Summary
Dynamic prediction uses patient information collected during follow-up to produce individualized survival predictions at given time points beyond treatment or diagnosis. This allows clinicians to obtain updated predictions of a patient’s prognosis that can be used in making personalized treatment decisions. Two commonly used approaches for dynamic prediction are landmarking and joint modeling. Landmarking does not constitute a comprehensive probability model, and joint modeling often requires strong distributional assumptions and computationally intensive methods for estimation. We introduce an alternative approximate approach for dynamic prediction that aims to overcome the limitations of both methods while achieving good predictive performance. We separately specify the marker and failure time distributions conditional on surviving up to a prediction time of interest and use standard variable selection and goodness-of-fit techniques to identify the best-fitting models. Taking advantage of its analytic tractability and easy two-stage estimation, we use a Gaussian copula to link the marginal distributions smoothly at each prediction time with an association function. With simulation studies, we examine the proposed method’s performance. We illustrate its use for dynamic prediction in an application to predicting death for heart valve transplant patients using longitudinal left ventricular mass index information.
Keywords: Dynamic prediction, Gaussian copula, Joint modeling, Landmarking, Longitudinal data, Survival analysis
1. Introduction
Personalized medicine focuses on tailoring a treatment to an individual based on an their particular risk. For survival outcomes, an estimate of this risk is traditionally obtained from a prediction model employed at some baseline time, such as diagnosis, and only uses information up to that point. However, there is increased interest in predicting the conditional survival of patients beyond baseline. During follow-up, new information such as updated biomarker measurements may become available for the patient. To obtain accurate, individualized survival predictions during follow-up, prediction models must incorporate this patient information that can change over time, and thus produce dynamic predictions. These predictions can then be used by clinicians to monitor a patient’s prognosis during follow-up to implement additional therapies or modify their screening schedule.
The statistical task is to develop a technique that produces survival predictions at
baseline, and incorporates additional marker information to obtain updated predictions for
patients that are still alive at future time points. Thus, dynamic prediction methods
require incorporating time-dependent marker information, 
, into a model
for the failure time, 
, to obtain the conditional distribution
, where
is the marker information
available up to time 
. The dynamic prediction of experiencing
the event of interest in the next 
interval given survival up to time
and up-to-date marker information is then
the conditional survival probability 
.
Two common approaches for obtaining these dynamic predictions are joint modeling and
landmarking.
1.1. Dynamic prediction with joint modeling and landmarking
Joint modeling specifies a model for the marker process, 
, and a
model for the failure time that links it to the marker process, 
, e.g., a
survival model with hazard 
(Wulfsohn and Tsiatis, 1997;Henderson and
others, 2000;Wang and Taylor,
2001). From these two models, the joint distribution 
can be
derived. Joint modeling produces a valid prediction function from which we can obtain
consistent conditional survival predictions that have a defined, meaningful relationship
with predictions obtained from the model at other time points (Jewell and Nielsen, 1993), thus making it suited for dynamic prediction
( Rizopoulos, 2011; Taylor and others, 2013; Rizopoulos and others, 2017). The dynamic predictions obtained
from joint modeling at time 
for surviving up to time
involve integrating the
conditional hazard 
from
to 
, which requires knowledge
of the distribution of future values of the marker process beyond the current measurement
. Thus, utilizing
joint modeling for prediction requires the full specification of the marker process, which
involves making specific distributional assumptions. In addition, the marker model may be
difficult to estimate when there are sparse longitudinal measurements, and
misspecification of this model can result in biased predictions (Rizopoulos and others, 2008b). A practical
disadvantage of this method is that it can require computationally intensive methods for
both estimation and the calculation of dynamic predictions (Rizopoulos, 2011).
Landmarking requires directly specifying a survival model for 
by looking at the
empirical failure time distribution at fixed time points, 
,
conditional on being alive at 
and having marker value
(van
Houwelingen, 2007;Zheng and Heagerty,
2005;Gong and Schaubel, 2013). Thus, at
each 
, we obtain the best-fitting model for
using information from individuals alive at
and their last available longitudinal
marker measurement, 
. Estimation of this empirical
distribution is accomplished by using a Cox regression to model the hazard
, where the baseline
hazard and covariate effects can be restricted to vary smoothly with
. The dynamic survival predictions can be
directly computed as 
.
The advantages of this method are that it avoids having to specify the distribution of the
marker process and can be easily implemented in standard software. A disadvantage of
landmarking is the numerous decisions required by the method. To conduct estimation,
landmarking requires prespecifying the prediction times of interest, referred to as
landmark times. For simple landmark models, computing dynamic predictions is restricted to
these time points. Since a model for the marker process is not specified, to perform
estimation landmarking also requires selecting an imputation method for marker values at
landmark times at which individuals do not have observations. As well, the landmarking
approach does not satisfy the consistency criteria described in Jewell and Nielsen (1993) since it directly models the conditional
hazard 
and does not derive it
from the joint distribution of failure time and marker processes, as in joint modeling. In
previous work (Suresh and others,
2017), we demonstrated that under a binary marker process, landmarking results in
a theoretically incorrect model; however, with increased flexibility it provides a
sufficient approximation to a joint model.
The advantage of using an ad hoc approach, such as landmarking, for dynamic prediction is that it is a simpler method that does not require assumptions about the marker distribution and does not impose a computational burden on estimation or calculating predicted probabilities. However, it lacks the consistency of a valid prediction function that is offered by a joint model. Thus, we propose an alternative approximate approach for dynamic prediction that aims to combine the advantages and mitigate the disadvantages of landmarking and joint modeling, while maintaining good predictive performance.
1.2. Copula approach for dynamic prediction
Copulas are multivariate cumulative distribution functions with uniform marginals, and by
Sklar’s theorem they provide a convenient approach to link marginals to construct a joint
distribution (Nelsen, 1999). We propose an
approximate method for dynamic prediction that requires specifying the marginal models
and 
for
individuals alive at time 
, and then uses a bivariate Gaussian
copula to model the joint distribution of 
and
conditional on being alive at
, 
. From this
joint distribution, we can directly compute the dynamic predictions. The copula allows us
to specify the marginal distributions of the marker data and time-to-event process and
then model their association separately. It is a flexible way of specifying this
association since there is no restriction on the marginal distributions, which do not have
to be specified parametrically.
As with landmarking, this method does not produce a comprehensive probability model;
however, we maintain a greater level of consistency in our predictions by specifying a
single model for 
, and then deriving the model for
, which will be consistently
defined for all 
. Unlike joint modeling, we do not
require a flexible specification of the marker process using random effects that can lead
to complex estimation. Instead, we specify the marginal distribution of the longitudinal
data at each 
, allowing the mean and variance of the
distribution to change smoothly with 
. We use two-stage estimation to first
estimate the parameters from the marginal models, and then hold them fixed in the joint
likelihood to estimate the association parameters. Estimation is conducted using
likelihood-based methods, which allow for standard methods of model checking and
validation.
Researchers have previously used copulas to facilitate joint modeling and dynamic
prediction. Rizopoulos and others
(2008a),Rizopoulos and others
(2008b) proposed a reparameterization of a shared random effects model by using a
copula to model the joint distribution of the random effects of the marker process and the
frailty term of the survival process. In our formulation, we avoid the complexity of
random effects estimation by using the copula to directly model the association between
simple, but flexible models, for the marginal distributions of the survival and marker
data. Ganjali and Baghfalaki (2015) uses a
multivariate copula to obtain a fully specified joint model for 
and
measured at fixed time points
, given by
.
This approach uses an individual’s entire longitudinal marker history to make predictions
but does not easily accommodate measurement times that vary by individual. Emura and others (2018) use a
joint-frailty copula model to specify the joint distribution between two survival
outcomes, time-to-death and time-to-tumor progression. From this joint model, they derive
the dynamic prediction conditional on the individual’s tumor progression status.
Similarly, we obtain the dynamic prediction by conditioning on an individual’s marker
value. In comparison, only one of our outcomes is time-to-event, thus for convenience of
derivation we do not use a survival copula in our model specification. Unlike these
approaches, our model formulation does not specify a joint model for
and 
, but rather models the
cross-sectional distribution between the survival and marker data at landmark times,
allowing that relationship to change smoothly over time.
In summary, we aim to describe a new method for dynamic prediction using a novel Gaussian copula approach. In Section 2, we introduce the model and discuss a two-stage approach for estimation in the situation of a continuous marker. Using a simulation study, we explore the performance of our method in Section 3. In Section 4, we illustrate the use of our method to obtain dynamic predictions of survival for heart valve replacement patients with longitudinal measurements of heart function. Section 5 is a discussion of the advantages and limitations of our method and future directions.
2. Method
Our proposed method for dynamic prediction specifies the marginal distributions of the marker data and the survival outcome and uses a copula to model the association between the two outcomes over time. The intuition behind this approach is that we can specify a model for each of the marginals that imposes fewer assumptions on the marginal distributions and for which we can assess goodness-of-fit, and then model their correlation using a copula with a time-varying association structure.
2.1. Copula model and estimation
Let 
denote the observed data, where 
is the true event time,
is the censoring time,
is the observed
event time, 
is the
censoring indicator, 
is the baseline covariate vector,
and 
is the 
longitudinal marker vector, with element 
denoting
the marker value at time 
, for subject
. We assume non-informative censoring, i.e.,
conditional on baseline covariates any additional dropout process is not related to the
event of interest or the longitudinal process. We discuss dependent censoring in Section A
of the supplementary material available at Biostatistics online.
The dynamic prediction of interest is the subject-specific predicted probability of
experiencing the survival event in the time interval 
],
, given that a new subject
has survived up to time
, i.e.,
 |
(2.1) |
where 
are the set of longitudinal measurements recorded up to time 
. In
practice, this prediction may depend only on the scalar marker measurement at time
, 
.
We are interested in specifying the marginal distributions of 
, the
time-to-event outcome, and 
, the longitudinal marker data, for each
landmark time 
. Thus, we restrict the models for
and 
to be conditional on the
patient being alive at time 
, and are specifically interested in
modeling the conditional survival time 
and the
cross-sectional marker data at 
, 
, denoted by
and 
,
respectively.
A Gaussian copula is then used to link the survival time distribution and the marker data distribution at all landmark time points, allowing us to compute the dynamic predictions from an overall model.
We consider the situation of a continuous marker process. Let 
and
be the marginal distributions of
the time-to-event outcome 
and the marker data
, respectively, conditional on the
individual being alive at time 
. Both of the marginals can be
conditional on baseline covariates 
, i.e., 
and
; however, we shall omit
them from the following model specification for brevity. The joint distribution
is then defined using a
Gaussian copula as
 |
(2.2) |
where 
is the standard normal
distribution, 
is the standard bivariate normal
distribution, and 
is the correlation,
which is specified as a smooth function of landmark time and possibly baseline covariates
. The joint density is given by
 |
where 
and
, and
and 
are
the marginal densities of 
and 
,
respectively. This is the likelihood contribution of individuals who at time
are alive and have observed marker value
, and at time 
have an
observed event. For individuals who are alive at time 
, but are censored at
time 
, the joint density is given by
 |
Let 
be the parameter
vector containing the respective marginal parameters 
and 
of
and 
, and
association parameters 
. The likelihood
contribution for individual 
at measurement time
is then
 |
(2.3) |
where 
is the time at which
individual 
has the event or was censored (i.e., last
observed time). Assuming working independence between measurements at different time
points for each individual, we construct the following pseudo-likelihood function
 |
Due to the number of parameters associated with both the marginal models and the
association structure, direct maximization of this pseudo-likelihood may still be
computationally difficult. Thus, we further simplify estimation by using the method of
inference functions for margins (IFM), where the marginal parameters,
and
, are first
estimated from the marginal models and then held fixed in the maximization of the
pseudo-likelihood over the association parameters 
(Joe and Xu, 1996). Specifically,
(a) The marginal parameter estimates,
and
, are
obtained separately from their respective marginal models using an appropriate
estimation method (e.g., maximum partial likelihood estimation for a Cox survival
model, least squares if using linear regression for the marker model). Estimation of
uses data
and estimation of 
uses data
.(b) The function
is maximized over 
to get
using
Nelder–Mead optimization.(c)
is the IFM estimate.
The standard errors for the marginal survival model can be obtained in the same way as with a standard Cox or parametric survival model (Andersen and Gill, 1982). Robust standard errors for the marginal marker model are computed using a sandwich estimator (Zeger and Liang, 1986). Following arguments presented in existing literature (Joe and Xu, 1996;Prenen and others, 2017;Song, 2007), we describe two-stage parametric variance estimation for the association parameters in Section B of the supplementary material available at Biostatistics online, with the extension to semiparametric variation following from arguments presented in Prenen and others (2017) and Spiekerman and Lin (1998). The analytic standard errors of the association parameter vector are complicated since they account for variability from the estimates from the marginal models. Thus, in practice the standard errors are computed using a resampling scheme, such as jackknife (Joe and Xu, 1996) or bootstrapping (Efron and Tibshirani, 1993).
Using the parameter estimates 
obtained
from the IFM method, we can compute the dynamic prediction of interest from (2.1) as,
 |
(2.4) |
where 
,
,
and 
.
Note that we compute this prediction conditioning on the scalar value of the marker at
time 
, 
. For the situation
where the marker is not observed at the prediction time, we instead use
in our prediction, where 
is the time at which the marker was
last observed. That is, we compute the quantile of the marker distribution at the time at
which the marker was observed and carry it forward to the prediction time of interest,
rather than carrying the marker value to a new time at which the marker distribution is
different.
2.2. Modeling copula components
In choosing models for the components of the copula, we consider simple, flexible, but possibly misspecified models that serve as a good approximation to the true distribution. This allows us to avoid making strong modeling assumptions and allows for easy estimation that can be readily implemented in standard software.
2.2.1. Modeling the marker data
Instead of specifying a mixed effects model for the evolution of the marker data over
time, as is the case with joint modeling, we specify the distribution of the
longitudinal data at each landmark time 
, where this model
has a mean and variance that can be specified as a function of landmark time
and baseline covariates
. We define the model
,
where 
is a vector of
regression coefficients, 
is a function of landmark time, baseline covariates, and regression coefficients, and
is an error term that is
independently distributed. We consider a linear regression for 
, where
we model 
as a smooth parametric function of landmark time and 
, where
is also a smooth function of 
and 
is a vector
of regression coefficients. The marginal parameters of 
are
then 
.
2.2.2. Modeling the failure time data
We model the time-to-event outcome, 
, and then compute the
conditional survival 
.
We propose modeling the event time using a Cox model that can be extended to accommodate
non-proportional hazards or additional flexibility, 
,
where 
is the baseline hazard,
is a vector of
regression coefficients, and 
is a
function of baseline covariates, regression coefficients, and possibly time, to allow
for non-proportional hazards or time-varying covariate effects. The marginal
distribution of the time-to-event outcome is then given by 
and the corresponding parameters to be estimated, 
,
are the parameter vector 
and the cumulative
baseline hazard 
.
2.2.3. Modeling the association
Once we define the marginals 
and 
, we
can use the Gaussian copula defined in (2.2) to describe the joint distribution at landmark time
. To model the association between the
two marginals, we reparameterize 
using Fisher’s z-transformation
.
We define 
as a function of landmark time 
, baseline covariates
, and regression coefficients
. Thus, from
the association function we can evaluate the extent of the correlation between the
time-to-event outcome and the marker process, and how that relationship changes with
landmark time.
2.2.4. Choosing the copula
We consider a Gaussian copula due to its tractable nature, and easy implementation in standard software. However, there are other choices of copulas that have differing strengths of dependence in the distribution tails. The Student’s t copula is symmetric and has an additional parameter for degrees of freedom that controls the strength of the tail dependence. This copula is both upper- and lower-tail dependent, which allows for joint extreme events and can be beneficial if we expect our distribution to have heavy tails. The Clayton and Gumbel copulas are Archimedean copulas that are lower-tail and upper-tail dependent, respectively. We present our method under these alternative copulas in Section C of the supplementary material available at Biostatistics online and consider the performance of the Student’s t copula in our simulation study.
3. Simulation study
We conduct a simulation study to assess the predictive performance of our proposed method in comparison to the existing methods of joint modeling and landmarking.
3.1. Performance comparison metrics
We evaluate the discrimination and calibration of dynamic predictions computed under the
different methods for the interval 
using dynamic analogs of
weighted area under the curve (AUC) and Brier score (BS), which account for censoring. We
denote these measures as 
and
, respectively, and use
the following definitions presented in Blanche and
others (2015),
 |
where 
and 
is the dynamic prediction of
interest given in (2.1). The inverse
probability of censoring weight estimators are then
 |
where 
,
,
and weights to account for censoring are given by 
,
where 
is the
Kaplan–Meier estimator of survival function for the censoring time at
.
Since BS depends on the cumulative incidence of death in 
,
we use a standardized version that produces an 
-type measure that
compares how well the predictions perform relative to predictions from a null model given
by the Kaplan–Meier estimate, 
, which does not
take into account subject-specific information. We denote this scaled measure as
.
To make comparisons between the different models, we compute the best-attainable AUC and
using the predicted probabilities from
the true models, denoted 
and
, respectively. We then
report the relative measures 
and 
for each
of the models, where values close to 0 indicate better performance. To ensure that our
method is not consistently predicting higher or lower than the true probabilities, we also
evaluate calibration using the root mean squared prediction errors (RMSPEs) between the
true conditional survival probabilities, 
, and the
predictions obtained from each of the different models considered, given by
 |
Five hundred simulations of 1000 subjects were run for each scenario. A random sample of 500 subjects were selected to create a training data set, to which the models were fit. The performance metrics were then computed for predictions from the remaining 500 subjects who compose the validation data set, and averaged across the simulations. The data sets included in this average are those for which all methods converged.
3.2. Simulation set-up
We simulate patients who have been followed for a period of 10 years, for whom
longitudinal biomarker measurements are available at baseline. We simulate marker
measurement under two patterns of observation: (i) the marker process is observed every
year for 9 years following baseline and (ii) the value of the marker is observed at
inspection times occurring according to a Poisson process with rate 0.5 and 1. We simulate
a binary baseline covariate 
that has 50% prevalence. We generate the
longitudinal marker measurements using the following linear mixed effects model
 |
where 
and 
. To
generate the survival times, we use the following joint model
 |
(3.1) |
with 
as the Weibull baseline hazard. Since 
describes how the
survival process is affected by the longitudinal biomarker, we consider different levels
of correlation between the two processes. The noisiness of the marker process is described
by 
, thus we simulate under
both a noisy and a more stable marker process. We also consider a missing at random (MAR)
scenario where the marker value at each time point is missing with probability 0.2 and 0.1
for those with 
and 
,
respectively.
To assess the robustness of the proposed copula method to more general situations, such as non-linear marker trajectories and complex dependencies, we also simulate under alternative joint model generating mechanisms. We generate the longitudinal marker measurements using the following linear mixed effects model with a non-linear function of time,
 |
where 
is the
B-spline basis for a cubic spline with boundary knots at 0 and 10 years and two internal
knots at 
3 and 6 years,
and 
. For the
joint models, when generating the data, we consider two types of association structure
between the survival and marker processes: (i) where the functional form depends on only
the marker value as given in (3.5)
or (ii) depends both on the marker value and the slope at 
,
 |
The parameters used in the various submodels to generate the data are given in Section E
of the supplementary material available at Biostatistics online, and the
simulation scenarios are described in Table E.1 of the supplementary material available at
Biostatistics online. We generate right-censoring from a Uniform(0,10)
distribution. Under the various scenarios considered, we vary the value of
to achieve censoring rates of
45–60%. We consider landmark times 
and present results for
predicting failure within a window of 
years beyond the
prediction time.
3.2.1. Models used for dynamic prediction
The joint, landmark, and copula models fit to the marker data are shown in Table 1. We consider a shared random effects model for the joint models, where (JM1) has a linear marker submodel and survival submodel with the marker value, (JM2) has a cubic spline marker submodel and survival submodel with the marker value, and (JM3) has a cubic spline marker submodel and survival model with both the marker value and slope. We fit (JM1) in Scenarios 1–4, (JM2) in Scenario 5, and (JM3) in Scenario 6. The landmark models considered are the super model (LM1) and the extended super model (LM2) (van Houwelingen, 2007). For additional flexibility, we also include an interaction between the marker value and the baseline covariates in the (LMInt*) models. For (LM1) and (LMInt1), we create a landmark data set for each prediction time of interest with administrative censoring at the prediction horizon, and stack them to form a super data set to which we fit the models. In the irregular inspection time scenarios, for (LM1) and (LMInt1) the marker value at each landmark time was imputed using last observation carried forward (LOCF). For (LM2) and (LMInt2), we used a longitudinal data set, which does not require imputation of the marker values for estimation. Details for the estimation and prediction of the joint and landmark models are given in Section D of the supplementary material available at Biostatistics online.
Table 1.
Summary of models fit in the simulation study
| Class | Model | Label |
|---|---|---|
| Joint Models |
|
(JM1) |
|
||
|
||
|
(JM2) | |
|
||
|
||
|
||
|
(JM3) | |
|
||
|
||
|
||
| Landmark Models |
|
(LM1) |
|
(LMInt1) | |
|
(LM2) | |
|
(LMInt2) | |
| Copula Models |
: Gaussian copula |
|
|
||
|
||
|
||
;  modeled
nonparametrically |
(CC1) | |
;  modeled as Weibull
hazard |
(CW1) | |
: Gaussian copula |
||
|
||
|
||
|
||
;  modeled
nonparametrically |
(CC2) | |
;  modeled as Weibull
hazard |
(CW2) | |
: Student’s t (df=4) |
||
|
||
|
||
|
||
;  modeled
nonparametrically |
(CC3) | |
;  modeled as Weibull
hazard |
(CW3) |
, 
is a B-spline basis for a
cubic spline with boundary knots at 0 and 10 years.
To identify the modeling structure for the copula components, we examine diagnostic
plots and test goodness-of-fit. We fit a population-averaged model to the longitudinal
data, and from the loess curve plotted to the marker trajectories, we identify that a
B-spline for landmark time with an interaction with the baseline covariate is the
best-fitting function for the marginal mean. Also, we allow the variance of the
population-averaged model to increase with time. We use this structure for the marginal
marker model in all of the copula models considered. We model the failure time data
parametrically (W: Weibull) or semiparametrically (C: Cox) and include the effect of the
baseline covariate 
. In (C*1), the association component is
a function of time, the baseline covariate, and their interaction. In models (C*2), the
association component is described more flexibly using a B-spline for landmark time.
Finally, (C*3) has the same marginal model components and association as (C*1), which
are instead joined using a Student’s t copula with 4 degrees of freedom to identify
whether the heavier tails of this copula provide a better fit to the data.
3.3. Results
We present the simulation results in Section E, Tables E.2–E.19, and Figures E.2–E.8 of
the supplementary material available at Biostatistics online. In Figure 1, we compare the performance of all three methods
under Scenario 1a (
,
, inspection rate = 0.5). The
copula model with semiparametric hazard (CC1) performs slightly better than the parametric
version (CW1) at earlier landmark times, with lower 
and RMSPE but
similar 
AUC. The copula models outperform the
landmark model (LM2) across all metric and have better performance than the (LM1) model at
earlier prediction times where the data are less sparse. The copula models perform
similarly to the joint model (JM1) at earlier landmark times; however, the joint model
performs better than the approximate methods at later time points.
Fig. 1.
Simulation estimates for Scenario 1a (
,
, inspection rate 0.5) for
AUC (top-left) and
(top-right), and RMSPE for
(bottom-left) and
(bottom-right) for predicted
probability 
from
copula models (CC1), (CW2), joint model (JM1), and landmark models (LM1), (LM2). Lower
values indicate better predictive performance.
We compare the performance of the different semiparametric copula models (Figure E.2 of the supplementary material available at Biostatistics online), noting that a comparison of the parametric copula models exhibits similar relationships. The copula models considered have the same AUC. In comparing (CC1) and (CC2), we find that changing the association structure results in similar performance. This suggests that choosing a flexible form for the association function is sufficient as long as well-fitting models are chosen for the marginal marker and failure time distributions. The (CC3) models that use the Student’s t copula have similar performance to the (CC1) and (CC2) model, with slightly better performance in scenarios with more frequent measurement times.
As the inspection rate increases (Scenarios *a vs. *b; Figure E.3 of the supplementary
material available at Biostatistics online), the RMSPE tended to decrease
for all the models, with the copula models still performing better than (LM2) and
(LMInt2), and marginally better or similar to (LM1) and (LMInt1) at earlier landmark
times. With a fixed inspection time (Scenarios *c; Figure E.4 of the supplementary
material available at Biostatistics online), the copula and landmark
models have similar AUC. Model (LM1) has lower RMSPE than the copula models at later
inspection times; however, it has comparable 
AUC and
. The parametric copula model
(CW1) has worsening performance at the later landmark times. The copula models outperform
(LM2) and (LMInt2), which have a smaller improvement in RMSPE and
compared to the other models for
all landmark times. Under the MAR assumption (Scenarios *d; Figure E.5 of the
supplementary material available at Biostatistics online), the RMSPE
increases for all the models; however, the relative performance between the models remains
the same.
As the association between the marker process and the hazard (
) decreases
(Scenario 1 vs. 2, and 3 vs. 4; Figure E.6 of the supplementary material available at
Biostatistics online), the semiparametric copula models have better
predictive performance than the landmarking models. With increasing measurement error
(
) (Scenarios 1 vs. 3, and 2
vs. 4; Figure E.7 of the supplementary material available at
Biostatistics online), the RMSPE for all the models increases. The
semiparametric copula models have similar RMSPE and BS as the (LM*1) models but higher
AUC, and the copula models perform better across all metrics than the (LM*2) models.
We evaluate the robustness of the copula method to capture dependence in more general situations. In Scenario 5, we generate data under a non-linear marker trajectory. The performance of the joint model (JM2) is superior to the other methods in terms of higher AUC, but the RMSPE and BS is similar to the copula and (LM*1) models at earlier landmark times (Figure 2). The (LM*2) models have poor performance across all of the metrics compared to the other models. The Student’s t copula models (C*3) have worse performance than the Gaussian copula models (C*1) and (C*2), which perform similarly. In Scenario 6, we simulate under a non-linear marker trajectory and a complex dependency, by including both the value of the marker and its slope in the dependence between the longitudinal and survival submodels, and find similar relative relationships in model performance (Figure E.8). Under both Scenario 5 and 6 (Tables E.18, E.19 of the supplementary material available at Biostatistics online), the joint models from which the data were generated, (JM2) and (JM3), respectively, had better performance than the landmarking or copula models, indicating the need to incorporate additional flexibility into these approaches when dealing with complex dependencies.
Fig. 2.
Simulation estimates for Scenario 5 (non-linear marker trajectory, inspection rate 1)
for 
AUC (top-left) and
(top-right), and RMSPE for
(bottom-left) and
(bottom-right) for predicted
probability 
from
copula models (CC1), (CW1), joint model (JM2), and landmark models (LM1), (LM2). Lower
values indicate better predictive performance.
In general, the copula method has good performance across all of the metrics considered. It has consistently better performance than the landmark model with baseline hazard as a function of landmark time, and outperforms the landmark model with stratified hazards when there are irregular measurement times. Under a linear marker trajectory and simple dependency, at earlier prediction times, the copula model performs similarly to the joint model from which the data is generated. It also appears to be robust to the choice of association function if the marginal models are well chosen, and is able to maintain good prediction with varying levels of measurement error and association between the marker and survival process.
4. Application to Aortic Heart Valve Study
To demonstrate the ability of the copula method to produce dynamic predictions, we use data from an observational study that followed 248 patients who received an aortic valve replacement with the aim of comparing the efficacy of two artificial heart valves: homograft or stentless (Lim and others, 2008;Philipson and others, 2017). Longitudinal measurements of the left ventricular mass index (LVMI) were collected after surgery (baseline time), with an average of 3.7 and a maximum of 10 measurements per patient. Long-term buildup of left ventricular muscle mass can result in a fatal heart attack, thus there is interest in using a patient’s changing LVMI to predict their future risk of death in a prediction window of 3 years. The baseline covariate information used in the models considered were: type of implanted aortic prosthesis (homograft: 53%, stentless: 47%); age (median: 68; interquartile range: 59–75); and gender (male: 71%, female: 29%).
Figure F.1a of the supplementary material available at Biostatistics
online depicts the survival curves by stent type and demonstrates a significantly higher
survival probability for those who received the stentless valve compared to those who
received the homograft valve. We examine the fit of a Cox model to the failure time data and
find no violation of the proportional hazards assumption for any of the baseline covariates.
Figure F.1b of the supplementary material available at Biostatistics online
depicts the longitudinal log(LVMI) observations per patient and from the loess curves we see
that there is a decrease in mean log(LVMI) in the first year, after which it appears to be
increasing with time. Thus, we consider a non-linear marker trajectory, with possible
interactions between the covariates and time. Selecting the best-fitting model using
backwards selection with Akaike information criterion (AIC), we identify the
population-averaged model for 
as the main effects model with a
quadratic spline effect for landmark time and constant variance 
. We
considered more flexible forms for the association function including interactions and
splines, but found that the results are similar to simpler forms. Thus, we fit the following
Gaussian copula model: 
,
,
, where
is a vector of the baseline covariates
age, valve type, and gender, 
is modeled nonparametrically, and
is a B-spline basis for a
quadratic spline with boundary knots at 0 and 11 years and an internal knot at
year. The parameter estimates and
bootstrapped standard errors for the copula model are given in Table F.1 of the
supplementary material available at Biostatistics online. From the marginal
model for 
, we find that age has a significant positive
effect on time to death, while the effect of gender and stent type were not significant.
From the marginal model for 
, females have a lower average log(LVMI)
than males, and those with the homograft valve have a higher average log(LVMI) than those
with the stent valve. There was a significant quadratic spline effect for time on average
log(LVMI). The association between the risk of death and increased log(LVMI) is negative
indicating decreased time to death and did not change with landmark time or baseline
covariates. In Figure 3, we depict the predicted
survival probabilities and the 95% bootstrap prediction intervals from the copula model for
two patients in the data set as their marker value changes. Individual A is a younger male,
who received the stentless valve, and has lower log(LVMI) that is increasing over time.
Individual B is older, received the homograft valve, and has a steady, but higher log(LVMI)
with a sudden increase at the last measurement. Since Individual A is a relatively low-risk
patient, we see that their predicted probability of death is low, with risk of death
increasing as their log(LVMI) increases. Individual B is at higher risk due to their
increased age at baseline, and their risk of death in the next 3 years increases greatly
after their log(LVMI) spikes suddenly at 4 years, and they eventually die at 5.4 years from
baseline. These predicted probability plots can be used by clinicians to monitor a patient’s
prognosis following valve replacement to identify if the patient’s changing log(LVMI) is
putting them at high risk for future death and further interventions must be
implemented.
Fig. 3.
Predicted survival probabilities and 95% bootstrap prediction interval for risk of death within 3 years from the copula model for two patients in the heart valve data set. Individual A (top) is male, 59 years old at baseline, received the stentless valve, and does not die before the end of the study. Individual B (bottom) is male, 78 years old at baseline, received the homograft valve and died at 5.4 years after baseline. The dashed line indicates time at which the most recent marker measurement (black dot) was recorded. The dotted line indicates time of death.
We also apply the joint modeling and landmarking approaches to the data for comparison. We fit a shared random effects model with the same quadratic spline for time in the longitudinal submodel as in the copula model and a random intercept and slope. More flexible spline functions were considered for the random effects specification, but resulted in convergence issues likely due to the limited sample size. To allow for irregular prediction times, we consider the landmark extended super model. The details of these models are given in Section F of the supplementary material available at Biostatistics online. We compare performance by computing AUC and BS using 5-fold cross-validation, repeated and averaged over 500 iterations. From Table 2, we see that the models perform similarly. The copula and joint model have similar AUC across landmark times and the joint model has slightly lower BS. Landmarking performs slightly worse in terms of AUC and BS than the other two methods. In Figures F.2 and F.3 of the supplementary material available at Biostatistics online, we compare the bootstrap prediction intervals from the joint and landmark models for the individuals presented in Figure 3, and find that the joint model has the narrowest intervals while the landmark model has slightly wider intervals than the copula model.
Table 2.
Cross-validated AUC and BS for the considered models in the aortic heart valve analysis at landmark times 0.5–3.5 years with a prediction horizon of 3 years
| 0.5 | 1.5 | 2.5 | 3.5 | ||
|---|---|---|---|---|---|
| AUC | Joint Model | 0.639 | 0.706 | 0.769 | 0.865 |
| Landmark | 0.635 | 0.707 | 0.761 | 0.860 | |
| Copula | 0.637 | 0.711 | 0.769 | 0.870 | |
| BS | Joint Model | 0.092 | 0.111 | 0.116 | 0.110 |
| Landmark | 0.097 | 0.114 | 0.122 | 0.116 | |
| Copula | 0.095 | 0.113 | 0.121 | 0.113 |
5. Discussion
Dynamic models that incorporate the effect of time-dependent covariates on survival are essential for making personalized clinical decisions about an individual’s care. While there are two popular statistical methods for dynamic prediction, landmarking and joint modeling, they both have limitations that we address by presenting an alternative approximate method that has useful advantages and good predictive performance.
We propose the use of a copula-based approach for incorporating longitudinally collected marker information in predicting an individual’s future survival. First, we specify from a well-established class of models the marginal distributions of the marker (e.g., linear regression, generalized linear model) and the failure time (e.g., Cox, cure model). This allows us to apply standard goodness-of-fit tests and variable selection techniques to identify the best-fitting marginal models. Second, we define the joint distribution of the survival time and marker conditional on being alive at a particular prediction time. With this formulation, we can easily derive the dynamic prediction of interest. We present our modeling framework for a continuous marker process and demonstrate its performance using a simulation study and a data application.
There are several advantages of our approach over the existing joint model and landmarking methods. In comparison to landmarking, the copula model does not require the creation of a landmark data set, instead only using marker information available at measurement times. This allows us to avoid prespecifying landmark times and imputing unobserved marker values at these times, which can introduce bias. We also do not need to fix a time horizon for prediction and can obtain predictions for a new patient at any continuous time point beyond baseline. Since the marginal model for time-to-event is internally consistent for all landmark times, we maintain a greater level of consistency in our predictions than landmarking. As in joint modeling, we specify a model for the marker; however, since we are modeling the population-averaged trajectory, rather than allowing for individual-specific random effects, we are able to specify a simpler model than a shared random effects or frailty model that can require complex and computationally intensive estimation. In principle, it is easier to check goodness-of-fit for marginal models; thus, we are able to minimize bias at this stage. Table F.3 of the supplementary material available at Biostatistics online presents a comparison of the different methods, listing some pros and cons of each approach.
A limitation of our proposed method is that it relies heavily on the availability of data at prediction times of interest for it to properly model the joint distribution between the marker and failure time. As well, with the different models for the marginals and the association, there are several parameters to be estimated. The two-stage approach results in large standard errors for the association parameters, due to the estimation variability of the marginal model parameters. In the simulation study, we found that the copula models’ performance was similar when comparing flexible association functions. Thus, rather than performing variable selection for the association parameters, we can consider choosing a flexible-enough association function or using cross-validation techniques to select an appropriate association structure. Finally, the use of a Gaussian copula is applicable only when linking two continuous outcomes, a survival time and a continuous marker value. Thus, to describe the association between a binary marker (e.g., the occurrence of an intermediate event) and a time-to-event outcome, our model specification must be adjusted using methods such as those introduced by Song and others (2009) and de Leon and Wu (2011). The predictive performance can be compared to Emura and others (2018) that uses a copula to incorporate time to the intermediate event rather than just the indication of experiencing the event.
We compare our copula models to standard landmark models and supermodels. In both
approaches, we impute the marker value at landmark times using LOCF (of the marker value or
the quantile). We can consider alternative imputation methods, such as fitting a mixed model
for 
as in Ferrer
and others (2019) that can increase the computational burden but
improve marker prediction. In the simulation study, we explore the performance of the copula
model under more complex longitudinal trajectories and dependence structures. As in Ferrer and others (2019), we find that
the performance of misspecified models that did not account for these additional
complexities were poor compared to the true joint models. Thus, extensions under the copula
framework are required to improve predictive performance under more complicated settings.
For example, to achieve a more accurate prediction, it might be of interest to incorporate
an individual’s longitudinal marker history 
, where
represents the individual’s
history of longitudinal marker values up to time 
. Joint modeling handles
this naturally. Rizopoulos and others
(2017) and Ferrer and others
(2019) extended the landmark model specification to consider parameterizations that
incorporate the marker relationship with survival in various ways, such as the slope of the
marker trajectory, 
. The copula model can also
potentially be extended to directly incorporate a summary of the marker history up to time
, by modeling a different aspect of the
marginal marker distribution instead of the marginal marker mean. For example, we could have
instead considered a marginal model for the change in the marker value from baseline,
. The ability to model
the marginals and their distributions separately allows for more complicated, better-fitting
models to be considered for the marginals, but still maintains simple estimation using the
two-stage method.
Using a copula framework provides the potential for several extensions to more complicated data structures. We can consider a multivariate Gaussian copula to accommodate multiple longitudinal markers or marker summaries. Thus, rather than specifying a full joint model for the different marker processes, it is easier to consider modeling their marginal behavior and using a flexible form for the copula association function to model their joint distributions. This model structure can also help identify the size and direction of the correlation between the various longitudinal markers. Such an approach can greatly increase the dimension of the parameter space, so care should be taken to choose parsimonious models for the marginal components and simpler association functions for the resulting correlation matrix.
While joint modeling and landmarking are popular in current literature, our copula-based approach provides an alternative method for dynamic prediction that has good predictive performance and easy estimation. By choosing more flexible and complicated models for the marginals, we could potentially further decrease the bias introduced by fitting a misspecified model. Future work will focus on extending the copula framework for dynamic prediction to address more complex data forms and applications.
6. Software
The R code is available on Github at http://github.com/ksuresh17/copula-dyn-pred.
Acknowledgments
Conflict of Interest: None declared.
Supplementary material
Supplementary material is available online at http://biostatistics.oxfordjournals.org.
Funding
National Institutes of Health (CA129102), in part.
References
- Andersen, P. K. and Gill, R. D. (1982). Cox’s regression model for counting processes: a large sample study. The Annals of Statistics 10, 1100–1120. [Google Scholar]
- Blanche, P., Proust-Lima, C., Loubère, L., Berr, C., Dartigues, J.-F. and Jacqmin-Gadda, H. (2015). Quantifying and comparing dynamic predictive accuracy of joint models for longitudinal marker and time-to-event in presence of censoring and competing risks. Biometrics 71, 102–113. [DOI] [PubMed] [Google Scholar]
- de Leon, A. R. and Wu, B. (2011). Copula-based regression models for a bivariate mixed discrete and continuous outcome. Statistics in Medicine 30, 175–185. [DOI] [PubMed] [Google Scholar]
- Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. New York: Chapman and Hall. [Google Scholar]
- Emura, T., Nakatochi, M., Matsui, S., Michimae, H. and Rondeau, V. (2018). Personalized dynamic prediction of death according to tumour progression and high-dimensional genetic factors: meta-analysis with a joint model. Statistical Methods in Medical Research 27, 2842–2858. [DOI] [PubMed] [Google Scholar]
- Ferrer, L., Putter, H. and Proust-Lima, C. (2019). Individual dynamic predictions using landmarking and joint modelling: Validation of estimators and robustness assessment. Statistical Methods in Medical Research 28, 3649–3666. [DOI] [PubMed] [Google Scholar]
- Ganjali, M. and Baghfalaki, T. (2015). A copula approach to joint modeling of longitudinal measurements and survival times using Monte Carlo expectation-maximization with application to AIDS studies. Journal of Biopharmaceutical Statistics 25, 1077–1099. [DOI] [PubMed] [Google Scholar]
- Gong, Q. and Schaubel, D. E. (2013). Partly conditional estimation of the effect of a time-dependent factor in the presence of dependent censoring. Biometrics 69, 338–347. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Henderson, R., Diggle, P. and Dobson, A. (2000). Joint modelling of longitudinal measurements and event time data. Biostatistics 1, 465–480. [DOI] [PubMed] [Google Scholar]
- Jewell, N. P. and Nielsen, J. P. (1993). A framework for consistent prediction rules based on markers. Biometrika 80, 153–164. [Google Scholar]
- Joe, H. and Xu, J. J. (1996). The estimation method of inference functions for margins for multivariate models. Technical Report, University of British Columbia. [Google Scholar]
- Lim, E., Ali, A., Theodorou, P., Sousa, I., Ashrafian, H., Chamageorgakis, T., Duncan, A., Henein, M., Diggle, P. and Pepper, J. (2008). Longitudinal study of the profile and predictors of left ventricular mass regression after stentless aortic valve replacement. The Annals of Thoracic Surgery 85, 2026–2029. [DOI] [PubMed] [Google Scholar]
- Nelsen, R. B. (1999). An Introduction to Copulas. New York: Springer Series in Statistics. [Google Scholar]
- Philipson, P., Sousa, I., Diggle, P. J., Williamson, P., Kolamunnage-Dona, R., Henderson, R. and Hickey, G. L. (2017). joineR: Joint Modelling of Repeated Measurements and Time-to-Event Data. Available from: http://cran.r-project.org.
- Prenen, L., Braekers, R. and Duchateau, L. (2017). Extending the Archimedean copula methodology to model multivariate survival data grouped in clusters of variable size. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 79, 483–505. [Google Scholar]
- Rizopoulos, D. (2011). Dynamic predictions and prospective accuracy in joint models for longitudinal and time-to-event data. Biometrics 67, 819–829. [DOI] [PubMed] [Google Scholar]
- Rizopoulos, D.s, Molenberghs, G. and Lesaffre, E.M.E.H. (2017). Dynamic predictions with time-dependent covariates in survival analysis using joint modeling and landmarking. Biometrical Journal 59, 1261–1276. [DOI] [PubMed] [Google Scholar]
- Rizopoulos, D., Verbeke, G., Lesaffre, E. and Vanrenterghem, Y. (2008a). A two-part joint model for the analysis of survival and longitudinal binary data with excess zeros. Biometrics 64, 611–619. [DOI] [PubMed] [Google Scholar]
- Rizopoulos, D., Verbeke, G. and Molenberghs, G. (2008b). Shared parameter models under random effects misspecification. Biometrika 95, 63–74. [Google Scholar]
- Song, P. X.-K. (2007). Correlated Data Analysis: Modeling, Analytics, and Applications. New York: Springer Series in Statistics. [Google Scholar]
- Song, P. X.-K., Li, M. and Yuan, Y. (2009). Joint regression analysis of correlated data using Gaussian copulas. Biometrics 65, 60–68. [DOI] [PubMed] [Google Scholar]
- Spiekerman, C. F. and Lin, D. Y. (1998). Marginal regression models for multivariate failure time data. Journal of the American Statistical Association 93, 1164–1175. [Google Scholar]
- Suresh, K., Taylor, J. M. G., Spratt, D. E., Daignault, S. and Tsodikov, A. (2017). Comparison of joint modeling and landmarking for dynamic prediction under an illness-death model. Biometrical Journal 59, 1277–1300. [DOI] [PMC free article] [PubMed] [Google Scholar]
- Taylor, J. M. G., Park, Y., Ankerst, D. P., Proust-Lima, C., Williams, S., Kestin, L., Bae, K., Pickles, T. and Sandler, H. (2013). Real-time individual predictions of prostate cancer recurrence using joint models. Biometrics 69, 206–213. [DOI] [PMC free article] [PubMed] [Google Scholar]
- van Houwelingen, H. C. (2007). Dynamic prediction by landmarking in event history analysis. Scandinavian Journal of Statistics 34, 70–85. [Google Scholar]
- Wang, Y. and Taylor, J. M. G. (2001). Jointly modeling longitudinal and event time data with application to acquired immunodeficiency syndrome. Journal of the American Statistical Association 96, 895–905. [Google Scholar]
- Wulfsohn, M. S. and Tsiatis, A. A. (1997). A joint model for survival and longitudinal data measured with error. Biometrics 33, 330–339. [PubMed] [Google Scholar]
- Zeger, S. L. and Liang, K.-Y. (1986). Longitudinal data analysis for discrete and continuous outcomes. Biometrics 42, 121–130. [PubMed] [Google Scholar]
- Zheng, Y. and Heagerty, P. J. (2005). Partly conditional survival models for longitudinal data. Biometrics 61, 379–391. [DOI] [PubMed] [Google Scholar]