JM (version 1.5-2)

aucJM: Time-Dependent AUCs for Joint Models

Description

Using the available longitudinal information up to a starting time point, this function computes an estimate of the prediction error of survival at a horizon time point based on joint models.

Usage

aucJM(object, newdata, Tstart, ...)
# S3 method for jointModel
aucJM(object, newdata, Tstart, Thoriz = NULL, 
    Dt = NULL, idVar = "id", simulate = FALSE, M = 100, ...)

Value

A list of class aucJM with components:

auc

a numeric scalar denoting the estimated prediction error.

Tstart

a copy of the Tstart argument.

Thoriz

a copy of the Thoriz argument.

nr

a numeric scalar denoting the number of subjects at risk at time Tstart.

classObject

the class of object.

nameObject

the name of object.

Arguments

object

an object inheriting from class jointModel.

newdata

a data frame that contains the longitudinal and covariate information for the subjects for which prediction of survival probabilities is required. The names of the variables in this data frame must be the same as in the data frames that were used to fit the linear mixed effects model (using lme()) and the survival model (using coxph()) that were supplied as the two first argument of jointModel. In addition, this data frame should contain a variable that identifies the different subjects (see also argument idVar).

Tstart

numeric scalar denoting the time point up to which longitudinal information is to be used to derive predictions.

Thoriz

numeric scalar denoting the time point for which a prediction of the survival status is of interest; Thoriz mast be later than Tstart and either Dt or Thoriz must be specified. If Thoriz is NULL is set equal to Tstart + Dt.

Dt

numeric scalar denoting the length of the time interval of prediction; either Dt or Thoriz must be specified.

idVar

the name of the variable in newdata that identifies the different subjects.

simulate

logical; if TRUE, a Monte Carlo approach is used to estimate survival probabilities. If FALSE, a first order estimator is used instead. See survfitJM for mote details.

M

a numeric scalar denoting the number of Monte Carlo samples; see survfitJM for mote details.

...

additional arguments; currently none is used.

Author

Dimitris Rizopoulos

Details

Based on a fitted joint model (represented by object) and using the data supplied in argument newdata, this function computes the following estimate of the AUC: $$\mbox{AUC}(t, \Delta t) = \mbox{Pr} \bigl [ \pi_i(t + \Delta t \mid t) < \pi_j(t + \Delta t \mid t) \mid \{ T_i^* \in (t, t + \Delta t] \} \cap \{ T_j^* > t + \Delta t \} \bigr ],$$ with \(i\) and \(j\) denote a randomly selected pair of subjects, and \(\pi_i(t + \Delta t \mid t)\) and \(\pi_j(t + \Delta t \mid t)\) denote the conditional survival probabilities calculated by survfitJM for these two subjects, for different time windows \(\Delta t\) specified by argument Dt using the longitudinal information recorded up to time t = Tstart.

The estimate of \(\mbox{AUC}(t, \Delta t)\) provided by aucJM() is in the spirit of Harrell's \(c\)-index, that is for the comparable subjects (i.e., the ones whose observed event times can be ordered), we compare their dynamic survival probabilities calculated by survfitJM. As with Harrell's \(c\)-index, \(\mbox{AUC}(t, \Delta t)\) does not fully take into account censoring, and therefore is expected to mask the true discriminative capability of the joint model under heavy censoring.

References

Antolini, L., Boracchi, P., and Biganzoli, E. (2005). A time-dependent discrimination index for survival data. Statistics in Medicine 24, 3927--3944.

Harrell, F., Kerry, L. and Mark, D. (1996). Multivariable prognostic models: issues in developing models, evaluating assumptions and adequacy, and measuring and reducing errors. Statistics in Medicine 15, 361--387.

Heagerty, P. and Zheng, Y. (2005). Survival model predictive accuracy and ROC curves. Biometrics 61, 92--105.

Rizopoulos, D. (2012) Joint Models for Longitudinal and Time-to-Event Data: with Applications in R. Boca Raton: Chapman and Hall/CRC.

Rizopoulos, D. (2011). Dynamic predictions and prospective accuracy in joint models for longitudinal and time-to-event data. Biometrics 67, 819--829.

Rizopoulos, D., Murawska, M., Andrinopoulou, E.-R., Lesaffre, E. and Takkenberg, J. (2013). Dynamic predictions with time-dependent covariates in survival analysis: A comparison between joint modeling and landmarking. under preparation.

See Also

survfitJM, dynCJM, jointModel

Examples

if (FALSE) {
# we construct the composite event indicator (transplantation or death)
pbc2$status2 <- as.numeric(pbc2$status != "alive")
pbc2.id$status2 <- as.numeric(pbc2.id$status != "alive")

# we fit the joint model using splines for the subject-specific 
# longitudinal trajectories and a spline-approximated baseline
# risk function
lmeFit <- lme(log(serBilir) ~ ns(year, 3),
    random = list(id = pdDiag(form = ~ ns(year, 3))), data = pbc2)
survFit <- coxph(Surv(years, status2) ~ drug, data = pbc2.id, x = TRUE)
jointFit <- jointModel(lmeFit, survFit, timeVar = "year", 
    method = "piecewise-PH-aGH")

# AUC using data up to year 5 with horizon at year 8
aucJM(jointFit, pbc2, Tstart = 5, Thoriz = 8)
}