# aucJM: Time-Dependent ROCs and AUCs for Joint Models In JMbayes: Joint Modeling of Longitudinal and Time-to-Event Data under a Bayesian Approach

## Description

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

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 aucJM(object, newdata, Tstart, ...) ## S3 method for class 'JMbayes' aucJM(object, newdata, Tstart, Thoriz = NULL, Dt = NULL, idVar = "id", simulate = FALSE, M = 100, ...) ## S3 method for class 'mvJMbayes' aucJM(object, newdata, Tstart, Thoriz = NULL, Dt = NULL, idVar = "id", M = 100, ...) rocJM(object, newdata, Tstart, ...) ## S3 method for class 'JMbayes' rocJM(object, newdata, Tstart, Thoriz = NULL, Dt = NULL, idVar = "id", simulate = FALSE, M = 100, ...) ## S3 method for class 'mvJMbayes' rocJM(object, newdata, Tstart, Thoriz = NULL, Dt = NULL, idVar = "id", M = 100, ...) 

## Arguments

 object an object inheriting from class JMbayes or mvJMbayes. 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 jointModelBayes. 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 must 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.

## 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, Δ t) = \mbox{Pr} \bigl [ π_i(t + Δ t \mid t) < π_j(t + Δ t \mid t) \mid \{ T_i^* \in (t, t + Δ t] \} \cap \{ T_j^* > t + Δ t \} \bigr ],

with i and j denote a randomly selected pair of subjects, and π_i(t + Δ t \mid t) and π_j(t + Δ t \mid t) denote the conditional survival probabilities calculated by survfitJM for these two subjects, for different time windows Δ t specified by argument Dt using the longitudinal information recorded up to time t = Tstart.

The estimate of \mbox{AUC}(t, Δ 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. For the subjects who due to censoring we do not know if they are comparable, they contribute in the AUC with the probability that they would have been comparable.

## 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.

## Author(s)

Dimitris Rizopoulos [email protected]

## 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. (2016). The R package JMbayes for fitting joint models for longitudinal and time-to-event data using MCMC. Journal of Statistical Software 72(7), 1–45. doi:10.18637/jss.v072.i07.

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.

survfitJM, dynCJM, jointModelBayes
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ## Not run: # 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 <- jointModelBayes(lmeFit, survFit, timeVar = "year") # AUC using data up to year 5 with horizon at year 8 aucJM(jointFit, pbc2, Tstart = 5, Thoriz = 8) plot(rocJM(jointFit, pbc2, Tstart = 5, Thoriz = 8)) ## End(Not run)