The document discusses fitting joint models using R packages jm and jmbayes, focusing on longitudinal and time-to-event data analysis. It presents a case study on aortic valve patients and outlines the research questions, joint modeling framework, and software implementation for analyzing the relationships between aortic gradient and survival/re-operation events. The goals are to introduce joint modeling methods and demonstrate the capabilities of the software for statistical analyses in biostatistics.

1. Fitting Joint Models in R using Packages JM and JMbayes
Dimitris Rizopoulos
Department of Biostatistics, Erasmus Medical Center, the Netherlands
d.rizopoulos@erasmusmc.nl
Joint Statistical Meetings
August 11th, 2015

2. 1.1 Introduction
• Often in follow-up studies different types of outcomes are collected
• Explicit outcomes
◃ multiple longitudinal responses (e.g., markers, blood values)
◃ time-to-event(s) of particular interest (e.g., death, relapse)
• Implicit outcomes
◃ missing data (e.g., dropout, intermittent missingness)
◃ random visit times
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3. 1.2 Illustrative Case Study
• Aortic Valve study: Patients who received a human tissue valve in the aortic position
◃ data collected by Erasmus MC (from 1987 to 2008);
77 received sub-coronary implantation; 209 received root replacement
• Outcomes of interest:
◃ death and re-operation → composite event
◃ aortic gradient
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4. 1.3 Research Questions
• Depending on the questions of interest, different types of statistical analysis are
required
• We will distinguish between two general types of analysis
◃ separate analysis per outcome
◃ joint analysis of outcomes
• Focus on each outcome separately
◃ does treatment affect survival?
◃ are the average longitudinal evolutions different between males and females?
◃ . . .
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5. 1.3 Research Questions (cont’d)
• Focus on multiple outcomes
◃ Complex effect estimation: how strong is the association between the longitudinal
outcome and the hazard rate of death?
◃ Handling implicit outcomes: focus on the longitudinal outcome but with dropout
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6. 1.3 Research Questions (cont’d)
In the Aortic Valve dataset:
• Research Question:
◃ Can we utilize available aortic gradient measurements to predict
survival/re-operation
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7. 1.4 Goals
• Methods for the separate analysis of such outcomes are well established in the
literature
• Survival data:
◃ Cox model, accelerated failure time models, . . .
• Longitudinal data
◃ mixed effects models, GEE, marginal models, . . .
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8. 1.4 Goals (cont’d)
• Goals of this talk:
◃ introduce joint models
◃ illustrate present software capabilities in R
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9. 2.1 Joint Modeling Framework
• To answer our questions of interest we need to postulate a model that relates
◃ the aortic gradient with
◃ the time to death or re-operation
• Problem: Aortic gradient is an endogenous time-dependent covariate (Kalbfleisch and
Prentice, 2002, Section 6.3)
◃ Measurements on the same patient are correlated
◃ Endogenous (aka internal): the future path of the covariate up to any time t > s
IS affected by the occurrence of an event at time point s
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10. 2.1 Joint Modeling Framework (cont’d)
• What is special about endogenous time-dependent covariates
◃ measured with error
◃ the complete history is not available
◃ existence directly related to failure status
• What if we use the Cox model?
◃ the association size can be severely underestimated
◃ true potential of the marker will be masked
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11. 2.1 Joint Modeling Framework (cont’d)
0 1 2 3 4 5
46810
Time (years)
AoGrad
Death
observed Aortic Gradient
time−dependent Cox
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12. 2.1 Joint Modeling Framework (cont’d)
• To account for the special features of these covariates a new class of models has been
developed
Joint Models for Longitudinal and Time-to-Event Data
• Intuitive idea behind these models
1. use an appropriate model to describe the evolution of the marker in time for each
patient
2. the estimated evolutions are then used in a Cox model
• Feature: Marker level is not assumed constant between visits
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13. 2.1 Joint Modeling Framework (cont’d)
Time
0.10.20.30.4
hazard
0.00.51.01.52.0
0 2 4 6 8
marker
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14. 2.1 Joint Modeling Framework (cont’d)
• We define a standard joint model
◃ Survival Part: Relative risk model
hi(t) = h0(t) exp{γ⊤
wi + αmi(t)},
where
* mi(t) = the true & unobserved value of aortic gradient at time t
* α quantifies the effect of aortic gradient on the risk for death/re-operation
* wi baseline covariates
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15. 2.1 Joint Modeling Framework (cont’d)
◃ Longitudinal Part: Reconstruct Mi(t) = {mi(s), 0 ≤ s < t} using yi(t) and a
mixed effects model (we focus on continuous markers)
yi(t) = mi(t) + εi(t)
= x⊤
i (t)β + z⊤
i (t)bi + εi(t), εi(t) ∼ N(0, σ2
),
where
* xi(t) and β: Fixed-effects part
* zi(t) and bi: Random-effects part, bi ∼ N(0, D)
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16. 3.1 Joint Models in R
• Joint models are fitted using function jointModel() from package JM. This
function accepts as main arguments a linear mixed model and a Cox PH model based
on which it fits the corresponding joint model
lmeFit <- lme(CD4 ~ obstime + obstime:drug,
random = ~ obstime | patient, data = aids)
coxFit <- coxph(Surv(Time, death) ~ drug, data = aids.id, x = TRUE)
jointFit <- jointModel(lmeFit, coxFit, timeVar = "obstime",
method = "piecewise-PH-aGH")
summary(jointFit)
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17. 3.1 Joint Models in R
• As before, the data frame given in lme() should be in the long format, while the
data frame given to coxph() should have one line per subject∗
◃ the ordering of the subjects needs to be the same
• In the call to coxph() you need to set x = TRUE (or model = TRUE) such that the
design matrix used in the Cox model is returned in the object fit
• Argument timeVar specifies the time variable in the linear mixed model
∗
Unless you want to include exogenous time-varying covariates or handle competing risks
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18. 3.1 Joint Models in R
• Argument method specifies the type of relative risk model and the type of numerical
integration algorithm – the syntax is as follows:
<baseline hazard>-<parameterization>-<numerical integration>
Available options are:
◃ "piecewise-PH-GH": PH model with piecewise-constant baseline hazard
◃ "spline-PH-GH": PH model with B-spline-approximated log baseline hazard
◃ "weibull-PH-GH": PH model with Weibull baseline hazard
◃ "weibull-AFT-GH": AFT model with Weibull baseline hazard
◃ "Cox-PH-GH": PH model with unspecified baseline hazard
GH stands for standard Gauss-Hermite; using aGH invokes the pseudo-adaptive
Gauss-Hermite rule
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19. 3.1 Joint Models in R
• Joint models under the Bayesian approach are fitted using function
jointModelBayes() from package JMbayes. This function works in a very similar
manner as function jointModel(), e.g.,
lmeFit <- lme(CD4 ~ obstime + obstime:drug,
random = ~ obstime | patient, data = aids)
coxFit <- coxph(Surv(Time, death) ~ drug, data = aids.id, x = TRUE)
jointFitBayes <- jointModelBayes(lmeFit, coxFit, timeVar = "obstime")
summary(jointFitBayes)
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20. 3.1 Joint Models in R
• JMbayes is more flexible (in some respects):
◃ directly implements the MCMC
◃ allows for categorical longitudinal data as well
◃ allows for general transformation functions
◃ penalized B-splines for the baseline hazard function
◃ . . .
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21. 3.1 Joint Models in R
• In both packages methods are available for the majority of the standard generic
functions + extras
◃ summary(), anova(), vcov(), logLik()
◃ coef(), fixef(), ranef()
◃ fitted(), residuals()
◃ plot()
◃ xtable() (you need to load package xtable first)
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22. 4.1 Association Structures
• The standard assumption is



hi(t | Mi(t)) = h0(t) exp{γ⊤
wi + αmi(t)},
yi(t) = mi(t) + εi(t)
= x⊤
i (t)β + z⊤
i (t)bi + εi(t),
where Mi(t) = {mi(s), 0 ≤ s < t}
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23. 4.1 Association structures (cont’d)
Time
0.10.20.30.4
hazard
0.00.51.01.52.0
0 2 4 6 8
marker
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24. 4.1 Association Structures (cont’d)
• The standard assumption is



hi(t | Mi(t)) = h0(t) exp{γ⊤
wi + αmi(t)},
yi(t) = mi(t) + εi(t)
= x⊤
i (t)β + z⊤
i (t)bi + εi(t),
where Mi(t) = {mi(s), 0 ≤ s < t}
Is this the only option? Is this the most optimal for
prediction?
JM & JMbayes – August 11, 2015 23/37

25. 4.2 Time-dependent Slopes
• The hazard for an event at t is associated with both the current value and the slope
of the trajectory at t (Ye et al., 2008, Biometrics):
hi(t | Mi(t)) = h0(t) exp{γ⊤
wi + α1mi(t) + α2m′
i(t)},
where
m′
i(t) =
d
dt
{x⊤
i (t)β + z⊤
i (t)bi}
JM & JMbayes – August 11, 2015 24/37

26. 4.2 Time-dependent Slopes (cont’d)
Time
0.10.20.30.4
hazard
0.00.51.01.52.0
0 2 4 6 8
marker
JM & JMbayes – August 11, 2015 25/37

27. 4.3 Cumulative Effects
• The hazard for an event at t is associated with area under the trajectory up to t:
hi(t | Mi(t)) = h0(t) exp
{
γ⊤
wi + α
∫ t
0
mi(s) ds
}
• Area under the longitudinal trajectory taken as a summary of Mi(t)
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28. 4.3 Cumulative Effects (cont’d)
Time
0.10.20.30.4
hazard
0.00.51.01.52.0
0 2 4 6 8
marker
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29. 4.4 Weighted Cumulative Effects
• The hazard for an event at t is associated with the area under the weighted trajectory
up to t:
hi(t | Mi(t)) = h0(t) exp
{
γ⊤
wi + α
∫ t
0
ϖ(t − s)mi(s) ds
}
,
where ϖ(·) appropriately chosen weight function, e.g.,
◃ Gaussian density
◃ Student’s-t density
◃ . . .
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30. 4.5 Association Structures in JM & JMbayes
• Both package give options to define the aforementioned association structures
◃ in JM via arguments parameterization & derivForm
◃ in JMbayes via arguments param & extraForm
• JMbayes also gives the option for general transformation functions, e.g.,
hi(t | Mi(t)) = h0(t) exp{γ⊤
wi + α1mi(t) + α2mi(t) × Treati +
α3m′
i(t) + α3(m′
i(t))2
},
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31. 5.1 Predictions – Definitions
• We are interested in predicting survival probabilities for a new patient j that has
provided a set of aortic gradient measurements up to a specific time point t
• Example: We consider Patients 20 and 81 from the Aortic Valve dataset
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32. 5.1 Predictions – Definitions (cont’d)
Follow−up Time (years)
AorticGradient(mmHg)
0
2
4
6
8
10
0 5 10
Patient 20
0 5 10
Patient 81
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33. 5.1 Predictions – Definitions (cont’d)
Follow−up Time (years)
AorticGradient(mmHg)
0
2
4
6
8
10
2 4 6 8 10 12
Patient 20
2 4 6 8 10 12
Patient 81
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34. 5.1 Predictions – Definitions (cont’d)
Follow−up Time (years)
AorticGradient(mmHg)
0
2
4
6
8
10
2 4 6 8 10 12
Patient 20
2 4 6 8 10 12
Patient 81
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35. 5.1 Predictions – Definitions (cont’d)
• What do we know for these patients?
◃ a series of aortic gradient measurements
◃ patient are event-free up to the last measurement
• Dynamic Prediction survival probabilities are dynamically updated as additional
longitudinal information is recorded
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36. 5.2 Predictions – JM & JMbayes
• Individualized predictions of survival probabilities are computed by function
survfitJM() – for example, for Patient 2 from the PBC dataset we have
sfit <- survfitJM(jointFit, newdata = pbc2[pbc2$id == "2", ])
sfit
plot(sfit)
plot(sfit, include.y = TRUE)
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37. 5.3 Dyn. Predictions – Illustration
Follow−up Time (years)
AorticGradient(mmHg)
0
2
4
6
8
10
0 5 10
Patient 20
0 5 10
Patient 81
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38. 5.3 Dyn. Predictions – Illustration (cont’d)
0 5 10 15
024681012
Time
Patient 20
0.0
0.2
0.4
0.6
0.8
1.0
AorticGradient(mmHg)
0 5 10 15
024681012
Time
0.0
0.2
0.4
0.6
0.8
1.0
Patient 81
Re−Operation−FreeSurvival
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39. 5.3 Dyn. Predictions – Illustration (cont’d)
0 5 10 15
024681012
Time
Patient 20
0.0
0.2
0.4
0.6
0.8
1.0
AorticGradient(mmHg)
0 5 10 15
024681012
Time
0.0
0.2
0.4
0.6
0.8
1.0
Patient 81
Re−Operation−FreeSurvival
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40. 5.3 Dyn. Predictions – Illustration (cont’d)
0 5 10 15
024681012
Time
Patient 20
0.0
0.2
0.4
0.6
0.8
1.0
AorticGradient(mmHg)
0 5 10 15
024681012
Time
0.0
0.2
0.4
0.6
0.8
1.0
Patient 81
Re−Operation−FreeSurvival
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41. 5.3 Prediction Survival – Illustration (cont’d)
0 5 10 15
024681012
Time
Patient 20
0.0
0.2
0.4
0.6
0.8
1.0
AorticGradient(mmHg)
0 5 10 15
024681012
Time
0.0
0.2
0.4
0.6
0.8
1.0
Patient 81
Re−Operation−FreeSurvival
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42. 5.3 Dyn. Predictions – Illustration (cont’d)
0 5 10 15
024681012
Time
Patient 20
0.0
0.2
0.4
0.6
0.8
1.0
AorticGradient(mmHg)
0 5 10 15
024681012
Time
0.0
0.2
0.4
0.6
0.8
1.0
Patient 81
Re−Operation−FreeSurvival
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43. 5.3 Dyn. Predictions – Illustration (cont’d)
0 5 10 15
024681012
Time
Patient 20
0.0
0.2
0.4
0.6
0.8
1.0
AorticGradient(mmHg)
0 5 10 15
024681012
Time
0.0
0.2
0.4
0.6
0.8
1.0
Patient 81
Re−Operation−FreeSurvival
JM & JMbayes – August 11, 2015 35/37

44. 6. Resources
• JM
◃ JSS paper (http://www.jstatsoft.org/v35/i09/)
◃ book for joint models (http://jmr.r-forge.r-project.org/)
• JMbayes
◃ JSS paper (http://arxiv.org/abs/1404.7625; to appear)
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45. Thank you for your attention!
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