Abstract
Unlike majority of current statistical models and methods focusing on mean response for highly skewed longitudinal data, we present a novel model for such data accommodating a partially linear median regression function, a skewed error distribution and within subject association structures. We provide theoretical justifications for our methods including asymptotic properties of the posterior and associated semiparametric Bayesian estimators. We also provide simulation studies to investigate the finite sample properties of our methods. Several advantages of our method compared with existing methods are demonstrated via analysis of a cardiotoxicity study of children of HIV-infected mothers.
Keywords: Dirichlet process, Median regression, Partial linear model, Semiparametric, Skewed error
1. Introduction
Existing methods for analysis of longitudinal continuous response (Verbeke and Molenberghs, 2009; Diggle and others, 2002) mostly focus on estimating the mean response function. These are appropriate when the outcome variable is either approximately Gaussian or symmetric. When the response variable is heavily skewed, a model based on median regression (e.g., Koenker and Bassett, 1978) may be more appropriate than mean regression-based models. The assumption of symmetric errors is not valid for many real-life biomedical and econometric studies. For example, in a Pediatric Pulmonary and Cardiac Complications (in short P2C2) study of children of HIV infected mothers, the longitudinal response of interest is the interval between consecutive R-waves measured via EKG (called RRBO) during clinic visits. This RRBO is known to be heavily skewed even after a log-transformation. In general, the main difficulties of using a transformation-based analysis of a heavily skewed response such as RRBO include determining a suitable transformation to symmetry, evaluating the covariate effects on RRBO from the fitted model of the transformed RRBO, and specifying prior opinions about the regression function of the transformed RRBO using the available prior opinion about the untransformed RRBO. To address these issues, we develop a flexible Bayesian model for skewed longitudinal response and focus on median regression function.
Our second important goal is to present a partial linear model (Härdle and Liang, 2007) to incorporate the effect of one of the covariates (say, time) on median as an unknown nonparametric function, while providing a good physical interpretation of the parametric effects of the rest of the covariates of interest (say, HIV status of a child for the P2C2 study). Existing partial linear models (e.g., Speckman, 1988) for longitudinal data often only allow either mean regression or parametric error density (Ho and Lin, 2010). The 
-estimation procedure of He and others (2002) uses a partial linear model for median. However, it does not specify the likelihood and any within-subject dependence structure. On the other hand, a likelihood-based Bayesian analysis can ensure accurate estimation, characterize all sources of uncertainty, and is suitable for prediction. Some authors (e.g., Kottas and Gelfand, 2001; Hanson and Johnson, 2002; Lin and others, 2012) implemented Bayesian median regression for univariate responses. In Section 2, we develop a novel Bayesian median regression method which incorporates a partial linear model for skewed longitudinal responses. Unlike existing Bayesian approaches for median regression of longitudinal responses using subject-specific random effects (e.g., Reich and others, 2010; Yue and Rue, 2011), our copula (Pitt and others, 2006)-based model can accommodate a wide range of within-subject dependencies while maintaining any desired form of the marginal error density. For example, in the P2C2 study, the copula structure allows us to maintain the desired stationarity of the marginal error density. Unlike our copula model, a random effects model has various shortcomings including assumptions of parametric random effects density and specifications of hyper-priors associated with the parameters of the random effects density. For longitudinal data, the class of correlation matrices can also be enriched using random effects by parameterizing it via partial autocorrelations (Lee and others, 2013). However, our main objective of using copula is to have flexible dependence structure while preserving the median zero property of the marginal residual density. For random effects models, it is not straightforward to enforce the median zero marginal error density unless we make the restrictive assumption of symmetric random effects density. We also demonstrate the ease of the specification of the priors of our model based on the available prior opinion about the marginal behavior of the response and the dependence structure. For example, to specify the prior process on marginal error density, we only use a “guess” (prior mean) of the error density along with a confidence/precision around the “guess”.
Our other novel contribution is to provide a rigorous theoretical justification for our semiparametric Bayesian method for longitudinal response in Section 3. Although consistency of the frequentist estimators of a partial linear model has been investigated previously (Härdle and Liang, 2007; Bhattacharya and Zhao, 1997), analogous results within a fully Bayesian paradigm are restricted to the parametric error distributions (Bickel and others, 2012), semiparametric mean regression for univariate response (Amewou-Atisso and others, 2003), and completely nonparametric regression and density estimation for multivariate response (Shen and others, 2013; Pati and others, 2013). Instead, our consistency results pertain to longitudinal response with within-subject dependencies. We obtain posterior consistency of the regression parameters as well as nonparametric effect of time using only minor regularity conditions on the tail of the error density and on the level of approximation of the nonparametric effect of time. Our minor sufficient conditions for posterior consistency assure a practitioner that our posterior inference is driven by observed data when the data-information is sufficiently large. In addition, existing Bayesian methods for partial linear model for even univariate skewed response suffer from a lack of easily implementable (via widely available softwares) computational tool. Our Bayesian method uses a Markov chain Monte Carlo (MCMC) procedure implementable via a free software such as JAGS (code is available from the authors). Section 2 describes the model and prior specifications. Section 3 studies the theoretical properties of the model. Section 4 presents simulation study results. Section 5 illustrates various advantages of our methods via the analysis of the so-called P2C2 study of children of HIV-infected mothers. Some remarks and discussions are in Section 6. Proofs are mostly deferred to supplementary material available at Biostatistics online.
2. Model and prior specification
Let 
for subject 
denote the observed vectors of the longitudinal responses measured at irregular time points 
with 
. The partial linear regression model for the response variable 
at time 
is given by
 |
(2.1) |
where 
is the 
covariate at 
, 
is the vector of regression parameters and 
is an unknown smooth function of time. Further, 
is the error vector with median zero and possibly skewed marginal density 
. This ensures that the median of 
is 
. For now, we assume that the marginal error density 
is stationary (appropriate for our motivating study). However, our model in (2.1) and corresponding methods can straightforwardly deal with non-stationary errors (discussed later). We approximate the true unknown smooth 
in (2.1) using a piecewise-polynomial B-spline function (Schumaker, 2007) with quantiles of observation time points 
selected as the knots of the basis functions 
for a suitably chosen large 
. As a consequence, the median 
is approximated by 
, for some vector 
. We later prove that 
can approximate the unknown 
arbitrarily well either by using a prior for 
or by allowing 
to grow at a certain rate with the sample size. We also demonstrate that such a specification facilitates desired asymptotic properties of the Bayesian estimators of 
and unspecified 
.
We model our skewed median 0 marginal error density 
, with cumulative distribution function (cdf) 
, in (2.1) as
 |
(2.2) |
where 
is the skewness parameter and 
is an unknown non-increasing density function with support 
. The class of densities in (2.2) is related to the split-density class of Geweke (1989). When 
, 
is left (right) skewed. This class of densities encompasses a large subset of skewed densities with unique median and mode at 
. Using the result of Feller (2008) that any decreasing density 
with support 
can be expressed as a mixture of uniforms, 
in (2.2) can be expressed as a mixture of uniform kernels:
 |
(2.3) |
where 
is an unknown cdf with support 
. Consequently, a flexible prior for 
will induce a flexible prior for 
. We later show that it is straightforward to determine priors for 
and 
using the prior opinion about 
. Kottas and Gelfand (2001) used a sub-class of (2.2) where 
was assumed to be a mixture of half Gaussian distributions.
We use a copula model (Sklar, 1959; Nelsen, 2007), a popular tool for constructing a multivariate density with pre-specified structures of marginal densities, to specify a multivariate density 
of error vector 
as
 |
(2.4) |
where 
is a 
-variate copula-density with corresponding marginal (univariate) cdf and density 
and 
respectively. A copula model can incorporate a very flexible class of dependence structure including non-linear dependence, and any multivariate density can be expressed as a copula. For ease of implementation, we focus on the Gaussian copula (Pitt and others, 2006), where 
is standard normal cdf and 
is the joint multivariate normal density with mean 
and correlation matrix 
. However, our methodology can accommodate any other parametric copula structure. We assume 
to be a known function of an unknown parameter vector 
. Examples of 
include the uniform correlation matrix with common 
for all 
and the exponentiated correlation matrix with 
. These 
and 
from two different subjects may even have different forms with no common parameters. For example, the two HIV status groups of P2C2 study have different within subject association, accommodated via using a uniform correlation and an exponentiated correlation matrix with different parameters for the two groups. Also note that non-stationarity of 
in (2.1) can be easily accommodated using different error distributions 
for time point 
. This joint density ensures the desired stationarity of the marginal density 
for RRBO while allowing a flexible within subject association.
Given observed data 
, the likelihood 
is proportional to 
, where 
. The prior distributions for 
, 
, 
, 
and 
are assumed to be mutually independent with: (i) 
and 
have different independent multivariate normal priors 
and 
, respectively; (ii) 
has a Gamma prior 
and 
has an appropriate prior 
on its range of possible values. (iii) The prior process 
of the unknown mixing distribution 
of (2.3) is Dirichlet (Ferguson, 1974) process 
, where 
is the prior mean (“prior guess”) of 
and 
is the pre-specified precision (or confidence) around 
. The resulting posterior density 
is proportional to 
.
A major practical advantage of our semiparametric Bayesian model is that the hyperparameter 
of the prior 
of 
in (2.3) is specified using the prior guess 
of the marginal error density 
, because 
corresponds to the unique prior guess 
for 
via 
. The proof is very similar to the result of Khintchine (1938). For example, to obtain an exponential “prior guess” for the density 
for 
, we need to choose a gamma distribution for 
. To obtain a heavy tailed 
as a guess, 
has to be set to an inverse-gamma density. The precision parameter 
of 
reflects the a priori uncertainty of the prior guess 
; small values of 
implies that the resulting 
for 
can be very different from the guess 
. To facilitate convenient implementation of the MCMC algorithm to sample from the posterior 
via standard software such as JAGS, we use a finite approximation of the constructive definition (Ishwaran and James, 2001) of the DP mixture prior for 
as 
, where 
are independent draws from 
, 
for 
are independent with 
density, 
and 
, 
, 
.
3. Prior properties and posterior consistency
Unlike the inference under parametric Bayesian models, semiparametric Bayesian methods do not guarantee that the influence of the prior on the posterior would be swamped away in presence of large data (Diaconis and Freedman, 1986), especially when our true function 
is only approximated by a finite spline bases during analysis. It is important and practical to study sufficient conditions for posterior consistency in such complicated Bayesian semiparametric models because they provide the conditions for the eventual identifiability of the parameters and functions of interest from observed data when the sample-size is sufficiently large. For the ease of exposition, we assume that the number of repeated observations 
is same across all subjects (
in (2.1)), the dimension 
of 
is 
and the within-subject dependence is based on Gaussian copula with a uniform correlation matrix 
for all 
.
As a first step of this practically important theory, we investigate whether the “support”, of our semiparametric prior distribution, denoted by 
, on the joint error density is large enough to contain all practically relevant true error densities. The support of the prior provides an idea of the flexibility of the prior process. We define the class 
to be the set of all univariate, possibly asymmetric, unimodal densities with median zero, and satisfying the condition of (2.2) for some 
. Define 
to be the set of all 
-dimensional multivariate densities under Gaussian copula and with unknown uniform correlation 
. We denote the prior 
on 
as 
, where 
and 
are the priors on 
and 
respectively. Recall that prior 
is defined via (2.3) with 
and 
. Next, we need to specify suitable topology, equivalently definitions of some meaningful neighborhoods of the true joint density, to define the support of the prior 
. The Kullback–Leibler support 
of any prior 
on a density space 
is defined by the subset 
of 
satisfying 
where 
. Similarly, the weak support 
of the prior 
is defined under weak topology. Define 
, 
. In the following Lemma 3.1, with proof in Appendix A of supplementary material available at Biostatistics online, we characterize the weak and the Kullback–Leibler support of 
.
Lemma 3.1 —
(1)
; (2)
if supp
,
is absolutely continuous with respect to the Lebesgue measure,
is the set of all distributions on
and
.
Next, we show that our B-splines approximation of 
, the set of continuous functions from 
to 
, is adequate to approximate any true continuous 
if we use appropriately chosen prior for the number of bases 
. Let 
denote the prior on 
based on linear combination of B-splines with Gaussian prior for the basis coefficients. We characterize the support of 
below in Lemma 3.2 whose proof is in Appendix B of supplementary material available at Biostatistics online.
Lemma 3.2 —
For any
,
.
Remark 3.3 —
When the true function
is a combination of B-splines, the prior for
induced through
can be even degenerate because for any
,
for large enough
.
We now provide sufficient conditions that guarantee that, as the sample size 
of the observed data 
increases, the posterior distributions of the scalar 
, multivariate error density 
, and nonparametric 
concentrate around any arbitrarily small pre-defined neighborhoods around their true values 
, 
and 
, respectively. The choices of topologies for these neighborhoods depend on practical considerations. We use a strong neighborhood around 
, the primary parameter of interest, and a weak neighborhood 
around 
, considered a nuisance function for partial linear model. We consider a neighborhood 
around 
because we can only hope to recover the true 
at the observed time points 
, unless we impose additional and somewhat unrealistic assumptions on the design and sampled time points (Amewou-Atisso and others, 2003). Consider 
, for arbitrary 
. We state our result on posterior consistency in Theorem 3.4 whose proof is in Appendix C of supplementary material available at Biostatistics online.
Theorem 3.4 —
Assume that
. Consider the prior
on
. Assume that the prior for
satisfies
with
for some constant
. Then
a.s. under
,
,
and
is the true distribution generating data
.
Remark 3.5 —
For example, when
, the condition
of Theorem 3.4 holds with
. When the true
is actually a linear combination of finite B-spline basis functions, one can achieve the same conclusion of Theorem 3.4 with a deterministic sequence
. Although our proof of the posterior consistency uses a Gaussian copula and equi-correlation
, our techniques can be used to prove the results for any correlation-matrix and other parametric copula model including heavy-tailed copula.
4. Simulation study
Using a variety of simulation studies, we investigate the finite sample properties of our Bayesian methods and compare those with some of the existing methods. We compare estimates of 
and 
obtained using the following classes of error distributions and associated methods: (i) 
-estimator (He and others, 2002) of median functional (called “
-estimator” in Table 1), (ii) Bayesian estimates with parametric skewed double exponential error density (called “SDE” in Table 1), (iii) Bayesian estimates with semiparametric skewed density of (2.3) (called “SPMSD” in Table 1). The parametric density of SDE model in (ii) is 
with 
prior for 
. To compare the performance of these three methods, we use the mean squared error 
for estimated 
, and the mean integrated squared error 
for estimated 
, where 
is the estimates of 
obtained via analyzing dataset 
. Similarly, 
for 
are the estimators of the spline coefficient vectors. The MISE is a measure of difference between the estimated and unknown 
at observed time points.
Table 1.
Approximate (via Monte Carlo) sampling mean and mean square error (MSE within parenthesis) of the estimated regression parameters (
), and approximate mean integrated squared error (MISE) for function 
. True parameter values are 
| Estimation models |
 ( ) |
 ( ) |
MISE |
|---|---|---|---|
Simulation study 
: common variance
| |||
 -estimator |
2.994 (1.209) |
 (1.320) |
19.020 |
| SDE | 2.985 (0.695) |
 (0.773) |
13.950 |
| SPMSD | 2.988 (0.591) |
 (0.717) |
7.412 |
Simulation study 
: heteroscedastic error
| |||
 -estimator |
3.002 (0.301) |
 (0.217) |
2.721 |
| SDE | 2.996 (0.176) |
 (0.158) |
2.816 |
| SPMSD | 3.000 (0.228) |
 (0.205) |
2.521 |
Simulation study 
: error with outliers
| |||
 -estimator |
2.995 (0.687) |
 (0.673) |
8.137 |
| SDE | 2.981 (0.581) |
 (0.550) |
6.161 |
| SPMSD | 2.985 (0.526) |
 (0.626) |
5.787 |
Simulation study 
: Gumbel error
| |||
 -estimator |
3.008 (5.952) |
 (6.483) |
80.260 |
| SDE | 2.913 (4.878) |
 (4.219) |
71.230 |
| SPMSD | 2.919 (4.282) |
 (3.902) |
34.290 |
We use standard diagnostic tests as well as trace plots of the MCMC samples to monitor the convergence of MCMC. Also, we get essentially identical posterior summaries with different MCMC starting points and for moderate changes of values of the hyperparameters. Each of the simulation settings uses 
replicates of simulated data sets, each with 
subjects and every subject measured at five random time points sampled from 
. All simulation models have the marginal regression structure 
, with skewed 
, 
and quadratic 
. We sample 
and 
independently from 
.
For Bayesian analysis, we use independent 
priors for 
and 
, 
prior for 
, and 
priors for 
and 
. We use quadratic splines for estimation of 
in all methods. For the 
-estimator, we select the knots using BIC as described in He and others (2002). For the SPMSD model with DP prior for 
, we use 
and 
for 
(fairly noninformative).
The first simulation model has skewed error 
with 
, where the diagonal elements 
and off diagonal elements 
for 
. This implies that 
has median 
and expectation 
. The summary of the results of simulation study 1 given in Table 1 shows that Bayesian methods using model of (2.1) with both parametric (SDE) and nonparametric error densities (SPMSD) outperform the 
-estimator.
Our second simulation setting aims to compare performances of the estimators when observations are generated from a model with heteroscedastic as well as skewed 
. This simulation model uses error 
dependent on the covariate 
using 
, where 
is simulated using the same error distribution of the first simulation model. In this setting, the performances of the estimates based on our model in (2.1) are, somewhat surprisingly, even robust to the heteroscedastic error density. As the residual density in (2.1) is highly flexible, the posterior means for the regression parameters are close to the true values, although the posterior distribution is different under misspecified error density. Also the parametric SDE method substantially outperforms the 
-estimation method for estimating 
.
Our third simulation study aims to evaluate the robustness of the methods to the presence of possible large outliers. The error vectors 
are simulated from the mixture of two distributions, 
for all 
with probability 
otherwise 
for all 
(resulting 
has outlier with probability 
), where 
is simulated from the multivariate density used for the first simulation model. In this study, the SPMSD method performs substantially better compared with the 
-estimator in estimating 
and 
. In the fourth simulation setting, we generate observations with marginal density of 
as the Gumbel with location 
and scale 
. This simulation study (also the previous one) aims to evaluate the robustness of our methods to the wrong specification of the error density in (2.3). In both simulation settings 3 and 4, the Bayesian methods outperform the 
-estimator with respect to the MSE of 
and the MISE of 
. Overall, it is clear that our semiparametric Bayesian estimators based on skewed uniform kernel error are very robust to the actual form of the density of 
. They perform better than the M-estimation method even when the modeling assumptions of M-estimation are more aligned with the true error density. In particular, the gain in the precision of the estimates from the uniform kernel skewed error model is substantial when the error density is highly skewed.
5. Analysis of P2C2 study
To monitor the progression of cardiac abnormalities in children born to HIV-infected women (Lipshultz and others, 1998), a cohort of 
infants born to HIV-infected women were followed from birth to 7 years of age in the P2C2 study. The continuous response of interest, RRBO (time between consecutive heartbeats), is measured at irregular different time points for different children. There are 
observations in total for all the 
children. There are 
children with only one observation, and 
children with as many as 
observations. The HIV status 
of a child (equal to 1 for 
HIV positive and equal to 0 for 
HIV negative children) is the main covariate of interest. This study has been previously analyzed by Parzen and others (2011) and Lipsitz and others (2009) using dichotomized values of RRBO. However, there has not been any previous analysis on the original continuous response of RRBO. The main goals of analysis include developing an easily interpretable and appropriate model of the effects of HIV status and age on highly skewed longitudinal response RRBO, estimation of the HIV status effect with good precision, and prediction of future RRBO of the same as well as new (similar) children.
Our model for RRBO 
of patient 
at age 
is the partial linear model of (2.1) with 
. For our Bayesian analysis, the unknown smooth function 
is approximated by cubic B-spline 
with (0.25, 0.375, 0.50, 0.625, 0.75) quantiles of age as the fixed knots. We compare M-estimation method with two Bayesian estimation methods: (1) Bayesian method with parametric skewed double-exponential error (called SDE method) and (2) Bayesian method with skewed semiparametric error (called SPMSD method) of (2.3). Based on the exploratory variogram plots in Figure 1, we use different structures of 
to model the within-patient Gaussian copula of (2.4) for two HIV status groups. We use a uniform correlation structure with 
for HIV-negative and 
for HIV-positive group for 
, where 
is the vector of association parameters.
Fig. 1.
Variogram plot.
We specify priors for 
and 
using prior opinion about the median RRBO of patients from two groups. For example, our 
prior for 
(regression parameter of HIV status) represents the prior opinion that two children with same age, but from different groups, can have a maximum difference of 1.96 in median RRBO (with 
95% prior probability). Simulation of possible trajectories of 
using priors for 
is used to evaluate whether they represent the prior opinion about possible change in median RRBO over time. We use independent 
priors for each 
, 
and 
; independent 
priors for 
and 
(different skewness parameters for two groups); and independent 
priors for 
and 
. For the sake of brevity, we omit the details and associated figures explaining our above choices of prior densities. We would like to emphasize that the Bayesian analysis presented here only serves the purpose of the illustration of the methodology. The priors chosen above may not correspond to the true prior opinions of the clinical investigators. However, our sensitivity analysis suggests very moderate influence of these priors on the estimation. For the DP prior of the mixing distribution 
in (2.3), the choice of 
is determined by the prior opinion about the error density 
. For example, a skewed double exponential “prior guess” of 
needs a corresponding Gamma density for 
. A small precision of 
used by us results in a very small confidence assigned to our “prior guess” 
of 
. This implies that actual 
can be very different from the guess 
. For the 
-estimator (He and others, 2002), we use BIC for knots selection. The estimators of the regression parameters and their corresponding estimated precision obtained from different methods are given in Table 2. We also present the estimators of the association parameters 
and of the skewness parameters 
and 
for two groups (available only for Bayesian analysis). We can see that the standard errors of the estimated effect of HIV status obtained from our two Bayesian methods are 30% (a substantial gain in precision) smaller than that those obtained from the 
-estimation method. All methods show strong evidence that HIV-positive status group has a lower median RRBO compared to the HIV-negative status group.
Table 2.
Results of the analysis of 
study with the estimators and the corresponding standard errors (called SE, given within parenthesis) of the regression parameters 
and other model parameters obtained from different methods
| Estimation method | SDE | SPMSD |
 -estimator |
|---|---|---|---|
| Parameters | Estimate (SD) | Estimate (SD) | Estimate (SD) |
|
0.440 (0.005) | 0.448 (0.006) | 0.428 (0.038) |
|
 0.041 (0.0039) |
 (0.0046) |
 0.042 (0.0063) |
 for 
|
0.715 (0.043) | 0.749 (0.037) | – |
 for 
|
0.839 (0.068) | 0.864 (0.058) | – |
|
0.234 (0.028) | 0.242 (0.028) | – |
|
0.168 (0.031) | 0.192 (0.035) | – |
In Figure 2, we illustrate the predictive and the goodness-of-fit performances of our semiparametric Bayesian method using plots of the predicted quartile functions (
, and 
, respectively, for the first, second, and third quartile functions) as well as the scatter plots of the responses from each group. The figure suggests different error densities for two groups. Also, for a grid of, say, 0.2 years over time, the estimated quantile functions do capture nearly intended proportions (
) of observations in all four ranges (
and 
) for each status group (suggesting excellent fit for our Bayesian methods). The results in Table 2 also give strong evidence of different skewness and different association structures for two groups.
Fig. 2.
Plot of observations and predictive quantiles (under semiparametric Bayesian model) for HIV groups. Age is scaled to the interval [0, 1].
6. Conclusions and remarks
In this paper, we have developed a novel semi-parametric Bayesian method and associated theoretical justifications for longitudinal data with skewed continuous responses. Our method can handle several challenges including any restriction on the marginal error density, a nonparametric effect of time, simple physical interpretation of the regression parameters, and prior specification based on prior opinion about the medians and the error density. Our MCMC computational tool is implementable even via freely available software such as WINBUGS and JAGS. Our models based on copula can even allow for different within-subject association structures, skewness levels, and scales for different treatment groups.
The results of our simulation studies in Table 1 show that our Bayesian estimators gives a much lower MSE for a finite sample compared with the 
-estimators of He and others (2002) even when the error density and correlation structure of the simulation model are very different from the model assumptions used for Bayesian estimation. Frequentist robust estimators such as M-estimators are aimed at consistent estimation of regression parameter under a very wide class of error densities. However, a likelihood-based Bayesian estimator such as ours often enjoys better precision than the competitors at finite samples, and is also particularly suitable for many biomedical studies where prediction is of practical importance. To validate the performance of the methods, we compared out-of-sample prediction errors from competing methods, and these comparisons reveal better predictive performance of our semiparametric Bayesian method. A logical way to choose the appropriate Bayesian model among various Bayesian models with different association structures would be to use the 
-criterion of Ibrahim and others (2001). In this paper, however, we are comparing among frequentist and Bayesian methods. For this purpose, we believe that diagnostic plots of Figure 2 and estimated precision of the regression parameters of Table 2 are two reasonable tools for comparing methods.
For the analysis of P2C2 study, our choices of two different structures of 
for two HIV-status groups are based on our preliminary data analysis and subject–area knowledge. In principle, our copula-based longitudinal model can allow a very wide class of association structures including uniform (equicorrelation), AR(1), and exponential, as long as the correlation matrix 
is positive definite. It is also straightforward to extend our method to allow other non-Gaussian copulas. Although the Gaussian copulas have several advantages including computational tractability and good physical interpretation, it has some limitations too. For example, a well-known criticism of Gaussian copula is the lack of good tail dependence. However, it is not a major concern when we are primarily interested in the inference about median and other quartile functions of the marginal density.
For the first time in the literature, we provide a theoretical justification of Bayesian methods for skewed longitudinal data under partial linear model. Our results guarantee consistent regression estimates under very mild regularity conditions and even when the unspecified time effect 
is approximated by a B-spline. For simplicity of exposition, our posterior consistency results are shown only for the uniform correlation (equicorrelation) structure. Our posterior consistency results can be extended straightforwardly for other positive definite 
. One possible criticism is that the class of residual densities in the support of our prior does not contain continuous asymmetric densities. However, our simulation studies 3 and 4 suggest that the assumption of split density for error has a negligible effect on the final estimates of 
even when the true 
is both continuous and asymmetric.
Instead of using a prior for 
(as suggested in Section 3), we used a fixed value of 
for the Bayesian analysis of P2C2 study. This value of 
is same as the one used for the frequentist analysis using method of He and others (2002) (which uses BIC to decide 
). We did this only to facilitate a comparison between Bayesian and frequentist analysis results while maintaining a comparable level of approximation of 
in both methods. Our results on Bayesian consistency are useful to even derive the rate of posterior convergence of the regression parameters. However, for the sake of brevity, we omit the proof of the optimal rate of convergence of our semiparametric Bayesian estimators. Another possible direction for future research is to develop computationally efficient heteroscedastic median zero error distributions to accommodate large outliers.
Funding
Dr Sinha acknowledges support from the National Cancer Institute (NCI) of the National Institutes of Health (R01CA69222 and R01CA60679). Dr Pati acknowledges support from the Office of Naval Research (ONR BAA 14-0001). Dr Lipsitz acknowledges support from the National Cancer Institute (NCI) of the National Institutes of Health (R01CA60679). Conflict of Interest: None declared.
Supplementary Materials
Supplementary material is available online at http://biostatistics.oxfordjournals.org
Supplementary Material
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