Bayesian Inference on Mixed-effects Models with Skewed Distributions for HIV longitudinal Data

Transcription

1 University f Suth Flrida Schlar Cmmns Graduate Theses and Dissertatins Graduate Schl January 2012 Bayesian Inference n Mixed-effects Mdels with Skewed Distributins fr HIV lngitudinal Data Ren Chen University f Suth Flrida, rchen@health.usf.edu Fllw this and additinal wrks at: Part f the Bistatistics Cmmns, and the Medicine and Health Sciences Cmmns Schlar Cmmns Citatin Chen, Ren, "Bayesian Inference n Mixed-effects Mdels with Skewed Distributins fr HIV lngitudinal Data" (2012). Graduate Theses and Dissertatins. This Dissertatin is brught t yu fr free and pen access by the Graduate Schl at Schlar Cmmns. It has been accepted fr inclusin in Graduate Theses and Dissertatins by an authrized administratr f Schlar Cmmns. Fr mre infrmatin, please cntact schlarcmmns@usf.edu.

2 Bayesian Inference n Mixed-Effects Mdels with Skewed Distributins fr HIV Lngitudinal Data by Ren Chen A dissertatin submitted in partial fulfillment f the requirements fr the degree f Dctr f Philsphy Department f Epidemilgy and Bistatistics Cllege f Public Health University f Suth Flrida Majr Prfessr: Yangxin Huang, Ph.D. Getachew A. Dagne, Ph.D. Yiliang Zhu, Ph.D. Patricia J. Emmanuel, M.D. Henian Chen, M.D., Ph.D. Date f Apprval: Nvember 30, 2012 Keywrds: mixed-effects mdel, HIV dynamics, Bayesian analysis, Markv chain Mnte Carl, skewed distributin Cpyright c 2012, Ren Chen

3 Dedicatin This wrk is dedicated t my mther, Zhenhua Zhang, my husband Ralph and my lvely daughter Victria, wh have been giving me their selfless supprt fr my study and research during all these years.

4 Acknwledgments First f all, I wuld like t express sincere appreciatin t my majr advisr, Dr. Yangxin Huang, wh has devted himself n mentring and nurishing me thrughut my study and research. I culd nt have gne s far withut his guidance and supprt. What I have learned frm him and his research enables me t vercme many hurdles and accmplish what I have dreamed f. I believe his prfund psitive impact will cntinue t benefit my future prfessinal wrk. I wuld als like t thank Dr. Henian Chen, my cmmittee member and directr f bistatistics cre in which I am wrking nw, wh prvided great supprt and career guides. Special thanks are given t Drs. Yiliang Zhu, Getachew A. Dagne, and Patricia J. Emmanuel fr psitive input, helpful suggestins cncerning the research presented in this dissertatin and fr taking valuable time ut f their busy schedules t serve n my dissertatin cmmittee. I als want t acknwledge and thank Dr. Jane Carver and all faculties and staff members at the Department f Epidemilgy and Bistatistics, Cllege f Public Health and Clinical and Transitinal Science Institute, Mrsani Cllege f Medicine, University f Suth Flrida. Last but nt the least, I wuld like t thank Dr. Michael Schell fr being the Chairman f my defense cmmittee.

5 Table f Cntents List f Tables... iii List f Figures... iv Abstract... vi 1. Intrductin / Literature Review Backgrund HIV dynamic mdels Statistical inference in HIV dynamics Skew-elliptical distributin Skew-t distributin Skew-nrmal distributin Specific aims Mixed-effects mdels with skewed distributins fr time-varying viral decay rate in HIV dynamics Intrductin HIV dynamic mdels with time-varying decay rate functin Bayesian mixed-effects mdels with skewed distributin Applicatin: AIDS clinical trial data AIDS clinical trial data and specific mdels Results Cnclusin and discussin Simultaneus Bayesian inference fr linear, nnlinear and semiparametric mixed-effects mdels with skew-nrmality and measurement errrs in cvariates Intrductin Bayesian inference n jint mdels with skew-nrmal distributins Measurement errr mdels with a skew-nrmal distributin Skew-nrmal Bayesian semiparametric nnlinear mixed-effects jint Mdels Analysis f AIDS clinical data Data and mdels Results f analysis Discussin Bivariate linear mixed-effects mdels with an applicatin t AIDS study using skew-elliptical distributins Intrductin Data and mdels with the skew-elliptical distributins Mtivating data set i

6 Bivariate linear mixed-effects mdels with ST distributin Bayesian Inference Data analysis Specific mdel and implementatin Mdel cmparisn results Estimatin results based n the ST Cmparisn between bivariate (CD4 and CD8) mdel and univariate (CD4 r CD8 ) mdel Cnclusin and discussin Overall discussin and cnclusins List f References Appendices Appendix A: WinBUGS Cde fr ST-Mdel IV-Equatin (2.12) in Chapter Appendix B: WinBUGS Cde fr Mdel I in Chapter Appendix C: WinBUGS Cde fr ST Bivariate Mdel- Equatin (4.5) in Chapter Appendix D: WinBUGS Cde fr ST Univariate Mdel-Equatin in Chapter Appendix E: Permissin f reprint fr Chapter Abut the authr... End Page ii

7 List f Tables Table 2.1. DIC, EPD and RSS amng the five mdels randm errrs are assumed t fllw ST Table 2.2. Fr Mdel IV, DIC values under different distributin assumptins Table 2.3. A summary f the estimated psterir values (based n A5055 data) Table 2.4. DIC values amng the five mdels in the 15 samples Table 2.5. Fr Mdel IV, amng the 15 samples in A398, DIC values under different distributin assumptins Table 3.1. A summary f the estimated PM f interested ppulatin (fixed-effects) and precisin parameters Table 4.1. Mdel cmparisn using DIC, EPD and RSS criteria Table 4.2. A summary f the estimated psterir mean (PM) f ppulatin (fixed-effects) parameters, as well as the crrespnding SD) and 95% CI Table 4.3.A summary f the estimated PM f dispersin matrix parameter, as well as the crrespnding SD and 95% CI Table 4.4. Bivariate and univariate mixed-effect mdels: a summary f the estimated PM f ppulatin (fixed-effects) parameters, as well as the crrespnding SD and 95% CI iii

8 List f Figures Figure 1.1. Diagram f HIV... 2 Figure 1.2. HIV replicatin... 3 Figure 1.3. A generalized graph f the relatinship between HIV cpies and CD4 cell... 4 Figure 1.4. The univariate skew-t and skew-nrmal density functins Figure 2.1. Prfile f viral lad in natural lg scale frm a clinical trial study-a Figure 2.2. The histgram f viral lad in natural lg scale fr 44 patients in a clinical trial study-a Figure 2.3. Prfiles f viral lad in natural lg scale fr fur randmly selected patients amng A5055 and A398, respectively Figure 2.4. Individual estimates f viral lad trajectries fr three randmly selected patients based n nrmal, t, SN and ST distributin assumptin in Mdel IV Figure 2.5. The bserved values versus fitted values f ln(rna) based n N, Student-t, SN r ST distributin fr randm errrs Figure 2.6. Q-Q plt Figure 2.7. Bxplt based n the DIC value frm 15 samples frm A Figure 2.8 Bxplt based n the DIC value frm 15 samples in A398, in Mdel IV with different distributin assumptins Figure 2.9. Prfile f viral lad in ln scale and decay rate in rebund and n rebund grup Figure 3.1. The histgrams f viral lad (in ln scale) and standardized CD4 cell cunt measured Figure 3.2. Prfiles f viral lad (respnse) in natural lg scale and CD4 cell cunt (cvariate)fr three randmly selected patients Figure 3.3. Prfiles f viral lad in ln scale frm an AIDS clinical trial study Figure 3.4. The individual estimates f viral lad trajectries fr iv

9 three randmly selected patients based n the BLME (left), BNLME (center) and BSNLME (right) mdels with a nrmal (dtted line) r SN (slid line) randm errrs Figure 3.5. The bserved values versus fitted values f ln(rna) based n the BLME (left), BNLME (center) and BSNLME (right) mdels with a nrmal r SN randm errr Figure 3.6. Crrelatins between baseline ln(rna) levels and the subject-specific first phase viral decay rates Figure 4.1. The histgram f CD4 and CD8 cell cunt (standardized scale) measured in peripheral bld fr 44 patients Figure 4.2. The trajectry prfiles f CD4 and CD8 cell cunt (standardized scale) measured in peripheral bld fr 44 patients Figure 4.3. The baseline ( ) and failure time( ) IC50 fr IDV/RTV drugs (tp panel), the minimum drug cncentratin (Cmin) fr tw drugs (middle panel) fr the 44 individual patients and adherence rates f IDV/RTV drugs (bttm panel) ver time fr the tw representative patients. SD and CV dente the standard deviatin and cefficient f variatin, respectively Figure 4.4. Bx plt f skewness parameter fr the SN and ST Mdels Figure 4.5. Marginal psterir densities estimates f parameter ν fr the ST Mdel Figure 4.6. The cefficient f time fr CD4 and CD8 cell cunt in rebund and n rebund grup v

10 Abstract Statistical mdels have greatly imprved ur understanding f the pathgenesis f HIV-1 infectin and guided fr the treatment f AIDS patients and evaluatin f antiretrviral (ARV) therapies. Althugh varius statistical mdeling and analysis methds have been applied fr estimating the parameters f HIV dynamics via mixed-effects mdels, a cmmn assumptin f distributin is nrmal fr randm errrs and randm-effects. This assumptin may lack the rbustness against departures frm nrmality s may lead misleading r biased inference. Mrever, sme cvariates such as CD4 cell cunt may be ften measured with substantial errrs. Bivariate clustered (crrelated) data are als cmmnly encuntered in HIV dynamic studies, in which the data set particularly exhibits skewness and heavy tails. In the literature, there has been cnsiderable interest in, via tangible cmputatin methds, cmparing different prpsed mdels related t HIV dynamics, accmmdating skewness (in univariate) and cvariate measurement errrs, r cnsidering skewness in multivariate utcmes bserved in lngitudinal studies. Hwever, there have been limited studies that address these issues simultaneusly. One way t incrprate skewness is t use a mre general distributin family that can prvide flexibility in distributinal assumptins f randm-effects and mdel randm errrs t prduce rbust parameter estimates. In this research, we develped Bayesian hierarchical mdels in which the skewness was incrprated by using skew-elliptical (SE) distributin and all f the inferences were carried ut thrugh Bayesian apprach via Markv chain Mnte Carl (MCMC). Tw real data set frm HIV/AIDS clinical trial were used t illustrate the prpsed mdels and methds. This dissertatin explred three tpics. First, with an SE distributin assumptin, we cmpared mdels with different time-varying viral decay rate functins. The effect f skewness n the mdel fitting was als evaluated. The assciatins between the estimated decay rates based n the best fitted mdel and clinical related variables such as baseline HIV viral lad, CD4 cell cunt and lngterm respnse status were als evaluated. Secnd, by jintly mdeling via a Bayesian apprach, we simultaneusly addressed the issues f utcme with skewness and a cvariate prcess with vi

11 measurement errrs. We als investigated hw estimated parameters were changed under linear, nnlinear and semiparametric mixed-effects mdels. Third, in rder t accmmdate individual clustering within subjects as well as the crrelatin between bivariate measurements such as CD4 and CD8 cell cunt measured during the ARV therapies, bivariate linear mixed-effects mdels with skewed distributins were investigated. Extended underlying nrmality assumptin with SE distributin assumptin was prpsed. The impacts f different distributins in SE family n the mdel fit were als evaluated and cmpared. Real data sets frm AIDS clinical trial studies were used t illustrate the prpsed methdlgies based n the three tpics and cmpare varius ptential mdels with different distributin specificatins. The results may be imprtant fr HIV/AIDS studies in prviding guidance t better understand the virlgic respnses t antiretrviral treatment. Althugh this research is mtivated by HIV/AIDS studies, the basic cncepts f the methds develped here can have generally brader applicatins in ther fields as lng as the relevant technical specificatins are met. In additin, the prpsed methds can be easily implemented by using the publicly available WinBUGS package, and this makes ur apprach quite accessible t practicing statisticians in the fields. vii

12 1 Intrductin / Literature Review 1.1. Backgrund The histry f human immundeficiency virus (HIV) and acquired immundeficiency syndrme (AIDS) can be traced back t In Califrnia and New Yrk, varius dctrs reprted that a small number f hmsexual men had been diagnsed with rare frms f Kapsi s sarcma and Pneumcystis carinii pneumnia, which are generally fund in peple with seriusly cmprmised immune systems. By mid 1982, it was clear that they were mre than islated incidents and in September f that year, Centers fr Disease Cntrl and Preventin (CDC) used the term AIDS as an fficial diagnsis fr this disease. Sn it was realized peple culd get HIV if they engaged in certain activities such as having unprtected sex, sharing needles, receiving a bld transfusin and if they were brn t a mther with HIV infectin. HIV infectin is cnsidered as a pandemic by the Wrld Health Organizatin (WHO). By the end f 2010 (UNAIDS 2011), an estimated 34 millin peple were living with HIV, up 17% frm Apprximately 16.8 millin are wmen and 3.4 millin are less than 15 years ld. The estimated prevalence f HIV varies dramatically amng regins: the mst affected regin is Sub- Saharan Africa and it accunts 68% HIV cases and 66% f HIV deaths; abut 5% f the adult ppulatin in this area is infected. Prevalence is the lwest in Western and Central Eurpe (0.2%) and East Asia (0.1%). With the significant expansin f HIV preventin prgrams and access t antiretrviral therapy, the number f new infectins and HIV/AIDS related deaths are decreasing. In 2010, there were 2.7 millin new HIV infectins, which was 15% less than in 2001 and 21% less than the number f new infectins that ccurred at the peak f the epidemic in 1997, and there were 1.8 millin AIDS related deaths, which was 18% less than in In the United States, since the beginning f the HIV and AIDS epidemic, ver half a millin peple have died f AIDS, and currently arund 1.2 millin peple are living with HIV, hwever 20% f them are unaware f their infectin. HIV belngs t a class f viruses knwn as retrviruses. Retrviruses use ribnucleic acid 1

13 (RNA) t encde their genetic infrmatin and RNA is translated int dexyribnucleic acid (DNA) during its life-cycle by a specific viral enzyme called reverse transcriptase. Viruses cannt grw r reprduce n their wn s they must infect cells f a living rganism in rder t survive and make new cpies. There are tw types f HIV, HIV-1 and HIV-2, and bth riginated thrugh the evlutin f simian immundeficiency virus (SIV). Althugh bth types can be transmitted by sexual cntact, bld, and frm mther t child, cmpared with HIV-2, HIV-1 is mre easily transmitted and patients with HIV-1 infectin will mre quickly prgress t AIDS. Therefre, it is respnsible fr the majrity f glbal HIV infectins and AIDS cases. Figure 1.1: Diagram f HIV. HIV virin is rughly spherical and has a diameter f abut 1/10,000 mm, which is 60 times smaller than a red bld cell. As Figure 1.1 shws, the basic structure f HIV includes: (i) a lipid membrane. It is the uter envelpe f the virus and cnsists f tw layers f lipids. Different prteins are embedded in this viral envelpe and frm spikes cnsisting f glycprtein (gp) 120 and transmembrane gp41. Gp120 is needed t attach the virin t the hst cell, and gp41 is critical fr the cell fusin prcess; (ii) the HIV matrix prteins. They lie between the envelpe and cre; (iii) the viral cre. It cntains the viral capsule prtein p24 which surrunds tw single strands f RNA and the enzymes needed fr HIV replicatin, such as reverse transcriptase, prtease, ribnuclease, 2

14 and integrase. Amng the nine virus genes cded n ne lng stand f RNA, three genes, gag, pl and env, cntain infrmatin needed t make structural prteins fr new virus particles. Figure 1.2: HIV replicatin. There are six steps invlved in HIV infectin and replicatin (Figure 1.2). Step 1: binding and entry. By binding specific receptrs n the surface f a target cell, such as CD4 psitive T cells (i.e., CD4 cells), macrphages and micrglial cells, HIV enters the hst cells. The CD4 receptr is necessary but nt sufficient t permit virus entry. The secndary receptrs are chemkine receptrs that bind t chemkines and are needed t facilitate the entering (Dragic et al., 1996); Step 2: reverse transcriptin. HIV uses an enzyme knwn as reverse transcriptase t cnvert its RNA int DNA; Step 3: integratin. HIV DNA enters the nucleus f the target cell and inserts itself int the cell s DNA, where it may hide and stay inactive fr years; Step 4: transcriptin. HIV DNA instructs the cell t make many cpies f the riginal virus, alng with sme mre specialized genetic materials fr making lnger prteins; Step 5: assembly. A special enzyme called prtease cuts the lnger HIV prteins int individual prteins. When these cme tgether with the virus genetic material, a new virus is assembled; Step 6: release. The virus pushes itself ut f the hst cell and takes with it part f the cell membrane. This uter part cvers the virus and cntains all f the structures necessary fr the virus t bind t a new CD4 cell and begin the virus life cycle prcess again. Knwing these 3

15 steps is critical in the develpment f medicatins that can interrupt the replicatin cycle. Current treatment strategy invlves a cmbinatin f drugs that target different steps f HIV s life cycle such as entry inhibitrs that prevent binding f HIV t the CD4 receptr, reverse transcriptase inhibitrs that prevent the HIV RNA frm being transcribed int DNA and prtease inhibitrs that prevent the assembly. Figure 1.3: A generalized graph f the relatinship between HIV cpies and CD4 cell cunt ver the average curse f untreated HIV infectin. HIV infectin generally can be brken int fur stages: primary infectin, clinical latency (asymptmatic) stage, symptmatic stage and AIDS (Figure 1.3). Stage 1: primary infectin. This stage can last fr a few weeks and patients are ften accmpanied by a shrt flu-like symptm, such as headache, nausea, sre thrat r fever. During this stage, the amunt f HIV in the peripheral bld increases sharply and the immune system starts t respnd t the virus by prducing HIV antibdies and cyttxic lymphcytes. This prcess is knwn as sercnversin. Since enzymelinked immunsrbent assay (ELISA), which is the mst cmmnly used methd t test fr HIV, uses bld, ral fluid r urine t detect HIV antibdies, the result may be negative if the ELISA test is dne befre sercnversin is cmplete. There is a crrespnding decrease in the number f CD4 cells and an increase in CD8 cells. Patients are extremely infectius during this stage. Stage 2: clinically asymptmatic stage. This stage lasts fr an average f ten years and patients are free 4

16 frm majr symptms, althugh sme may have swllen glands. During this stage, the immune system is able t munt an effective respnse, s the viral lad starts t decrease and then stays at a cnstant lw level. The number f CD4 cells rises and then slwly falls. Peple remain infectius and HIV antibdies are detectable in the bld s the antibdy test will shw a psitive result. Althugh the viral lad remains at a cnstant lw level fr years, virus replicatin is very active during this perid. Stage 3: symptmatic HIV infectin. Eventually, the immune system is severely damaged r burned ut by years f activity. HIV mutates and becmes mre pathgenic leading mre immune cells destructin, while the bdy fails t keep up with replacing the lst cells. Symptmatic HIV infectin is mainly caused by pprtunistic infectins that the nrmal immune system usually wuld prevent. This stage f HIV infectin is ften characterized by multi-system diseases and infectins ccurring in almst all bdy systems. Withut any effective treatment, the immune suppressin will cntinue t wrsen. Stage 4: AIDS. Once the CD4 cell cunt is less than 200/mL r CD4 cell percentage is less than 15, AIDS will be diagnsed. The CD4 cell, the majr target cell fr HIV, is a T lymphcytes. Under the micrscpe, lymphcytes can be divided int large and small lymphcytes. Large lymphcytes include natural killer (NK) cells, while small lymphcytes cnsist f T cells that mature frm thymus and B cells that are bursa-derived. T cells are invlved in cell-mediated immunity whereas B cells are primarily respnsible fr humral immunity (relating t antibdies). The CD4 cell is a subset f T cells that express the cluster f differentiatin 4 (CD4) and it is als knwn as T helper cell. These cells assist ther white bld cells in immunlgic prcesses. The nrmal CD4 cells accunt fr 32% t 68% f ttal number f lymphcytes and range between /mL. Withut effective HIV treatment, the hallmark decrease in CD4 cells that ccurs during AIDS results in such a weakened immune system that the bdy can n lnger fight infectins r certain cancers, and eventually death ensues. The mechanisms f CD4 cell death in HIV infectin are still nt fully understd and are ne f the mst cntrversial issues in AIDS research. The mechanisms by which HIV can directly induce infected cell death include plasma membrane disruptin r increased permeability due t cntinuus budding f the virin (Facui, 1988), increasing cellular txicity due t build up f un-integrated liner viral DNA (Levy, 1993) and inactivatin f anti-appttic genes (Nie et al., 2002). Hwever, a lngstanding questin in HIV bilgy is hw HIV viruses kill s many CD4 cells, despite the fact that mst f them appear t be bystander cells that are nt infected (Embretsn et al., 1993). Re- 5

17 cent data demnstrate that the majrity uninfected CD4 cells in peripheral bld and lymph ndes underg three types f apptsis (Varbanv et al., 2006), which is a tightly regulated prgrammed cell death (Evan et al., 1998). Several HIV prteins, such as Env and Vpr, have been fund t be able t up-regulate Fas/FasL gene expressin either n the infected cells r neighbring uninfected cells (Kaplan and Sieg, 1998), and these tw genes will send signal f apptsis t these cells. CD8 cell is anther type f T cell. It destrys virally infected cells and tumr cells s it is als knwn as cyttxic T cell (T c cells r CTLs). A healthy adult usually has 150 1,000/mL CD8 cells and the nrmal rati f CD4/CD8 is In cntrast t CD4 cells, CD8 cells ften increase in peple with HIV and the significance has nt been well understd. Researches have revealed (Chevret et al., 1992; Krantz et al., 2011) that elevated ttal CD8 cell cunt was assciated with greater risk f future virlgic failure. The CD4/CD8 Rati is used t help in diagnsing HIV, mnitring HIV prgress, and making treatment decisins. HIV diagnstic test is dne by either detecting hst antibdies made against different HIV prteins r by directly detecting the whle virus r cmpnents f virus (Iweala, 2004). Tests that detect hst antibdies that are specific t the virus include ELISA, Western blt, the immunflurescence assay (IFA), and the detuned assay. Fr screening purpses, ELISA is usually used first, and in rder t minimize the risk f false psitive results, a cnfirmatry test, such as Western blt r IFA, shuld be cnducted befre a patient is given the diagnsis f HIV infectin. Detuned assay is used t distinguish recent HIV infectin within the past days frm lder HIV infectins (Parekh et al., 2002). These tests may be negative during the acute infectin r befre sercnversin is cmpleted. In cntrast, three types f tests can directly detect the virus r parts f the virus as sn as peple becme infected with HIV. These tests include p24 antigen detectin, peripheral bld mnnuclear cell culture and RNA nucleic acid-based assays, such as reverse transcriptin fllwed by plymerase chain reactin (RT-PCR) and hybridizatin-based assays. Undetectable viral lad is usually defined as less than 50 cpies/ml. Until recently, this was the lwest detectable level fr the cmmnly used tests in rutine viral lad mnitring. There are nw sme ultra-sensitive tests that can measure less than 20 cpies/ml and even 1 cpy/ml f plasma (Palmer et al., 2003). It takes an average f 10 years after HIV infectin t develp AIDS, and the viral lad generally remains unchanged if measured repeatedly during thse years. Originally, many peple thught 6

18 the rate f HIV replicatin and disease prcess wuld be slw, which is nt true. In 1995 and 1996, several imprtant papers (H et al., 1995; Perelsn et al., 1996; Wei et al., 1995) published in prestigius jurnals shwed that HIV replicatin and the disease prcess are very vibrant. On average, plasma virins have a mean lifespan f 0.3 days (half-life = 0.24 days), and the average ttal HIV-1 prductin is per day, the minimum duratin f the HIV-1 life cycle in viv is 1.2 days, and the average HIV-1 generatin time is 2.6 days (generatin time is defined as the time frm release f a virin until it infects anther cell and causes the release f a new generatin f viral particles.) Because the high viral replicatin rate may result in a high mutatin rate, H (1995) prpsed the treatment strategy f Hit Hard, Hit Early. Hit Hard requires simultaneusly cmbining different medicatins in the treatment, while Hit Early means the treatment shuld start as early as HIV infectin has been cnfirmed. Althugh the s called ccktail treatment apprach prpsed by H is still the mst cmmnly used treatment strategy, Hit Early was abandned quickly when clinicians realized the adverse effects utweighed the benefits. The treatment shuld be Hit HIV-1 hard, but nly when necessary (Harringtn et al., 2000). Based n 2012 U.S. Department f Health and Human Services Panel n Antiretrviral Guidelines fr Adults and Adlescents (Guidelines, 2012), the initiatin f antiretrviral therapy (ART) is ptinal if the CD4 cell cunt is > 500 /ml, mderately recmmended if the CD4 cell cunt is 350 t 500 /ml and strngly recmmended if the value is < 350 /ml. Regardless f the CD4 cell cunt, ART is strngly recmmended if patients have certain cnditins such as pregnancy, histry f an AIDS defining illness r hepatitis B (HBV) c-infectin. The usual highly active antiretrviral therapy (HAART) cmbines three r mre different medicatins such as tw nucleside reverse transcriptase inhibitrs (NRTIs) and a prtease inhibitr (PI), a nn-nucleside reverse transcriptase inhibitr (NNRTI) r ther such cmbinatins. These HAART regimens have been prven t be able t reduce the amunt f active viruses and in sme cases can lwer the number f active viruses until it is undetectable by current bld testing techniques HIV dynamic mdels The basic mdel fr HIV infectin includes three parts: target uninfected cell T, virus V and infected cell T. The equatins that describe the basic mdel f viral dynamics befre the treatment are: 7

19 dt dt dt dt dv dt = ρ dt kvt = kvt δt = ηt cv where T is prduced at a rate f ρ and dies at rate d, virus V is cleared frm the bdy at rate c and infects the target cells T t T at rate f k, infected cell T dies at rate δ and prduces new virus particles at a cnstant rate η. This is a system f nnlinear rdinary differential equatins (ODE) withut any clsed frm slutin, hwever, we can derive varius apprximatins and btain an understanding f the system. Befre infectin, V = 0,T = 0 and uninfected cells T are at equilibrium ast = ρ/d. Dente by t = 0 is the time when infectin ccurs. Suppse infectin ccurs with a certain amunt f virus, s the initial cnditins aret 0 = ρ/d,t 0 = 0 andv 0. Similar as the cnditin that spread f an infectius disease in a ppulatin, whether r nt the virus can grw and establish an infectin depends n a crucial quantity called basic reprductive rati R. R is defined as the number f newly infected cells arising frm ne infected cell when almst all cells are uninfected and R = ρkη dδc. If R < 1 then the virus will nt spread, because every infected cell will n average prduce less than ne ther infected cell. If starting with N infected cells, then n average, we expect rughly ln N/ ln(1 R) runds f replicatins befre the virus ppulatin dies ut. If n the ther hand, R > 1, then, n average, every infected cell will prduce mre than ne newly infected cell. The chain will generate an explsive multiplicatin f virus as V(t) = V 0 exp(rt), where r is the expnential grwth rate f the virus ppulatin and it is given by the larger rt f the equatin r 2 +(δ + c)r +δc(1 r 2 ) = 0, the apprximatin f r = δ(r 1), which means each infected cell prduces R newly infected cells befre dying. Virus grwth will nt cntinue indefinitely because the supply f uninfected cells is limited. (1.1) During the shrt time since initiatin f HAART treatment, the viral lad decrease sharply. This change with time can be expressed by the differential equatin as, dv/dt = P λv, where P is the viral prductin rate, λ is the decay rate f viral lad, and V is the HIV viral lad in plasma. If assuming a pretreatment steady state exists, dv/dt = 0, and a perfect treatment effect that n new infectin r new virin prduced, the HIV dynamics can be expressed as a simple ne-expnential equatin (H et al., 1995): V(t) = V(0)exp( λt) (1.2) 8

20 where V(t) is the viral lad at time t and V(0) is the viral lad at the baseline. Equatin (1.2) can nly reasnably describe the behavir f the viral dynamics during 1 2 weeks after the initializatin f treatment. Assuming a perfect prtease inhibitr treatment effect (Perelsn et al., 1996), which means n new infectius virins (V I ) but sme nninfectius virins (V NI ) will still be prduced, the HIV dynamics can be expressed as the fllwing system f ODE: dt dt dv I dt dv NI dt = kv I T δt = cv I = NδT cv NI where N is the number f new virins prduced per infected cell during its life time. Under the assumptin f cnstant supply f target cellt and quasi-steady state befre treatment (dt /dt = 0 and dv/dt = 0), a clse frm slutin t the system f ODE (1.3) can be btained: (1.3) V(t) = V 0 exp( λt)+ λv 0 λ δ [λv 0 λ δ {exp( δt) exp( λt)} δtexp( λt)] (1.4) where V(t) = V I (t) +V NI (t), Perelsn et al.(1996) applied equatin (1.4) t mre frequent measured HIV-1 RNA data during the first week f treatment. By nnlinear least-squares regressin, the estimated half-life f free virins is abut six hurs and it is 1.6 days fr prductively infected cells. Perelsn et al.(1997) further extended the ODE (1.3) in rder t include a lnger perid f treatment that a biphasical decay rate f plasma HIV-1 RNA was bserved: an initial rapid expnential decline f nearly 2-lgs (first phase), fllwed by a slwer expnential decline (secnd phase). Tw mre target cells are added in the mdel: (i) lng-lived infected cells, macrphages (M), will be infected int M with a rate f k M, prduce virins at rate f p and die with a rate f µ M ; (ii) latently infected lymphcytes (L) will be prduced by a rate cnstant fk and die at a rate f µ L. The HIV dynamics can be expressed as: dt dt dl dt dm dt dv dt = kvt +αl δt = fkvt µ L L = k M VM µ M M = NδT +pm cv where latent infected cells L can becme prductively infected cells at rate f α. With the similar assumptins used fr equatin (1.4), a clsed frm slutin t the system f ODE (1.5) is, (1.5) 9

21 V(t) = V 0 [Aexp( δt)+bexp( µ L t)+cexp( µ M t)+(1+a+b +C)] (1.6) where A, B and C are functins f system parameters. Even with additinal peripheral bld mnnuclear cells infrmatin, this equatin is t cmplicated t identify all parameters, therefre, sme parameters are assumed t be knwn and replaced by the values frm previus studies. The first six weeks since the treatment was used in equatin (1.6) and the half-life f prductively infected CD4 cells, lng-lived infected cells and latently infected cells were estimated as 1.1 days, 14.1 days and 8.5 days, respectively. Perfect treatment effect may nt be a very reasnable assumptin, especially after shrt perid f treatment. Wu and Ding (1999) prpsed a system f ODE that included a prtease inhibitr efficacy parameter f γ, 0 γ 1, while γ = 0 means the PI medicatins have n effect and γ = 1 means perfect effect. The riginal ODE they prpsed included many parameters that either can be negligible if they are assciated with the faster decays r can be apprximated by cnstants if they are slw enugh in the mdeling time perid r if they are impssible t be accurately estimated based n the HIV-1 viral lad available. The simplified system f ODE Wu and Ding prpsed is: d dt T = kv I T δt d dt V I = (1 γ)p cv I d dt V NI = γp +P +NδT cv NI where P is the virus prduced rate by prductively infected cells, such as CD4 cell, P accunts fr virus prduced frm mysterius infected cells such as Langerhans cells and micrglial cells, r lng-lived infected cells such as macrphages and latent infected cells, and k,t,δ,v I,V NI,N and c have the same meaning as ODE (1.3). A clsed frm slutin t the system f ODE (1.7) is, V(t) = exp(p 1 λ 1 t)+exp(p 2 λ 2 t)+(p 3 +P 4 t)exp( ct) where V(t) = V I (t) +V NI (t), λ 1 = δ and it is the first-phase viral decay rate that may represent the minimum turnver rate f prductively infected cells, such as CD4, λ 2 is a pssibly cmpund clearance rate f lng-lived and latently infected cells and the value depends n the infectin rate and destryed rate by HIV virus. Because c has been estimated t be very rapid (less than 6 hurs f half life), it can be negligible cmpared with ther terms. Thus, the equatin can be further simplified as a tw-expnential equatin: (1.7) 10

22 V(t) = exp(p 1 λ 1 t)+exp(p 2 λ 2 t) (1.8) where P 1 and P 2 is initial viral prductin rate frm prductively infected cells, lng-lived and latently infected cells, respectively. Nnlinear mixed-effects (NLME) mdeling can be used in the estimatin f the parameters in equatin (1.8). NLME mdeling will pl individual data tgether t estimate the ppulatin parameters first, then estimate the individual parameters by the empirical Bayesian methd (Vnesh and Chinchili, 1996). Althugh the ccktail HAART treatment can suppress HIV in 60 t 90% f cases, 30 t 60% f patients will end up as being cnsidered treatment failure eventually because f the viral lad rebund (Havlir et al., 2000). Hwever, all f the equatins intrduced s far require the decay rate t be cnstant s they can t be applied t rebund values. Several extensins have been develped in rder t catch up viral lad respnse that include rebund data and three representatives are fllwing: (i) Extended frm the ODE (1.1), Huang et al.(2003) prpsed a viral dynamic mdel with a time varying treatment efficacy functin γ(t) as, dt dt dt dt dv dt = ρ dt [1 γ(t)]ktv = [1 γ(t)]ktv δt = NδT cv where γ(t) represents a time varying treatment efficacy and it can be mdeled as a functin f drug expsure and drug sensitivity. (ii) Extended frm ne expnential equatin (1.2) by replacing the cnstant decay rate with a time varying decay functin (Wu, 2004): V(t) = V(0)exp( λ(t)t) (iii) Extended frm tw expnential (1.8) by replacing the secnd cnstant decay rate with a time varying decay rate functin as (Wu and Zhang, 2002): V(t) = exp(p 1 λ 1 t)+exp(p 2 λ 2 (t)t) 11

23 Amng these three extensins, the first ne is a system f nnlinear ODE withut a clsed frm, s cmpared with the ther tw, the cmputatin is even mre challengeable and the mdel may nt cnverge, therefre, we will fcus n either the ne expnential r tw expnential equatin in the Chapter 2 and 3. HIV prgress status is usually measured via HIV viral lad r CD4 cell cunt, which are bth surrgate bimarkers. CD4 cell cunt is mre ften used as an endpint fr lng fllw-up trials r advanced patients ppulatin, but fr trials with shrt fllw-up perids, viral lad is ften used as a primary endpint t quantify treatment effect, where CD4 cell cunt is viewed as a cvariate t help predict virlgic respnses. Hwever, we shuld be aware the pssible issues f using either HIV viral lad r CD4 cell cunt as the utcme. The pssible trublesme aspects f using the viral lad as the primary utcme include (i) if the viral lad is measured by RT-PCR which is based n the viral fragments, the result may verestimate the number f infectius virus by an average factr f 60,000 (Nwak et al., 1991); the lack crrelatin between f viral lad and infectin was als nted in sme publicatins (Perelsn et al., 1993; 1999), where n evidence f virus by culture amng the patients with detectable viral lad; (ii) the lack f crrelatin between viral lad and CD4 level such that the changes in viral lad were nly able t explain as little as 4% f change in the CD4 cell cunt (Rdriguez et al., 2006). Althugh CD4 cell cunt seems t be a better HIV prgressin indicatr, especially fr the study with a lnger fllw-up perid, predictin may be risky since CD4 cell cunt mdels are ften empirical (Wu and Ding, 1999; Wu, 2002). On the ther hand, treating bth viral lad and CD4 cell cunt as a bivariate respnse (Sy et al., 2007) may be cmplicated, because the HIV dynamic mdel fr viral lad is nnlinear and CD4 cell cunt cntains missing data Statistical inference in HIV dynamics Varius statistical inferences and analysis methds have been applied in HIV dynamics. Linear and nnlinear regressin via least-squares (LS) estimatin can be applied t very frequent measurements during the first 1 2 weeks after the treatment is initiated (H et al., 1995; Perelsn et al., 1996; 1997; Wei et al., 1995). Because frequent viral lad measurement is nly achievable in small clinical studies and nly subjects withut any missing values can be included in LS, this methd is cnsidered t be less pwerful than sme ther inferences. 12

24 Because viral lad are measured repeatedly since the treatment, the values btained frm the same subject may be crrelated but can be assumed t be independent if btained frm different subjects. One pwerful tl t handle such lngitudinal data is mixed-effects mdeling, in which within-subject and between-subject variatins are bth cnsidered (Laird and Ware, 1982). Linear mixed-effects (LME) and nnlinear mixed-effects (NLME) mdeling appraches have been prpsed in HIV dynamics (Wu et al., 1998; 2004; Wu and Ding, 1999). Semiparametric nnlinear mixed-effects (SNLME) mdeling (Liu and Wu, 2007; Wu and Zhang, 2002; Wu et al., 2004) is prpsed in rder t allw the decay rate t vary with time s the rebund viral lad can be included. Jint mdel apprach via Mnte Carl EM algrithm can be applied t the NLME with cvariate measurement errrs and nn-ignrable missing respnses (Liu and Wu, 2007; Wu, 2002; 2004). Estimatin f NLME is cmplex because usually the likelihd has n clsed frm slutin, even fr simple mdels. The Bayesian apprach based n Markv chain Mnte Carl (MCMC) algrithm has been prpsed fr cmplex ODE and NLME (Huang et al., 2006; Huang and Dagne, 2011; 2012a; 2012b; Putter et al., 2002; Wu et al., 2005). T avid the numerical cmputatin f multiple integrals invlved in the likelihd, likelihd apprximatin such as linearizatin, Laplace apprximatin, Stchastic apprximatin EM algrithm (SAEM) have been applied in HIV dynamics (Ding and Wu, 2000; Guedj et al., 2007; Kuhn and Lavielle, 2005; Wu, 2004). Anther cmplexity f viral lad analysis is left censring which ccurs when viral lads are belw a limit f qualificatin (LOQ), and if ignred, the censring may induce biased parameter estimates. Different appraches have been prpsed t address this prblem (Fitzgerald et al., 2002; Hughes, 1999; Lavielle et al., 2011; Samsn et al., 2006; Thiébaue et al., 2005). The mdel randm errrs and randm-effects in mixed-effect mdels are usually assumed t have a nrmal distributin and that assumptin may nt be satisfied in HIV viral lad and CD4 cell cunt, s the estimatin can be biased. Skewed distributin can be applied in rder t cnsider this nn-ignrable departure frm nrmality (Huang et al., 2006; Huang and Dagne, 2012a; 2012b; Dagne and Huang, 2012). CD4 and CD8 cell cunt can be used as surrgate bimarkers fr HIV disease prcess. Shah et al.(1997) used an EM algrithm t fit a bivariate linear randm-effects mdel. Sy et al.(1997) used the Fisher scring methd t fit a bivariate linear randm-effects mdel including an integrated Orstein-Uhlenbeck prcess (IOU). IOU is a stchastic prcess that includes Brwnian mtin as a 13

25 special limiting case Skew-elliptical distributins Linear and nnlinear mixed-effect mdels are pwerful tls fr analyzing repeated measures and clustered data. In these mdels, randm-effects are included in rder t accunt crrelatin. Usually either randm-effects r mdel errrs r bth are assumed t fllw a nrmal distributin. Althugh nrmality assumptin may be reasnable fr many situatins, the skewness can still be bvius even after the variables have been transfrmed. Ignring the departure frm nrmality may cause biases r misleading results (Ghsh et al., 2007; Verbeke and Lesaffre, 1996). Ideally, we hpe t use a mre generalized distributin family that (i) has high flexibility in shapes and with a wide range f skewness and kurtsis; (ii) is mathematically tractable, which means it can retain nice prperties f riginal family such that parameters can be directly linked t sme aspects f knwn prbability density functin (pdf); (iii) allws us t easily apply the distributins in the existing sftware. Skew-elliptical (SE) distributin is a parametric class f prbability distributins that is extended frm elliptical distributin by including an additinal shape parameter fr skewness. This class, which is usually btained by using transfrmatin and cnditining, cntains many standard families such as multivariate skew-nrmal (SN), skew-t (ST), Student-t and nrmal distributins. Different versins f the multivariate SE distributins have been prpsed. The versin prpsed by Azzalini et al.(1996; 1999) is based n cnditining ne suitable randm variable being greater than zer; SE distributin prpsed by Jnes and Faddy (2003) is scaled inverse χ distributin; Fernandez and Steel (1998) develped a frm that tw Student-t distributins (with different scale parameters) in psitive and negative dmains are cmbined t frm an SE distributins; We adpt a class f multivariate SE distributins prpsed by Sahu et al.(2003), which is btained by using transfrmatin and cnditining, cntains multivariate ST, SN, Student-t and nrmal distributin as special cases. A k-dimensinal randm vectr Y fllws a k-variate SE distributin if its pdf is given by f(y µ,σ, ;m (k) ν ) = 2 k f(y µ,a;m (k) ν )P(V > 0) (1.9) where A = Σ + 2, µ is a lcatin parameter vectr, Σ is a cvariance matrix, is a skewness diagnal matrix with the skewness parameter vectr δ = (δ 1,δ 2,...,δ k ) T, V fllws the el- 14

26 ( ) liptical distributin El A 1 (y µ),i k A 1 ;m (k) ν and the density generatr functin ν (u) = Γ(k/2) m ν(u) 0 rk/2 1 m, with m ν(u)dr ν(u) being a functin such that 0 r k/2 1 m ν (u)dr m (k) π k/2 exists. The functin m ν (u) prvides the kernel f the riginal elliptical density and may depend n the parameter ν. We dente this SE distributin by SE(µ,Σ, ;m (k) ν ). Tw examples f m ν (u), leading t imprtant special cases used thrughut the paper, arem ν (u) = exp( u/2) and m ν (u) = (u/ν) (ν+k)/2, where ν > 0. These tw expressins lead t the multivariate SN and ST distributins, respectively. In the latter case, ν crrespnds t the degree f freedm parameter Skew-t distributin We briefly discuss a multivariate ST distributin intrduced by Sahu et al.(2003) in this sectin. A k-dimensinal randm vectr Y fllws ak-variate ST distributin if its pdf is given by f(y µ,σ,,ν) = 2 k t k,ν (y µ,a)p(v > 0) (1.10) we dente thek-variatetdistributin with parametersµ,aand degrees f freedmν byt k,ν (µ,a) and the crrespnding pdf by t k,ν (y µ,a) hencefrth, V fllws the t distributin t k,ν+k. We dente this distributin by ST k,ν (µ,σ, ). In particular, whenσ = σ 2 I k and = δi k, equatin (1.10) simplifies t { f(y µ,σ 2,δ,ν) = 2 k (σ 2 +δ 2 ) k/2 Γ((ν+k)/2) Γ(ν/2)(νπ) k/2 T k,ν+k [ {ν+(σ 2 +δ 2 ) 1 (y µ) T (y µ) ν+k } 1+ (y µ)t (y µ) (ν+k)/2 ν(σ 2 +δ 2 ) } 1/2 δ(y µ) σ σ 2 +δ 2 ] where T k,ν+k ( ) dentes the cumulative distributin functin (cdf) f t k,ν+k (0,I k ). Hwever, unlike in the SN distributin belw, the ST density can nt be written as the prduct f univariate ST densities. HereY are dependent but uncrrelated. The mean and cvariance matrix f the ST distributin ST k,ν (µ,σ 2 I k, ) are given by E(Y ) = µ+(ν/π) 1/2 Γ((ν 1)/2) Γ(ν/2) δ, cv(y ) = [ σ 2 I k + 2] ν ν 2 ν π [ ] Γ{(ν 1)/2} 2 2 Γ(ν/2) The ST distributin f Y has tw types f stchastic representatin as fllws, and each prvides a cnvenience device fr randm number generatin and implementatin purpse. (i). By the prpsitin f Sahu et al.(2003), Y = µ+ X 0 +Σ 1/2 X 1 (1.11) where X 0 and X 1 are tw independent randm vectrs fllwing t k,ν (0,I k ). Let w = X 0, then w fllws ak-dimensinal standard t distributin t k,ν (0,I k ) truncated in the spacew > 0 (i.e., the standard half-t distributin). Thus, a hierarchical representatin f (1.11) is given by 15

27 Y w t k,ν+k (µ+ w,ωσ), w t k,ν (0,I k )I(w > 0) (1.12) where ω = (ν +w T w)/(ν +k). (ii) By Prpsitin 1 f Arellan-Valle et al.(2007), the ST f Y has anther cnvenient stchastic representatin as fllws Y = µ+ X 0 +ξ 1/2 Σ 1/2 X 1 (1.13) where X 0 and X 1 are tw independent N k (0,I k ) randm vectrs. Let w = X 0, then w fllws a k-dimensinal standard nrmal distributin N k (0,I k ) truncated in the space w > 0. Thus, fllwing Sahu et al.(2003), a hierarchical representatin f 1.13 is given by Y w,ξ N k (µ+ w,ξ 1 Σ), w N k (0,I k )I(w > 0), ξ Γ(ρ/2,ρ/2) (1.14) Nte that the ST distributin presented in (1.12) r (1.14) can be reduced t the fllwing three special distributins: (a). An SN distributin SN k (µ,σ, ) as ν and ξ 1 with prbability f 1 (based n equatin f 1.14) r asν with prbability f 1 (based n equatin f 1.12); (b). A Student-t distributin t k,ν (µ,σ) as = 0; (c). A nrmal distributin N k (µ,σ) if bth cnditins f (a) and (b) are satisfied. In rder t better understand the shape f an ST distributin, plts f an ST density as a functin f the skewness parameter with δ = 3,0,3 are shwn in Figure 1.4(a). (a): ST density with mean=0 (b): SN density with mean=0 Density functin f(t) delta=0 delta=3 delta= 3 Density functin f(t) delta=0 delta=3 delta= Time (t) Time (t) Figure 1.4: The univariate skew-t (df ν = 4) and skew-nrmal density functins with precisin σ 2 = 1 and skewness parameter δ = 0, -3 and 3, respectively. 16

28 Skew-nrmal distributin We briefly discuss a multivariate SN distributin intrduced by Sahu et al.(2003) in this sectin. A k-dimensinal randm vectr Y fllws ak-variate SN distributin, if its pdf is given by f(y µ,σ, ) = 2 k A 1/2 φ k {A 1/2 (y µ)}p(v > 0), (1.15) where V N k { A 1 (y µ),i k A 1 }, and φ k ( ) is the pdf f N k (0,I k ). We dente the abve distributin by SN k (µ,σ, ). An appealing feature f equatin (1.15) is that it gives independent marginal when Σ = diag(σ1 2,σ2 2,...,σ2 k ). The pdf (1.15) thus reduces t f(y µ,σ, ) = [ { } }] k 2 y i=1 φ i µ i Φ, σ 2 i +δi 2 σ 2 i +δ 2 i { δ i σ i y i µ i σ 2 i +δ 2 i where φ( ) and Φ( ) are the pdf and cdf f the standard nrmal distributin, respectively. The mean and cvariance matrix are given by E(Y ) = µ+ 2/πδ, cv(y ) = Σ+(1 2/π) 2 It is nted that when = 0, the SN distributin reduces t usual nrmal distributin. In additin, the SN distributin is a special case f the ST distributin. That is, the ST distributin reduces t the SN distributin when the degree f freedm is large. In rder t better understand the shape f an SN distributin, plts f an SN density as a functin f the skewness parameter with δ = 3,0, and 3 are shwn in Figure 1.4(b) Specific aims A cmmn assumptin in mixed-effect mdel fr randm errrs and randm-effects is nrmal distributin. This assumptin may lack rbustness against departure frm nrmality and can be greatly affected by utliers t, therefre, the results may be smewhat misleading. In HIV/AIDS studies, the viral lad, CD4 and CD8 cell cunt can exhibit bvius skewness, even after transfrmatin. It will be valuable t explre whether a general skewed distributin such as ST r SN will bring a better mdel fitting. Als due t the nature f HIV dynamics, the related mdels can be very cmplicated and assciated intensive cmputatin burden in the inference. Nn-cnvergence f the algrithms may exist under the framewrk f likelihd estimatin. Besides these issues, there are at least three specific questins that have nt been satisfactrily answered: First, it is imprtant t use entire HIV viral lad data t have a better understand abut the disease prgress and t cmpare the effect f different medicatins. Hwever, amng all f thse 17

29 mdels that can be applied t include the rebund data, it is unclear which ne is preferred, r whether different distributins will affect the mdel fit, r whether the estimated parameters can be gd predictrs fr sme lng-term result as as treatment failure. Secnd, in rder t explain individual difference in HIV dynamics, cvariates, such as CD4, are ften used in the mdel. Hwever, CD4 values may be measured with substantial errrs r at a different schedule as the viral lad measurement. Als, LME, NLME and SNLME can be used fr shrt, middle and lng term f HIV dynamics data, respectively. Althugh they have sme f the same parameters such as the first decay rate which is the minimal turn ver f the prductively infected cells, it is unclear whether this estimatin btained frm different mdels is cnstant, and if nt, which mdel will yield mre reasnable estimatins. Third, using HIV viral lad as a surrgate t predict the disease prgress might be prblematic. Fr example, the amunt f infectius virus may be verestimated, therefre, the CD4 cell cunt seems t be a better indicatr. Hwever, the mechanism by which the CD4 cell cunt change during the HIV prgress is nt clear. Althugh using bivariate utcmes f CD4 and CD8 cell cunt appear t be superir t any f these cell cunt alne r their rati (Ir et al., 1990), the distributin f CD4 and CD8 cell cunt shws skewness with heavy tails, and n mdel has been prpsed t cnsider CD4 and CD8 as utcme simultaneusly with skewed distributin assumptin. Via the Bayesian apprach and assuming an SE distributin, this dissertatin research is rganized as fllw: Aim 1. Related t the first questin f multiple mdels fr entire HIV viral lad fllw-up, in Chapter 2, we explred different mdels with time-varying decay rate functin in rder t find which ne has the best fit. We als assumed different distributins in each mdel t check the effect f skewness n the mdel fit. After finding the best fitted mdel, we explred the applicatins f the estimated decay rate, such as their assciatin with decay rate, CD4 cell cunt and viral lad rebund status. T the best f ur knwledge, n time-varying decay rate functin was checked r had been fund t have any significant assciatin with the lng-term utcme such as viral lad rebund, althugh sme research fund the cnstant decay rate may reflect 18