatonal study on women, where the orrelated responses are the negatve bnomal response of jont damage and ontnuous responses of body mass ndex. Smultaneous effets of some ovarates on both responses are nvestgated. Keywords: Latent varable models, Fatorzaton Models, Mxed Correlated Responses, Lkelhood Dsplaement, Body Mass Index, Jont Damage. AMS-MSC (2010) No.: 62F03 1. Introduton Some medal sene data nlude orrelated dsrete and ontnuous outomes. The example s n the study of the effet of type of aommodaton on body mass ndex as ontnuous response and jont damage as negatve bnomal response (vde, our applaton n Seton 4), where body mass ndex (BMI) and jont damage are orrelated responses n an observatonal study on women. Furthermore, separate analyses gve based estmates for the parameters and msleadng nferene. Consequently, we need to onsder a method n whh these varables an be modeled jontly, for example one may use the fatorzaton of the jont dstrbuton of the outomes or ntrodue an unobserved (latent) varable to model the orrelaton among the multple outomes. Many researhers have nvestgated the mxed orrelated data, for example, Olkn and Tate 404

2 AAM: Intern. J., Vol. 8, Issue 2 (Deember 2013) 405 (1961), Hekman (1978), Poon and Lee (1987), Catalano and Ryan (1992), Ftzmaure and Lard (1995), Sammel et al. (1997), Ln et al. (2000), Gueorgueva and Agrest (2001), Gueorgueva and Sanora (2006), MClluh (2007), Deleon and Carrer (2007) Yang et al. (2007) and Bahram Saman et al. (2008). The man dea of the fatorzaton method s to wrte the lkelhood as the produt of the margnal dstrbuton of one outome and ondtonal dstrbuton of the seond outome gven the frst outomes. Cox and Wermuth (1992), Ftzmaure and Lard (1995) and Catalano and Ryan (1992) dsussed and extended two possble fatorzatons for modelng a ontnuous and bnary outome as funtons of ovarates. Several models usng latent varables have been proposed to analyze multple nonommensurate outomes as funtons of ovarates. Sammel et al. (1997) dsussed a model where the outomes are assumed to be a physal manfestaton of a latent varable and ondtonal on ths latent varable. Another approah based on latent varables was proposed by Dunson (2000). A major dfferene between ths approah and Sammel s approah relates to the assoaton between the responses and the ovarates. In Dunson s approah, the ovarates are not nluded n the model through the latent varable but rather ntrodued separately. Pnto and Normand (2009) used an dea smlar to the saled multvarate mxed model proposed by Ln et al. (2000). They ntrodued a new latent varable model by onstranng the parameters of latent model proposed by Dunson (2000) for dentfablty wthout restrtons on the orrelaton. Yang and Kang (2011) nvestgate the nferental method for mxed Posson and ontnuous longtudnal data wth non-gnorable mssng values. The am of ths paper s to use and extend an approah smlar to that of Sammel et al. (1997) and Dunson (2000), for modelng of a negatve bnomnal and a ontnuous varable, by fatorzaton of the jont dstrbuton and the use of latent modelng of bvarate negatve bnomal and ontnuous outomes. In Seton 2, the models and lkelhoods are gven. In Seton 3, smulaton studes are used to ompare onssteny, effeny and overage of the multvarate approah wth those of the unvarate approah. In Seton 4, the models are used on a medal data set where jont damage and body mass ndex (BMI) are orrelated responses n an observatonal study on women. In these models jont damage s a negatve bnomal response and BMI s ontnuous response and age, the amount of total body alum (Ca), job status (employee or housekeeper) and type of aommodaton (house or apartment) are explanatory varables. We shall nvestgate the effets of these explanatory varables on responses smultaneously. The nfluene of a small perturbaton of orrelaton parameter of the model on the lkelhood dsplaement s also studed. Fnally, n Seton 5, the paper onludes wth some remarks. 2. Models for Mxed Correlated Negatve Bnomal and Contnuous Responses Let denote a ontnuous response and denote a negatve bnomal response for the th of ndvduals and and denote 1 and 1 vetors of ovarates assoated wth

3 406 F. Raze et al. eah response, respetvely Unvarate Models One ommon approah to model multple responses as funtons of ovarates s to gnorable the orrelaton between the responses and ft a separate model to eah response varable. We are usng a lnear regresson model for ontnuous response and a negatve bnomal regresson (,) model,, 1 1, log, 0,, 1 where,...,,,..., and σ 0 s a dsperson parameter A Fatorzaton Models We proposed a model for a orrelated negatve bnomal and a ontnuous responses based on the fatorzaton of the jont dstrbuton of the responses,,. The model for the two responses s wrtten as:,,,,,, 0,, 2 where s the parameter for the regresson oeffent of on. Large absolute values of ndate a strong orrelaton between the two responses. f 0, the two responses are ndependent gven the ovarates. Maxmum lkelhood estmates for the parameters of the fatorzaton method an be obtaned wth ommonly used algorthms for maxmzng the lkelhood. The log-lkelhood funton under the fatorzaton model (2) s n l y, y logfy y, x, x logfy x, 1 Γ y σ n ln y σ ln σμ, 1 Γ σ Γ y 1 σlnσ y lnμ lnfy x.

4 AAM: Intern. J., Vol. 8, Issue 2 (Deember 2013) 407 The vetor of parameters and, the parameters of and should be estmated. The fatorzaton of the jont dstrbuton of and an also be onsder n reverse order:,. The model for the two responses s wrtten as:,,,,, 0,, 2 where s the parameter for the regresson oeffent of on A Latent Varable Model We presented a latent varable model where t s assumed that the observed responses are physal manfestatons of a latent varable. Condtonal on ths latent varable, the responses are assumed to be ndependent and are modeled as funtons of fxed ovarates and a subjetspef latent varable. Let denote the latent varable. The responses are modeled as funton of the latent varable,,,,, 0,, 0,, 3 where s a subjet -spef latent vatable. The latent varable shared by both responses ndues the orrelaton and t s assumed that gven the latent varable, the two responses are ndependent. Also s ndependent of and. However,,, and are not dentfable. There are four parameters to be estmated but only nformaton from the, and,. We have to restrt at least two parameters to obtan an dentfable model. Here, we assume 1 and. We an rewrte 3 and obtan the fnal expresson for a latent model for two responses:,,,,,, 0,1, 0,. 3 The log lkelhood for the model s wrtten as:

5 408 F. Raze et al. l y,y logfy,y x,x, n 1 n log fy x,bfy x,bf b db, 1 Γ y σ ' n exp β λb log ' 1 Γ σ Γ y 1 exp β λb σ ' exp β λb σ 1, ' exp β λb σ ' 2 y 2 x β λb exp b 2 2σ exp 2 db. 2 2πσ 2π The vetor of parameters and, the parameters of, and should be estmated. y, 3. Smulaton Study We used a smulaton study to nvestgate estmates obtaned by the unvarate model, fatorzaton model and latent varable model. In ths seton, smulaton study s used to llustrate the applaton of our proposed models. In ths smulaton, ε, Y and were generated from a normal dstrbuton, negatve bnomal dstrbuton and a normal dstrbuton. We thus generated data sets wth dfferent ases (1, 2 and 3). For eah ase, we generated 1000 samples. The data generated from the followng ases: Case 1: Case 2: Y β β β β β ε, Y Y NB μ,0.66, ε N 0,5. logμ β b0 β b1 1 β b2 2 β b3 3 β b4 4 η Y E Y, E Y β 0 β 1 1 β 2 2 β 3 3 β 4 4, Y Y β 0 β 1 1 β 2 2 β 3 3 β 4 4 ξ Y E Y ε, E Y μ, Y NBμ,0.66, logμβ b0 β b1 1 β b2 2 β b3 3 β b4 4, ε N 0,5.

6 AAM: Intern. J., Vol. 8, Issue 2 (Deember 2013) 409 Case 3: β 0 Y β 1 1 β 2 2 β 3 3 β 4 4 λbε, Y NB μ,0.66, logμβ b0 β b1 1 β b2 2 β b3 3 β b4 4 λb, ε N 0,5, b N 0,1. Also and are generated from 100,2, and are generated from gamma2,2 and,, and are generated from Bernll0.5. The vetor of oeffents assoated wth the ovarate was hosen,,,, 30,0.1, 0.130,1.715,0.981,,,,, 2,0.003, 0.223,0.39, and,,,, 5,0.66,0.435,0.305,0.25. The data generated from eah ase s modeled usng that ase and unvarate approah (gnorng the orrelaton between the outomes),, 0.66,, 0,5. The models were ftted usng nlm from R to assure that the same numeral algorthms were used to maxmze the lkelhoods. Table 1. Results of the smulatons study for ase 1 Case 1 Model (2) Un. model Parameter Real value Est. S.E Est. S.E. NJ Constant Age Ca Job TA BMI Constant Age Ca Job TA

7 410 F. Raze et al. Table 2. Results of the smulatons study for ase 2 Case 2 Model (2 ) Un. model Parameter Real value Estmate S.E. Este S.E. NJ Constant Age Ca Job TA BMI Constant Age Ca Job TA Table 3. Results of the smulatons study for ase 3 Case 3 Model (3) Un. model Parameter Real value Est. S.E. Est. S.E. NJ Constant Age Ca Job TA BMI Constant Age Ca Job TA Tables 1-3 ontans the average estmated values of σ2,,,,,,,,,,,σ and (for model (3)), (for model (2 )) and (for model (2)) for n = The results are summarzed as follows. The parameter estmates by the model (2), model (2 ) and model (3) are lose to the true values of the parameters. 4. Applaton and Senstvty Analyss 4.1. Applaton In ths seton, we use the Mxed orrelated models n (2) and (3) for the medal data set desrbe n the followng subseton. The medal data set s obtaned from an observatonal study on women n the Taleghan hosptal of Tehran, Iran. These data reord the number of jont

8 AAM: Intern. J., Vol. 8, Issue 2 (Deember 2013) 411 damage (NJ) as negatve bnomal responses and body mass ndex (BMI) as ontnuous responses for 163 patents. These patents are heavy body. Jont damage s a dsease of bone n whh the bone mneral densty (BMD) s redued, bone mro arhteture s dsrupted and the amount and varety of non-ollage nous protens n bone s altered. BMI s a statstal measure of the weght of body mass ndex. A person body mass ndex may be aurately alulated usng any of the formulas suh as where W s weght and H s heght. Also, The heavy body an result n damages to jonts of knee and ankle, et. These two varables, jont damage and BMI orrelated varables, and they have to be modeled. Explanatory varables whh affet these varables are: (1) amount of total body alum (Ca), (2) job status (Job, employee or housekeeper), (3) type of the aommodaton (Ta, house or apartment) and (4) age. We used a test to nvestgate over dsperson for ount response. Devane and Pearson Ch- Square dvded by the degrees of freedom are used to detet over dsperson or under dsperson n the Posson regresson. Values greater than 1 ndate over dsperson, that s, the true varane s bgger than the mean, values smaller than 1 ndate under dsperson, the true varane s smaller than the mean. Evdene of under dsperson or over dsperson ndates nadequate ft of the Posson model. We an test for over dsperson wth a lkelhood rato test based on Posson and negatve bnomal dstrbutons. Ths test tests equalty of the mean and the varane mposed by the Posson dstrbuton aganst the alternatve that the varane exeeds the mean. For the negatve bnomal dstrbuton, the varane of ount response ( ) s 2, Var Y E Y ke Y ount ount ount where 0, the negatve bnomal dstrbuton redues to Posson when 0. The null hypothess s : 0 and the alternatve hypothess s : 0). Use the (lkelhood rato) test, that s, ompute statst, -2( L L (Posson) - LL (negatve bnomal)), where LL s log(lkelhood). The asymptot dstrbuton of the LR statst has probablty mass of one half at zero and one half Ch-sq dstrbuton wth 1 df (see Cameron and Trved, 1998). To test the null hypothess at the sgnfane level, use the rtal value of Ch-sq dstrbuton orrespondng to sgnfane level, that s rejet H 0 f statst,. In ths data, we alulated LL (Posson for NJ)= , LL (negatve bnomal for NJ) = and -2(LL (Posson) - LL (negatve bnomal)) = (wth 1 d.f. and P- value=0.003). So, NJ has over dsperson and NJ s negatve bnomal dstrbuton. Results of usng three models (model 1, 2 and 3) are gven n Table 4. We used the unvarate model (model 1), the fatorzaton model (model 2) and the latent varable model (model 3) as desrbed Seton 2.4) to estmate the parameters of the models. Unvarate model (model 1) shows sgnfant no effet of ovarates on BMI and the number of jont damage. For the fatorzaton models (model 2) shows sgnfant effet of amount of total body alum and job status on the frequeny of jont damage. From these effets we an nfer

9 412 F. Raze et al. that the amount of total body alum have a negatve mpat on the frequeny of number of jont damage. Job status has a postve mpat on the frequeny of number of jont damage. ndates that the nrease of dsperson has a postve mpat on the frequeny of number of jont damage. In these models, orrelaton parameter s strongly sgnfant. It shows a postve orrelaton between BMI and the number of jont damage. The estmated varane of BMI ( obtaned by the fatorzaton model s less than those of unvarate model. The fatorzaton model (model 2) gves the same results as the latent varable model (model 3). In the latent varable model, orrelaton parameter s strongly sgnfant. The better performane of the latent varable model over the fatorzaton model. Table 4. Estmaton results of the four models (NJ: Negatve Bnomal Response and BMI: Contnuous Response) of real data, parameter estmates hghlghted n bold are sgnfant at 5 % level.) Model Model (1) Model (2) Model (3) Parameter Est. S.E. Est. S.E. Est. S.E. NJ Constant Age Ca Job TA BMI Constant Age Ca Job TA loglke Senstvty Analyss Lkelhood dsplaement s a very mportant onept as t provdes a general approah to study the problem of nfluene. The method of loal nfluene was ntrodued by Cook (1986) and modfed by Bllor and Loynes (1993) as a general tool for assessng the nfluene of loal departures from the assumptons underlyng the statstal models. Perturbatons of the model nfluene key results of the analyss are to ompare the results derved from the orgnal and perturbed models. The nfluene graphs ntrodued n ths Seton are smply deves to faltate suh omparsons when the behavor of the parameter estmates s of nterest. Ths artle shows that loal-nfluene analyss of perturbatons of the orrelaton parameters of models. The log-lkelhood for the unperturbed and perturbed models are denoted by and, respetvely. The perturbed lkelhood q 1 obtaned after the lkelhood have been perturbed by an amount where s a vetor whh s restrted to some open subset of. Then the lkelhood dsplaement LDω s defned by

10 AAM: Intern. J., Vol. 8, Issue 2 (Deember 2013) 413 Generally, one ntrodue perturbatons nto the model through the q 1 vetor whh s q restrted to some open subset of R and θ s p 1vetor of unknown parameters. Cook (1986) proposed the maxmum normal urvature. The s defned by Cmax maxlc l, where s the lfted lne n the dreton l an be easly alulated by T T 1 C 2 l Δ L Δl, 4.2 l 2 l θ ω where Δ θθ,ω ˆ and defne as the p n matrx wth Δ 0 as ts th olumn ω η and denote the p p matrx of seond-order dervatves of lθ ω0, where there s an n, wth respet to θ, also evaluated at ˆ. Obvously, C an be alulated for any dreton l. One evdent hoe s the vetor l ontanng one n the th poston and zero elsewhere, orrespondng to the perturbaton of the th weght only. The orrespondng loal nfluene T measure, denoted by, then beomes 1 Cl 2 Δ L Δ. Another mportant dreton s the dreton of maxmal normal urvature. Also, s the largest Egen value of T 1 Δ L Δ and l s the orrespondng egenvetor. Let see how we an use ths approah for our purposes. Condton for ndependent responses ( 0) and ondton for Posson dstrbuton for NJ s ( 1) whh gves the followng ondton for not havng over dsperson 10 n the model (3). We an use maxmal normal urvature for the effet of perturbaton from ndependent responses to orrelated responses and the perturbaton from Posson dstrbuton to negatve bnomal dstrbuton (or over dsperson). Let,. Here, 0,0 for eah model and 2. Denote the log-lkelhood funton by, where s the ontrbuton of the th ndvdual to the log-lkelhood and s the parameter vetor. Here, s the log-lkelhood funton whh orresponds to ndependent responses. Suppose an be perturbed around 0. Let be MLE estmator for obtaned by maxmzng and let denote the MLE estmator for under. Now one ompare and as loal nfluene. Strongly dfferent estmates show that the estmaton proedure s hghly senstve to suh modfaton. We an quantfy the dfferenes usng maxmal normal urvature defned as (4.2). To searh for Senstvty analyss we fnd. Ths s onfrmed by the urvature

11 414 F. Raze et al Ths urvature ndates extreme loal sensvty. These urves show a hgh urvature aroundω, so that the dfferng values of ω affets the model (3) results, hene fnal results of the model (3), s hghly senstve to orrelated responses and negatve dstrbuton for NJ. 5. Conluson We presented dfferent approahes to model orrelated negatve bnomal and ontnuous outomes. We proposed new multvarate varable models. We also mplemented lkelhood approah based. Smulaton results suggest that the four approahes lead to onsstent estmates of the regresson parameters. Ths suggests that the orrelaton between the outomes wll not be worse than the assumpton of ndependene. In ontrast to the fatorzaton approah, the latent varable model presented s easly extended to several ontnuous and/or several negatve bnomal outomes by nludng addtonal latent varables as long as the outomes are postvely orrelated. However, some of the assumptons of the model, suh as the dstrbuton of the latent varables, are not easly assessed. In the presene of mssng observatons n one of the outomes, the fatorzaton approah only uses the omplete ases or t requres the EM-algorthm to nlude all the ases n the analyss (Ftzmaure and Lard, 1997). Ths s not the ase wth the latent model. If the mssng data s mssng at random or mssng ompletely at random (Lttle and Shluhte, 1987), ths stuaton an be easly aommodated due to the ondtonal ndependene of the outomes gven the latent varable. Furthermore, the latent varable model s easly ftted usng standard software. REFERENCES Bahram Saman, E., Ganjal, M. and Khodaddad, A. (2008). A Latent Varable Model for Mxed Contnuous and Ordnal Responses. Journal of Statstal Theory and Applatons, (3), Bllor, N. and Loynes, R.M. (1993). Loal Influene: A New Approah, Comm. Statst.-Theory Meth., Catalano, P. and Ryan, L. M. (1992). Bvarate latent varable models for lustered dsrete and ontnuous outomes. Journal of the Ameran Statstal Assoaton, (3), Cameron, A.C. and Trved, P. K. (1998). Regresson Analyss of Count Data, Cambrdge: Cambrdge Unversty Press. Cox, D. R. and Wermuth, N. (1992) Response models for mxed bnary and quanttatve varables, Bometrka, (3), Cook, R. D. (1986). Assessment of Loal Influene (wth dsusson). Journal. Royal Statst. So., Ser. B.,, De Leon, A. R. and Carr re, K. C. (2007). General mxed-data model: Extenson of general loaton and grouped ontnuous models. Canadan Journal of Statsts, (4), Dunson D. B. (2000). Bayesan latent varable models for lustered mxed outomes, Journal of the Royal Statstal Soety, Seres B: Statstal Methodology, (2), Ftzmaure, G. M. and Lard, N. M. (1995). Regresson models for Bvarate dsrete and ontnuous outome wth lusterng, Journal of the Ameran Statstal Assoaton, Ftzmaure, G. M. and Lard, N. M. (1997). Regresson models for mxed dsrete and ontnuous

12 AAM: Intern. J., Vol. 8, Issue 2 (Deember 2013) 415 responses wth potentally mssng value. Bometrs, Gueorgueva, R.V. and Agrest, A. (2001). Correlated Probt Model for Jont Modelng of Clustered Bnary and Contnuous Response, Journal of the Ameran Statstal Assoaton, Gueorgueva, R.V. and Sanaora, G. (2006). Jont Analyss of Repeatedly Observed Contnuous and Ordnal Measures of Dsease Severty, Statsts n Medne (8), Hekman, J. J. D. (1978). Endogenous varable n a smultaneous Equaton system, Eonometral, (6), Ln., Ryan L., Sammel M., Zhang D., Padungtod C. and u. (2000). A saled lnear mxed model for multple outomes. Bometrs, (2), Lttle, R. J. and Shluhter M. (1987). Maxmum lkelhood estmaton for mxed ontnuous and ategoral data wth mssng values. Bometrka, MCulloh, C. (2007). Jont modelng of mxed outome type usng latent varables, statstal methods n Medal Researh, (1), Olkn L. and Tate R. F. (1961). Multvarate orrelaton models wth mxed dsrete and ontnuous varables, Annals of Mathematal Statsts, Sammel M. D., Ryan L. M and Legler J. M. (1997). Latent varable models for mxed dsrete and ontnuous outomes, Journal of the Royal Statstal Soety, Seres B: Methodologal, Pnto, A. T. and Normand, S. L. T. (2009). Correlated bvarate ontnuous and b- nary outomes: Issues and applatons. Statsts n Medne, Poon, W. Y. and Lee, S. Y. (1987). Maxmum lkelhood estmaton of multvarate polyseral and polyhrom orrelaton oeffents. Psyhometrka (3): Yang, Y., Kang, J., Mao, K. and Zhang, J. (2007), Regresson models for mxed Posson and ontnuous longtudnal data, Statsts n Medne; Yang, Y. and Kang, J. (2011). Jont analyss of mxed Posson and ontnuous longtudnal data wth non-gnorable mssng values, Computatonal Statsts Data Analyss,