Some state space models of hiv epidemic and its applications for the estimation of hiv infection and incubation

In this paper we have developed some state space models for the HIV epidemic for populations at risk for AIDS. By using these state space models, we have developed a general Bayesian procedure for estimating simultaneously the unknown parameters and the state variables. The unknown parameters include the immigration and recruitment rates, the death and retirement rates, the incidence of HIV infection ( and hence the HIV infection distribution ) and the incidence of HIV incubation ( and hence the HIV incubation distribution). The state variables are the numbers of susceptible people (S people), HIV-infected people (I people) and AIDS incidence over time. The basic approach is through multi-level Gibbs sampler combined with the weighted bootstrap method. We have applied the methods to the Swiss AIDS homosexual and IV drug data to estimate simultaneously the unknown parameters and the state variables. Our results show that in both populations, both the HIV infection and HIV incubation have multi-peaks indicating the mixture nature of these distributions. Our results have also shown that the estimates of the death and retirement rates for I people are greater than those of S people, suggesting that the infection by HIV may have increased the death and retirement rates of the individuals.     

The influence of behavioural heterogeneity on the population level effect of potential prophylactic human immunodeficiency virus type 1 vaccines

Theoretical results indicate that extensive coverage with low efficacy type 1 human immunodeficiency virus (HIV) vaccines could substantially reduce the incidence of HIV in developing countries. There is a non-linear relationship between effective vaccine coverage and HIV prevalence such that improved efficacy brings diminishing returns. The relative contribution of HIV-associated mortality and behavioural heterogeneity to this non-linear relationship is explored using deterministic mathematical models. If the duration of risk of acquiring HIV is long relative to the HIV incubation period then infection-associated mortality can generate the non-linear relationship. However, in its absence the same relationship results from behavioural heterogeneity. Models of HIV vaccination alongside other interventions generate qualitative results that suggest that targeted interventions lead to less redundancy in control efforts.     

Partitioning Pearson's Chi-Squared Statistic for a Completely Ordered Three-Way Contingency Table

The paper presents a partition of the Pearson chi-squared statistic for triply ordered three-way contingency tables. The partition invokes orthogonal polynomials and identifies three-way association terms as well as each combination of two-way associations. This partition provides information about the structure of each variable by identifying important bivariate and trivariate associations in terms of location (linear), dispersion (quadratic) and higher order components. The significance of each term in the partition, and each association within each term can also be determined.
The paper compares the chi-squared partition with the log-linear models of Agresti (1994) for multi-way contingency tables with ordinal categories, by generalizing the model proposed by Haberman (1974).     

Segmental modeling of changing immunologic response for CD4 data with skewness,missingness and dropout

In clinical practice, the profile of each subject's CD4 response from a longitudinal study may follow a ‘broken stick’ like trajectory, indicating multiple phases of increase and/or decline in response. Such multiple phases (changepoints) may be important indicators to help quantify treatment effect and improve management of patient care. Although it is a common practice to analyze complex AIDS longitudinal data using nonlinear mixed-effects (NLME) or nonparametric mixed-effects (NPME) models in the literature, NLME or NPME models become a challenge to estimate changepoint due to complicated structures of model formulations. In this paper, we propose a changepoint mixed-effects model with random subject-specific parameters, including the changepoint for the analysis of longitudinal CD4 cell counts for HIV infected subjects following highly active antiretroviral treatment. The longitudinal CD4 data in this study may exhibit departures from symmetry, may encounter missing observations due to various reasons, which are likely to be non-ignorable in the sense that missingness may be related to the missing values, and may be censored at the time of the subject going off study-treatment, which is a potentially informative dropout mechanism. Inferential procedures can be complicated dramatically when longitudinal CD4 data with asymmetry (skewness), incompleteness and informative dropout are observed in conjunction with an unknown changepoint. Our objective is to address the simultaneous impact of skewness, missingness and informative censoring by jointly modeling the CD4 response and dropout time processes under a Bayesian framework. The method is illustrated using a real AIDS data set to compare potential models with various scenarios, and some interested results are presented.     

Longitudinal poisson regression with disturbed random intercept

Consider repeated event-count data from a sequence of exposures, during each of which a subject can experience some number of events, which is reported at ‘visits’ following each exposure. Within-subject heterogeneity not accounted for by visit-varying covariates is called ‘visit-level’ heterogeneity. Using generalized linear mixed models with log link for longitudinal Poisson regression, I model visit-level heterogeneity by cumulatively adding ‘disturbances’ to the random intercept of each subject over visits to create a ‘disturbed-random-intercept$rsquo; model. I also create a ‘disturbed-random-slope’ model, where the slope is over visits, and both intercept and slope are random but only the slope is disturbed. Simulation studies compare fixed-effect estimation for these models in data with 15 visits, large visit-level heterogeneity, and large multiplicative overdispersion. These studies show statistically significant superiority of the disturbed-random-intercept model. Examples with epidemiological data compare results of this model with those from other published models.     

Estimation of Stratified Mark‐Specific Proportional Hazards Models with Missing Marks

Abstract. An objective of randomized placebo‐controlled preventive HIV vaccine efficacy trials is to assess the relationship between the vaccine effect to prevent infection and the genetic distance of the exposing HIV to the HIV strain represented in the vaccine construct. Motivated by this objective, recently a mark‐specific proportional hazards (PH) model with a continuum of competing risks has been studied, where the genetic distance of the transmitting strain is the continuous ‘mark’ defined and observable only in failures. A high percentage of genetic marks of interest may be missing for a variety of reasons, predominantly because rapid evolution of HIV sequences after transmission before a blood sample is drawn from which HIV sequences are measured. This research investigates the stratified mark‐specific PH model with missing marks where the baseline functions may vary with strata. We develop two consistent estimation approaches, the first based on the inverse probability weighted complete‐case (IPW) technique, and the second based on augmenting the IPW estimator by incorporating auxiliary information predictive of the mark. We investigate the asymptotic properties and finite‐sample performance of the two estimators, and show that the augmented IPW estimator, which satisfies a double robustness property, is more efficient.     

Mixed-effects Models with Skewed Distributions for Time-varying Decay Rate in HIV Dynamics

After initiation of treatment, HIV viral load has multiphasic changes, which indicates that the viral decay rate is a time-varying process. Mixed-effects models with different time-varying decay rate functions have been proposed in literature. However, there are two unresolved critical issues: (i) it is not clear which model is more appropriate for practical use, and (ii) the model random errors are commonly assumed to follow a normal distribution, which may be unrealistic and can obscure important features of within- and among-subject variations. Because asymmetry of HIV viral load data is still noticeable even after transformation, it is important to use a more general distribution family that enables the unrealistic normal assumption to be relaxed. We developed skew-elliptical (SE) Bayesian mixed-effects models by considering the model random errors to have an SE distribution. We compared the performance among five SE models that have different time-varying decay rate functions. For each model, we also contrasted the performance under different model random error assumptions such as normal, Student-t, skew-normal, or skew-t distribution. Two AIDS clinical trial datasets were used to illustrate the proposed models and methods. The results indicate that the model with a time-varying viral decay rate that has two exponential components is preferred. Among the four distribution assumptions, the skew-t and skew-normal models provided better fitting to the data than normal or Student-t model, suggesting that it is important to assume a model with a skewed distribution in order to achieve reasonable results when the data exhibit skewness.     

Sampling from the posterior distribution in generalized linear mixed models

Generalized linear mixed models provide a unified framework for treatment of exponential family regression models, overdispersed data and longitudinal studies. These problems typically involve the presence of random effects and this paper presents a new methodology for making Bayesian inference about them. The approach is simulation-based and involves the use of Markov chain Monte Carlo techniques. The usual iterative weighted least squares algorithm is extended to include a sampling step based on the Metropolis–Hastings algorithm thus providing a unified iterative scheme. Non-normal prior distributions for the regression coefficients and for the random effects distribution are considered. Random effect structures with nesting required by longitudinal studies are also considered. Particular interests concern the significance of regression coefficients and assessment of the form of the random effects. Extensions to unknown scale parameters, unknown link functions, survival and frailty models are outlined.     

Marcinkiewicz–Zygmund and ordinary strong laws for empirical distribution functions and plug-in estimators

Both Marcinkiewicz–Zygmund strong laws of large numbers (MZ-SLLNs) and ordinary strong laws of large numbers (SLLNs) for plug-in estimators of general statistical functionals are derived. It is used that if a statistical functional is ‘sufficiently regular’, then an (MZ-)SLLN for the estimator of the unknown distribution function yields an (MZ-)SLLN for the corresponding plug-in estimator. It is in particular shown that many L-, V- and risk functionals are ‘sufficiently regular’ and that known results on the strong convergence of the empirical process of α-mixing random variables can be improved. The presented approach does not only cover some known results but also provides some new strong laws for plug-in estimators of particular statistical functionals.     

Gaussian models for degradation processes-part I: Methods for the analysis of biomarker data

We present two stochastic models that describe the relationship between biomarker process values at random time points, event times, and a vector of covariates. In both models the biomarker processes are degradation processes that represent the decay of systems over time. In the first model the biomarker process is a Wiener process whose drift is a function of the covariate vector. In the second model the biomarker process is taken to be the difference between a stationary Gaussian process and a time drift whose drift parameter is a function of the covariates. For both models we present statistical methods for estimation of the regression coefficients. The first model is useful for predicting the residual time from study entry to the time a critical boundary is reached while the second model is useful for predicting the latency time from the infection until the time the presence of the infection is detected. We present our methods principally in the context of conducting inference in a population of HIV infected individuals.     

Data and projections of HIV/AIDS cases in Portugal: an unstoppable epidemic?

The size of the affected population with HIV/AIDS is a vital question asked by healthcare providers. A statistical procedure called Back-calculation has been the most widely used method to answer that question. Recent discussions suggest that this method is gradually becoming less appropriate for reliable incidence and prevalence estimates, as it does not take into account the effect of treatment. In spite of this, in the current paper that method and a worst-case scenario are used to assess the quality of previous projections and obtain new ones. The first problem faced was the need to account for reporting delays, no reporting and underreporting. The adjusted AIDS incidence data were then used to obtain lower bounds on the size of the AIDS epidemic, using the back-calculation methodology. A Weibull and Gamma distribution was considered for the latency period distribution. The EM algorithm was applied to obtain maximum likelihood estimates of the HIV incidence. The density of infection times was parameterized as a step function. The methodology is applied to AIDS incidence in Portugal for four different transmission categories (injecting drug users, heterosexual, homo/bisexual and other) to obtain short-term projections (2002–2005) and an estimate of the minimum size of the epidemic.     

Regression models for binary longitudinal responses

Some conditional models to deal with binary longitudinal responses are proposed, extending random effects models to include serial dependence of Markovian form, and hence allowing for quite general association structures between repeated observations recorded on the same individual. The presence of both these components implies a form of dependence between them, and so a complicated expression for the resulting likelihood. To handle this problem, we introduce, as a first instance, what Follmann and Wu (1995) called, in a different setting, an approximate conditional model, which represents an optimal choice for the general framework of categorical longitudinal responses. Then we define two more formally correct models for the binary case, with no assumption about the distribution of the random effect. All of the discussed models are estimated by means of an EM algorithm for nonparametric maximum likelihood. The algorithm, an adaptation of that used by Aitkin (1996) for the analysis of overdispersed generalized linear models, is initially derived as a form of Gaussian quadrature, and then extended to a completely unknown mixing distribution. A large scale simulation work is described to explore the behaviour of the proposed approaches in a number of different situations.     

Some properties and a characterization of trivariate and multivariate binomial distributions

By considering a trivariate binomial distribution, the regression equations are obtained and a set of necessary and sufficient conditions are given for the regression to be linear. Under the limiting conditions, the expressions approach those of trivariate Poisson distribution derived by Mahamunulu (1967). A characterization of the trivariate binomial distribution is also established based on the distribution of the sum of two trivariate random vectors. The results are shown to extend to the multidimensional case in a natural way.     

A copula model for bivariate hybrid censored survival data with application to the MACS study

A copula model for bivariate survival data with hybrid censoring is proposed to study the association between survival time of individuals infected with HIV and persistence time of infection with an additional virus. Survival with HIV is right censored and the persistence time of the additional virus is subject to interval censoring case 1. A pseudo-likelihood method is developed to study the association between the two event times under such hybrid censoring. Asymptotic consistency and normality of the pseudo-likelihood estimator are established based on empirical process theory. Simulation studies indicate good performance of the estimator with moderate sample size. The method is applied to a motivating HIV study which investigates the effect of GB virus type C (GBV-C) co-infection on survival time of HIV infected individuals.     

Comparison of Mixed-Effects Models for Skew-Normal Responses with an Application to AIDS Data: A Bayesian Approach

The potency of antiretroviral agents in AIDS clinical trials can be assessed on the basis of a viral response such as viral decay rate or change in viral load (number of HIV RNA copies in plasma). Linear, nonlinear, and nonparametric mixed-effects models have been proposed to estimate such parameters in viral dynamic models. However, there are two critical questions that stand out: whether these models achieve consistent estimates for viral decay rates, and which model is more appropriate for use in practice. Moreover, one often assumes that a model random error is normally distributed, but this assumption may be unrealistic, obscuring important features of within- and among-subject variations. In this article, we develop a skew-normal (SN) Bayesian linear mixed-effects (SN-BLME) model, an SN Bayesian nonlinear mixed-effects (SN-BNLME) model, and an SN Bayesian semiparametric nonlinear mixed-effects (SN-BSNLME) model that relax the normality assumption by considering model random error to have an SN distribution. We compare the performance of these SN models, and also compare their performance with the corresponding normal models. An AIDS dataset is used to test the proposed models and methods. It was found that there is a significant incongruity in the estimated viral decay rates. The results indicate that SN-BSNLME model is preferred to the other models, implying that an arbitrary data truncation is not necessary. The findings also suggest that it is important to assume a model with an SN distribution in order to achieve reasonable results when the data exhibit skewness.     

Inference for Bivariate Survival Data by Copula Models Adjusted for the Boundary Effect

Copula models describe the dependence structure of two random variables separately from their marginal distributions and hence are particularly useful in studying the association for bivariate survival data. Semiparametric inference for bivariate survival data based on copula models has been studied for various types of data, including complete data, right-censored data, and current status data. This article discusses the boundary effect on these inference procedures, a problem that has been neglected in the previous literature. Specifically, asymptotic distribution of the association estimator on the boundary of parameter space is derived for one-dimensional copula models. The boundary properties are applied to test independence and to study the estimation efficiency. Simulation study is conducted for the bivariate right-censored data and current status data.     

Learning causal structure from mixed data with missing values using Gaussian copula models

We consider the problem of causal structure learning from data with missing values, assumed to be drawn from a Gaussian copula model. First, we extend the ‘Rank PC’ algorithm, designed for Gaussian copula models with purely continuous data (so-called nonparanormal models), to incomplete data by applying rank correlation to pairwise complete observations and replacing the sample size with an effective sample size in the conditional independence tests to account for the information loss from missing values. When the data are missing completely at random (MCAR), we provide an error bound on the accuracy of ‘Rank PC’ and show its high-dimensional consistency. However, when the data are missing at random (MAR), ‘Rank PC’ fails dramatically. Therefore, we propose a Gibbs sampling procedure to draw correlation matrix samples from mixed data that still works correctly under MAR. These samples are translated into an average correlation matrix and an effective sample size, resulting in the ‘Copula PC’ algorithm for incomplete data. Simulation study shows that: (1) ‘Copula PC’ estimates a more accurate correlation matrix and causal structure than ‘Rank PC’ under MCAR and, even more so, under MAR and (2) the usage of the effective sample size significantly improves the performance of ‘Rank PC’ and ‘Copula PC.’ We illustrate our methods on two real-world datasets: riboflavin production data and chronic fatigue syndrome data.

     

Joint modeling of correlated binary outcomes: HIV-1 and HSV-2 co-infection

Herpes Simplex Virus Type 2 (HSV-2) facilitates the sexual acquisition and transmission of HIV-1 infection and is highly prevalent in most regions experiencing severe HIV epidemics. In sub-Saharan Africa, where HIV infection is a public health burden, the prevalence of HSV-2 is substantially high. The high prevalence of HSV-2 and the association between HSV-2 infection and HIV-1 acquisition could play a significant role in the spread of HIV-1 in the region. The objective of our study was to identify risk factors for HSV-2 and HIV-1 infections among men in sub-Saharan Africa. We used a joint response model that accommodates the interdependence between the two infections in assessing their risk factors. Simulation studies show superiority of the joint response model compared to the traditional models which ignore the dependence between the two infections. We found higher odds of having HSV-2/HIV-1 among older men, in men who had multiple sexual partners, abused alcohol, or reported symptoms of sexually transmitted infections. These findings suggest that interventions that identify and control the risk factors of the two infections should be part of HIV-1 prevention programs in sub-Saharan Africa where antiretroviral therapy is not readily available.     

Stochastic geometry models in high-level vision

We survey the use of Markov models from stochastic geometry as priors in ‘high-level’ computer vision, in direct analogy with the use of discrete Markov random fields in ‘low-level’ vision. There are analogues of the Gibbs sampler, ICM and simulated annealing, and connections with existing methods in computer vision.     

索赔次数的开放式混合泊松分布研究

本文建立了索赔次数的多风险类别混合泊松模型。首先,考虑索赔次数的零膨胀、厚尾性和异质性等特征,建立风险类别待定的开放式混合泊松模型,开放式结构使该模型对实际数据的多样特征和风险类别具有良好的自适应性;其次,定义混合权重参数的iSCAD惩罚函数,实现对权重参数的筛选;最后,借助EM算法求得模型参数,实现对各风险类别下索赔次数的估计。借助iSCAD惩罚函数,给出最优混合数,避免传统混合模型中主观选择的弊端,克服传统混合模型中结构复杂、参数估计没有显式表达式、估计结果不便于解释等问题。基于三组风险特征多样数据的实证分析,本文发现该模型可以显著改进现有模型的拟合效果。