Complex longitudinal data are commonly analyzed using nonlinear mixed-effects (NLME) models with a normal distribution. However, a departure from normality may lead to invalid inference and unreasonable parameter estimates. Some covariates may be measured with substantial errors, and the response observations may also be subjected to left-censoring due to a detection limit. Inferential procedures can be complicated dramatically when such data with asymmetric characteristics, left censoring, and measurement errors are analyzed. There is relatively little work concerning all of the three features simultaneously. In this paper, we jointly investigate a skew-

Modeling of longitudinal data is an active area of biostatistics and statistics research that has received a lot of attention in the recent years. Various statistical modeling and analysis methods have been suggested in the literature for analyzing such data with complex features (Higgins et al. [

Firstly, our model relaxes the normality assumption for random errors and random-effects by using flexible skew-normal and skew-

The histograms (a,b) of viral load measured from RNA levels (in natural

Histogram of viral load in ln scale

Profiles of viral load in ln scale

Profiles of CD4 cell count

Histogram of CD4 cell count

Secondly, an outcome of a longitudinal study may be subject to a detection limit because of low sensitivity of current standard assays (Perelson et al. [

Thirdly, another feature of a longitudinal data set is the existence of time-varying covariates which suffer from random measurement errors. This is usually the case in a longitudinal AIDS study where CD4 cell counts are often measured with substantial measurement errors. Thus, any statistical inference without considering measurement errors in covariates may result in biased results (Liu and Wu [

Our research was motivated by the AIDS clinical trial considered by Acosta et al. [

For longitudinal data, it is not clear how asymmetric nature, left censoring due to BDL, and covariate measurement error may interact and simultaneously influence inferential procedures. It is the objective of this paper to investigate the effects on inference when all of the three typical features exist in the longitudinal data. To achieve our objective, we employ a fairly general framework to accommodate a large class of problems with various features. Accordingly, we explore a flexible class of skew-elliptical (SE) distributions (see the Appendix for details) which include skew-normal (SN) and skew-

The remaining of the paper is structured as follows. In Section

In this section, we present the models and methods in general forms so that our methods may be applicable to other areas of research. An approach we present in this paper treats censored values as realizations of a latent (unobserved) continuous variable that has been left-censored. This idea was popularized by Tobin ([

For the response process with left-censoring, we consider the following NLME model with an ST distribution which incorporates possibly mismeasured time-varying covariates

It is noticed that we assume that an

Covariate models have been investigated extensively in the literature (Higgins et al. [

Nonparametric mixed-effects model (

In a longitudinal study, such as the AIDS study described previously, the longitudinal response and covariate processes are usually connected physically or biologically. Statistical inference based on the commonly used two-step method may be undesirable since it fails to take the covariate estimation into account (Higgins et al. [

We assume that

Let

Let

In general, the integrals in (

We now analyze the data set described in Section

Models for covariate processes are needed in order to incorporate measurement errors in covariates. CD4 cell counts often have nonnegligible measurement errors, and ignoring these errors can lead to severely misleading results in a statistical inference (Carroll et al. [

For the initial stage of viral decay after treatment, a biologically reasonable viral load model can be formulated by the uniexponential form (Ho et al. [

Since CD4 cell counts are measured with errors, we assume that the individual-specific and time-varying parameters

In this section, we analyze the AIDS data set described in Section

We investigate the following

The progress in the Bayesian posterior computation due to MCMC procedures has made it possible to fit increasingly complex statistical models (Lunn et al. [

To carry out the Bayesian inference, we need to specify the values of the hyperparameters in the prior distributions. In the Bayesian approach, we only need to specify the priors at the population level. The values of the hyperparameters were mostly chosen from previous studies in the literature (Liu and Wu [

The population posterior mean (PM), the corresponding standard deviation (SD), and

A summary of the estimated posterior mean (PM) of population (fixed-effects) and scale parameters, the corresponding standard deviation (SD) and lower limit (

Method | Model | DIC | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

JM | SN | PM | −0.95 | 0.15 | 863.0 | ||||||||

0.06 | |||||||||||||

SD | |||||||||||||

ST | PM | ||||||||||||

SD | 0.26 | ||||||||||||

PM | 1242.3 | ||||||||||||

SD | |||||||||||||

NV | SN | PM | — | — | — | −2.10 | — | 1083.5 | |||||

— | — | — | −4.18 | — | |||||||||

— | — | — | — | ||||||||||

SD | — | — | — | — | |||||||||

TS | SN | PM | |||||||||||

SD |

To assess the goodness-of-fit of the proposed JM method, the diagnosis plots for the SN, ST, and

The goodness-of-fit. (a–c): Residuals versus fitted values of

In order to further investigate whether SN model under JM method can provide better fit to the data than ST model, the DIC values are obtained and found to be 863.0 for SN model and 985.6 for ST model. The DIC value for SN model is smaller than that of ST model, confirming that SN model is better than ST model in fitting the proposed joint model. As mentioned before, it is hard sometimes to tell which model is “correct” but which one fits data better. The model which fits the data better may be more appealing in order to describe the mechanism of HIV infection and CD4 changing process. Thus, based on the DIC criterion, the results indicate that SN model is relatively better than either ST model or

For comparison, we used the “naive” (NV) method to estimate the model parameters presented in the lower part of Table

Comparing the JM method against the two-step (TS) method from the lower part of Table

The estimated results based on the JM method for SN model in Table

A summary of the estimated posterior mean (PM) of skewness and degree of freedom parameters, the corresponding standard deviation (SD), and lower limit (

Method | Model | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

JM | SN | PM | — | — | — | — | ||||||||

— | — | — | — | |||||||||||

— | — | — | — | |||||||||||

SD | — | — | — | — | ||||||||||

ST | PM | |||||||||||||

−0.14 | ||||||||||||||

SD | ||||||||||||||

NV | SN | PM | — | — | — | — | — | — | — | — | ||||

— | — | — | — | −1.71 | — | — | — | — | ||||||

— | — | — | — | — | — | — | — | |||||||

SD | — | — | — | — | — | — | — | — | ||||||

TS | SN | PM | — | — | — | — | ||||||||

−0.48 | −8.53 | — | — | — | — | |||||||||

— | — | — | — | |||||||||||

SD | — | — | — | — |

In summary, the results indicate that the SN model under the JM method is a better suited model for viral loads and CD4 covariate with measurement errors. Looking now at the estimated population initial stage of viral decay after treatment bases on the JM method, we get

Attempts to jointly fit the viral load data and CD4 cell counts with measurement errors are compromised by left censoring in viral load response due to detection limits. We addressed this problem using Bayesian nonlinear mixed-effects Tobit models with skew distributions. The models were fitted based on the assumption that the viral dynamic model (

Our results suggest that both ST (skew-

The proposed NLME Tobit joint model with skew distributions can be easily fitted using MCMC procedure by using the WinBUGS package that is available publicly and has a computational cost similar to the normal version of the model due to the features of its hierarchically stochastic representations. Implementation via MCMC makes it straightforward to compare the proposed models and methods with various scenarios for real data analysis in comparison with symmetric distributions and asymmetric distributions for model errors. This makes our approach quite powerful and also accessible to practicing statisticians in the fields. In order to examine the sensitivity of parameter estimates to the prior distributions and initial values, we also conducted a limited sensitivity analysis using different values of hyperparameters of prior distributions and different initial values (data not shown). The results of the sensitivity analysis showed that the estimated dynamic parameters were not sensitive to changes of both priors and initial values. Thus, the final results are reasonable and robust, and the conclusions of our analysis remain unchanged (see Huang et al. [

The methods of this paper may be extended to accommodate various subpopulations of patients whose viral decay trajectories after treatment may differ. In addition, the purpose of this paper is to demonstrate the proposed models and methods with various scenarios for real data analysis for comparing asymmetric distributions for model errors to a symmetric distribution, although a limited simulation study might have been conducted to evaluate our results from different model specifications and the corresponding methods. However, since this paper investigated many different scenarios-based models and methods with real data analysis, the complex natures considered, especially skew distributions involved, will pose some challenges for such a simulation study which requires additional efforts, and it is beyond the purpose of this paper. We are currently investigating these related problems and will report the findings in the near future.

Different versions of the multivariate skew-elliptical (SE) distributions have been proposed and used in the literature (Sahu et al. [

Since the SN distribution is a special case of the ST distribution when the degree of freedom approaches infinity, for completeness, this section is started by discussing the multivariate ST distribution that will be used in defining the ST joint models considered in this paper. For detailed discussions on properties and differences among various versions of ST and SN distributions, see the references above. We consider a multivariate ST distribution introduced by Sahu et al. [

An

The mean and covariance matrix of the ST distribution

According to Lemma 1 of Azzalini and Capitanio [

Specifically, if a

The authors are grateful to the Guest Editor and three reviewers for their insightful comments and suggestions that led to a marked improvement of the paper. They gratefully acknowledge A5055 study investigators for allowing them to use the clinical data from their study. This research was partially supported by NIAID/NIH Grant R03 AI080338 and MSP/NSA Grant H98230-09-1-0053 to Y. Huang.