Hyun J. Lim^{1*}, Prosanta Mondal^{1} and Stuart Skinner^{2}

*Correspondence: Hyun J. Lim hyun.lim@usask.ca

1. Department of Community Health & Epidemiology, College of Medicine, University of Saskatchewan 107 Wiggins Road, Saskatoon, SK S7N 5E5, Canada.

2. Department of Medicine, College of Medicine, University of Saskatchewan 103 Hospital Drive, Saskatoon, SK, S7N 0W8, Canada.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Many clinical studies generate a dataset having longitudinal repeated biomarker measurement data and time to an event data, which often depend on each other. In such studies, characteristics of the pattern of a biomarker change, and the association between the primary survival endpoint and features of the longitudinal profiles are commonly of interest. Often separate analyses using a mixed effects model for the longitudinal outcome and a survival model for the time to event outcome are performed. However, separate models are overly simplified because they do not consider the association between two components of such data and so produce misleading conclusions. An alternative approach is two-stage modeling which allows a separate biomarker pathway for each patient but the parameter estimates are still biased. Joint modeling is the most sophisticated complex approach but enables the repeated biomarker measurements and survival processes to be modelled while accounting for the interrelationship between the two processes. We demonstrate the use of joint modeling in analysis of an HIV dataset with CD4+ count measurements and survival time. In the joint model, we combine a linear Gaussian random effects sub-model for the repeated CD4+ count measurements and Cox or Weibull survival sub-model, linked through their shared dependence on the latent variable. Our study showed that the hazard rate of death depended on the longitudinal progression of CD4+ counts, i.e., a patient's baseline CD4+ count and the rate of change in CD4+ counts significantly impact on his or her survival time.

**Keywords:** Joint model, random effects model, survival model, longitudinal data, event time data, HIV

Patients in many clinical studies are followed over a period of time and biomarkers are collected repeatedly at multiple time points. The data also indicates the time, at which an event of particular interest occurred, eg., relapse of a disease or death. For example, in HIV studies patients are measured for biomarkers such as the CD4+ lymphocyte count or the viral RNA (viral load) until specific outcomes such as AIDS or death. Other applications include cognitive performance and survival in geriatric studies, systolic blood pressure and a coronary event, prostate-specific antigen biomarker and prostate cancer recurrence, and hemoglobin level and survival in type 2 diabetes [1-7]. In such studies, a common primary objective is to investigate the effect of treatment on survival and/or biomarker process. But patterns of change in a biomarker or the association between the primary endpoint and features of the longitudinal profiles are also of interest. The traditional approach uses two models separately, a random effects model that describes the process of the repeated measurements over time and a survival model that examines time to the event. However, separate models are overly simplified because they do not consider the association between the two components of such data, and in failing to take into account all the available information in an integrated manner, provide invalid inferences. One of the alternative approaches is two-stage modeling. In the first stage of two-stage modeling, the repeated biomarker measurements are modeled using a random effect model so that subject-specific values of the covariate may be estimated. The modeled value of individual prediction from the random effects model is substituted into the survival model as timevarying covariate in the second stage. This two-stage approach accounts for some of the measurement error and allows the pathway of the covariates for each patient to be estimated [8]. However, parameter estimates of the two-stage model are still biased and inefficient, even though much of the bias of the uncorrected values is reduced [9]. Since either separate models or the two-stage approach is inappropriate when the longitudinal repeated measurements and survival outcome have an association, others have proposed joint modeling and its extensions, which have obtained great attention in the literature [9-13].

Joint modeling is a sophisticated, complex approach but it enables both longitudinal repeated biomarker measurements and survival processes to be modelled together while taking account of association between them. By including the random effects model for longitudinal data in the survival model, the patterns of a biomarker's performance and the relationship between its progression and survival time can be characterized. In this way, joint modeling provides less biased estimates and more efficient inferences than separate models or the two-stage approach [9,12,13]. The main advantage of joint modeling is that the effect of a covariate on the longitudinal process can be separated from its effect on survival. Modeling of a covariate effect (eg., treatment) can be described and visualized as in Figure 1.

Figure 1 : **Covariate (treatment) effect on longitudinal biomarker measurements and survival
outcome.**

For jointly modeling longitudinal data and event time data, many have proposed a linear or nonâ€“linear random effects model for longitudinal measurements and a semiparametric or parametric survival model for event time data, where a set of random effects is assumed to induce their interdependence [9,14-18]. Wulsohn and Tsiatis [9] proposed a two-step procedure for fitting their model in HIV study (Wulsohn- Tsiatis model). First, they assumed a growth curve random components model with normal errors for true CD4+ counts using the modified EM algorithm for estimation. They then substituted these estimates into the Cox proportional hazards model to obtain estimates of the survival parameters. Other authors have also assumed piecewise constant hazards, parametric Weibull, or accelerated failure time models [17,19-21]. Henderson et al., developed a more general version of the Wulsohn-Tsiatis model and proposed the use of an unobserved or latent bivariate Gaussian process to link longitudinal data and survival data, assuming that the longitudinal and event processes were conditionally independent given the one process covariates [15,22].

Our paper focuses on the application of available methodologies and on interpretation of the results when these are utilized in the analysis of an HIV dataset. In our HIV study, the main objective is to use longitudinal CD4+ measurements to improve prediction of survival prognosis. In the next section, we introduce a description of our HIV study. In Section 3, methods of joint modeling for analysis are reviewed. In Section 4, the conventional separate models and the joint models are applied to the HIV dataset to measure the relative effects of covariates. We examine whether the baseline CD4+ value and the slope of CD4+ counts reveal association with mortality risk. It should be noted, however, our intent is to illustrate the application of joint modeling rather than give universally valid estimates for HIV study outcomes. Section 5 contains a discussion about the results and conclusions.

**Clinical background**

CD4+ counts and viral load are considered important
biomarkers of HIV disease progression. CD4+ cells are an
important part of the immune system, which begin to deplete
as the virus infects the body; the amount of decline indicates
the degree of immunosuppression. Thus CD4+ count is
considered as the primary indicator for prognostic information
and a guide for antiretroviral therapy in HIV infected individuals
[23,24]. Successful treatment for HIV patients depends on
the therapy's ability to restore immune functions. Studies
have shown that disease progression among patients with
HIV infection is delayed substantially when CD4+counts
increase in response to treatment [24,25]. Identification of
factors influencing disease progression is vital to effectively
care for patients and to improve their survival and quality of
life. Detection of not only clinical features and risk factors but
also characteristics associated with a more rapid progression
of CD4+ counts can help identify patients who may benefit
from closer and more frequent clinical follow-up and earlier
treatment intervention [24]. We anticipate that changes in
biomarkers (CD4+ counts or viral load) will have an impact
on the risk of major HIV complications.

**Study population**

This study was a retrospective, longitudinal case-based
investigation. Data was obtained solely by reviewing medical
records of HIV patients diagnosed at the Royal University
Hospital or the West Side Community Clinic in Saskatoon,
Canada between January 1, 2005 and June 1, 2010. A total of
321 adult (age â‰¥ 18 years) patients who had at least one CD4+
measurement after the first report of HIV diagnosis regardless
of clinical stage were eligible for the study. Of them, 1129 CD4+ count measurements were observed with median of
3 measurements. The number of measurements per patient
varied and the measurement intervals were unequally spaced.
Individual CD4+ count profiles of 10 randomly selected patients
are shown in Figure 2. Plotting of CD4+ counts was grouped
according to their date of measurement. The first measurement
done within 180 days after study enrollment (first clinic visit)
was considered the baseline value. We created a person-period
data set for each patient covering 3-month intervals for the
first year and then 6-month intervals afterward as CD4+ counts
are generally measured every 3 ~ 6 months in clinical practice
as the standard of medical care. Our choice of the 3-month or
6-month period for CD4+ counts was supported by clinical
practice guidelines and CD4+ data availability in our study.
When the CD4+ count was not observed during any particular
interval period, we considered the value as missing. We did
not anticipate that missing CD4+ counts would depend on
the nature of the unobserved CD4+ counts, and thus they are
assumed missing at random. For any patients with 2 or more
CD4+ counts in a given interval, the average of those CD4+
count measurements was used for that interval. Subjects were
followed from the time of entry into the study until death,
loss to follow-up, or the end of the study (August 31, 2011).

Figure 2 : **Individual profiles of CD4+ counts (of 10 randomly
selected patients).**

CD4+ count measurements were limited to 12 per patient in the subsequent analysis as very few subjects had more than 12 measurements. The maximum of 12 measurements is a practical limitation of the data, rather than a methodological one. The other outcome variable considered was time to death, which was defined as time between date of entry into the cohort and date of death, date last seen, or the end of the study. In our study, a 0.05 alpha level was used for statistical significance. Analyses in this study were performed using SAS version 9.2 (The SAS Institute, Cary, NC) or R package.The study received ethical approval from the University of Saskatchewan institutional review board committee.

Let *T* be a random variable representing survival time that
has the distribution function, *F(t)*. The survival function *S(t)* at time *t* is defined to be the probability that the survival
time is greater than *t*, where *S(t) = P(T > t)* = 1 - *F(t)*. Let *T _{i}* be
the time when the event occurs for the

Let *Z _{ij}* be the observed measurement for the ith subject at
the

A joint model is comprised of two sub-models, one for the
longitudinal process * Z_{i}* (

The subject-specific random effects model has become a
popular tool to analyse various types of longitudinal data as it
adequately describes the pattern of repeated measurements
over time. In this model, the average progression is described
using some function of time, and subject-specific deviations
from this average evolution are introduced by using random
effects [26,27]. We assume that the longitudinal measurement
of the covariate**Z**** _{i}** follows a linear mixed effects model (growth
curve model) with random intercept Î¸

where Î± is the regression coefficient of covariate vector
** W**. Î¸U

Here the random effect Î¸i
account for the heterogeneity among
the subjects and describes how the profile of the *i*th subject
deviates from the average profile. Conditional on bivariate
Gaussian latent variable Î¸* _{i}* = (Î¸

The coefficient **Î±** assesses the covariate effect (for example,
treatment) on the longitudinal biomarker. Possible extensions
from a linear mixed effects model allow for more complex
relationships such as polynomial specifications of ** Z_{i}** and

where *h _{0}*(

where *Y _{i}* (

The likelihood method is a widely used approach for the
parameter estimation in the joint model [9,10,16,18,21]. Assuming
that censoring and timing of longitudinal measurements
are non-informative, the likelihood function *L* with the
observed data for each subject (*X _{i}* ,

where

and

The parameters Î¸, **V**, *Ïƒ ^{2}_{Îµ}*, and Î² are estimated using parametric
likelihood and

where **Y**_{j} (**u**) = * I* (

A total of 321 HIV infected patients were eligible for the study
(Table 1). Of them, 165 (51.4%) were males and 216 (71%)
were Aboriginal. The mean age was 35 years (SD = 10.5). At
baseline, 30% had a history of incarceration, 79% had Hepatitis
C antibodies (HCV), 81% had history of injection-drug use
(IDU), and 58% had ever been on antiretroviral therapy (ART).
The mean CD4+ count at baseline was 388 cells/mm^{3} (SD=
229; median=353; interquartile range: 234-536); the mean
viral load was 4.3 log_{10} copies/ml (SD= 1.0; median=4.4;
interquartile range: 3.9-4.9). The median follow-up time
was 20 months (interquartile range: 9-36 months) and 25
(7.8%) patients died in the study cohort. The overall mean
CD4+ counts measurements showed decreasing trend. The
overall survival probability was 0.97 at 1 year, 0.95 at 2 years
and 0.92 at 3 years, respectively. To normalize CD4+ counts,
we used the square root of the CD4+ counts in the models.

Table 1 : **Demographic and clinical characteristics of the
study patients. (N=321).**

**Results of random effects model**

Figure 3 shows the observed mean CD4+ counts profile
as a function of time within separate ART strata. The ART
treatment group started with a lower mean CD4+ count
than the ART naÃ¯ve group, but the ART naÃ¯ve groups had a
decreasing mean response while the ART treatment group
mean was sustained. The mean response towards the end
of the study was unstable because of the small number of
patients. At each time point, mean responses were calculated
only among those patients who have not yet dropped out
of the study. The correct interpretation of the two mean
response profiles is subtle because of the potential interdependence
between a patient's predisposition to drop-out
and their associated CD4+ counts. From the univariate model
for CD4+ count measurement, HCV and ART treatment were
significant (p-values are 0.02 and <0.0001, respectively) while
gender, age, ethnicity, clinic site, history of incarceration,
and history of IDU were not. Since HCV and ART treatment
were significantly associated with longitudinal CD4+ counts
measurement, these covariates were included in the random
effect model. The random effect model for longitudinal CD4+
count repeated measurements is:

Figure 3 : **Observed mean CD4+ counts profile by antiretroviral
therapy (ART) group.**

The results of the random effect model using the unstructured
covariance matrix Î£ are summarized in Table 2. The estimated
average regression coefficient of *Time* for the ART treatment
naÃ¯ve group is âˆ’0.088 (95% CI: -0.14, -0.04; p=0.0008),
suggesting a significant decrease in CD4+ count over the study period. Average regression coefficient for the ART
treatment group is 0.001 and indicates that CD4+ did not
decrease over the study period. However, the difference in
slope between two groups is statistically significant (95% CI:
0.03 â€“ 0.15; p=0.005). Patients who were co-infected with HCV
at study entry had significantly lower CD4+ counts than those
without HCV co-infection (p=0.002).

Table 2 : **Estimation of coefficient, 95% confidence interval (C.I),
and p-value from separate models: longitudinal random effects
model, Cox survival model, Weibull survival model, respectively.**

**Results of survival model**

Figure 4 shows the Kaplan-Meier survival curve estimates by
ART groups. The two groups are similar during the first nine
months but subsequent survival in the ART treatment group
is significantly higher than in the ART naÃ¯ve group. From
univariate analysis for survival time, history of IDU, HCV, and
ART treatment were significant (p-values are 0.04, 0.02 and
0.002, respectively) while gender, age, ethnicity, clinic site,
and history of incarceration were not. ART treatment and HCV were included in the model because both were significantly
associated with survival time as well as CD4+ repeated
measurements. No interaction was observed between ART
treatment and HCV. The Cox regression model in the absence
of random effects is

Figure 4 : **Survival curve by antiretroviral therapy (ART)
group.**

Under this Cox model, as expected, the ART treatment group had significantly lower hazard rates than the ART naÃ¯ve group after controlling for HCV (HR=0.34; 95% CI: 0.145 â€“ 0.786; p=0.01). HCV-positive status also resulted in a worse survival than HCV-negative status in the adjusted model, but the difference is marginally significant (p=0.07). The results from the Cox model are similar to the results obtained from the Weibull model. In the Weibull model, the relative hazard rate for patients with ART treatment to ART naÃ¯ve is exp(-1.109) = 0.33 as compared to 0.34 under the Cox model (Table 2).

**Results of joint model**

Assuming the CD4+ longitudinal value is given by the growth
curve model with random intercept and slope, the Cox
proportional hazards model for subject *i* is

where *h _{0}*(

Table 3 : **Estimation of coefficient, 95% confidence interval
(C.I.), and p-value from the joint model with (i) random effects
sub-model and Cox survival sub-model and (ii) random effects
sub-model and Weibull survival sub-model.**

For all estimations in this study, we used SAS PROC MIXED, PROC PHREG, PROC NLMIXED and R package (http://rwiki.sciviews.org/doku.php?id5packages:cran:jm) [31,32].

In our study we addressed the relationship between the trajectory of CD4+ counts over time and the risk of death among HIV patients using joint modeling with a longitudinal random effects sub-model and a Cox or Weibull survival submodel. The joint model revealed that the hazard of death depended on a longitudinal process of an intercept and a slope, i.e., a patient's baseline CD4+ count and the rate of change in CD4+ counts significantly impact on his or her survival time. This observation suggested that the rate of CD4+ count change confers risks of mortality, even after adjustment for HCV co-infection as a risk factor. The parameter estimates of the joint model were different from those of the separate models; this difference might be due to the model correction accounting for the correlation between the longitudinal CD4 + counts and survival time.

Simple methods like separate modeling or two-stage modeling are useful when we explore optional models and select potential covariates. With simple methods, if survival for patients is far longer than the last follow-up time, interpretation should be cautious because extrapolation of the predicted value beyond the observed date may be risky. However, the joint model is able to account for the healthy survivor effect by exploiting these random effects, which account for the dispersion among individual times to death [10,33]. As long as there is sufficient information on the longitudinal repeated measurements, the estimates of the joint model are robust and efficient [Hsieh 2006]. Thus, even if the association with the longitudinal measurements is not of interest, under heavy censoring a joint modelling approach exploits the association to give more efficient inferences for parameters of the survival distribution [34].

As Henderson et al., [22] mentioned, joint modeling is flexible methodology for handling combined longitudinal and event history data. But when the goals of a study concern population-level inferences, and especially when the time-to-event process is not of direct interest, random effect joint modelling may be an over-elaboration [34]. When the focus of interest is the time-to-event process, joint modelling enables longitudinal measurements on each subject to be used as time varying explanatory variables while recognizing that they may be measured with a non-negligible error. When the association parameter between the longitudinal and survival data is not significant, the joint model analysis should have the same results as would be obtained from separate analyses for each component. However, joint modelling is a valuable technique not only in its efficient use of all available data and its ability to obtain accurate inference, but it is highly recommended when survival time and longitudinal measurements have the same clinical meaning. It is especially applicable to problems involving biomarkers where the focus is on using longitudinal measurements to improve prediction of survival prognosis. If survival time and longitudinal measurements have a different clinical meaning or are not comparable, joint modeling is not appropriate because the result may lead to the conclusion that a covariate effect with worse survival is superior [35].

One of the concerns many authors have raised in making inference on the longitudinal process is that occurrence of the event may induce an informative censoring [36-39]. Subjects with advanced disease may have fewer CD4+ count measurements because they experience death earlier, potentially leading to biased estimation. Valid inference requires a framework in which potential underlying relationships between the death event and the longitudinal process are explicitly acknowledged [10]. The joint model assumes that censoring is independent of the random effects. But if the underlying relationship and censoring process that leads to drop-out is not correctly modeled, the estimate of parameters may be biased [9,20]. Much more work remains to be done for joint models with informative censoring and missing data.

A limitation of this study is the short duration of follow-up time, which might affect the estimates of the covariates. In the data, the median follow-up was 20 months and only 8% of the study patients died. When the follow-up duration is not long enough, it has an impact on the number of CD4+ measurements, possibly leading to less reliable estimation of the random effect model [40]. The second limitation is that our study was a retrospective cohort, limiting the availability and quality of data collected. We only included HIV positive individuals who attended clinics. Thus the results are likely biased towards a more stable study population and could be potentially biased against rapid disease progressors. For subjects with 2 or more CD4+ values in a given interval, the average of those CD4+ values for that interval was used for simplicity. We did not anticipate that missing CD4+ counts would depend on the unobserved CD4+ counts and thus we assumed missing at random in order to analyze the data. We used generalized linear mixed models which are valid under the assumption of missingness at random [27,41] and believe this assumption to be appropriate for our study population. In our study, a linear growth function was also assumed for CD4+ count trajectories so that the characteristics of intercept and slope could be related to the survival outcome; however, a linear trend may not be adequate to describe the time course of the CD4+ counts. If the CD4+ count trajectories were not linear, model misspecification may lead to biased results. With no general, user-friendly commercial software available as of yet for this technique, the use of joint modeling is limited in practice.

In summary, the analysis of the data using a joint model with latent variables that link the longitudinal models and the survival models together will result in unbiased and more efficient estimates. Joint model is intuitive and enable prognostic information to be collected from longitudinal measurements. With such prognostic information, clinicians make better informed decisions for specific patients based on their longitudinal biomarkers of disease. Thus we recommend that the joint modeling of CD4+ count progression and survival time should be performed to obtain a clear picture of the effect of specific covariate. This model provides useful predictive information on future CD4+ counts when the current value and the rate change of CD4+ count are known. As a flexible modeling approach, joint modeling can be used for patientspecific treatment strategies and future clinical interventions.

The authors declare that they have no competing interests. The data collected for this study was obtained solely from patients records.

Authors' contributions |
HJL |
PM |
SS |

Research concept and design | âˆš | -- | -- |

Collection and/or assembly of data | -- | -- | âˆš |

Data analysis and interpretation | âˆš | âˆš | -- |

Writing the article | âˆš | -- | -- |

Critical revision of the article | âˆš | -- | -- |

Final approval of article | âˆš | âˆš | âˆš |

Statistical analysis | -- | âˆš | -- |

We thank Ms. Stephanie Konrad who assisted data collection.

Editors: Jimmy Efird(EIC) East Carolina University, USA.

Zhongxue Chen(Editor), Indiana University Bloomington, USA.

Received: 12-Sep-2013 Revised: 19-Sep-2013

Accepted: 21-Sep-2013 Published: 28-sep-2013

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Lim HJ, Mondal P and Skinner S. **Joint modeling of longitudinal and event time data: application to HIV study**. *J Med Stat Inform*. 2013; **1**:1. http://dx.doi.org/10.7243/2053-7662-1-1

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