# A Review on Joint Models in Biometrical Research

**ABSTRACT** In some fields of biometrical research joint modelling of longitudinal measures and event time data has become very popular. This article reviews the work in that area of recent fruitful research by classifying approaches on joint models in three categories: approaches with focus on serial trends, approaches with focus on event time data and approaches with equal focus on both outcomes. Typically longitudinal measures and event time data are modelled jointly by introducing shared random effects or by considering conditional distributions together with marginal distributions. We present the approaches in an uniform nomenclature, comment on sub-models applied to longitudinal measures and event time data outcomes individually and exemplify applications in biometrical research.

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**ABSTRACT:**Joint longitudinal-survival models are useful when repeated measures and event time data are available and possibly associated. The application of this joint model in aging research is relatively rare, albeit particularly useful, when there is the potential for nonrandom dropout. In this article we illustrate the method and discuss some issues that may arise when fitting joint models of this type. Using prose recall scores from the Swedish OCTO-Twin Longitudinal Study of Aging, we fitted a joint longitudinal-survival model to investigate the association between risk of mortality and individual differences in rates of change in memory. A model describing change in memory scores as following an accelerating decline trajectory and a Weibull survival model was identified as the best fitting. This model adjusted for random effects representing individual variation in initial memory performance and change in rate of decline as linking terms between the longitudinal and survival models. Memory performance and change in rate of memory decline were significant predictors of proximity to death. Joint longitudinal-survival models permit researchers to gain a better understanding of the association between change functions and risk of particular events, such as disease diagnosis or death. Careful consideration of computational issues may be required because of the complexities of joint modeling methodologies.GeroPsych. 12/2011; 24(4):177-185.

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Neuhaus, Augustin, Heumann, Daumer:

A Review on Joint Models in Biometrical Research

Sonderforschungsbereich 386, Paper 506 (2006)

Online unter: http://epub.ub.uni-muenchen.de/

Projektpartner

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University of Munich

Discussion Paper 506 - SFB 386

A Review on Joint Models in Biometrical Research

A. Neuhaus†1, T. Augustin‡, C. Heumann‡, M. Daumer†

†Sylvia Lawry Centre for MS Research, Hohenlindenerstr. 1, D-81677 Munich

‡Department of Statistics, Ludwigstr. 33, D-80539 Munich

Abstract

In some fields of biometrical research joint modelling of longitudinal measures and event

time data has become very popular. This article reviews the work in that area of recent

fruitful research by classifying approaches on joint models in three categories: approaches

with focus on serial trends, approaches with focus on event time data and approaches with

equal focus on both outcomes. Typically longitudinal measures and event time data are

modelled jointly by introducing shared random effects or by considering conditional dis-

tributions together with marginal distributions. We present the approaches in an uniform

nomenclature, comment on sub-models applied to longitudinal measures and event time

data outcomes individually and exemplify applications in biometrical research.

Key words: Joint model, shared random effects, mixed effects model, survival analysis.

1Introduction

Clinical trials and epidemiological studies often collect more than one outcome for each

subject. In addition to the outcome for which the study was primarily initiated, secondary

and tertiary outcomes are collected during an investigation. These data are often time to

event data or repeated measurements. Various approaches have been described in the past

decades to handle repeated measurements and survival data separately, but in the situation

that both outcomes were selected on one subject, classical modelling does not consider

dependencies between the two types of responses. A powerful method to overcome this

problem is a joint modelling of survival and repeated measurements. Well known examples

in which repeated measurements and event time data are generated are studies in the field of

the acquired immunodeficiency syndrome (e.g. Tsiatis et al., 1995). In these studies disease

1neuhaus@slcmsr.org

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Discussion Paper 506 - SFB 3862

markers (e.g. viral load or CD4 counts) are measured repeatedly and disease specific events

(e.g. seroconversion or death) are documented at the same time. A joint examination of

the longitudinal process of such disease markers and the time to event is possible using

joint model approaches. This example also indicates that the issue of surrogate markers is a

natural area for the application of joint models, since the course of the longitudinal disease

marker process might serve as surrogate for the event.

This article classifies approaches on joint modelling, spread around the literature, in the

categories ‘focus on serial trend’, ‘focus on event times’ and ‘equal focus on both outcomes’.

We concentrate on methodological aspects of joint model approaches, details on forming

estimates are only noted marginally.

Throughout the paper, we assume that k subjects are observed, each with possibly differ-

ent visit schedules, i.e. at different time points, ti1 := 0,ti2,...,tini. Thus altogether ni

individual observations are collected for the ith subject (i = 1,...,k). In addition to the

outcome measured longitudinally, the time to a specific event is recorded.

We use the following notation for the ith subject to harmonise diverse approaches:

yi= (yi1,yi2,...,yini)?

ti= (ti1,ti2,...,tini)?

(ni× 1) vector containing longitudinal observations

(ni× 1) vector with corresponding observation time

points, with ti1:= 0

(ni× 1) vector of errors independent from yi

(ni× p) matrix of possibly time-varying covariates with

Xi[t] the corresponding step function which is xijif

tij≤ t < ti(j+1)for j = 1,...,ni− 1 and xiniif tini≤ t

(p × 1) fixed effects corresponding to X = (X?

(ni× q) matrix of covariates with Zi[t] defined in the

same way as Xi[t]

(q × 1) random effects corresponding to Zi

event time of survival outcome

censoring time

min(τi,ci)

censoring indicator with δi= I(τi≤ ci)

?i= (?i1,?i2,...,?ini)?

β

Xi=

x?

i1

...

x?

ini

1,X?

2,...,X?

k)?

Zi

bi

τi

ci

τ∗

i

δi

Ideally, the complete longitudinal process yiis known and measurement times are non-

informative. The latter means that the tijs are not affected by the trend or values of yi.

Therefore the tis might differ in length and schedules for different subjects. If the tis are

identical for all subjects panel data are available and corresponding models can be applied.

Two approaches are available to arrive at a joint distribution for repeated measures yiand

survival outcome τi: (1) the introduction of shared random effects and (2) the use of mixture

and selection models. In the first approach random effects biare used to connect yiand τi.

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Discussion Paper 506 - SFB 3863

Conditioning on these random effects provides assumed independence of yiand τi. That is

f(yi,τi|bi) = f(yi|bi)f(τi|bi).

(1)

Assuming a certain distribution f(bi) for the random effects gives the joint distribution

f(y,τ) for k independent subjects:

f(y,τ) =

k?

i=1

?

f(yi,τi|bi)f(bi)dbi=

k?

i=1

?

f(yi|bi)f(τi|bi)f(bi)dbi.

(2)

Unless explicitly stated we rely on the common assumption bi∼ N(0,Σb).

The second approach to arrive at the joint distribution is based on a factorisation of y and

τ using conditional and marginal distributions. That is

f(y,τ)=

f(y|τ)f(τ)

f(τ|y)f(y).

or(3)

f(y,τ)=(4)

These models are known as mixture and selection models (Little, 1993).

In both approaches joint distributions are constructed on basis of sub-models for the lon-

gitudinal process and the survival outcome. A variety of models can be fitted to both

outcomes. Longitudinal measurements are easiest described by a linear mixed effects model

yi= Xiβ+Zibi+?i. Possible extensions allow for more complex relationships, for example

polynomial specifications of Xiand Zior any functions f(Xi,β) and f(Zi,bi).

In general survival models are constructed within the class of multiplicative hazards models

and are built without random effects. The hazard rate λi, conditioned on covariates at time

t, has the form:

λi(t|Xi[t]) = λ0(t)c(Xi[t]?β).

Mostly known representatives are the Cox proportional hazards model and the Weibull

model. In both models c(.) is specified as exp(.). Whereas the baseline hazard rate λ0(t) is

left completely unspecified in the Cox model it is taken as αµtα−1in the Weibull model (e.g

Klein & Moeschberger, 2003). The latter model is also a representative of another model class

used for specification of the survival outcome within joint models, the accelerated failure

time (AFT) models in which covariates are assumed to have linear influence on log(τi),

that is log(τi) = Xi[t]?β + ei, with eithe error. The distribution of the error specifies the

model (e.g. extreme value distribution which leads to the Weibull regression model). The

conditional hazard rate of an AFT model has the form

(5)

λi(t|Xi[t]) = λ0(texp(Xi[t]?β))exp(Xi[t]?β)(6)

where λ0is specified by the error e as mentioned above. Extensions to semiparametric ap-

proaches are possible by leaving λ0unspecified (Lin & Zhiliang, 1995).

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Discussion Paper 506 - SFB 3864

Subsequent sections of this paper are organised as follows: In Section 2, we describe joint

model approaches with focus on serial trends in which the pattern of repeated measure-

ments given a survival outcome are of main concern. Models with focus on event times

are described in Section 3. These models specify how the longitudinal measurements affect

survival outcomes. Section 4 reviews approaches that jointly focus on serial trends and event

times. Within these approaches covariate effects on longitudinal measurements and survival

outcome are jointly estimated. In Section 5, we give examples in which joint models have

been applied in practice. A brief look at further approaches and extensions is presented in

Section 6.

2Joint models with focus on serial trends

Models with primary focus on serial trends are applied when the description of repeated

measurements is of main concern. Informative dropouts or events are considered within these

models to avoid biased estimates for the longitudinal process. We describe two approaches,

one based on a shared random effects model and the other one is based on a mixture model,

to handle such data.

The approach introduced by Vonesh et al. (2006) is based on shared random effects; the joint

density is factorised as in (2). Within this factorisation the class of generalised non-linear

mixed-effects models is assumed for the sub-model yi|bi.

The conditional distribution of τi|biis modelled using multiplicative hazards models that

include subject-specific intercept and time trends via a function gi(.) that may also depend

on fixed effects,

λi(t|bi) = λ0(t)exp[gi(β,bi,Xi[t],Zi[t])].

Vonesh et al. specify λ0(t) in two ways, similar to a Weibull or a piecewise exponential model.

In the latter case the time scale is partitioned in disjoint exogenously given intervals. The

baseline hazard rate is assumed to be constant within each interval but may vary from

interval to interval. That is for p disjoint intervals λ0(t) =?p

in the model.

The estimation of the unknown parameters is done via the likelihood

(7)

h=1λ0hI(t ∈ (th−1,th]).

Replacing gi(.) by gih(.) in (7) additionally allows the inclusion of time-dependent covariates

L =

k?

i=1

?

f(yi|bi)f(τ∗

i|bi)δiSi(τ∗

i|bi)1−δif(bi)dbi

where S(τ∗

Since the integral has no closed form solution, numerical integration or alternatively the

Laplace approximation is needed.

i|bi) is the conditional survivor function.

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Discussion Paper 506 - SFB 3865

Hogan & Laird (1997) assume a mixture model for longitudinal measurements given time to

event. The joint density function for the mutually independent pairs (yi,τi) is obtained from

the factorisation given by (3). Thereby no parametric form is assumed for the cumulative

function F(τ), the Kaplan-Meier product-limit estimator replaces F(τ), which implicitly

means that survival times are homogenous.

f(y|τ) is described in two stages and the corresponding model contains fixed effects, random

effects and information on the event time. To implement these components we assume that

the first and second column of Zicontains 1 and tirespectively. This allows the construction

of a subject-specific intercept and time trend. Furthermore, a (q × p) matrix Wi(τi) is

introduced to cover information on the event time.

We choose the following structure for Wi(τi) so that Xi= ZiWi(τi): let m be the number

of covariate effects that are modelled as both random and fixed effects. Thus

Wi(τi) = (Em, ˜ τi,˜Xi)

with Em a (q × m) matrix containing canonical unit vectors el, l ∈ {1,2,...,q}, ˜ τi =

(g(τi),0,...,0)?with g(.) allowing for transformations of τiand˜Xia (q×(p−m−1)) matrix

containing covariates and interaction terms corresponding to fixed effects only. Thereby the

jth column is (˜ xij1,0,...,0) in constant covariates and (0, ˜ xij2,0,...,0) in covariates with

time-interactions.

Let yiτidenote the longitudinal observations for a known τi.

ear mixed effects model with subject-specific random intercept and time trends (αiτi=

(α1iτi,...,αqiτi)?) is constructed for yiτi. In the second step the αiτis are described via

Wi(τi). That is

(I)

yiτi= Ziαiτi+ ?i

(II)

αiτi= Wi(τi)β + bi.

In the first step a lin-

The combination of (I) and (II) can be formulated as a standard linear mixed effects model

that specifies f(y|τ)

yiτi= Ziαiτi+ ?i = Zi(Wi(τi)β + bi) + ?i= Xiβ + Zibi+ ?i.

(8)

The EM algorithm is used to obtain ML estimates based on f(y,τ) (see Hogan & Laird,

1997).

3Joint models with focus on event times

The aim of joint models with focus on event time data is the description on how the longitu-

dinal measurements affect the survival outcome. Thereby the longitudinal process is mostly

considered as time-dependent covariate within a survival model which is specified as Cox

proportional hazards model or accelerated failure time model.

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Discussion Paper 506 - SFB 3866

Tsiatis et al. (1995) assume the yito be measured with error and use a two-stage approach in

which information on the longitudinal process is incorporated in a Cox proportional hazards

model. First, the yiare modelled with a linear mixed effects model. The resulting model is

used to estimate the status of the longitudinal process at time t. In the second stage, these

so-called empirical Bayes estimates replace the observations measured with error and serve

as time-dependent covariate in the Cox model.

Wulfson & Tsiatis (1997) improved this approach by modelling the process of the time-

dependent covariate and the survival data simultaneously, assuming linear growth for yi

which might be measured with an error. Using the assumption bi∼ N(0,Σb), yij can be

written as

yij= Xi[tij]?β + b0i+ b1itij+ ?ij

(9)

with mutually independent error ?ij ∼ N(0,σ2

of the random intercept b0i and slope b1i. The Cox proportional hazards model with a

covariate described by the random effects model given above is written as

?). Furthermore ?ij is taken as independent

λ(t|bi,yi) = λ(t|bi) = λ0(t)exp{(Xi[t]?β + b0i+ b1it)γ}.

Estimates for this semiparametric approach are obtained by the EM algorithm (Wulfson &

Tsiatis, 1997). Hsieh et al. (2006) made a simulation study to examine the robustness and

efficiency of these estimates. They concluded that as long as there is sufficient information

on the longitudinal outcome, the estimates are both, robust and efficient.

Tsiatis & Davidian (2001) and Song et al. (2002b) generalised this model by setting the

normality assumption of the random effects aside. A further generalisation to multiple

time-dependent covariates was introduced by Song et al. (2002a). Dang et al. (2007) give a

full likelihood approach in which a bivariate growth curve from two longitudinal measures

of an individuum is used to predict recurrences of an event with a Cox model.

Another approach introduced by Tseng et al. (2005) is based on an accelerated failure time

model combined with a linear mixed effects model for the longitudinal observations. The

advantage of the class of accelerated failure time models is that they are applicable to

situations in which the proportional hazards assumption on the event time data fails. The

longitudinal process yi(t) is considered as a time-varying covariate and replaces Xi[t] in (6).

Tseng et al. (2005) assume a step function, but no parametric form for λ0(.), and takes

λ0(.) as constant between two consecutive event times. The restriction to a step function is

necessary to enable the application of the EM-algorithm.

Assuming noninformative censoring and a measurement schedule tij, that is independent of

the random effects and covariate history, the joint likelihood L(θ) = L(β,b,Σb,σ2

event time data and longitudinal measurements (as described in (9)) is

?,λ0) for

L(θ) =

k?

i=1

?

ni

?

j=1

f(yij|bi,ti,σ2

?)

f(τ∗

i,δi|bi,ti,λ0,β)f(bi|Σb)dbi

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Discussion Paper 506 - SFB 3867

where f(yij|bi,ti,σ2

?) and f(bi|Σb) are the densities of yijand N(0,Σb) respectively and

f(τ∗

i,δi|bi,ti,λ0,β) = {λ0(˜ τ∗

i(β,yi))exp[yi(τ∗

i)β]}δiexp

?

−

?˜ τ∗

i(β,yi)

0

λ0(t)dt

?

.

with ˜ τ∗

parameter estimates are described in detail in Tseng et al. (2005).

i(β,yi) =?τ∗

i

0

exp[yi(s)β]ds. The specific steps of the EM-algorithm to arrive at the

4Joint models with equal focus on both outcomes

The models described in the previous sections consider one of the jointly measured out-

comes as primary and explain this with the other outcome and covariates. In the following

we describe models that quantify covariates effects on longitudinal measurements and sur-

vival outcome jointly. For solving this task different approaches have been proposed, for

instance a common latent stochastic process that is shared by both outcomes (Henderson

et al., 2000), shared random effects that affect outcomes differently (Zeng & Cai, 2005), and

joint models within the framework of hierarchical generalised linear models (Ha et al., 2003):

Henderson et al. (2000) introduce a model class for joint modelling of repeated measurements

and event time data, including recurrent events. The idea of this model is similar to shared

random effects models. Here, the random effect turns into a dynamic bivariate Gaussian

process B(t) = {B(1)(t),B(2)(t)}. It is assumed that the longitudinal observations and

the event process are conditionally independent given B(t) and covariates. The association

between both outcomes is described by the cross-correlation between B(1)(t) and B(2)(t).

Thus, the joint distribution of both outcomes for a subject i is modelled via a latent zero-

mean Gaussian process Bi(t) = {B(1)

The sub-model for the repeated measurements of a subject i is assumed to be of the form

i(t),B(2)

i(t)} that is realised for each i independently.

yi= X(1)

iβ(1)+ B(1)

i(ti) + ?i

with X(1)

repeated measurements. The number of events is specified by a counting process Ni(t) that

allows the modelling of recurrent events. The semi-parametric model for the event process

and corresponding hazard function is

i

(time-varying) covariates and β(1)the corresponding coefficients associated to the

λi(t) = Hi(t)λ0(t)exp{X(2)

i[t]?β(2)+ B(2)

i(t)}.

Thereby, Hi(t) is the zero-one process that indicates whether individuum i is at risk or not.

λ0(t) is the baseline hazard with unspecified form.

Henderson et al. propose B(1)(t) and B(2)(t) to be of the form

B(1)(t) = b1+ b2t

and

B(2)(t) = γ1b1+ γ2b2+ γ3B(1)(t) + b3

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Discussion Paper 506 - SFB 3868

with (b1,b2) bivariate normal with mean zero and b3independent of (b1,b2) normally dis-

tributed with mean zero. The shape of B(2)is an extension of the usual proportionality

assumption between B(1)and B(2).

A general expression of the components B(l)

i(t),l = 1,2 of the bivariate Gaussian process is

B(l)

i(t) = Z(l)

i[t]?bli+ V(l)

i

(t)

where V(l)

function

i

(t) is a stationary Gaussian process with mean 0, variance σ2

vland correlation

rl(u) = cov{V(l)

i

(t),V(l)

i

(t − u)}/σ2

vl.

Parameters of for the longitudinal and the counting process can be estimated using the

EM-algorithm.

Similar to Henderson et al. (2000), Zeng & Cai (2005) introduce shared random effects to

obtain a joint model. Random effects are designed in a way that they affect both outcomes

differently. The longitudinal measurements are described by a linear mixed effects model

yi= Xiβ + Zibi+ ?i and the survival outcome by a multiplicative hazards model that

contains random effects which partly affect both outcomes and partly the survival outcome

only. That is˜bi= φbi+ciwith cithe subject-specific effect which only affects the survival

outcome and φ a scale parameter corresponding to the random effect that affects both

outcomes, but with different intensities. The hazard rate for the survival outcome is λ(t) =

λ0(t)exp(˜

linear mixed effects model. Zeng & Cai (2005) set cito zero since their application allows

the assumption that any unobserved factor that affects the longitudinal measurements also

affects the survival outcome. The joint likelihood is built as in (2) by integrating over the

random effects and parameters σ2

EM-algorithm to maximise the likelihood.

A similar approach has been proposed by Lin et al. (2002a). The authors use a frailty model

for the survival outcome and a mixed effects model to explain the longitudinal measure-

ments. The joint likelihood contains covariates having an effect on both outcomes. Likewise

Ratcliffe et al. (2004) analyse survival and longitudinal data when data clustering is present.

They use a Cox frailty model for the survival outcome and a mixed effects model for the

longitudinal one. A common cluster-level random effect links both outcomes and allows the

simultaneous analysis of survival and longitudinal measurements.

Xi[t]?˜β +˜ Zi[t]?˜bi) where˜

Xi[t] and˜ Zi[t] might differ from Xi[t] and Zi[t] in the

?,Σb,β,˜β,φ and Λ(t) =?t

0λ(s)ds are estimated using the

Ha et al. (2003) introduce a joint model approach within the framework of hierarchical gen-

eralised linear models (HGLM). Shared random effects link the linear mixed model proposed

for the longitudinal measurements and the Weibull frailty model proposed for the survival

outcome. As in Zeng & Cai (2005) both outcomes are affected differently by the random

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Discussion Paper 506 - SFB 3869

effects. Since frailty models can be presented in form of Poisson HGLM (Ha & Lee, 2003)

two HGLMs together with a random effect provide the basis for this joint model. The vector

of longitudinal measurements of subject i is assumed to follow

yi|bi∼ N(Xiβ + (γ∗

1bi)1,ρ2I)

with bi∼ N(0,1) the random effects with scale parameter γ∗

with all elements set to 1.

The hazard function of τi|biwith Weibull baseline hazard is given by

λi(t|bi) = αtα−1exp(˜

1=√γ1(γ1> 0), 1 a vector

Xi[t]?˜β + γ2bi)

with γ2∈ (−∞,∞) the scale parameter for the random effect and α the shape parameter

of the Weibull distribution.

The estimation of the unknown parameters β,˜β,α,ρ2,γ1and γ2is done via a hierarchical

likelihood approach using

h = h(β,˜β,α,ρ2,γ1,γ2) =

?

i

l1i+

?

i

l2i+

?

i

l3i

with l1ithe logarithm of the conditional density function for yigiven bi, l2iis that for τ∗

and δigiven biand l3iis the logarithm of the density for bi.

i

5Applications in biometrical research

A large number of studies generate repeated measurements and event time data, which often

depend on each other. Joint model approaches allow a modelling of both outcomes by taking

those dependencies into account. This technique has been applied in the following settings:

In acquired immunodeficiency syndrome (AIDS) studies viral load and CD4 counts are

important disease markers. The longitudinal process of these markers together with time to

seroconversion or death has been investigated using models with focus on event time data

(Tsiatis et al., 1995; Wang & Taylor, 2001). Guo & Carlin (2004) transferred the idea of

Henderson et al. (2000) into a Bayesian framework and modelled CD4 counts and time to

death simultaneously via a latent bivariate Gaussian process.

The issue of surrogacy of a disease marker for an endpoint can also be addressed with joint

models (Taylor & Wang, 2002). This has been done for example in prostate cancer where

the prostate-specific antigen (PSA) level was evaluated as surrogate for survival (Renard

et al., 2003). Furthermore Lin et al. (2002b) found subpopulations that differ in the longi-

tudinal PSA trajectories among prostate cancer patients using latent class models for joint

modelling.

In patients with cystic fibrosis the pattern of pulmonary function was explored by Schluchter

et al. (2002). Since patients with poorest lung function are oftentimes censored by death,

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Discussion Paper 506 - SFB 386 10

a joint model with focus on serial trend regarding informative dropout was used to avoid

biased estimates.

Additional to disease-specific markers many studies also measure quality of life or depression

measures together with survival data. Bowman & Manatunga (2005) focus on the serial

trend of a depression score that includes the risk of study discontinuation to determine a

treatment effect. In data from a pancreatic cancer study the treatment effect was measured

in terms of quality and quantity of life (Billingham & Abrams, 2002). The focus of the

analysis was survival which then was adjusted for the quality of life.

Joint model approaches with equal focus on repeated measures and survival outcome were

for example applied to data that arose from a drug therapy of schizophrenia patients. The

course of scores describing psychiatric disorder together with the time of withdraw were

analysed jointly (Henderson et al., 2000). Furthermore Ha et al. (2003) investigated the

serum creatinine level in patients with renal transplants together with the failure of kidney

graft. The quality of life together with the time to the first episode of severe hypoglycaemia

was analysed in patients that participated in the Diabetes Control and Complications Trial

(Rochon & Gillespie, 2001).

6 A brief look at further models and some extensions

Besides the presented models Bayesian approaches that address the simultaneous analysis

of longitudinal measurements and event time data are reasonable. For details on these ap-

proaches see Ibrahim et al. (2004) and Brown & Ibrahim (2003). Chi & Ibrahim (2006) give

further extensions to the likelihood approach for models with focus on event times based on

Bayesian inference. Their model allows both outcomes, longitudinal and survival, to be mul-

tidimensional. Multivariate longitudinal processes are modelled using a multivariate mixed

effects model which captures both, the dependence among the longitudinal measures over

time and the dependence between different longitudinal processes. Multivariate event time

data are described by a survival model with proportional hazards structure that additionally

includes a frailty to account for correlations between event times.

Tsiatis & Davidian (2004) give additional insight into the structure of likelihoods used to

estimate model parameters within joint models. Extensions are particularly necessary when

approaches are applied to different types of data. In case of non-continuous event times

Rochon & Gillespie (2001) propose to combine a generalised estimating equation (GEE)

model for the longitudinal outcome with a GEE for the survival endpoint. But this approach

restricts the longitudinal process to equally-spaced intervals.

Many approaches use a linear mixed effects model to describe the longitudinal process. How-

ever, this linearity assumption is not always appropriate in practice. Therefore it would be

important to extent joint model approaches to more general trajectories for the longitudinal

outcome. Splines with high-dimensional basis functions might be a starting point at this

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Discussion Paper 506 - SFB 38611

place. Further extensions would also be necessary if one outcome is discrete. That is either a

discrete longitudinal outcome, a discrete time scale or both. An example is the disease mul-

tiple sclerosis, which motivated this review. There disability status is rated according to an

ordinal scale (Kurtzke, 1983). In further research we aim to expand joint model approaches

to data from multiple sclerosis patients to increase the understanding of interactions between

exacerbations, permanent disability and demographic factors.

Predominantly joint model approaches focus on single event times. In Henderson et al.

(2000) an embedded counting process allows to include recurrent events in the model. To

examine to which extent recurrent events have an effect on the longitudinal disease pro-

cess analogous ways have to be followed within the model class focusing on longitudinal

processes. As a starting point Hogan et al. (2004) use varying coefficients random effects

models to allow the coefficients that describe the trajectories to depend on a single event

time.

Acknowledgement

Financial support from the German Research Foundation (DFG), Collaborative Research

Center (SFB) 386 - Statistical Analysis of Discrete Structures (Projects B2 and C2), is

gratefully acknowledged.

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Chi, Y.-Y. & Ibrahim, J. G. (2006).

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