Censored linear regression models for irregularly observed longitudinal data using the multivariate-t distribution

Abstract

In acquired immunodeficiency syndrome (AIDS) studies it is quite common to observe viral load measurements collected irregularly over time. Moreover, these measurements can be subjected to some upper and/or lower detection limits depending on the quantification assays. A complication arises when these continuous repeated measures have a heavy-tailed behavior. For such data structures, we propose a robust structure for a censored linear model based on the multivariate Student’s t-distribution. To compensate for the autocorrelation existing among irregularly observed measures, a damped exponential correlation structure is employed. An efficient expectation maximization type algorithm is developed for computing the maximum likelihood estimates, obtaining as a by-product the standard errors of the fixed effects and the log-likelihood function. The proposed algorithm uses closed-form expressions at the E-step that rely on formulas for the mean and variance of a truncated multivariate Student’s t-distribution. The methodology is illustrated through an application to an Human Immunodeficiency Virus-AIDS (HIV-AIDS) study and several simulation studies.

Full text

Censored Linear Regression Models for Irregularly Observed
Longitudinal Data using the Multivariate-tDistribution
Aldo M. Garaya, Luis M. Castrob, Jacek Leskowcand Victor H. Lachosa
aDepartamento de Estat´ıstica, Universidade Estadual de Campinas, Brazil
bDepartamento de Estat´ıstica, Universidad de Concepci´on, Chile
cInstitute of Mathematics, Cracow Technical University, Cracow, Poland
Abstract: In AIDS studies it is quite common to observe viral load measurements collected irregularly
over time. Moreover, these measurements can be subjected to some upper and/or lower detection limits
depending on the quantification assays. A complication arises when these continuous repeated measures
have a heavy-tailed behavior. For such data structures, we propose a robust structure for a censored
linear model based on the multivariate Student-tdistribution. To compensate for the autocorrelation
existing among irregularly observed measures, a damped exponential correlation structure is employed.
An efficient EM (expectation maximization) type algorithm is developed for computing the maximum
likelihood estimates, obtaining as a by-product the standard errors of the fixed effects and the log-
likelihood function. The proposed algorithm uses closed-form expressions at the E-step, that rely on
formulas for the mean and variance of a truncated multivariate Student-tdistribution. The methodology
is illustrated through an application to an HIV-AIDS study and several simulation studies.
Key words and phrases: Censored data, ECM Algorithm, longitudinal data, HIV viral load, outliers.
1 Introduction
In many biomedical studies and clinical trials, the use of longitudinal models has shown a significant
growth in recent years and becomes a powerful tool for modeling correlated outcomes. In clinical
trials of anti-retroviral therapy in AIDS studies, HIV-1 RNA (viral load) measures are collected
over a period of treatment to determine rates of changes in the amount of actively replicating virus.
These measures are used as a key primary endpoint because the viral load monitoring during the
therapy is mostly available, a failure in the treatment can be defined virologically, and a new
regimen of therapy is recommended as soon as virological rebound occurs 1. Since for each patient
the viral load measures are collected over time, the correlation structure among responses must be
taken into account.
Longitudinal models allow us to estimate viral load trajectories as well as to quantify the
correlation structure between viral load measurements2,3. However, in practice, the statistical
modeling of viral loads can be challenging due to the following features. First, the measurements
can be subject to a upper/lower limit of quantification. As a result, the viral load responses are
either left or right censored depending upon the diagnostics assay used. In general, the range of limit
detection varies from 400 copies/ml for the earlier assays to 40 copies/ml for the more sophisticated
1

assays of the recent times. Second, the viral load are usually recorded at irregular occasions because
the timings of measurements often vary from one patient to another, and typically measurement
times are associated with the course of the disease. Third, the viral load measurements often
contain influential and/or outlying observations, which can cause misleading results in parameter
estimates as well as their standard errors when a Gaussian assumption is considered. Therefore,
one of the greatest challenges related to longitudinal data modeling in AIDS research is to consider
the inherent features of viral load measurements simultaneously.
In the statistical and biomedical literature, linear and nonlinear mixed effects models based
on Gaussian assumptions are routinely used to model longitudinal data4–6. Nevertheless, such an
assumption could be not realistic because of the presence of atypical observations (outliers). To
deal with this weakness, some alternatives based on heavy-tailed distributions have been proposed.
For example, Pinheiro et al.7proposed a Student-tlinear mixed model that has demonstrated its
robustness against outliers. Other authors focused their research interest in developing strategies
for fitting both linear and nonlinear mixed effects models under heavy-tailed distributions such as
the Student-t, slash and contaminated normal distributions8–12. The regression and mixed effects
models for censored responses under heavy-tailed distributions have been studied in detail in the
last years.13–16
Recently, a variety of heavy-tailed statistical models have been proposed for longitudinal data
considering not only the correlation structure induced by the random effects term but also by other
types of correlation in the error term. For example, Wang17 studied the multivariate Student-
tlinear mixed model (t-LMM) for outcome variables recorded at irregular occasions by using a
parsimonious damping exponential correlation (DEC) structure. This type of correlation structure,
proposed by Mu˜noz et al.18, takes into account the autocorrelation generated by the within-subject
dependence among irregular occasions. Wang and Fan 19 considered the multivariate Student-t
linear mixed with AR(p) dependence structure (a particular case of DEC structure) for the within-
subject errors in the case of multiple outcomes. However, and as was mentioned by Goldstein et
al.20 and Browne and Goldstein21, in cases when the repeated measures are collected close in time
or correlations among the measures do not decay quickly, random effects modes may not adequately
account for the dependency and a more complex correlation structure must be specified.
Following Wang17, the aim of this paper is to consider the DEC structure for the across-occasion
covariance matrix of the random errors under censored responses. As a consequence, the robust so
called Student-tmultivariate linear censored (t-MLC) model with DEC structure is defined and a
fully likelihood-based approach is conducted, including the implementation of an exact expectation
conditional maximization (ECM) algorithm for the maximum likelihood (ML) estimation. As in
Matos et al.22, we show that the E-step reduces to computing the first two moments of certain
truncated multivariate Student-tdistribution, with the computation of the likelihood function and
the asymptotic standard errors as a by-product of the E-step. General formulas for these moments
were derived by using the results given in Ho et al.23 . The likelihood function is used for monitoring
convergence and the model selection is conducted through the Akaike information criterion (AIC)
2

and Bayesian information criterion (BIC).
The rest of the paper is organized as follows. Section 2 describes a motivating real life data set
of HIV-AIDS infected patients. In Section 3 we introduce some notation related to the truncated
Student-tdistribution. Then, the t-MLC is presented. In Section 4, the related likelihood-based
inference are presented including the imputation procedure of censored cases. The method for the
prediction of future observations is presented in Section 5. Section 6 presents two simulation studies
for comparing the performance of our method with other normality-based one. The advantage of the
proposed method is presented through the analysis of a case studies of HIV viral load in Section
7. Section 8 concludes with a short discussion of issues raised by our study and some possible
directions for a future research.
2 Motivating example:
Unstructured treatment interruption (UTI) data
In this section, we present a longitudinal dataset corresponding to UTI - unstructured antiretroviral
therapy treatment interruption - in HIV-infected adolescents in four institutions in the US. In this
case, the HIV-1 RNA measures are subject to censoring below the lower limit of detection of the
assay (50 copies/mL) and the presence of outlying and influential observations is noted.
23456
Month of TI
log10 HIV−1
0 3 6 9 12 15 18 21 24
Ind=20
Ind=35
Ind=19
(a)
−3 −2 −1 0 1 2 3
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
(b)
Quantiles of Standard Normal
Residuals
Figure 1: UTI data. (a) Individual profiles (in log10 scale) for HIV viral load at different follow-up
times. Trajectories for three censored individuals are indicated in different colors. The abbreviation
“Ind” stands for “Individual” . (b) Normal Q–Q plot for model residuals obtained by using the
lmec package of R.
This dataset consist on the measurements of HIV-1 RNA measures after UTI in 72 adolescents
3

log10RNA
month 0 month 1 month 3 month 6 month 9 month 12 month 18 month 24
month 0 0.4877 0.4100 0.4052 0.4820 0.4435 0.3441 0.6529
month 1 0.4877 0.9145 0.8551 0.8455 0.6978 0.7090 0.6140
month 3 0.4100 0.9145 0.9255 0.8638 0.7209 0.7601 0.6301
log10RNA month 6 0.4052 0.8551 0.9255 0.8238 0.6490 0.6548 0.5314
month 9 0.4820 0.8455 0.8638 0.8238 0.9185 0.7642 0.8061
month 12 0.4435 0.6978 0.7209 0.6490 0.9185 0.6646 0.6897
month 18 0.3441 0.7090 0.7601 0.6548 0.7642 0.6646 0.8947
month 24 0.6529 0.6140 0.6301 0.5314 0.8061 0.6897 0.8947
Table 1: Observed correlation of log10RNA for a single response over different times.
from US. UTI was defined as discontinuation of all antiretroviral drugs for any period of time, after
which treatment was resumed. The reasons for interruption might have been diverse. For example,
Saitoh et al.24 mentioned (a) the medication fatigue, (b) patients inability to take antiretroviral
medications (c) toxicity associated with the use of antiretroviral medications, (d) adverse effects.
The data set presents about 7% of observations below the detection limits of assay quantifica-
tions (left-censored). The viral load were monitored from the closest time points at 0,1,3,6,9,12,18
and 24 months after the treatment interruption, that is, they were irregularly collected over time.
A more detailed explanation of the data can be founded in Saitoh et al.24 and Vaida and Liu25.
The individual profiles of viral load at different follow-up times after UTI appear in Figure 1 (panel
a). This figure also presents the normal quantile–quantile (Q-Q) plot for the residuals (panel b)
obtained by fitting a censored (Gaussian) mixed effect model proposed by Vaida and Liu 25 us-
ing the lmec package of R. The Q–Q plot exhibits a heavy-tailed behavior, suggesting that the
normality assumption for the within-subject errors might be inappropriate. The non-normality of
the distribution gives an indication that some atypical observations or outliers might exist in the
data. Outliers may cause misleading results in parameter estimates as well as their standard errors
and they could have an enormous impact on statistical inferences. In addition, Table 1 shows the
observed correlation for the responses in different time points. Clearly, the data do not appear to
satisfy an uncorrelated structure of zero covariances or a compound symmetry assumption of equal
variances and covariances across time.
3 Model Specification
3.1 Preliminaries
In this section, we present some useful results associated to the p-variate Student-tdistribution
that will be needed for implementing the EM algorithm. We start with the probability density
function (pdf) of a Student-trandom vector Y∈Rpwith location vector µ, scale matrix Σand ν
4

degrees of freedom. Its pdf is given by
tp(y|µ,Σ, ν) = Γ(p+ν
2)
Γ(ν
2)πp/2ν−p/2|Σ|−1/21 + δ
ν−(p+ν)/2
,
where Γ(·) is the standard gamma function and δ= (y−µ)>Σ−1(y−µ) is the Mahalanobis
distance. The notation adopted for the Student-tpdf is tp(µ,Σ, ν).
The cumulative distribution function (cdf) is denoted by Tp(·|µ,Σ, ν). It is important to stress
that if ν > 1, the mean of yis µand if ν > 2, the covariance matrix is given by ν(ν−2)−1Σ.
Moreover, as νtends to infinity, Yconverges in distribution to a multivariate normal with mean
µand covariance matrix Σ.
An important property of the random vector Yis that it can be written as a mixture of a
normal random vector and a positive random variable, i.e,
Y=µ+U−1/2Z,
where Zis a normal random vector, with zero-mean vector and covariance Σ, independent of U,
which is a positive random variable with a gamma distribution Gamma(ν /2, ν /2).
The distribution of Yconstrained to lie within the right-truncated hyperplane
A={y∈Rp|y≤a},(1)
where y= (y1, . . . , yp)>and a= (a1, . . . , ap)>, is a truncated Student-tdistribution, denoted by
T tp(µ,Σ, ν;A), with pdf given by f(y|µ,Σ, ν ;A) = tp(y|µ,Σ, ν)
Tp(a|µ,Σ, ν)IA(y),where IA(y) is the indicator
function of A.
As was mentioned at the beginning of this section, the following properties of the multivariate
Student-tand truncated Student-tdistributions are useful for the implementation of the EM-
algorithm. We start with the marginal-conditional decomposition of a Student-trandom vector.
Details of the proofs are provided in Arellano-Valle and Bolfarine26.
Proposition 1. Let Y∼tp(µ,Σ, ν)and Ybe partitioned as Y>= (Y>
1,Y>
2)>, with dim(Y1) =
p1,dim(Y2) = p2,p1+p2=p, and where Σ= Σ11 Σ12
Σ21 Σ22 !and µ= (µ>
1,µ>
2)>, are the
corresponding partitions of Σand µ. Then, we have
(i)Y1∼tp1(µ1,Σ11, ν ); and
(ii)the conditional cdf of Y2|Y1=y1is given by
P(Y2≤y2|Y1=y1) = Tp2y2|µ2.1,e
Σ22.1, ν +p1,
Gamma(a, b) denotes a gamma distribution with a/b mean.
5

where e
Σ22.1=ν+δ1
ν+p1Σ22.1, δ1= (y1−µ1)>Σ−1
11 (y1−µ1),Σ22.1=Σ22 −Σ21Σ−1
11 Σ12,
and µ2.1=µ2+Σ21Σ−1
11 (y1−µ1).
The following results provide the truncated moments of a Student-trandom vector. The proofs
of Proposition 2 and 3 are given in Matos et al.22. The proof of Proposition 4 is given in Ho et
al.23.
Proposition 2. If Y∼T tp(µ,Σ, ν ;A)with Aas (1), then the k-th moment of Y,k= 0,1,2, is
Eν+p
ν+δr
Y(k)=cp(ν, r)Tp(a|µ,Σ∗, ν + 2r)
Tp(a|µ,Σ, ν)EWhW(k)i,W∼T tp(µ,Σ∗, ν + 2r;A),
where cp(ν, r) = ν+p
νrΓ((p+ν)/2)Γ((ν+ 2r)/2)
Γ(ν/2)Γ((p+ν+ 2r)/2) , δ = (Y−µ)>Σ−1(Y−µ),a= (a1, . . . , ap)>,
Σ∗=ν
ν+ 2rΣ,Y(0) = 1,Y(1) =Y,Y(2) =YY>, and ν+ 2r > 0.
Having established a formula for the k-order moments of Y, we now present a result on condi-
tional moments of the partition of Y.
Proposition 3. Let Y∼T tp(µ,Σ, ν;A)with Aas (1). Consider the partition Y>= (Y>
1,Y>
2)
with dim(Y1) = p1,dim(Y2) = p2,p1+p2=p, and the corresponding partition of the parameters
µ,Σ,a(ay1,ay2) and A(Ay1,Ay2). Then, under the notation of Proposition 1, the conditional
k-th moment of Y2is
Eν+p
ν+δr
Y(k)
2|Y1=dp(p1, ν, r)
(ν+δ1)r
Tp2(ay2|µ2.1,e
Σ∗
22.1, ν +p1+ 2r)
Tp2(ay2|µ2.1,e
Σ22.1, ν +p1)EWhW(k)i,
where W∼T tp2(µ2.1,e
Σ∗
22.1, ν +p1+2r;Ay2), δ = (Y−µ)>Σ−1(Y−µ),δ1= (Y1−µ1)>Σ−1
11 (Y1−
µ1),ay2= (a1, . . . , ap2)>,e
Σ∗
22.1=ν+δ1
ν+ 2r+p1Σ22.1,ν+p1+ 2r > 0and dp(p1, ν, r) =
(ν+p)rΓ((p+ν)/2)Γ((p1+ν+ 2r)/2)
Γ((p1+ν)/2)Γ((p+ν+ 2r)/2).
In the following Proposition, we establish relationships between the expectation and covariance
of Yand W.
Proposition 4. Let Y∼T tp(µ,Σ, ν;A∗), with A∗={y∈Rp|a∗<y≤b∗},where a∗=
(a∗
1, . . . , a∗
p)>,b∗= (b∗
1, . . . , b∗
p)>,Σ = ΛRΛand Λ = Diag (σ11, . . . , σpp)is a p×pdiagonal
matrix with each element being positive. We have that W= Λ−1(Y−µ)∼T tp(0,R, ν ;A),where
a= Λ−1(a∗−µ)and b= Λ−1(b∗−µ). Therefore,
E[Y] = µ+ ΛE[W]
E[YY>] = µµ>+ ΛE[W]µ>+µE[W>]Λ + ΛE[WW>]Λ>,
where E[W]and E[WW>]are given in Ho et al. 23.
6

3.2 The statistical model
Our multivariate Student-tlinear (t-ML) model, for longitudinal responses, is defined by:
Yi=Xiβ+i,(2)
with i∼tni{0,Σi, ν}, where Yi= (Yi1, . . . , Yini)>is a ni×1 vector of continuous responses for
sample unit imeasured at particular time points ti= (ti1, . . . , tni)>,Xiis the ni×pdesign matrix
corresponding to the p×1 vector of fixed effects βand iis the ni×1 vector of random errors.
As was noted in Section 2, the measurements of the viral load for each subject present evidence of
serial correlation. Therefore, to obtain accurate parameter estimates, we consider a parsimonious
structure on the dispersion matrix Σi=σ2Ei, where the matrix Eiincorporates a time-dependence
structure. Thus, we adopt the DEC structure for Σi, as proposed by Mu˜noz et al.18 . This correla-
tion structure allows us to deal with unequally spaced and unbalanced observations and is defined
as
Σi=σ2Ei=σ2Ei(φ,ti) = σ2hφ|tij −tik|φ2
1i,(3)
for i= 1, . . . , n, j, k = 1, . . . , ni, . The correlation parameter φ1describes the autocorrelation
between observations separated by the absolute length of two time points, and the damping pa-
rameter φ2allows the acceleration of the exponential decay of the autocorrelation function defining
a continuous-time autoregressive (AR) model. It is important to stress that considering the DEC
structure it is possible to obtain different correlation structures. For example, for a given nonneg-
ative φ1:
•If φ2= 0, then the Eiis the compound symmetry structure.
•If 0 < φ2<1, then we have Eiwith a decay rate between that of compound symmetry and
the first-order AR (AR(1)) model.
•If φ2= 1, then the Eiis an AR(1) model.
•If φ2>1, then we have Eiwith a decay rate faster than AR(1) model.
•If φ2→ ∞,then the Eiis the first-order moving average (MA(1)) model.
For a more detailed discussion about the DEC structure, we refer to Mu˜noz et al.18 and Wang17.
From a practical point of view and in order to avoid computational problems, the parameter
space of φ1and φ2is confined within {(φ1, φ1):0< φ1<1, φ2>0}, that is, we restrict our
attention to nonnegative values of φ1and φ2. Such assumption is quite common in most biomedical
and epidemiological applications. Finally, we consider the approach proposed by Vaida and Liu25
and Matos et al.22 to modeling the censored responses. Thus, the observed data for the ith subject
is given by (Vi,Ci),where Virepresents the vector of uncensored reading or censoring level and
7

Cithe vector of censoring indicators. In other words,
yij ≤Vij if Cij = 1,and yij =Vij if Cij = 0,(4)
so that, considering (4) along with (2)-(3) defined the multivariate Student-tlinear censored (t-
MLC) model. Notice that a left censoring structure causes a right truncation of the distribution,
since we only know that the true observation yij is less than or equal to the observed quantity
Vij. Moreover, the right censored problem can be represented by a left censored problem by
simultaneously transforming the response yij and censoring level Vij to −yij and −Vij.
3.3 The likelihood function
To obtain the likelihood function of the t-MLC model, first we treat separately the observed and
censored components of yi,i.e.,yi= (yo>
i,yc>
i)>, with Cij = 0 for all elements in yo
i, and Cij = 1
for all elements in yc
i. Analogous, we write Vi=vec(Vo
i,Vc
i), where vec(·) denotes the function
which stacks vectors or matrices of the same number of columns, with Σi=Σoo
iΣoc
i
Σco
iΣcc
i. Then, using
Proposition 1, we have that yo
i∼tno
i(Xo
iβ,Σoo
i, ν) and yc
i|yo
i,∼tnc
i(µco
i,Sco
i, ν +no
i),where
µco
i=Xc
iβ+Σco
iΣoo−1
i(yo
i−Xo
iβ),Sco
i=ν+Q(yo
i)
ν+no
iΣcc.o
i,(5)
with Σcc.o
i=Σcc
i−Σco
iΣoo−1
iΣoc
iand Q(yo
i) = (yo
i−Xo
iβ)>Σoo−1
i(yo
i−Xo
iβ). Therefore, the
likelihood function for subject iis
Li(θ|y) = f(Vi|Ci,θ) = f(yc
i≤Vc
i|yo
i=Vo
i,θ)f(yo
i=Vo
i|θ),
=Tnc
i(Vc
i|µco
i,Sco
i, ν +no
i)tno
i(Vo
i|Xo
iβ,Σoo
i, ν) = Li.
Straightforwardly, the log-likelihood function for the observed data is given by `(θ|y) = Pn
i=1 log Li.
It is important to note that this function can be computed at each step of the EM-type algorithm
without additional computational burden since the Li’s have already been computed at the E-
step. We assume that the degrees of freedom parameter of the Student-tdistribution is fixed. For
choosing the most appropriate value of this parameter, we will use the log-likelihood profile10,27.
This assumption is based on the work by Lucas28, in which the author showed that the protection
against outliers is preserved only if the degrees of freedom parameter is fixed. Consequently, the
parameter vector for the t-MLC model is θ= (β>, σ2, φ1, φ2)>.
4 The EM algorithm
We describe in detail how to carry out ML estimation for the proposed t-MLC model. The EM
algorithm, originally proposed by Dempster et al.29, is a very popular iterative optimization strategy
commonly used to obtain ML estimates for incomplete data problems. This algorithm has many
8

attractive features such as the numerical stability and the simplicity of implementation and its
memory requirements are quite reasonable30 . However, ML estimation for the t-MLC model is
complicated because of the censoring and the DEC structure, and the EM algorithm is less advisable
due to the computational difficulty at the M-step. To overcome this problem, we used an extension
of the EM algorithm, called the ECM algorithm 31. A key feature of this algorithm is that it
preserves the stability of the EM and has a typically faster convergence rate than the original EM.
In order to propose the ECM algorithm for our t-MLC model, firstly we define y= (y>
1,...,y>
n)>,
u= (u1, . . . , un)>,V=vec(V1,...,Vn), and C=vec(C1,...,Cn) such that we observe (Vi,Ci)
for the i-th subject. Now, we treat uand yas hypothetical missing data, and augmenting with the
observed data V,Ccorresponding to the censoring mechanism. Consequently, we set the complete-
data vector as yc= (C>,V>,y>,u>)>. As is well known, the ECM algorithm must be applied to
the complete data log-likelihood function given by
`c(θ|yc) =
n
X
i=1
`i(θ|yc),
where
`i(θ|yc) = −1
2hnilog σ2+ log |Ei|+ui
σ2(yi−Xiβ)>E−1
i(yi−Xiβ)i+h(ui|ν) + c,
cis a constant that does not depend on θand h(ui|ν) is the Gamma(ν/2, ν/2) pdf. Finally, the
ECM algorithm for the t-MLC model can be summarized through the following two steps.
E-step:
Given the current value θ=b
θ(k), the E-step provides the conditional expectation of the complete
data log-likelihood function
Q(θ|b
θ(k)) =
n
X
i=1
Qi(θ|b
θ(k)),(6)
where
Qi(β, σ2,φ|b
θ(k)) = −ni
2log σ2−1
2log |Ei| − 1
2σ2A(k)
i(β,φ),
with
A(k)
i(β,φ) = trd
uy2
i
(k)E−1
i−2βX>
iE−1
ic
uy(k)
i+bu(k)
iβX>
iE−1
iXiβi.
Note that, since νis fixed, there is no need to obtain Ehh(ui|ν)|V,C,b
θ(k)i.
CM-step:
9

In this step, Q(θ|b
θ(k)) is conditionally maximized with respect to θand a new estimate b
θ(k+1)
is obtained. Specifically, we have that
b
β(k+1) = n
X
i=1 bu(k)
iX>
ib
E(k)
i−1Xi!−1n
X
i=1
X>
ib
E(k)
i−1c
uy(k)
i,(7)
c
σ2(k+1) =1
N
n
X
i=1
A(k)
i(b
β(k+1),b
φ(k)) (8)
b
φ(k+1) =argmaxφ(−1
2
n
X
i=1
[log(|Ei|) + A(k)
i(b
β(k+1),φ)]),(9)
where N=Pn
i=1 ni. The optim routine in Rpackage32 can be used to perform a two-dimensional
search of (φ1, φ2) in (9) subject to box constraints: φ1∈[0,1) and φ2∈(0,∞).
The algorithm is iterated until a suitable convergence rule is satisfied. In this case, we adopt
the distance involving two successive evaluations of the log-likelihood |`(b
θ(k+1))/`(b
θ(k))−1|as a
convergence criterion. It is important to stress that from equations (7)-(9), the E-step reduces to
the computation of d
uy2
i,c
uyi, and bui. These expected values can be determined in closed form,
using Propositions 1-4, as follows:
1. If the subject ihas only censored components, from Proposition 2
d
uy2
i=Ehuiyiy>
i|Vi,Ci,b
θ(k)i=Tni(Vi|b
µ(k)
i,b
Σ∗(k)
i, ν + 2)
Tni(Vi|b
µ(k)
i,b
Σ(k)
i, ν)
EhWiW>
ii,
c
uyi=Ehuiyi|Vi,Ci,b
θ(k)i=Tni(Vi|b
µ(k)
i,b
Σ∗(k)
i, ν + 2)
Tni(Vi|b
µ(k)
i,b
Σ(k)
i, ν)
E[Wi],
bui=Ehui|Vi,Ci,b
θ(k)i=Tni(Vi|b
µ(k)
i,b
Σ∗(k)
i, ν + 2)
Tni(Vi|b
µ(k)
i,b
Σ(k)
i, ν)
,
where Wi∼T tni(b
µ(k)
i,b
Σ∗(k)
i, ν + 2; Ai), b
µ(k)
i=Xib
β(k),b
Σ∗(k)
i=ν
ν+ 2 b
Σ(k)
i,b
Σ(k)
i=[
σ2(k)b
E(k)
i
and Ai={Wi∈Rni|wi≤Vi}where wi= (wi1, . . . , wini)>and Vi= (Vi1, . . . , Vini)>.
2. If the subject ihas only non-censored components, then,
d
uy2
i=ν+ni
ν+Q(yi)yiy>
i,c
uyi=ν+ni
ν+Q(yi)yi,bui=ν+ni
ν+Q(yi),
where Q(yi)=(yi−Xiβ)>Σ−1
i(yi−Xiβ).
3. If the subject ihas censored and uncensored components, then from Proposition 3 and given
10

that {yi|Vi,Ci},{yi|Vi,Ci,yo
i}, and {yc
i|Vi,Ci,yo
i}are equivalent processes, we have that
d
uy2
i=Ehuiyiy>
i|yo
i,Vi,Ci,b
θ(k)i= yo
iyo>
ibuibuiyo
ib
wc>
i
buib
wc
iyo>
ibuic
w2c
i!,
c
uyi=Ehuiyi|yo
i,Vi,Ci,b
θ(k)i=vec(yo
ibui,b
wc
i),
bui=Ehui|yo
i,Vi,Ci,b
θ(k)i=no
i+ν
ν+Q(yo
i)Tni(Vi|µco
i,e
Sco
i, ν +no
i+ 2)
Tni(Vi|µco
i,Sco
i, ν +no
i),
where e
Sco
i=ν+Q(yo
i)
ν+2+no
iΣcc.o
i,b
wc
i=E[Wi],and c
w2c
i=EWiW>
i, with Wi∼
T tnc
i(µco
i,e
Sco
i, ν +no
i+ 2; Ac
i) and Σcc.o
i,µco
i,and Sco
iare as in (5).
As was mentioned in Proposition 4, formulas for E[W] and E[WW>], where W∼T tp(µ,Σ, ν;A),
can be found in Ho et al.23. For the the computation of multivariate Student-tcdf we used the pmvt
function of the mvtnorm package33 from Rsoftware. Additional details about the ECM algorithm
for our proposed t-MLC model can be found in Appendix.
4.1 The expected information matrix
Louis34 proposed a technique for computing the observed information matrix within the EM al-
gorithm framework. Using this method, and from the results given by Lange et al.27, we can
find an asymptotic approximation for the variances of the fixed effects in the t-MLC model. This
approximation is given by
Jββ =V ar(b
β) = n
X
i=1
ν+ni
ν+ni+ 2X>
iΣ−1
iXi−
n
X
i=1
X>
iΣ−1
i(Bi+Di)Σ−1
iXi!−1
,(10)
where Bi=V ar ν+ni
ν+Q(yi)(yi−Xiβ)|Vi,Ciand Di=2
ν+ni(d
uyi∗−bui∗Xiβ)(d
uyi∗−
bui∗Xiβ)>, with yi∼T tni(Xiβ,Σi, ν;Ai), where Ai={yi∈Rni|yi≤ai}, with yi= (yi1, . . . , yini)>
and ai= (ai1, . . . , aini)>.
Note that clearly Biand Didepend on the following quantities:
d
uy2
i
∗=E"ν+ni
ν+Q(yi)2
yiy>
i|Vi,Ci,b
θ#,d
uyi∗=E"ν+ni
ν+Q(yi)2
yi|Vi,Ci,b
θ#and
bui∗=E"ν+ni
ν+Q(yi)2
|Vi,Ci,b
θ#.
After some algebraic manipulations, we have three possible scenarios for the calculation of these
quantities:
11

•If individual ihas only non-censored components, then
d
uy2
i
∗=ν+ni
ν+Q(yi)2
yiy>
i,d
uyi∗=ν+ni
ν+Q(yi)2
yi,bui∗=ν+ni
ν+Q(yi)2
,
where Q(yi)=(yi−Xiβ)>Σ−1
i(yi−Xiβ).
•If individual ihas only censored components then from Proposition 2
d
uy2
i
∗=cni(ν, 2)Tni(Vi|b
µi,b
Σ∗
i, ν + 4)
Tni(Vi|b
µi,b
Σi, ν)EhWiW>
ii,
d
uyi∗=cni(ν, 2)Tni(Vi|b
µi,b
Σ∗
i, ν + 4)
Tni(Vi|b
µi,b
Σi, ν)E[Wi],bui∗=cni(ν, 2)Tni(Vi|b
µi,b
Σ∗
i, ν + 4)
Tni(Vi|b
µi,b
Σi, ν),
where Wi∼T tni(b
µi,b
Σi, ν + 4,Ai), c
µi=Xib
β,b
Σ∗
i=ν
ν+ 4 b
Σiand
cni(ν, 2) = ν+ni
ν2



Γν+ni
2Γν+ 4
2
Γν
2Γν+ni+ 4
2


.
•If individual ihas censored and uncensored components, then from Proposition 3 and the
fact that {yi|Vi,Ci},{yi|Vi,Ci,yo
i}and {yc
i|Vi,Ci,yo
i}are equivalent processes, we have:
d
uy2
i
∗=
bui∗yo
iyo>
ibui∗yo
ic
wc
i
>
bui∗c
wc
iyo>
ibui∗\
wc
iwc>
i
,d
uyi∗=vec(buiyo
i,c
wc
i),
bui∗=dni
(ν+Q(yo
i))2Tnc
i(Vi|µco
i,e
Sco
i, ν +no
i+ 4)
Tnc
i(Vi|µco
i,Σco
i, ν +no
i),
where
dni= (ν+ni)2
Γni+ν
2Γno
i+ν+4
2
Γno
i+ν
2Γni+ν+4
2
,
e
Sco
i=ν+Q(yo
i)
ν+4+no
iΣcc.o
i,c
wc
i=E[Wi] and \
wc
iwc>
i=E[WiW>
i], with Wi∼T tnc
i(µco
i,e
Sco
i, ν +
no
i+ 2,Ac
i) and µco
i,Σcc.o
iand Sco
ias defined in Subsection 3.3.
Asymptotic confidence intervals for the fixed effects and hypothesis tests as well are obtained
assuming that the ML estimates b
βhas approximately a Np(β,J−1
ββ) distribution. In practice, Jββ
is usually unknown and it needs to be replaced by its ML estimates Jb
β
b
β.
12

4.2 Imputation of censored components
Let y(c)
ithe true unobserved response vector for the censored components of the ith subject. Now,
as a byproduct of the EM algorithm we can obtain the predictor of the censored components,
denoted by e
y(c)
i, as follows
e
y(c)
i=Ehyi|yo
i,Vi,Ci,b
θi,(11)
which is obtained considering two possible cases:
1. If subject ihas only censored components
e
y(c)
i=Ehyi|Vi,Ci,b
θi,
where yi|Vi,Ci,b
θ∼T tni(Xib
β,b
Σi, ν;Ai), with Ai={yi∈Rni|yi≤ai},yi= (yi1, . . . , yini)>
and ai= (ai1, . . . , aini)>. This expression is obtained using Proposition 4.
2. If subject ihas censored and uncensored components, then from Proposition 3 with r= 0
and k= 1, we have that
e
y(c)
i=Ehyc
i|yo
i,Vi,Ci,b
θi,
with yc
i|yo
i,∼T tnc
i(b
µco
i,b
Sco
i, ν+no
i;Ai) where Ai={yi∈Rni|yi≤ai}with yi= (yi1, . . . , yini)>,
ai= (ai1, . . . , aini)>and
b
µco
i=Xc
ib
β+b
Σco
ib
Σoo−1
i(yo
i−Xo
ib
β),b
Sco
i=ν+Q(yo
i)
ν+no
ib
Σcc.o
i,
with b
Σcc.o
i=b
Σcc
i−b
Σco
ib
Σoo−1
ib
Σoc
iand Q(yo
i)=(yo
i−Xo
ib
β)>b
Σoo−1
i(yo
i−Xo
ib
β).
5 Prediction of future values
The problem related to the prediction of future values has a great impact in many practical appli-
cations. Rao35 pointed out that the predictive accuracy of future observations can be taken as an
alternative measure of “goodness-of-fit”. In order to propose a strategy for generating predicted
values from our t-MLC model, we used the approach proposed by Wang17. Thus, let yi,obs be
an observed response vector of dimension ni,obs ×1 for a new subject iover the first portion of
time and yi,pred the corresponding ni,pred ×1 response vector over the future portion of time. Let
¯
Xi= (Xi,obs,Xi,pred ) be the (ni,obs +ni,pred)×pdesign matrix corresponding to ¯
yi= (y>
i,obs,y>
i,pred).
To deal with the censored values existing in yi,obs, we used the imputation procedure presented
in the Subsection 4.2. Therefore, when the censored values are imputed, a complete data set,
13

denoted by yi,obs∗, is obtained. The reason to use the imputation procedure is that we avoid to
compute truncated conditional expectations of multivariate Student-tdistribution originated by
the censoring scheme. Hence, we have that
¯
y∗
i=y>
i,obs∗,y>
i,pred>∼tni,obs +ni,pred (Xiβ,Σi, ν ),
where the matrix Σidefined in (3), can be represented by Σi= Σobs∗,obs∗
iΣobs∗,pred
i
Σpred,obs∗
iΣpred,pred
i!. As was
mentioned in Wang 17 and Rao 36 , the best linear predictor of yi,pred with respect to the minimum
mean squared error (MSE) criterion is the conditional expectation of yi,pred given yi,obs∗, which,
from Proposition 1, is given by
b
yi,pred(θ) = Xi,predβ+Σpred,obs∗
iΣobs∗,obs∗−1
i(yi,obs∗−Xi,obs∗β).(12)
Therefore, yi,pred can be estimated directly by substituting b
θinto (12), leading to \
b
yi,pred =
b
yi,pred(b
θ).
6 Simulation studies
In order to study the performance of the proposed method, we present two simulations studies.
The first one shows the performance of t-MLC with DEC structure on the imputation procedure.
The second one shows the asymptotic behavior of the ML estimates for our proposed model.
For both simulation schemes, we considered the t-MLC model defined in the Subsection 3.2,
with parameters setting at β1= 2.5 , β2= 4, σ2= 4, φ1= 0.8 and φ2= 1. In addition, the time
points are set as ti= (1,3,5,7,10,14)>, for i= 1, . . . , n.
6.1 Imputation performance
As was mentioned above, the goal of this simulation study is to compare the performance of the
t-MLC models with DEC structure under two scenarios: a) when the parameters φ1and φ2are
unknown and estimated from the data, called unspecified (U) structure, and b) when Ei=Iniwith
Inirepresenting the identity matrix of order ni×ni, that is, when an uncorrelated (UNC) structure
is considered. For these purposes, we proceeded as follows:
1. We generated M= 100 data sets of size n= 300 from the t-MLC model with a DEC structure
Ei= 0.8|tij −tik|, under four different settings of censoring proportions say, γ= 5%,15%,25%
and 35%. It is important to note that, the goal here is to study the effect of the level of
censoring in the estimation under misspecification of the correlation structure.
2. All the censored observations were imputed using the mechanism described in Subsection 4.2
and considering both, U and UNC structure.
14

Correlation structure
Censoring Unspecified - U Uncorrelated - UNC
level MAE MSE MAE MSE
5% 1.120052 2.744973 1.199131 3.075949
15% 1.293753 3.106423 1.563213 4.340442
25% 1.409025 3.902546 1.684068 5.475168
35% 1.568360 4.647703 1.830202 6.170776
Table 2: Simulated data. Arithmetics means of the MAE and MSE over M= 100 datasets.
In order to compare performance of the U and UNC structures through the EM-imputation
defined in (11), we utilized two empirical discrepancy measures called the mean absolute error
(MAE) and mean square error (MSE)17,37. They are defined by
MAE = 1
kX
i,j
|yij −eyij|and MSE = 1
kX
i,j
(yij −eyij)2,(13)
where yij is the original simulated value (before being considered as a censored observation) and
eyij is the EM-imputation, for i= 1,...,300 and j= 1,...,6.Note that for γ= 5% we have that
k= 90, for γ= 15%, k= 270, for γ= 25% k= 270 and for γ= 35%, k= 630.
Arithmetic means of MSE and MAE over the 100 datasets are displayed in Table 2 and Figure
2. We can see that in all cases, the U structure presents the smallest MSE and MAE than the UNC
one, as expected.
Censoring level
MAE
(a)
1.0 1.2 1.4 1.6 1.8 2.0
5% 15% 25% 35%
U
UNC
Censoring level
MSE
(b)
234567
5% 15% 25% 35%
U
UNC
Figure 2: Simulated data. Arithmetics means of (a) MAE and (b) MSE over M= 100 datasets
under the t-MLC model with U and UNC structures.
15

6.2 Asymptotic Properties
In this simulation study, we analyze the absolute bias (Bias) and mean square error (MSE) of
the regression coefficient estimates obtained from the t-MLC model for six different sample sizes
(n= 50,100,200,300,400 and 600). These measures are defined by
Bias (θi) = 1
M
M
X
j=1 b
θ(j)
i−θiand MSE (θi) = 1
M
M
X
j=1 b
θ(j)
i−θi2,
where b
θ(j)
iis the ML estimate of the parameter θifor the jth sample.
The idea of this simulation is to provide empirical evidence about consistency of the ML esti-
mators under the t-MLC model with DEC structure. For each sample size, we generate M= 100
dataset with 5% of censoring proportion. In this simulation scheme the parameter φ2is fixed at
1 and thus a continuous-time AR(1) model is considered. Using the ECM algorithm, the absolute
bias and mean squared error for each parameter over the 100 datasets were computed.
From Figure 3 we can see that the absolute bias and the MSE tend to zero as the sample size
increase. As is expected, under the t-MLC model the EM algorithm provide estimates with good
asymptotic properties.
7 Application
We applied our method to the UTI data described in Section 2. This dataset consist of 362 ob-
servations, 26 were below the detection limits (50 or 400 copies/mL). The UTI data was analyzed
previously by Lachos et al.13, where it was observed that inferences based on Gaussian assumptions
are questionable. Consequently, we revisited this dataset with the aim of carrying out robust infer-
ence by considering the Student-tmodel. We consider our proposed t-MLC model with the DEC
correlation structure Σi=σ2Eidefined in Subsection 3.2 and for the sake of model comparison,
we also fit the normal MLC (N-MLC) counterparts, which can be treated as the reduced t-MLC
as νtends to infinity. Here we have that yij is the log10 HIV-1 RNA for subject iat time tj, with
t1= 0, t2= 1, t3= 3, t4= 6, t5= 9, t6= 12, t7= 18, and t8= 24.
It is important to stress that the t-LMM model related to the UTI data, mentioned in the
introduction, corresponds to a linear mixed model with a random intercept. In that model, Matos
et al.22 consider that the random effect follows a Student-tdistribution. It is well known that this
type of model may not adequately account of the serial correlation generated by the longitudinal
data arising from medical studies such as HIV studies, since the correlation structure induced by
the mixed model with random intercept generates a model with compound symmetric correlation
structure. Moreover, if we misspecified the distributional assumption for the random effects, we
may severely affect the likelihood-based inference and predictions as well McCulloch and Neuhaus38.
For those reasons, our approach do not consider the use of random effects. Instead, we consider
16

Samples Sizes (n)
Bias
β1
0.00 0.05 0.10 0.15 0.20 0.25 0.30
50 100 200 300 400 600
Samples Sizes (n)
MSE
β1
0.02 0.04 0.06 0.08 0.10 0.12 0.14
50 100 200 300 400 600
Samples Sizes (n)
Bias
β2
0.005 0.010 0.015 0.020 0.025 0.030 0.035
50 100 200 300 400 600
Samples Sizes (n)
MSE
β2
0.0005 0.0010 0.0015
50 100 200 300 400 600
Samples Sizes (n)
Bias
φ1
0.005 0.010 0.015 0.020 0.025 0.030
50 100 200 300 400 600
Samples Sizes (n)
MSE
φ1
0.0002 0.0004 0.0006 0.0008 0.0010 0.0012
50 100 200 300 400 600
Figure 3: Simulated data. Bias (first column) and MSE (second column) of parameter estimates
in the t-MLC model under 5% of censoring.
17

the DEC structure to deal with the correlation generated by the HIV data. Moreover, using the
DEC structure we can also obtain the compound symmetry structure (φ2= 0) as a particular
case, which is equivalent to the model considered by Matos et al.22 . From our point of view, the
DEC structure is a flexible and parsimonious way to deal with correlation in longitudinal data. It
avoids specifying the probability distribution for the random effects and allow us to obtain different
correlation structures.
We consider four cases of correlation structure induced by the specification of the matrix Ei,
namely, (a) the uncorrelated (UNC) structure, (b) the continuous-time AR(1) structure, (c) the
MA(1) structure and (d) the unknown (U) structure (when φ1and φ2are unknown).
The degrees of freedom parameter νwas fixed at the value that maximizes the t-MLC likelihood
function. Figure 4 shows that the likelihood function reaches the maximum at ν= 10, indicating
the lack of adequacy of the normal assumption for the UTI data. The ML estimates of the other
parameters were obtained using the ECM algorithm described in Section 4, with starting values
obtained through the library lmec25.
5 10 15 20
−382 −380 −378 −376 −374 −372 −370
ν
Log−likelihood
Figure 4: UTI data. Plot of the profile log-likelihood of the degrees of freedom ν.
Table 3 presents the ML estimates and standard errors of the regression parameters βfor the
t-MLC and N-MLC models. Although the estimates are quite similar in both cases, the standard
errors are in general smaller under the Student-t, indicating that our proposed censored model
(t-LMC) produces more precise estimates.
Note that we fitted eight models, resulting from the combinations of the four correlations struc-
tures, say UNC, AR(1), MA(1) and U, and two distributional assumptions (normal and Student-t).
The value of the log-likelihood function and the AIC and BIC criteria for these 8 models are pre-
sented in Table 4, where we can see that the t-MLC outperform consistently the normal counterpart
in all cases. In particular, these criteria indicate a preference of the unspecified correlation structure
(U structure), that is, when the parameters φ1and φ2of the matrix Eiare estimated from the
18

N-MLC
Parameters UNC AR(1) MA(1) U
Est SE Est SE Est SE Est SE
β13.6160 0.0153 3.6334 0.0162 3.6194 0.0150 3.6196 0.0156
β24.1527 0.0172 4.2095 0.0168 4.1825 0.0166 4.1834 0.0164
β34.2381 0.0184 4.2502 0.0182 4.2384 0.0181 4.2568 0.0169
β44.3727 0.0187 4.3224 0.0189 4.3729 0.0184 4.3738 0.0170
β54.3650 0.0248 4.4680 0.0237 4.3652 0.0245 4.5791 0.0195
β64.2326 0.0313 4.3781 0.0303 4.2327 0.0309 4.5819 0.0221
β74.3258 0.0444 4.3749 0.0463 4.3260 0.0438 4.6879 0.0275
β84.5620 0.0818 4.5762 0.0842 4.5620 0.0807 4.8061 0.0418
σ21.0631 1.1498 1.0486 1.1053
φ1– 0.8251 0.4068 0.7027
φ2– 1.00 ∞0.0286
t-MLC
Parameters UNC AR(1) MA(1) U
Est SE Est SE Est SE Est SE
β13.6511 0.0120 3.6410 0.0155 3.6578 0.0120 3.6330 0.0153
β24.2386 0.0146 4.3022 0.0172 4.2706 0.0144 4.2697 0.0171
β34.3149 0.0156 4.3312 0.0187 4.3246 0.0156 4.3290 0.0177
β44.4715 0.0159 4.4297 0.0195 4.4792 0.0159 4.4715 0.0178
β54.5268 0.0210 4.5476 0.0248 4.5293 0.0209 4.6359 0.0206
β64.3923 0.0267 4.4435 0.0317 4.3963 0.0266 4.6238 0.0235
β74.5012 0.0373 4.4660 0.0475 4.5092 0.0377 4.7082 0.0295
β84.6896 0.0692 4.6481 0.0863 4.5092 0.0687 4.7998 0.0455
σ20.8092 1.0272 0.8003 1.0103
φ1– 0.7754 0.2752 0.6629
φ2– 1.00 ∞0.0222
ν10.00 10.00 10.00 10.00 –
Table 3: UTI data. ML estimation (Est) and standard errors (SE) for the regression coefficients
under the normal and t-MLC models with different DEC structures.
data.
The regression coefficients βj, for j= 1,...,8,increase gradually under the two models. This is
the evidence of the negative effect of the antiretroviral therapy interruption on the viral load levels.
In another words, the viral load increments consistently along the time when the antiretroviral
therapy begins to be interrupted. For the best model (t-MLC), the coefficients increase from 3.63
at the beginning of the study to 4.79 at the end of this. Note that considering an asymptotic 95%
confidence interval, the estimates of all regression coefficients are significant. The estimate of the
within-subject (σ2) scale parameter (in log10 scale) is 1.01.
N-MLC t-MLC
Criteria UNC AR(1) MA(1) U UNC AR(1) MA(1) U
log-likelihood -524.166 -463.043 -516.507 -411.926 -484.165 -421.249 -476.647 -369.129
AIC 1066.333 946.087 1053.014 845.852 986.331 862.498 973.295 760.259
BIC 1101.357 985.004 1091.931 888.660 1021.357 901.415 1012.212 803.067
AICcor r 1066.844 946.714 1053.641 846.607 986.843 863.125 973.922 761.014
Table 4: UTI data. Comparison between the normal and t-MLC models using different model
selection criteria.
19

Outlying observations may affect the estimation of the parameters under assumptions of normal-
ity. Our t-MLC model with DEC structure accommodates these discrepant observations attributing
to them small weights in the estimation procedure. The estimated weights (bui, i = 1,...,72) for
the t-MLC model with Ustructure are presented in Figure 5. In this Figure, we observe that
observations ]20, ]35, ]41 and ]42 present smaller weights, verifying the robust aspects of the ML
estimation under the Student-tdistribution. These results agree with those obtained by Lachos et
al.13 under the Bayesian paradigm.
0 10 20 30 40 50 60 70
0.4 0.6 0.8 1.0 1.2 1.4 1.6
Index
Weights
(a)
20 35
41
42
0 10 20 30 40 50 60 70
0.4 0.6 0.8 1.0 1.2 1.4 1.6
Index
Weights
(b)
42
35 20
41
0 5 10 15 20 25 30
0.0 0.5 1.0 1.5
(c)
Mahalanobis distance
Weights
20 35
41
42
Figure 5: UTI data. Estimated weights buifor the t-MLC model under U structure.
Now we turn our attention to the one-step-ahead and two-step-ahead forecast of future obser-
vations using the approach proposed in Section 5 for the UTI data.
As a simple illustration, we considered in the analysis the cases who were measured on at
least five (41 individuals or 56 %) and six occasions (29 individuals in total) and we predicted the
last two measures. As in the simulation scheme presented in Subsection 6.1, we considered the
20

MAE and MSE measures for comparing the performance of the prediction under different DEC
structures. Table 5 shows the comparison between the predicted values (one-step-ahead and two-
step-ahead) with the real ones, under the t-MLC model considering four different DEC structures,
say, AR(1), MA(1), UNC and U. Figure 6 shows the performance of prediction for individuals with
different types of trajectories under three different correlation structures, namely the MA(1) (red
line), AR(1) (green line) and U (blue line) for the individuals: ]4, ]15, ]41, ]43, ]47, ]59 and ]61.
We can see from these results once again how the U structure outperforms the other correlation
structures from a predictive point of view, i.e. the U structure generates predictive values close to
the real ones.
For Individuals with ni>5(41 individuals)
U AR(1) MA(1) UNC
Forecast MAE MSE MAE MSE MAE MSE MAE MSE
one step 0.3218993 0.1768571 0.4968972 0.3456072 0.7392737 0.8093977 0.7239416 0.8128087
two step 0.4143890 0.2827286 0.5987888 0.6146529 0.7311850 0.9008162 0.7240895 0.9038104
For Individuals with ni>6(29 individuals)
U AR(1) MA(1) UNC
Forecast MAE MSE MAE MSE MAE MSE MAE MSE
one step 0.3308357 0.1912845 0.4388853 0.2702304 0.6237170 0.5197352 0.6030431 0.5267867
two step 0.3721411 0.2159791 0.5222417 0.5049302 0.6417997 0.7027740 0.6307786 0.7051547
Table 5: UTI data. Evaluation of the prediction accuracy for the t-MLC model with different
DEC structures.
Moreover, the two-step-ahead prediction performance of the N-MLC under the U structure
provides values of the MAE and MSE equal to 0.3786 and 0.2321 respectively. Thus, from Table 5
we can conclude that the t-MLC model generates better predictive results under the U structure
than the normal one.
8 Conclusions
We have proposed a robust approach to linear regression models with censored observations based
on the multivariate Student-tdistribution, called the t-MLC model. This offers a flexible alter-
native for dealing with longitudinal censored data in the presence of outliers and/or influential
observations. For modeling the autocorrelation existing among irregularly observed measures, a
damped exponential correlation structure was adopted as proposed by Mu˜noz et al.18. A novel
ECM algorithm to obtain the ML estimates is developed by exploring the statistical properties
of the multivariate truncated Student-tdistribution. Our proposed algorithm has a closed-form
expression for the E-step, based on formulas for the mean and variance of the truncated Student-t
distribution. We applied our methods to a recent AIDS study (freely downloadable from R), con-
cluding that when the antiretroviral therapy is interrupted, the HIV-1 RNA levels in blood increase
consistently along the period of evaluation. We also perform two simulation studies, showing the
21

Observation(Condition)
Log 10 HIV−1
Ind=47 (ni=6)
2.0 2.5 3.0 3.5 4.0 4.5 5.0
1(NC) 2(NC) 3(NC) 4(NC) 5(NC) 6(NC)
U
AR(1)
MA(1)
Real Data
Observation(Condition)
Log 10 HIV−1
Ind=59 (ni=6)
1(NC) 2(NC) 3(NC) 4(NC) 5(NC) 6(NC)
2.0 2.5 3.0 3.5 4.0 4.5
U
AR(1)
MA(1)
Real Data
Observation (Condition)
Log 10 HIV−1
Ind=43 (ni=7)
2.5 3.0 3.5 4.0 4.5 5.0
1(NC) 2(NC) 3(NC) 4(NC) 5(NC) 6(NC) 7(NC)
U
AR(1)
MA(1)
Real Data
Observação
Log 10 HIV−1
Indiv=15 (ni=7)
U
AR(1)
MA(1)
Real Data
3.0 3.5 4.0 4.5 5.0
1(NC) 2(NC) 3(NC) 4(NC) 5(NC) 6(NC) 7(NC)
Observation (Condition)
Log 10 HIV−1
Indiv=4 (ni=8)
U
AR(1)
MA(1)
Real Data
4.6 4.8 5.0 5.2
1(C) 2(C) 3(C) 4(C) 5(NC) 6(NC) 7(C) 8(NC)
Observation (Condition)
Log 10 HIV−1
Ind=41 (ni=8)
U
AR(1)
MA(1)
Real Data
1 2 3 4 5
1(C) 2(C) 3(C) 4(C) 5(NC) 6(NC) 7(C) 8(NC)
Observation (Condition)
Log 10 HIV−1
Indiv=61 (ni=8)
U
AR(1)
MA(1)
Real Data
3.0 3.5 4.0 4.5 5.0 5.5 6.0
1(NC) 2(NC) 3(NC) 4(NC) 5(NC) 6(NC) 7(NC) 8(NC)
Figure 6: UTI data. Evaluation of the prediction performance for seven random subjects. The
abbreviation ”Ind” stands for ”Individual”. nidenotes the number of measurements. C and NC
indicate if the measurement was censored or not respectively.
superiority of t-MLC model on the provision of more adequate results when the available data has
censored components. Furthermore, the simulation results demonstrate that our method gives very
competitive performance in terms of imputation when a DEC structure is considered. From these
results it is encouraging that the use of the t-MLC model with DEC structure offer a better fit,
protection against outliers and more precise inferences.
22

An established way to validate the model (2) is to use the bootstrap technique (see e.g. Efron
and Tibshirani39 or Chernick40). Here, the appropriate way to use bootstrap will be to bootstrap
observations (Yi, Xi), i = 1, . . . , n. This means drawing randomly with replacement from the set
of indices {1, . . . , n}and obtaining a “new” sample of the same size nas the original sample.
Since the bootstrap sampling is with replacement, it is quite likely here to draw twice or more the
observation with the same index i, where 1 ≤i≤n. For each of the new sample we can recalculate
the estimates of the parameter βin the model. After creating Bsamples we can have a data-driven
approximation of the sampling distribution of the estimate ˆ
βand a better idea on variability of ˆ
β.
This now can be compared with the chosen pdimensional Student-tdistribution to check whether
this parametric model is reflecting data-driven sampling distribution of ˆ
βand its variance.
Although the t-MLC considered here has shown great flexibility for modeling symmetric data,
its robustness against outliers can be seriously affected by the presence of skewness. Recently,
Lachos et al.41 (see also Bandyopadhyay et al.14) proposed a remedy to accommodate skewness
and heavy-tailedness simultaneously, using scale mixtures of skew-normal (SMSN) distributions.
We conjecture that our methods can be used under MLC models, and should yield satisfactory
results at the expense of additional complexity in the implementation. An in-depth investigation
of such extensions is beyond the scope of the present paper, but it is an interesting topic for further
research.
Acknowledgment
We thank the editor, associate editor and two referees whose constructive comments led to an
improved presentation of the paper. Victor H. Lachos and Aldo M. Garay would like to acknowl-
edge the support of the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq-
Brazil) and the Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo (Grants 2013/21468-0
and 2014/02938-9 from FAPESP-Brazil). Luis M. Castro acknowledges funding support by Grant
FONDECYT 1130233 from the Chilean government and Grant 2012/19445-0 from FAPESP-Brazil.
Jacek Leskow would like to acknowledge the support of the grant of Polish National Center for Sci-
ence, grant number UMO-2013/10/M/ST1/00096. Moreover, while working on this paper Jacek
Leskow was also supported by the Grant 2014/11831-3 from FAPESP-Brazil.
23

Appendix: Details of the ECM algorithm
In this Appendix, we derive the ECM algorithm equations (7)–(9) for the t-MLC model. Let
y= (y>
1,...,y>
n)>,u= (u1, . . . , un)>,V=vec(V1,...,Vn), and C=vec(C1,...,Cn) such
that we observe (Vi,Ci) for the i-th subject. Treating uand yas hypothetical missing data, and
augmenting with the observed data V,C, we set yc= (C>,V>,y>,u>)>.
Denoting the complete-data likelihood by L(·|C>,V>,y>,u>) and pdf’s in general by f(·), we
have that for θ= (β>, σ2, φ1, φ2)>
Lθ|C>,V>,y>,u>=f(y|V,C,u)f(u)
=f(y|u)f(u) =
n
Y
i=1
f(yi|ui)h(ui|ν).
Dropping unimportant constants, the complete-data log-likelihood function is given by
`c(θ|yc) = log {L[θ|yc]}= log (n
Y
i=1
f(yi|ui)h(ui|ν))
=
n
X
i=1
log n(2π)−p/2u1/2
i|Σi|−1/2exp ui
2(yi−Xiβ)>Σ−1
i(yi−Xiβ)o
+
n
X
i=1
log {h(ui|ν)},
=−1
2
n
X
i=1 hnilog σ2+ log |Ei|+ui
σ2(yi−Xiβ)>E−1
i(yi−Xiβ)i
+
n
X
i=1
log {h(ui|ν)}+c,
where cis a constant that is independent of the parameter vector θand h(ui|ν) is the gamma density
(Gamma(ν/2, ν /2)). Our EM-type algorithm (ECM) for the t-MLC model can be summarized in
the following way
E-step:
Given the current value θ=b
θ(k), the E-step calculates the conditional expectation of the complete
data log-likelihood function
Q(θ|b
θ(k)) =
n
X
i=1
Qi(θ|b
θ(k)),
=
n
X
i=1 −ni
2log σ2−1
2log |Ei| − 1
2σ2A(k)
i(β,φ),
with
A(k)
i(β,φ) = trd
uy2
i
(k)E−1
i−2βX>
iE−1
ic
uy(k)
i+bu(k)
iβX>
iE−1
iXiβi.
24

Note that in this case we do not consider the computation of E[h(ui|ν)|V,C,b
θ(k)] because νis
fixed.
CM-step:
The conditional maximization (CM) step then conditionally maximizes Q(θ|b
θ(k)) with respect to
θ= (β>, σ2, φ1, φ2)>and obtains a new estimate b
θ(k+1)
∂Q(θ|θ(k))
∂β=1
σ2
n
X
i=1 X>
ib
E(k)
i−1c
uy(k)
i−bu(k)
iX>
ib
E(k)
i−1Xiβ;
∂Q(θ|θ(k))
∂σ2=−N
2σ2+1
2σ4
n
X
i=1
A(k)
i(b
β(k+1),b
φ(k)).
Thus, the solutions of ∂Q(θ|θ(k))
∂β= 0 and ∂Q(θ|θ(k))
∂σ2= 0 are
b
β(k+1) = n
X
i=1 bu(k)
iX>
ib
E(k)
i−1Xi!−1n
X
i=1
X>
ib
E(k)
i−1c
uy(k)
i,
c
σ2(k+1) =1
N
n
X
i=1
A(k)
i(b
β(k+1),b
φ(k)),
where N=Pn
i=1 ni. We estimate φby maximizing the marginal log-likelihood, circumventing the
(in general) complicated task of computing ∂Ei
∂φ. This strategy were used, for instance, by Wang
and Fan37 and Wang 17. Then,
b
φ(k+1) =argmaxφ(−1
2
n
X
i=1
[log(|Ei|) + A(k)
i(b
β(k+1),φ)]),
The algorithm is iterated until the distance involving two successive evaluations of the log-likelihood,
|`(b
θ(k+1))/`(b
θ(k))−1|, is sufficiently small.
25

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