Inference for a linear regression model with an interval-censored covariate

Interval-censored observations of a response variable are a common occurrence in medical studies, and usually result when the response is the elapsed time until some event whose occurrence is periodically monitored. In this paper we consider a multivariate regression setting in which the explanatory variable is interval censored. Use of an ad hoc method of analysis for such data, such as taking the midpoint of the interval-censored covariate and applying ordinary least-squares, is not in general valid. We develop a likelihood approach, together with a two-step conditional algorithm, to jointly estimate the regression coefficients as well as the marginal distribution of the covariate. The resulting estimators are asymptotically normal. The performance of the method is assessed via simulations, and illustrated using data from a recent HIV/AIDS clinical trial to assess the association between waiting time between indinavir failure and subsequent viral load at enrolment. Extensions of the procedure to other parametric distributions are discussed.     

Analysis of responses in migraine modelling using hidden Markov models

Markov-type models have been used in the analysis of disease progression. Although standard errors of model parameters are usually estimated, available software often does not permit the construction of confidence intervals around predictions of the dependent or response variable. A method is presented to calculate means and confidence intervals of model-predicted responses in time governed by a non-homogeneous hidden Markov model in continuous time. The Kolmogorov equations serve as the basis for the calculations. The method is realised in S-Plus and is applied to the prediction of headache responses in clinical studies of anti-migraine treatment. Means and confidence intervals are calculated by numerically solving differential equations that are non-linear in the explanatory variable. Results indicate that uncertainty on predicted drug responses is larger than that on predicted placebo responses and that pain-free responses are less precisely predicted than pain-relief responses. This is due to the uncertainty in the drug-specific parameters which is not present in predicted placebo responses.     

Modelling the association between adherence and viral load in HIV-infected patients

The primary objective of this paper is to investigate the effect of adherence to prescribed antiretroviral therapy on virologic response measured repeatedly over time in HIV-infected patients. To this end observations on plasma viral load (HIV RNA) assessed in copies/ml are categorized into four clinically meaningful states, [0--50[, [50--400[, [400--2000[, [2000 and up. A time-dependent continuation ratio model is used to analyse longitudinal ordinal responses. The main challenge lies in modelling dependencies over time and using information contained in the data efficiently to establish a dynamic relation between observed patient adherence and viral load. Among the several measures of adherence investigated, two specifically account for long periods of time without intake. One is derived from the third moment of the inter-dose interval distribution, while the second reflects internal drug exposure using pharmacokinetic parameters. The approach is applied to a clinical trial involving 35 patients who were followed over 12 months. Results demonstrate a significant relation between patient adherence and virologic response.     

Maximum likelihood estimation of correction for dilution bias in simple linear regression using replicates from subjects with extreme first measurements

The least-squares estimator of the slope in a simple linear regression model is biased towards zero when the predictor is measured with random error. A corrected slope may be estimated by adding data from a reliability study, which comprises a subset of subjects from the main study. The precision of this corrected slope depends on the design of the reliability study and estimator choice.Previous work has assumed that the reliability study constitutes a random sample from the main study. A more efficient design is to use subjects with extreme values on their first measurement. Previously, we published a variance formula for the corrected slope, when the correction factor is the slope in the regression of the second measurement on the first. In this paper we show that both designs improve by maximum likelihood estimation (MLE). The precision gain is explained by the inclusion of data from all subjects for estimation of the predictor's variance and by the use of the second measurement for estimation of the covariance between response and predictor. The gain of MLE enhances with stronger true relationship between response and predictor and with lower precision in the predictor measurements. We present a real data example on the relationship between fasting insulin, a surrogate marker, and true insulin sensitivity measured by a gold-standard euglycaemic insulin clamp, and simulations, where the behavior of profile-likelihood-based confidence intervals is examined. MLE was shown to be a robust estimator for non-normal distributions and efficient for small sample situations. Copyright (c) 2008 John Wiley & Sons, Ltd.     

Seasonal influences on immunological parameters in HIV-infected homosexual men: searching for the immunomodulating effects of sunlight

In view of the capacity of ultraviolet radiation (UVR) to induce suppression of various immunological parameters and to enhance the viral replication of HIV, we investigated whether seasonal influences on immunological parameters that are relevant for HIV infection could be identified. As the sunny season is associated with high levels of ambient UVR, a decline of immunological parameters and an increase of the HIV viral load during the summer months might ensue. We analysed the immunological data of the HIV-infected homosexual men who participated in the Amsterdam Cohort Study on HIV infection and AIDS (1984-1996; n = 556). The effect of season on the individual development of various immunological parameters in time was examined by means of a random effects model for repeated measurements. Lower levels in the mean number of CD4+ T cells and the mean CD4+/CD8+ ratio were found during summer and spring, respectively (P = 0.0001/0.0001). For the CD8+ T cells, high mean values were observed both in April and September (P = 0.0001). The highest T-cell reactivity values were found during the summer (P = 0.0001). No effect of season on the viral load was established. The seasonal effect on CD4+ T cells seemed to be more pronounced at a more advanced stage of the HIV infection. It is concluded that the lower CD4+ T-cell counts during summer support the notion that solar UVR may have a suppressive effect on the cellular immunity of HIV-infected persons. However, whether this observation can be attributed to the effect of ambient UVR solely is questionable, as the other immunological parameters follow different seasonal courses and other reports suggest that both internal and environmental factors influence immunological parameters.     

Multiple imputation analysis of case-cohort studies

The usual methods for analyzing case-cohort studies rely on sometimes not fully efficient weighted estimators. Multiple imputation might be a good alternative because it uses all the data available and approximates the maximum partial likelihood estimator. This method is based on the generation of several plausible complete data sets, taking into account uncertainty about missing values. When the imputation model is correctly defined, the multiple imputation estimator is asymptotically unbiased and its variance is correctly estimated. We show that a correct imputation model must be estimated from the fully observed data (cases and controls), using the case status among the explanatory variable. To validate the approach, we analyzed case-cohort studies first with completely simulated data and then with case-cohort data sampled from two real cohorts. The analyses of simulated data showed that, when the imputation model was correct, the multiple imputation estimator was unbiased and efficient. The observed gain in precision ranged from 8 to 37 per cent for phase-1 variables and from 5 to 19 per cent for the phase-2 variable. When the imputation model was misspecified, the multiple imputation estimator was still more efficient than the weighted estimators but it was also slightly biased. The analyses of case-cohort data sampled from complete cohorts showed that even when no strong predictor of the phase-2 variable was available, the multiple imputation was unbiased, as precised as the weighted estimator for the phase-2 variable and slightly more precise than the weighted estimators for the phase-1 variables. However, the multiple imputation estimator was found to be biased when, because of interaction terms, some coefficients of the imputation model had to be estimated from small samples. Multiple imputation is an efficient technique for analyzing case-cohort data. Practically, we suggest building the analysis model using only the case-cohort data and weighted estimators. Multiple imputation can eventually be used to reanalyze the data using the selected model in order to improve the precision of the results.     

Joint longitudinal hurdle and time‐to‐event models: an application related to viral load and duration of the first treatment regimen in patients with HIV initiating therapy

Shared parameter joint models provide a framework under which a longitudinal response and a time to event can be modelled simultaneously. A common assumption in shared parameter joint models has been to assume that the longitudinal response is normally distributed. In this paper, we instead propose a joint model that incorporates a two‐part ‘hurdle’ model for the longitudinal response, motivated in part by longitudinal response data that is subject to a detection limit. The first part of the hurdle model estimates the probability that the longitudinal response is observed above the detection limit, whilst the second part of the hurdle model estimates the mean of the response conditional on having exceeded the detection limit. The time‐to‐event outcome is modelled using a parametric proportional hazards model, assuming a Weibull baseline hazard. We propose a novel association structure whereby the current hazard of the event is assumed to be associated with the current combined (expected) outcome from the two parts of the hurdle model. We estimate our joint model under a Bayesian framework and provide code for fitting the model using the Bayesian software Stan. We use our model to estimate the association between HIV RNA viral load, which is subject to a lower detection limit, and the hazard of stopping or modifying treatment in patients with HIV initiating antiretroviral therapy. Copyright © 2016 John Wiley & Sons, Ltd.     

On modelling metabolism-based biomarkers of exposure: a comparative analysis of nonlinear models with few repeated measurements

Establishing and characterizing exposure-biomarker relationships is an important problem in molecular epidemiology. The problem is difficult due to several complicating features, namely, the biomarker response is a nonlinear function of exposure and unknown parameters; variation in exposure and biomarker levels occurs both within and between subjects; and errors tend to be heteroscedastic. To overcome some of the statistical challenges in analysing such data, it is common for the investigator to make several assumptions about the data structure. For example, it is common to assume that the natural logarithm of right-skewed, biomarker measurements lead to homoscedasicity and normality so the effect of outliers is minimized and Gauss-Markov theory is applicable. In this paper, we compare a lognormal maximum likelihood estimator (MLE) to generalized estimating equations (GEE) for drawing statistical inference in a nonlinear model of a benzene biomarker (benzene oxide-albumin adducts) as a function of benzene exposure. We explore the characteristic properties of the lognormal MLE under a certain type of model misspecification and compare its small sample performance to the estimating equation approach in simulation studies. We show that the multiplicative lognormal model can lead to severe biases for modest deviations from the true outcome (biomarker) distribution. Furthermore, the lognormal MLE can exhibit very poor small sample properties even under the true model. All methods are applied in a novel data analysis from a study of benzene-exposed workers in China.     

Bayesian bent‐cable growth mixture tobit models for longitudinal data with skewness and detection limit: application to AIDS studies

This paper presents a new Bayesian methodology for identifying a transition period for the development of drug resistance to antiretroviral drug or therapy in HIV/AIDS studies or other related fields. Estimation of such a transition period requires an availability of longitudinal data where growth trajectories of a response variable tend to exhibit a gradual change from a declining trend to an increasing trend rather than an abrupt change. We assess this clinically important feature of the longitudinal HIV/AIDS data using the bent‐cable framework within a growth mixture Tobit model. To account for heterogeneity of drug resistance among subjects, the parameters of the bent‐cable growth mixture Tobit model are also allowed to differ by subgroups (subpopulations) of patients classified into latent classes on the basis of trajectories of observed viral load data with skewness and left‐censoring. The proposed methods are illustrated using real data from an AIDS clinical study. Copyright © 2016 John Wiley & Sons, Ltd.     

A mixture model for the joint analysis of latent developmental trajectories and survival

A general joint modeling framework is proposed that includes a parametric stratified survival component for continuous time survival data, and a mixture multilevel item response component to model latent developmental trajectories given mixed discrete response data. The joint model is illustrated in a real data setting, where the utility of longitudinally measured cognitive function as a predictor for survival is investigated in a group of elderly persons. The object is partly to determine whether cognitive impairment is accompanied by a higher mortality rate. Time-dependent cognitive function is measured using the generalized partial credit model given occasion-specific mini-mental state examination response data. A parametric survival model is applied for the survival information, and cognitive function as a continuous latent variable is included as a time-dependent explanatory variable along with other explanatory information. A mixture model is defined, which incorporates the latent developmental trajectory and the survival component. The mixture model captures the heterogeneity in the developmental trajectories that could not be fully explained by the multilevel item response model and other explanatory variables. A Bayesian modeling approach is pursued, where a Markov chain Monte Carlo algorithm is developed for simultaneous estimation of the joint model parameters. Practical issues as model building and assessment are addressed using the DIC and various posterior predictive tests.     

Markov modelling of changes in HIV-specific cytotoxic T-lymphocyte responses with time in untreated HIV-1 infected patients

HIV-specific cytotoxic CD8(+) T-lymphocytes (CTL) appear to be the cornerstone of the immune response to HIV infection. Recent studies show that CTL activity reflects patients' anti-HIV immune status and slows disease progression. However, the dynamics of the diversity of this response also appears as a key parameter for immune control but the dynamics of this diversity is largely undocumented. We modelled changes in CTL responses against the seven principal HIV proteins over time. We also studied the influence of plasma viral load on temporal changes in HIV protein recognition by memory CTL. The generic model we developed is based on a continuous time homogeneous Markov process with reversible states. Those states are defined by the number of proteins recognized by memory CTL in a given patient at a given time. This approach was developed within a Bayesian framework. Full Bayesian inference is implemented using Markov chain Monte Carlo simulations (MCMC). The Gibbs sampling algorithm was used to estimate the marginal posterior distributions of the transition intensities between stages of CTL responses. We applied our model to data of 152 HIV-infected patients included in the IMMUNOCO cohort. The model suggested that the diversity of HIV protein recognition by memory CTL in treatment-naive patients decreases as the disease progresses. Namely, the loss of T cytotoxic responses is globally faster than their acquisition. Indeed, these patients' T cytotoxic responses were characterized by marked individual turnover and a gradual loss of multiple protein recognition over time, this loss accelerating as viral load increased.     

A flexible nonlinear mixed effects model for HIV viral load rebound after treatment interruption

Characterization of HIV viral rebound after the discontinuation of antiretroviral therapy is central to HIV cure research. We propose a parametric nonlinear mixed effects model for the viral rebound trajectory, which often has a rapid rise to a peak value followed by a decrease to a viral load set point. We choose a flexible functional form that captures the shapes of viral rebound trajectories and can also provide biological insights regarding the rebound process. Each parameter can incorporate a random effect to allow for variation in parameters across individuals. Key features of viral rebound trajectories such as viral set points are represented by the parameters in the model, which facilitates assessment of intervention effects and identification of important pretreatment interruption predictors for these features. We employ a stochastic expectation-maximization (StEM) algorithm to incorporate HIV-1 RNA values that are below the lower limit of assay quantification. We evaluate the performance of our model in simulation studies and apply the proposed model to longitudinal HIV-1 viral load data from five AIDS Clinical Trials Group treatment interruption studies.     

Lack of association of herpes simplex virus type 2 seropositivity with the progression of HIV infection in the HERS cohort

Many studies have chronicled the "epidemiologic synergy" between human immunodeficiency virus (HIV) and herpes simplex virus type 2 (HSV-2). HIV adversely affects the natural history of HSV-2 and results in more frequent and severe HSV-2 reactivation. Few longitudinal studies, however, have examined whether HSV-2 is associated with increased HIV plasma viral loads or decreased CD4 counts. The authors estimated the effect of HSV-2 seropositivity on HIV RNA viral load and on CD4 count over time among 777 HIV-seropositive US women not receiving suppressive HSV-2 therapy in the HIV Epidemiology Research Study (1993-2000). Linear mixed models were used to assess the effect of HSV-2 on log HIV viral load and CD4 count/mm(3) prior to widespread initiation of highly active antiretroviral therapy. Coinfection with HSV-2 was not associated with HIV RNA plasma viral loads during study follow-up. There was a statistically significant association between HSV-2 seropositivity and CD4 count over time, but this difference was small and counterintuitive at an increase of 8 cells/mm(3) (95% confidence interval: 2, 14) per year among HSV-2-seropositive women compared with HSV-2-seronegative women. These data do not support a clinically meaningful effect of baseline HSV-2 seropositivity on the trajectories of HIV plasma viral loads or CD4 counts.     

轮廓似然函数及其应用

目的介绍轮廓似然方法及其两个应用。方法拟合logistic回归模型,计算轮廓似然的置信区间,并与Wald置信区间进行比较;采用轮廓似然方法解决多余参数过多的模型拟合问题。结果轮廓似然可以解决偏态分布下Wald置信区间失效的问题;最大轮廓似然估计与最大似然估计的结果是一致的,而轮廓似然仅需要较弱的假设条件。结论参数呈非正态分布时,轮廓似然置信区间要优于Wald置信区间;轮廓似然作为最大似然估计的替代方法,可以解决最大似然无法计算或计算困难的问题,也可以提高模型的适用性。     

Assessing uncertainty in reference intervals via tolerance intervals: application to a mixed model describing HIV infection

We define the reference interval as the range between the 2.5th and 97.5th percentiles of a random variable. We use reference intervals to compare characteristics of a marker of disease progression between affected populations. We use a tolerance interval to assess uncertainty in the reference interval. Unlike the tolerance interval, the estimated reference interval does not contains the true reference interval with specified confidence (or credibility). The tolerance interval is easy to understand, communicate and visualize. We derive estimates of the reference interval and its tolerance interval for markers defined by features of a linear mixed model. Examples considered are reference intervals for time trends in HIV viral load, and CD4 per cent, in HIV-infected haemophiliac children and homosexual men. We estimate the intervals with likelihood methods and also develop a Bayesian model in which the parameters are estimated via Markov-chain Monte Carlo. The Bayesian formulation naturally overcomes some important limitations of the likelihood model.     

Estimating the response rate in the presence of measurement error

In clinical research, it is often of interest to estimate the response rate (i.e. the proportion of subjects who achieve a clinically meaningful threshold) for a particular variable. The standard estimator of the response rate is generally biased in the presence of measurement error. The estimation accounting for the measurement error utilizing fully nonparametric (NP) methods is complicated and may not be efficient. Therefore, we propose a model-based approach assuming a parametric model for the true value and only the first few moments for the measurement error. The estimator for the true response rate and the variance for the estimator are derived. An innovative method using bootstrap simulation is proposed to check the model assumption. Simulations show that the proposed estimator outperforms a fully NP estimator if the model assumption for X holds. This method is applied to address a commonly occurring question in osteoporosis regarding response to treatment in terms of longitudinal changes in bone mineral density (BMD). Bootstrap simulations showed that the model utilized is appropriate. The proposed method can also be applied in other fields of clinical research.     

中药乾坤宁治疗艾滋病疗效分析

目的：探索中药治疗艾滋病的作用。方法：随机选择在1992年～1999年间经WB确认的60例感染者，进行治疗前后的自身对照，每天服3次乾坤宁．每次6片．每3个月采集外周血查病毒载量和CD4细胞数。同时选择同期感染又不愿做治疗的10例感染者进行追踪观察。结果：治疗组病死率为13.3％，未接受治疗者病死率为80％。治疗组中5例死亡，24例不明原因退出治疗，对一直坚持治疗的31例进行分析，结果显示病毒载量有下降趋势．CD4有增高趋势。结论：乾坤宁在增强机体免疫功能和抑制病毒方面有一定作用，值得深入研究。     

Simultaneous inference for longitudinal data with detection limits and covariates measured with errors, with application to AIDS studies

In AIDS studies such as HIV viral dynamics, statistical inference is often complicated because the viral load measurements may be subject to left censoring due to a detection limit and time-varying covariates such as CD4 counts may be measured with substantial errors. Mixed-effects models are often used to model the response and the covariate processes in these studies. We propose a unified approach which addresses the censoring and measurement errors simultaneously. We estimate the model parameters by a Monte-Carlo EM algorithm via the Gibbs sampler. A simulation study is conducted to compare the proposed method with the usual two-step method and a naive method. We find that the proposed method produces approximately unbiased estimates with more reliable standard errors. A real data set from an AIDS study is analysed using the proposed method.     

Design of HIV viral dynamics studies

HIV viral dynamics studies involve repeated measurement of viral load in HIV-infected individuals, to asses short-term rates of viral load change in response to interventions such as initiation or withdrawal of antiviral therapy. Such studies are an important source of information on HIV pathogenesis. This paper concerns some statistical issues arising in their design. Using a linear random-effects model to incorporate between-patient differences in rates of viral load change, I discuss the choice of number of individuals and frequency of observation per individual. I suggest an approach for calculating the optimal sample size and observation frequency, based on minimizing the total number of viral load measurements that one needs to undertake. The conclusion, using this approach, is that over a period of linear change in viral load, three to five measurements per individual is generally appropriate. I also examine the observation frequency when the number of available individuals is limited, in which case it is shown that one can use a higher frequency of measurement per individual to achieve adequate power or precision. Finally, I consider sources of data for prior specification of variance components, together with conservative designs that are insensitive to a lack of prior information about between-patient differences. © 1998 John Wiley & Sons, Ltd.     

Impact du paludisme sur l’infection par le VIH

Malaria and HIV are two major public health issues, especially in sub-Saharan Africa. HIV infection increases the incidence of clinical malaria, inversely correlated with the degree of immunodepression. The effect of malaria on HIV infection is not as well established. Malaria, when fever and parasitemia are high, may be associated with transient increases in HIV viral load. The effect of subclinical malaria on HIV viral load is uncertain. During pregnancy, placental malaria is associated with higher plasma and placental HIV viral loads, independently of the severity of immunodeficiency. However, the clinical impact of these transient increases of HIV viral load remains unknown. Although some data suggests that malaria might enhance sexual and mother-to-child transmissions, no clinical study has confirmed this. Nevertheless pregnant women and children with malaria-induced anemia are also exposed to HIV through blood transfusions. Integrated HIV and malaria control programs in the regions where both infections overlap are necessary.