jointModel: Joint Models for Longitudinal and Survival Data

Description

This function fits shared parameter models for the joint modelling of normal longitudinal responses and time-to-event data under a maximum likelihood approach. Various options for the survival model are available.

Usage

jointModel(lmeObject, survObject, timeVar, 
  parameterization = c("value", "slope", "both"),
  method = c("weibull-PH-GH", "weibull-PH-aGH", "weibull-AFT-GH", 
    "weibull-AFT-aGH", "piecewise-PH-GH", "piecewise-PH-aGH", 
    "Cox-PH-GH", "Cox-PH-aGH", "spline-PH-GH", "spline-PH-aGH", 
    "ch-Laplace"),
  interFact = NULL, derivForm = NULL, lag = 0, scaleWB = NULL,
  init = NULL, control = list(), ...)

Arguments

lmeObject

an object inheriting from class lme (see also Note).

survObject

an object inheriting from class coxph or class survreg. In the call to coxph() or survreg(), you need to specify the argument x = TRUE such that the design matrix is contained in

timeVar

a character string indicating the time variable in the linear mixed effects model.

parameterization

a character string indicating the type of parameterization. See Details

method

a character string specifying the type of joint model to fit. See Details.

interFact

a list with components value a formula for the interaction terms corresponding to the value parameterization, slope a formula for the interaction terms corresponding to the slope parameterizati

derivForm

a list with components fixed a formula representing the derivative of the fixed-effects part of the liner mixed model with respect to time, indFixed a numeric vector indicating which fixed effects of lmeObject

lag

a numeric scalar denoting a lag effect in the time-dependent covariate represented by the mixed model; default is 0.

scaleWB

a numeric scalar denoting a fixed value for the scale parameter of the Weibull hazard; used only when method = "weibull-AFT-GH" or method = "weibull-PH-GH". The default NULL means that the scale parameter

init

a list of user-specified initial values. The initial values list must have the following components: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[obje

control

a list of control values with components: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],

...

options passed to the control argument.

Value

See jointModelObject for the components of the fit.

Details

Function jointModel fits joint models for longitudinal and survival data (more detailed information about the formulation of these models can be found in Rizopoulos (2010)). For the longitudinal responses the linear mixed effects model represented by the lmeObject is assumed. For the survival times let $w_i$ denote the vector of baseline covariates in survObject, with associated parameter vector $\gamma$, $m_i(t)$ the value of the longitudinal outcome at time point $t$ as approximated by the linear mixed model (i.e., $m_i(t)$ equals the fixed-effects part + random-effects part of the linear mixed effects model for sample unit $i$), $\alpha$ the association parameter for $m_i(t)$, $m_i'(t)$ the derivative of $m_i(t)$ with respect to $t$, and $\alpha_d$ the association parameter for $m_i'(t)$. Then, for method = "weibull-AFT-GH" a time-dependent Weibull model under the accelerated failure time formulation is assumed. For method = "weibull-PH-GH" a time-dependent relative risk model is postulated with a Weibull baseline risk function. For method = "piecewise-PH-GH" a time-dependent relative risk model is postulated with a piecewise constant baseline risk function. For method = "spline-PH-GH" a time-dependent relative risk model is assumed in which the log baseline risk function is approximated using B-splines. For method = "ch-Laplace" an additive model on the log cumulative hazard scale is assumed (see Rizopoulos et al., 2009 for more info). Finally, for method = "Cox-PH-GH" a time-dependent relative risk model is assumed where the baseline risk function is left unspecified (Wulfsohn and Tsiatis, 1997). For all these options the linear predictor for the survival submodel is written as $$\eta = \gamma^\top w_i + \alpha m_i{min(t-k, 0)},$$ when parameterization = "value", $$\eta = \gamma^\top w_i + \alpha_s m_i'{min(t-k, 0)},$$ when parameterization = "slope", and $$\eta = \gamma^\top w_i + \alpha m_i{min(t-k, 0)} + \alpha_s m_i'{min(t-k, 0)},$$ when parameterization = "both", where in all the above the value of $k$ is specified by the lag argument and $m_i'(t) = d m_i(t) / dt$. If interFact is specified, then $m_i{min(t-k, 0)}$ and/or $m_i'{min(t-k, 0)}$ are multiplied with the design matrices derived from the formulas supplied as the first two arguments of interFact, respectively. In this case $\alpha$ and/or $\alpha_s$ become vectors of association parameters. For method = "spline-PH-GH" it is also allowed to include stratification factors. These should be included in the specification of the survObject using function strata(). Note that in this case survObject must only be a 'coxph' object. For all survival models except for the time-dependent proportional hazards model, the optimization algorithm starts with EM iterations, and if convergence is not achieved, it switches to quasi-Newton iterations (i.e., BFGS in optim() or nlminb(), depending on the value of the optimizer control argument). For method = "Cox-PH-GH" only the EM algorithm is used. During the EM iterations, convergence is declared if either of the following two conditions is satisfied: (i) $L(\theta^{it}) - L(\theta^{it - 1}) < tol_3 { | L(\theta^{it - 1}) | + tol_3 }$, or (ii) $\max { | \theta^{it} - \theta^{it - 1} | / ( | \theta^{it - 1} | + tol_1) } < tol_2$, where $\theta^{it}$ and $\theta^{it - 1}$ is the vector of parameter values at the current and previous iterations, respectively, and $L(.)$ is the log-likelihood function. The values for $tol_1$, $tol_2$ and $tol_3$ are specified via the control argument. During the quasi-Newton iterations, the default convergence criteria of either optim() or nlminb() are used. The required integrals are approximated using the standard Gauss-Hermite quadrature rule when the chosen option for the method argument contains the string "GH", and the (pseudo) adaptive Gauss-Hermite rule when the chosen option for the method argument contains the string "aGH". For method = "ch-Laplace" the fully exponential Laplace approximation described in Rizopoulos et al. (2009) is used. The (pseudo) adaptive Gauss-Hermite and the Laplace approximation are particularly useful when high-dimensional random effects vectors are considered (e.g., when modelling nonlinear subject-specific trajectories with splines or high-order polynomials).

References

Henderson, R., Diggle, P. and Dobson, A. (2000) Joint modelling of longitudinal measurements and event time data. Biostatistics 1, 465--480. Hsieh, F., Tseng, Y.-K. and Wang, J.-L. (2006) Joint modeling of survival and longitudinal data: Likelihood approach revisited. Biometrics 62, 1037--1043. Rizopoulos, D. (2011). Pseudo-adaptive Gaussian quadrature integration in joint models for longitudinal and time-to-event data. In progress. Rizopoulos, D. (2010) JM: An R package for the joint modelling of longitudinal and time-to-event data. Journal of Statistical Software 35 (9), 1--33. http://www.jstatsoft.org/v35/i09/ Rizopoulos, D. (2010). Dynamic prediction and prospective accuracy in joint models for longitudinal and time-to-event data. Biometrics, accepted. Rizopoulos, D., Verbeke, G. and Lesaffre, E. (2009) Fully exponential Laplace approximations for the joint modelling of survival and longitudinal data. Journal of the Royal Statistical Society, Series B 71, 637--654. Rizopoulos, D., Verbeke, G. and Molenberghs, G. (2010) Multiple-imputation-based residuals and diagnostic plots for joint models of longitudinal and survival outcomes. Biometrics 66, 20--29. Tsiatis, A. and Davidian, M. (2004) Joint modeling of longitudinal and time-to-event data: an overview. Statistica Sinica 14, 809--834. Wulfsohn, M. and Tsiatis, A. (1997) A joint model for survival and longitudinal data measured with error. Biometrics 53, 330--339.

Examples

Run this code

# linear mixed model fit (random intercepts)
fitLME <- lme(log(serBilir) ~ drug * year, random = ~ 1 | id, data = pbc2)
# survival regression fit
fitSURV <- survreg(Surv(years, status2) ~ drug, data = pbc2.id, x = TRUE)
# joint model fit, under the (default) Weibull model
fitJOINT <- jointModel(fitLME, fitSURV, timeVar = "year")
fitJOINT
summary(fitJOINT)

# linear mixed model fit (random intercepts + random slopes)
fitLME <- lme(log(serBilir) ~ drug * year, random = ~ year | id, data = pbc2)
# survival regression fit
fitSURV <- survreg(Surv(years, status2) ~ drug, data = pbc2.id, x = TRUE)
# joint model fit, under the (default) Weibull model
fitJOINT <- jointModel(fitLME, fitSURV, timeVar = "year")
fitJOINT
summary(fitJOINT)

# we also include an interaction term of log(serBilir) with drug
fitJOINT <- jointModel(fitLME, fitSURV, timeVar = "year",
    interFact = list(value = ~ drug, data = pbc2.id))
fitJOINT
summary(fitJOINT)


# a joint model in which the risk for and event depends both on the true value of
# marker and the true value of the slope of the longitudinal trajectory
lmeFit <- lme(sqrt(CD4) ~ obstime * drug, random = ~ obstime | patient, data = aids)
coxFit <- coxph(Surv(Time, death) ~ drug, data = aids.id, x = TRUE)

# to fit this model we need to specify the 'derivForm' argument, which is a list
# with first component the derivative of the fixed-effects formula of 'lmeFit' with
# respect to 'obstime', second component the indicator of which fixed-effects 
# coefficients correspond to the previous defined formula, third component the 
# derivative of the random-effects formula of 'lmeFit' with respect to 'obstime', 
# and fourth component the indicator of which random-effects correspond to the 
# previous defined formula
dForm <- list(fixed = ~ 1 + drug, indFixed = c(2, 4), random = ~ 1, indRandom = 2)
jointModel(lmeFit, coxFit, timeVar = timeVar, method = "spline-PH-GH",
  parameterization = "both", derivForm = dForm)

# linear mixed model fit
fitLME <- lme(sqrt(CD4) ~ obstime * drug - drug, 
    random = ~ 1 | patient, data = aids)
# cox model fit
fitCOX <- coxph(Surv(Time, death) ~ drug, data = aids.id, x = TRUE)
# joint model fit with a spline approximate baseline hazard
fitJOINT <- jointModel(fitLME, fitCOX, 
    timeVar = "obstime", method = "spline-PH-GH")
fitJOINT
summary(fitJOINT)

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