Dissertations / Theses on the topic 'Time-dependent covariates'

I propose a new approach, based on time-dependent covariates, to assess the impact of within-subject changes in predictors on subsequent mortality, and apply it to reevaluate the impact of changes in serum cholesterol and smoking status on the coronary heart mortality in the Framingham Heart Study. Time-dependent covariates, representing updated risk factor value or its changes from either the baseline or the most recent measurement are included in two types of multivariable Cox regression analyses. The results reveal that in order to avoid confounding of the effects of changes in risk factor, the model should include a time-dependent variable identifying subjects who developed coronary disease during the follow-up. After adjusting for this variable, a within-subject decrease in cholesterol was associated with a significant reduction of corollary mortality, in contrast to the results of previous studies that did not prevent such confounding.

Survival models are used in analysing time-to-event data. This type of data is very common in medical research. The Cox proportional hazard model is commonly used in analysing time-to-event data. However, this model is based on the proportional hazard (PH) assumption. Violation of this assumption often leads to biased results and inferences. Once non-proportionality is established, there is a need to consider time-varying effects of the covariates. Several models have been developed that relax the proportionality assumption making it possible to analyse data with time-varying effects of both baseline and time-updated covariates. I present various approaches for handling time-varying covariates and time-varying effects in time-to-event models. They include the extended Cox model which handles exogenous time-dependent covariates using the counting process formulation introduced by cite{andersen1982cox}. Andersen and Gill accounts for time varying covariates by each individual having multiple observations with the total-at-risk follow up for each individual being further divided into smaller time intervals. The joint models for the longitudinal and time-to-event processes and its extensions (parametrization and multivariate joint models) were used as it handles endogenous time-varying covariates appropriately. Another is the Aalen model, an additive model which accounts for time-varying effects. However, there are situations where all the covariates of interest do not have time-varying effects. Hence, the semi-parametric additive model can be used. In conclusion, comparisons are made on the results of all the fitted models and it shows that choice of a particular model to fit is influenced by the aim and objectives of fitting the model. In 2002, an AntiRetroviral Treatment (ART) service was established in the Cape Town township of Gugulethu, South Africa. These models will be applied to an HIV/AIDS observational dataset obtained from all patients who initiated ART within the programme between September 2002 and June 2007.

In many longitudinal studies, individual characteristics associated with their repeated measures may be covariates for the time to an event of interest. Thus, it is desirable to model both the survival process and the longitudinal process together. Statistical analysis may be complicated with missing data or measurement errors in the time-dependent covariates. This thesis considers a nonlinear mixed-effects model for the longitudinal process and the Cox proportional hazards model for the survival process. We provide a method based on the joint likelihood for nonignorable missing data, and we extend the method to the case of time-dependent covariates. We adapt a Monte Carlo EM algorithm to estimate the model parameters. We compare the method with the existing two-step method with some interesting findings. A real example from a recent HIV study is used as an illustration.

In survival studies the values of some covariates may change over time. It is natural to incorporate such time dependent covariates into the model to be used in the survival analysis. A standard approach is to use the semi parametric extended Cox proportional hazard model. An alternative is to extend a standard parametric model, such as a Weibull regression model, to include time-dependent covariates. However, the use of such simple parametric models may be too restrictive. Therefore in this thesis we further extend the Weibull regression model with time dependent covariates by using splines to give greater flexibility. The use of Cox, simple parametric and Weibull spline models is illustrated with and without time dependent covariates on two large survival data sets supplied by NHS Blood and Transplant. One data set involves times to graft failure of patients who have undergone a corneal transplant and contains many fixed covariates and one time-dependent covariate with at most one change point. The other data set concerns time to death of heart transplant patients and contains many fixed covariates and a time-dependent covariate with possibly many change points. A simulation study is used to evaluate and compare likelihood-based methods of inference for the competing models. In the first stage attention is focused on selection of the number of knots in the Weibull spline model in the simple case with no covariates. Stage two examines the results of inferences from the Weibull splines model with fixed covariates. Stage three compares the results of inferences for parameters in the extended Cox model and two simple parametric models with time-dependent covariates. Finally, stage four examines the Weibull splines model with time-dependent covariates.

Generalized estimating equations (GEE) are popularly utilized for the marginal analysis of longitudinal data. In order to obtain consistent regression parameter estimates, these estimating equations must be unbiased. However, when certain types of time-dependent covariates are presented, these equations can be biased unless an independence working correlation structure is employed. Moreover, in this case regression parameter estimation can be very inefficient because not all valid moment conditions are incorporated within the corresponding estimating equations. Therefore, approaches using the generalized method of moments or quadratic inference functions have been proposed for utilizing all valid moment conditions. However, we have found that such methods will not always provide valid inference and can also be improved upon in terms of finite-sample regression parameter estimation. Therefore, we propose a modified GEE approach and a selection method that will both ensure the validity of inference and improve regression parameter estimation. In addition, these modified approaches assume the data analyst knows the type of time-dependent covariate, although this likely is not the case in practice. Whereas hypothesis testing has been used to determine covariate type, we propose a novel strategy to select a working covariate type in order to avoid potentially high type II error rates with these hypothesis testing procedures. Parameter estimates resulting from our proposed method are consistent and have overall improved mean squared error relative to hypothesis testing approaches. Finally, for some real-world examples the use of mean regression models may be sensitive to skewness and outliers in the data. Therefore, we extend our approaches from their use with marginal quantile regression to modeling the conditional quantiles of the response variable. Existing and proposed methods are compared in simulation studies and application examples.

The thesis concerns regression models related to the competing risks setting in survival analysis and deals with both the case of known specific causes and the case of unknown (even if present) specific causes of the event of interest. In the first part, dealing with events whose specific cause is known, competing risks modelling has been applied to a breast cancer study and some of the dynamic aspects such as time-dependent variables are tackled within the context of the application. The aim of the application was to detect an optimal chemotherapy dosage for different typologies of patients with advanced breast cancer in order to control the risk of cardiotoxicity. The attention was concentrated on the cumulative incidence probability of getting cardiotoxicity in a well-defined time period, conditional on risk factors. This probability was estimated as a function of the time-dependent covariate dosage. Within the context of the application, some problems of goodness-of-fit related to time-dependent covariates are discussed. The previous application gave rise to investigating the role of time-dependent covariates in competing risks regression models. There exist various types of time-dependent covariates, which differ in their random or deterministic development in time. For so-called internal covariates, predictions based on the model are not allowed, or they meet with difficulties. We describe a general overview of the state of the art, problems and future directions. Moreover, a possible extension of the competing risks model, that allows us to include a simple random binary time-dependent variable, in a multi-state framework, is presented. Inclusion of the sojourn time of an individual in a certain state as a time-dependent covariate into the model, is also studied. In the second part of the thesis, dealing with events whose specific cause is unavailable, regression models for relative survival are discussed. We study the nonparametric additive excess hazards models, where the excess hazard is on additive form. We show how recent developments can be used to make inferential statements about this models, and especially to test the hypothesis that an excess risk effect is time-varying in contrast to being constant over time. We also show how a semiparametric additive risk model can be considered in the excess risk setting. These two additive models are easy to fit with estimators on explicit form and inference including tests for time-constant effects can be carried out based on a resampling scheme. We analyze a real dataset using different approaches and show the need for more flexible models in relative survival. Finally, we describe a new suggestion for goodness-of-fit of the additive and proportional models for relative survival, which avoids some disadvantages of recent proposals in the literature. The method consists of statistical and graphical tests based on cumulative martingale residuals and it is illustrated for testing the proportional hazards assumption in the semiparametric proportional excess hazards model.

The purpose of this paper is to generalize regression models for repeated categorical data based on maximizing a conditional likelihood. Some existing methods, such as those proposed by Duncan (1985), Fischer (1989), and Agresti (1993, and 1997) are special cases of this latent variable approach, used to account for dependencies in clustered observations. The generalization concerns the incorporation of rather general data structures such as subject-specific time-dependent covariates, a variable number of observations per subject and time periods of arbitrary length in order to evaluate treatment effects on a categorical response variable via a linear parameterization. The response may be polytomous, ordinal or dichotomous. The main tool is the log-linear representation of appropriately parameterized Rasch-type models, which can be fitted using standard software, e.g., R. The proposed method is applied to data from a psychiatric study on the evaluation of psychobiological variables in the therapy of depression. The effects of plasma levels of the antidepressant drug Clomipramine and neuroendocrinological variables on the presence or absence of anxiety symptoms in 45 female patients are analyzed. The individual measurements of the time dependent variables were recorded on 2 to 11 occasions. The findings show that certain combinations of the variables investigated are favorable for the treatment outcome. (author´s abstract)
Series: Research Report Series / Department of Statistics and Mathematics

Joint modeling of a single event time response with a longitudinal covariate dates back to the 1990s. The three basic types of joint modeling formulations are selection models, pattern mixture models and shared parameter models. The shared parameter models are most widely used. One type of a shared parameter model (Joint Model I) utilizes unobserved random effects to jointly model a longitudinal sub-model and a survival sub-model to assess the impact of an internal time-dependent covariate on the time-to-event response. Motivated by the Muscular Dystrophy Surveillance, Tracking and Research Network (MD STARnet), we constructed a new model (Joint Model II), to jointly analyze correlated bivariate time-to-event responses associated with an internal time-dependent covariate in the Frequentist paradigm. This model exhibits two distinctive features: 1) a correlation between bivariate time-to-event responses and 2) a time-dependent internal covariate in both survival models. Developing a model that sufficiently accommodates both characteristics poses a challenge. To address this challenge, in addition to the random variables that account for the association between the time-to-event responses and the internal time-dependent covariate, a Gamma frailty random variable was used to account for the correlation between the two event time outcomes. To estimate the model parameters, we adopted the Expectation-Maximization (EM) algorithm. We built a complete joint likelihood function with respect to both latent variables and observed responses. The Gauss-Hermite quadrature method was employed to approximate the two-dimensional integrals in the E-step of the EM algorithm, and the maximum profile likelihood type of estimation method was implemented in the M-step. The bootstrap method was then applied to estimate the standard errors of the estimated model parameters. Simulation studies were conducted to examine the finite sample performance of the proposed methodology. Finally, the proposed method was applied to MD STARnet data to assess the impact of shortening fractions and steroid use on the onsets of scoliosis and mental health issues.

Dans le contexte de la modélisation aléatoire des événements récurrents, un modèle statistique particulier est exploré. Ce modèle est fondé sur la théorie des processus de comptage et est construit dans le cadre d'analyse de défaillances dans les réseaux d'eau. Dans ce domaine nous disposons de données sur de nombreux systèmes observés durant une certaine période de temps. Les systèmes étant posés à des instants différents, leur âge est utilisé en tant qu'échelle temporelle dans la modélisation. Le modèle tient compte de l'historique incomplet d'événements, du vieillissement des systèmes, de l'impact négatif des défaillances précédentes sur l'état des systèmes et des covariables. Le modèle est positionné parmi d'autres approches visant à l'analyse d'événements récurrents utilisées en biostatistique et en fiabilité. Les paramètres du modèle sont estimés par la méthode du Maximum de Vraisemblance (MV). Une covariable dépendante du temps est intégrée au modèle. Il est supposé qu'elle est extérieure au processus de défaillance et constante par morceaux. Des méthodes heuristiques sont proposées afin de tenir compte de cette covariable lorsqu'elle n'est pas observée. Des méthodes de simulation de données artificielles et des estimations en présence de la covariable temporelle sont proposées. Les propriétés de l'estimateur (la normalité, le biais, la variance) sont étudiées empiriquement par la méthode de Monte Carlo. L'accent est mis sur la présence de deux directions asymptotiques : asymptotique en nombre de systèmes n et asymptotique en durée d'observation T. Le comportement asymptotique de l'estimateur MV constaté empiriquement est conforme aux résultats théoriques classiques. Il s'agit de l'asymptotique en n. Le comportement T-asymptotique constaté empiriquement n'est pas classique. L'analyse montre également que les deux directions asymptotiques n et T peuvent être combinées en une unique direction : le nombre d'événements observés. Cela concerne les paramètres classiques du modèle (les coefficients associés aux covariables fixes et le paramètre caractérisant le vieillissement des systèmes). Ce n'est en revanche pas le cas pour le coefficient associé à la covariable temporelle et pour le paramètre caractérisant l'impact négatif des défaillances précédentes sur le comportement futur du système. La méthodologie développée est appliquée à l'analyse des défaillances des réseaux d'eau. L'influence des variations climatiques sur l'intensité de défaillance est prise en compte par une covariable dépendante du temps. Les résultats montrent globalement une amélioration des prédictions du comportement futur du processus lorsque la covariable temporelle est incluse dans le modèle
In the context of stochastic modeling of recurrent events, a particular model is explored. This model is based on the counting process theory and is built to analyze failures in water distribution networks. In this domain the data on a large number of systems observed during a certain time period are available. Since the systems are installed at different dates, their age is used as a time scale in modeling. The model accounts for incomplete event history, aging of systems, negative impact of previous failures on the state of systems and for covariates.The model is situated among other approaches to analyze the recurrent events, used in biostatistics and in reliability. The model parameters are estimated by the Maximum Likelihood method (ML). A method to integrate a time-dependent covariate into the model is developed. The time-dependent covariate is assumed to be external to the failure process and to be piecewise constant. Heuristic methods are proposed to account for influence of this covariate when it is not observed. Methods for data simulation and for estimations in presence of the time-dependent covariate are proposed. A Monte Carlo study is carried out to empirically assess the ML estimator's properties (normality, bias, variance). The study is focused on the doubly-asymptotic nature of data: asymptotic in terms of the number of systems n and in terms of the duration of observation T. The asymptotic behavior of the ML estimator, assessed empirically agrees with the classical theoretical results for n-asymptotic behavior. The T-asymptotics appears to be less typical. It is also revealed that the two asymptotic directions, n and T can be combined into one unique direction: the number of observed events. This concerns the classical model parameters (the coefficients associated to fixed covariates, the parameter characterizing aging of systems). The presence of one unique asymptotic direction is not obvious for the time-dependent covariate coefficient and for a parameter characterizing the negative impact of previous events on the future behavior of a system.The developed methodology is applied to the analysis of failures of water networks. The influence of climatic variations on failure intensity is assessed by a time-dependent covariate. The results show a global improvement in predictions of future behavior of the process when the time-dependent covariate is included into the model

This thesis develops two semiparametric methods for censored survival data when the covariates involved are time-dependent. Respectively in the two parts of this thesis, we introduce an interquantile regression model and a censored quantile regression model that account for the commonly observed time-dependent covariates in survival analysis. The proposed quantile-based techniques offer a greater model flexibility comparing to the Cox proportional hazards model and the accelerated failure time model. The first half of this thesis introduces a censored interquantile regression model with time-dependent covariates. Conventionally, censored quantile regression stipulates a specific, pointwise conditional quantile of the survival time given covariates. Despite its model flexibility and straightforward interpretation, the pointwise formulation oftentimes yields rather unstable estimates across neighbouring quantile levels with large variances. In view of this phenomenon, we propose a new class of censored interquantile regression models with time-dependent covariates that can capture the relationship between the failure time and the covariate processes of a target population that falls within a specific quantile bracket. The pooling of information within a homogeneous neighbourhood facilitates more efficient estimates hence more consistent conclusion on statistical significances of the variables concerned. This new formulation can also be regarded as a generalization of the accelerated failure time model for survival data in the sense that it relaxes the assumption of global homogeneity for the error at all quantile levels. By introducing a class of weighted rank-based estimation procedure, our framework allows a quantile-based inference on the covariate effect with a less restrictive set of assumptions. Numerical studies demonstrate that the proposed estimator outperforms existing alternatives under various settings in terms of smaller empirical bias and standard deviation. A perturbation-based resampling method is also developed to reconcile the asymptotic distribution of the parameter estimates. Finally, consistency and weak convergence of the proposed estimator are established via empirical process theory. In the second half of this thesis, we propose a class of censored quantile regression models for right censored failure time data with time-dependent covariates that only requires a standard conditionally independent censorship. Upon a quantile based transformation, a system of functional estimating equations for the quantile parameters is derived based on the martingale construction. While time-dependent covariates naturally arise in time to event analysis, the few existing literature requires either an independent censoring mechanism or a fully observed covariate process even after the event has occured. The proposed formulation extends the existing censored quantile regression model so that only the covariate history up to the observed event time is required as in the Cox proportional hazards model for time-dependent covariates. A recursive algorithm is developed to evaluate the estimator numerically. Asymptotic properties including uniform consistency and weak convergence of the proposed estimator as a process of the quantile level is established. Monte Carlo simulations and numerical studies on the clinical trial data of the AIDS Clinical Trials Group is presented to illustrate the numerical performance of the proposed estimator.

碩士
國立成功大學
統計學系碩博士班
101
This study focuses on assessing statistical software R for survival data in the presence of time-dependent covariates. The common Cox model is considered to illustrate the association between the hazard function and the potential factors. The time-dependent covariates, for instance some biomarkers, of each patient are recorded at each clinical visit. The survival time is subject to usual right-censorship. In the simulation studies, many different types of time-dependent covariates, including linear process, nonlinear process, and process with individual effect, are assumed in the generating procedure of survival times. In addition to the usual fitting process in R with the function 'coxph' with time-dependent covariates, models with initial values of covariates taken at the onset time are also considered in the simulations. By comparing the simulation results, we can explore the impact of including the whole observed time-dependent covariates in model fitting. This method is then applied on a Chronic Kidney Disease (CKD) study in the National Cheng Kung University Hospital to detect the possible risk factors.

碩士
國立臺北大學
統計學系
96
Management business is closely associated with society, since if a large company went bankrupt it could have a negative effect on the society. It is crucial to establish a financial warning system to prevent financial institutions from blindly going down the path of financial distress. This paper investigates the forecasting accuracy by employing time-dependent covariates Cox model, and stratified time-dependent covariates Cox model in survival analysis using U.S. firms data during 1989-2006. There are two main findings in this research. First, the time-dependent covariates Cox model with Shumway's variables is used to incorporate the relation between covariates and the survival duration of each firm at each point in time for predicting bankruptcy. The forecasting ability of this model is as accurate as discrete-time hazard model. Second, the stratified time-dependent covariates Cox model stratified with confounded factor, firm size, can further improve the forecasting accuracy for predicting bankruptcy.

abstract: Longitudinal studies contain correlated data due to the repeated measurements on the same subject. The changing values of the time-dependent covariates and their association with the outcomes presents another source of correlation. Most methods used to analyze longitudinal data average the effects of time-dependent covariates on outcomes over time and provide a single regression coefficient per time-dependent covariate. This denies researchers the opportunity to follow the changing impact of time-dependent covariates on the outcomes. This dissertation addresses such issue through the use of partitioned regression coefficients in three different papers. In the first paper, an alternative approach to the partitioned Generalized Method of Moments logistic regression model for longitudinal binary outcomes is presented. This method relies on Bayes estimators and is utilized when the partitioned Generalized Method of Moments model provides numerically unstable estimates of the regression coefficients. It is used to model obesity status in the Add Health study and cognitive impairment diagnosis in the National Alzheimer’s Coordination Center database. The second paper develops a model that allows the joint modeling of two or more binary outcomes that provide an overall measure of a subject’s trait over time. The simultaneous modelling of all outcomes provides a complete picture of the overall measure of interest. This approach accounts for the correlation among and between the outcomes across time and the changing effects of time-dependent covariates on the outcomes. The model is used to analyze four outcomes measuring overall the quality of life in the Chinese Longitudinal Healthy Longevity Study. The third paper presents an approach that allows for estimation of cross-sectional and lagged effects of the covariates on the outcome as well as the feedback of the response on future covariates. This is done in two-parts, in part-1, the effects of time-dependent covariates on the outcomes are estimated, then, in part-2, the outcome influences on future values of the covariates are measured. These model parameters are obtained through a Generalized Method of Moments procedure that uses valid moment conditions between the outcome and the covariates. Child morbidity in the Philippines and obesity status in the Add Health data are analyzed.
Dissertation/Thesis
Doctoral Dissertation Statistics 2020

abstract: When analyzing longitudinal data it is essential to account both for the correlation inherent from the repeated measures of the responses as well as the correlation realized on account of the feedback created between the responses at a particular time and the predictors at other times. A generalized method of moments (GMM) for estimating the coefficients in longitudinal data is presented. The appropriate and valid estimating equations associated with the time-dependent covariates are identified, thus providing substantial gains in efficiency over generalized estimating equations (GEE) with the independent working correlation. Identifying the estimating equations for computation is of utmost importance. This paper provides a technique for identifying the relevant estimating equations through a general method of moments. I develop an approach that makes use of all the valid estimating equations necessary with each time-dependent and time-independent covariate. Moreover, my approach does not assume that feedback is always present over time, or present at the same degree. I fit the GMM correlated logistic regression model in SAS with PROC IML. I examine two datasets for illustrative purposes. I look at rehospitalization in a Medicare database. I revisit data regarding the relationship between the body mass index and future morbidity among children in the Philippines. These datasets allow us to compare my results with some earlier methods of analyses.
Dissertation/Thesis
Arizona Medicare Data on Rehospitalization
Philippine Data on Children's Morbidity
M.S. Statistics 2012

碩士
東海大學
統計學系
99
This study is to investigate the factors associated with the survival status of the elderly in Taiwan, we used five waves of the Survey of Health and Living Status of the Elderly in Taiwan, held from 1989 to 2003, to explore the effects on the survival status of the elderly. Based on Cox proportional hazard model with time-dependent covariates, there are eight variables (age, gender, ethnic group, Activities of Daily Living (ADL), self-rated health, physical function, smoking and marital status) strongly related to the survival status of the elderly. Moreover, from Cox model, the plots of ln[-lnS(t)] against ln(t) of each covariate are straight lines means the data can be fitted with Weibull distribution. In addition, consider with the unobservable random effect from the individuals, this study introduced frailty into the Weibull model with time-dependent covariates, to explore the effect of factors, e.g. demographic characteristics, health status, health behavior, home condition, and social participation, on the survival status of the elderly in Taiwan.

碩士
國立中央大學
統計研究所
95
The Department of Health in Taiwan began to freely provide the treatment of HAART (highly active antiretroviral therapy) for the AIDS patients in the appointed hospitals all over the country ever since April, 1997 and up to now, reach a decade period. We are interested in the different effects of HAART to the 136 AIDS patients before and after the onset of AIDS. To investigate this research problem, we focus on three marginal approaches, the AG (Andersen and Gill, 1982) model, WLW (Wei, Lin, and Wiessfeld, 1989) model and PWP (Prentice, Williams and Petersen, 1981) model. In addition to compare the performance of the three approaches, we also study the effect of CD4 count to both survival times.

碩士
國立臺灣大學
流行病學研究所
97
In many clinical trials and medical studies, the course of disease for each individual is monitored during the follow-up period. Information of disease progression including the occurrence of events and biological markers associated with the development of disease as well as its death is often collected in the study. Proportional hazards models incorporating the information of disease progression as time-dependent covariates are frequently used to investigate the effect of disease progression on survival. Here we extend Xu and O’Quigley’s approach to predict the probability of subsequent survival given the past information of disease progression under such time-dependent covariate model. The performance of proposed approach is evaluated by a simulation study.

Joint modeling of longitudinal and survival data has received much attention and is becoming increasingly useful. In clinical studies, longitudinal biomarkers are used to monitor disease progression and to predict survival. These longitudinal measures are often missing at failure times and may be prone to measurement errors. In previous studies these two types of data are frequently analyzed separately where a mixed effects model is used for longitudinal data and a survival model is applied to event outcomes. The argument in favor of a joint model has been the efficient use of the data as the survival information goes into modeling the longitudinal process and vice versa. In this thesis, we present joint maximum likelihood methods, a two stage approach and time dependent covariate methods that link longitudinal data to survival data. First, we use simulation studies to explore and assess the performance of these methods with bias, accuracy and coverage probabilities. Then, we focus on four time dependent methods considering models that are unadjusted and adjusted for time. Finally, we consider mediation analysis for longitudinal and survival data. Mediation analysis is introduced and applied in a research framework based on genetic variants, longitudinal measures and disease risk. We implement accelerated failure time regression using the joint maximum likelihood approach (AFT-joint) and an accelerated failure time regression model using the observed longitudinal measures as time dependent covariates (AFT-observed) to assess the mediated effect. We found that the two stage approach (TSA) performed best at estimating the link parameter. The joint maximum likelihood methods that used the predicted values of the longitudinal measures, similar to the TSA, provided larger estimates. The time dependent covariate methods that used the observed longitudinal measures in the survival analysis underestimated the true estimates. The mediation results showed that the AFT-joint and the AFT-observed underestimated the mediated effect. Comparison of the methods in Framingham Heart Study data revealed similar patterns. We recommend adjusting for time when estimating the association parameter in time dependent Cox and logistic models. Additional work is needed for estimating the mediated effect with longitudinal and survival data.

碩士
國立中央大學
統計研究所
96
Highly Active Anti-Retroviral Therapy, or HAART, is highly beneficial to many HIV-infected individuals. The Department of Health in Taiwan has began to provide the treatment of HAART for the AIDS patients since April, 1997. However, so far in Taiwan, there are very few cases using mathematical models to analyse the efficacy of HAART to HIV patients. Conseqently, to investigate the problem, we use marginal model and frailty model, two methods of multivariate survival data analysis. We want to find the different effect of HAART in two time periods, HIV infection to onset of AIDS and onset of AIDS to death and the different effects of CD4 / CD8 ratio.