Help for package FastJM

Type:

Package

Title:

Semi-Parametric Joint Modeling of Longitudinal and Survival Data

Version:

1.6.0

Date:

2026-03-28

Maintainer:

Shanpeng Li <lishanpeng0913@ucla.edu>

Encoding:

UTF-8

Description:

Implements scalable joint models for large-scale competing risks time-to-event data with one or multiple longitudinal biomarkers using the efficient algorithms developed by Li et al. (2022) <doi:10.1155/2022/1362913> and <doi:10.48550/arXiv.2506.12741>. The time-to-event process is modeled using a cause-specific Cox proportional hazards model with time-fixed covariates, while longitudinal biomarkers are modeled using linear mixed-effects models. The association between the longitudinal and survival processes is captured through shared random effects. The package enables analysis of large-scale biomedical data to model biomarker trajectories, estimate their effects on event risks, and perform dynamic prediction of future events based on patients' longitudinal histories. Functions for simulating survival and longitudinal data for multiple biomarkers are included, along with built-in example datasets. The package also supports modeling a single biomarker with heterogeneous within-subject variability via functionality adapted from the 'JMH' package.

License:

GPL (≥ 3)

NeedsCompilation:

yes

Imports:

Rcpp (≥ 1.0.7), dplyr, nlme, caret, pec, future, future.apply, rlang (≥ 0.4.11)

LinkingTo:

Rcpp, RcppEigen

Depends:

R (≥ 3.5.0), survival, utils, MASS, statmod, magrittr, stats

RoxygenNote:

7.3.2

LazyData:

true

Packaged:

2026-03-28 08:33:43 UTC; shanpengli

VignetteBuilder:

knitr

Suggests:

testthat (≥ 3.0.0), spelling, knitr, rmarkdown

Language:

en-US

Config/testthat/edition:

Author:

Shanpeng Li [aut, cre], Emily Ouyang [ctb], Gang Li [ctb]

Repository:

CRAN

Date/Publication:

2026-03-28 16:00:02 UTC

Time-dependent AUC for joint models

Description

Time-dependent AUC for joint models

Usage

AUCJMMLSM(
  seed = 100,
  object,
  landmark.time = NULL,
  horizon.time = NULL,
  obs.time = NULL,
  method = c("Laplace", "GH"),
  quadpoint = NULL,
  maxiter = 1000,
  n.cv = 3,
  survinitial = TRUE,
  opt = "nlminb",
  initial.para = FALSE,
  LOCF = FALSE,
  LOCFcovariate = NULL,
  clongdata = NULL,
  metric = c("AUC", "Cindex"),
  ...
)

Arguments

seed

a numeric value of seed to be specified for cross validation.

object

object of class 'JMMLSM'.

landmark.time

a numeric value of time for which dynamic prediction starts..

horizon.time

a numeric vector of future times for which predicted probabilities are to be computed.

obs.time

a character string of specifying a longitudinal time variable.

method

estimation method for predicted probabilities. If Laplace, then the empirical empirical estimates of random effects is used. If GH, then the pseudo-adaptive Gauss-Hermite quadrature is used.

quadpoint

the number of pseudo-adaptive Gauss-Hermite quadrature points if method = "GH".

maxiter

the maximum number of iterations of the EM algorithm that the function will perform. Default is 10000.

n.cv

number of folds for cross validation. Default is 3.

survinitial

Fit a Cox model to obtain initial values of the parameter estimates. Default is TRUE.

opt

Optimization method to fit a linear mixed effects model, either nlminb (default) or optim.

initial.para

Initial guess of parameters for cross validation. Default is FALSE.

LOCF

a logical value to indicate whether the last-observation-carried-forward approach applies to prediction. If TRUE, then LOCFcovariate and clongdata must be specified to indicate which time-dependent survival covariates are included for dynamic prediction. Default is FALSE.

LOCFcovariate

a vector of string with time-dependent survival covariates if LOCF = TRUE. Default is NULL.

clongdata

a long format data frame where time-dependent survival covariates are incorporated. Default is NULL.

metric

a string to indicate which metric is used.

...

Further arguments passed to or from other methods.

Value

a list of matrices with conditional probabilities for subjects.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Concordance for joint models

Description

Concordance for joint models

Usage

ConcordanceJMMLSM(
  seed = 100,
  object,
  opt = "nlminb",
  n.cv = 3,
  maxiter = 10000,
  initial.optimizer = "BFGS",
  initial.para = TRUE,
  ...
)

Arguments

seed

a numeric value of seed to be specified for cross validation.

object

object of class 'JMMLSM'.

opt

Optimization method to fit a linear mixed effects model, either nlminb (default) or optim.

n.cv

number of folds for cross validation. Default is 3.

maxiter

the maximum number of iterations of the EM algorithm that the function will perform. Default is 10000.

initial.optimizer

Method for numerical optimization to be used. Default is BFGS.

initial.para

Initial guess of parameters for cross validation. Default is FALSE.

...

Further arguments passed to or from other methods.

Value

a list of matrices with conditional probabilities for subjects.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Concordance for joint models

Description

Concordance for joint models

Usage

Concordancejmcs(
  seed = 100,
  object,
  opt = "nlminb",
  n.cv = 3,
  maxiter = 10000,
  initial.optimizer = "BFGS",
  initial.para = TRUE,
  ...
)

Arguments

seed

a numeric value of seed to be specified for cross validation.

object

object of class 'jmcs'.

opt

Optimization method to fit a linear mixed effects model, either nlminb (default) or optim.

n.cv

number of folds for cross validation. Default is 3.

maxiter

the maximum number of iterations of the EM algorithm that the function will perform. Default is 10000.

initial.optimizer

Method for numerical optimization to be used. Default is BFGS.

initial.para

Initial guess of parameters for cross validation. Default is FALSE.

...

Further arguments passed to or from other methods.

Value

a list of matrices with conditional probabilities for subjects.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Calculating evaluation metrics for joint models

Description

Calculating evaluation metrics for joint models

Usage

DynPredAccjmcs(
  seed = 100,
  object,
  landmark.time = NULL,
  horizon.time = NULL,
  obs.time = NULL,
  method = c("Laplace", "GH"),
  quadpoint = NULL,
  maxiter = NULL,
  n.cv = 3,
  survinitial = TRUE,
  quantile.width = 0.25,
  initial.para = FALSE,
  LOCF = FALSE,
  LOCFcovariate = NULL,
  clongdata = NULL,
  metrics = c("AUC", "Cindex", "Brier", "MAE", "MAEQ"),
  ...
)

Arguments

seed

a numeric value of seed to be specified for cross validation.

object

object of class 'jmcs'.

landmark.time

a numeric value of time for which dynamic prediction starts.

horizon.time

a numeric vector of future times for which predicted probabilities are to be computed.

obs.time

a character string of specifying a longitudinal time variable.

method

estimation method for predicted probabilities. If Laplace, then the empirical empirical estimates of random effects is used. If GH, then the pseudo-adaptive Gauss-Hermite quadrature is used.

quadpoint

the number of pseudo-adaptive Gauss-Hermite quadrature points if method = "GH".

maxiter

the maximum number of iterations of the EM algorithm that the function will perform. Default is 10000.

n.cv

number of folds for cross validation. Default is 3.

survinitial

Fit a Cox model to obtain initial values of the parameter estimates. Default is TRUE.

quantile.width

a numeric value of width of quantile to be specified. Default is 0.25.

initial.para

Initial guess of parameters for cross validation. Default is FALSE.

LOCF

LOCFcovariate

a vector of string with time-dependent survival covariates if LOCF = TRUE. Default is NULL.

clongdata

a long format data frame where time-dependent survival covariates are incorporated. Default is NULL.

metrics

a list to indicate which metric is used.

...

Further arguments passed to or from other methods.

Value

a list of matrices with conditional probabilities for subjects.

Joint Modeling for Continuous outcomes

Description

Joint modeling of longitudinal continuous data and competing risks

Usage

JMMLSM(
  cdata,
  ydata,
  long.formula,
  surv.formula,
  variance.formula,
  random,
  maxiter = 1000,
  epsilon = 1e-04,
  quadpoint = NULL,
  print.para = FALSE,
  survinitial = TRUE,
  initial.para = NULL,
  method = "adaptive",
  opt = "nlminb",
  initial.optimizer = "BFGS"
)

Arguments

cdata

a survival data frame with competing risks or single failure. Each subject has one data entry.

ydata

a longitudinal data frame in long format.

long.formula

a formula object with the response variable and fixed effects covariates to be included in the longitudinal sub-model.

surv.formula

a formula object with the survival time, event indicator, and the covariates to be included in the survival sub-model.

variance.formula

an one-sided formula object with the fixed effects covariates to model the variance of longitudinal sub-model.

random

a one-sided formula object describing the random effects part of the longitudinal sub-model. For example, fitting a random intercept model takes the form ~ 1|ID. Alternatively. Fitting a random intercept and slope model takes the form ~ x1 + ... + xn|ID.

maxiter

the maximum number of iterations of the EM algorithm that the function will perform. Default is 10000.

epsilon

Tolerance parameter. Default is 0.0001.

quadpoint

the number of Gauss-Hermite quadrature points to be chosen for numerical integration. Default is 15 which produces stable estimates in most dataframes.

print.para

Print detailed information of each iteration. Default is FALSE, i.e., not to print the iteration details.

survinitial

Fit a Cox model to obtain initial values of the parameter estimates. Default is TRUE.

initial.para

a list of initialized parameters for EM iteration. Default is NULL.

method

Method for proceeding numerical integration in the E-step. Default is adaptive.

opt

Optimization method to fit a linear mixed effects model, either nlminb (default) or optim.

initial.optimizer

Method for numerical optimization to be used. Default is BFGS.

Value

Object of class JMMLSM with elements

ydata

the input longitudinal dataset for fitting a joint model. It has been re-ordered in accordance with descending observation times in cdata.

cdata

the input survival dataset for fitting a joint model. It has been re-ordered in accordance with descending observation times.

PropEventType

a frequency table of number of events.

beta

the vector of fixed effects for the mean trajectory in the mixed effects location and scale model.

tau

the vector of fixed effects for the within-subject variability in the mixed effects location and scale model.

gamma1

the vector of fixed effects for type 1 failure for the survival model.

gamma2

the vector of fixed effects for type 2 failure for the survival model. Valid only if CompetingRisk = TRUE.

alpha1

the vector of association parameter(s) for the mean trajectory for type 1 failure.

alpha2

the vector of association parameter(s) for the mean trajectory for type 2 failure. Valid only if CompetingRisk = TRUE.

vee1

the vector of association parameter(s) for the within-subject variability for type 1 failure.

vee2

the vector of association parameter(s) for the within-subject variability for type 2 failure. Valid only if CompetingRisk = TRUE.

H01

the matrix that collects baseline hazards evaluated at each uncensored event time for type 1 failure. The first column denotes uncensored event times, the second column the number of events, and the third columns the hazards obtained by Breslow estimator.

H02

the matrix that collects baseline hazards evaluated at each uncensored event time for type 2 failure. The data structure is the same as H01. Valid only if CompetingRisk = TRUE.

Sig

the variance-covariance matrix of the random effects.

iter

the total number of iterations until convergence.

convergence

convergence identifier: 1 corresponds to successful convergence, whereas 0 to a problem (i.e., when 0, usually more iterations are required).

vcov

the variance-covariance matrix of all the fixed effects for both models.

sebeta

the standard error of beta.

setau

the standard error of tau.

segamma1

the standard error of gamma1.

segamma2

the standard error of gamma2. Valid only if CompetingRisk = TRUE.

sealpha1

the standard error of alpha1.

sealpha2

the standard error of alpha2. Valid only if CompetingRisk = TRUE.

sevee1

the standard error of vee1.

sevee2

the standard error of vee2. Valid only if CompetingRisk = TRUE.

seSig

the vector of standard errors of covariance of random effects.

loglike

the log-likelihood value.

EFuntheta

a list with the expected values of all the functions of random effects.

CompetingRisk

logical value; TRUE if a competing event are accounted for.

quadpoint

the number of Gauss Hermite quadrature points used for numerical integration.

LongitudinalSubmodelmean

the component of the long.formula.

LongitudinalSubmodelvariance

the component of the variance.formula.

SurvivalSubmodel

the component of the surv.formula.

random

the component of the random.

call

the matched call.

Examples

require(FastJM)
data(ydata)
data(cdata)
## fit a joint model
## Not run: 
fit <- JMMLSM(cdata = cdata, ydata = ydata, 
              long.formula = Y ~ Z1 + Z2 + Z3 + time,
              surv.formula = Surv(survtime, cmprsk) ~ var1 + var2 + var3,
              variance.formula = ~ Z1 + Z2 + Z3 + time, 
              quadpoint = 6, random = ~ 1|ID, print.para = FALSE)
              
## make dynamic prediction of two subjects
cnewdata <- cdata[cdata$ID %in% c(122, 152), ]
ynewdata <- ydata[ydata$ID %in% c(122, 152), ]
survfit <- survfitJMMLSM(fit, seed = 100, ynewdata = ynewdata, cnewdata = cnewdata, 
                         u = seq(5.2, 7.2, by = 0.5), Last.time = "survtime",
                         obs.time = "time", method = "GH")
oldpar <- par(mfrow = c(2, 2), mar = c(5, 4, 4, 4))
plot(survfit, include.y = TRUE)
par(oldpar)

## End(Not run)

A metric of prediction accuracy of joint model by comparing the predicted risk with the empirical risks stratified on different predicted risk group.

Description

A metric of prediction accuracy of joint model by comparing the predicted risk with the empirical risks stratified on different predicted risk group.

Usage

MAEQJMMLSM(
  seed = 100,
  object,
  landmark.time = NULL,
  horizon.time = NULL,
  obs.time = NULL,
  method = c("Laplace", "GH"),
  quadpoint = NULL,
  maxiter = 1000,
  survinitial = TRUE,
  n.cv = 3,
  quantile.width = 0.25,
  opt = "nlminb",
  initial.para = FALSE,
  LOCF = FALSE,
  LOCFcovariate = NULL,
  clongdata = NULL,
  ...
)

Arguments

seed

a numeric value of seed to be specified for cross validation.

object

object of class 'JMMLSM'.

landmark.time

a numeric value of time for which dynamic prediction starts..

horizon.time

a numeric vector of future times for which predicted probabilities are to be computed.

obs.time

a character string of specifying a longitudinal time variable.

method

estimation method for predicted probabilities. If Laplace, then the empirical empirical estimates of random effects is used. If GH, then the standard Gauss-Hermite quadrature is used.

quadpoint

the number of standard Gauss-Hermite quadrature points if method = "GH".

maxiter

the maximum number of iterations of the EM algorithm that the function will perform. Default is 10000.

survinitial

Fit a Cox model to obtain initial values of the parameter estimates. Default is TRUE.

n.cv

number of folds for cross validation. Default is 3.

quantile.width

a numeric value of width of quantile to be specified. Default is 0.25.

opt

Optimization method to fit a linear mixed effects model, either nlminb (default) or optim.

initial.para

Initial guess of parameters for cross validation. Default is FALSE.

LOCF

LOCFcovariate

a vector of string with time-dependent survival covariates if LOCF = TRUE. Default is NULL.

clongdata

a long format data frame where time-dependent survival covariates are incorporated. Default is NULL.

...

Further arguments passed to or from other methods.

Value

a list of matrices with conditional probabilities for subjects.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

A metric of prediction accuracy of joint model by comparing the predicted risk with the counting process.

Description

A metric of prediction accuracy of joint model by comparing the predicted risk with the counting process.

Usage

PEJMMLSM(
  seed = 100,
  object,
  landmark.time = NULL,
  horizon.time = NULL,
  obs.time = NULL,
  method = c("Laplace", "GH"),
  quadpoint = NULL,
  maxiter = 1000,
  n.cv = 3,
  survinitial = TRUE,
  opt = "nlminb",
  initial.para = FALSE,
  LOCF = FALSE,
  LOCFcovariate = NULL,
  clongdata = NULL,
  ...
)

Arguments

seed

a numeric value of seed to be specified for cross validation.

object

object of class 'JMMLSM'.

landmark.time

a numeric value of time for which dynamic prediction starts..

horizon.time

a numeric vector of future times for which predicted probabilities are to be computed.

obs.time

a character string of specifying a longitudinal time variable.

method

estimation method for predicted probabilities. If Laplace, then the empirical empirical estimates of random effects is used. If GH, then the pseudo-adaptive Gauss-Hermite quadrature is used.

quadpoint

the number of pseudo-adaptive Gauss-Hermite quadrature points if method = "GH".

maxiter

the maximum number of iterations of the EM algorithm that the function will perform. Default is 10000.

n.cv

number of folds for cross validation. Default is 3.

survinitial

Fit a Cox model to obtain initial values of the parameter estimates. Default is TRUE.

opt

Optimization method to fit a linear mixed effects model, either nlminb (default) or optim.

initial.para

Initial guess of parameters for cross validation. Default is FALSE.

LOCF

LOCFcovariate

a vector of string with time-dependent survival covariates if LOCF = TRUE. Default is NULL.

clongdata

a long format data frame where time-dependent survival covariates are incorporated. Default is NULL.

...

Further arguments passed to or from other methods.

Value

a list of matrices with conditional probabilities for subjects.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Anova Method for Fitted Joint Models

Description

Performs a likelihood ratio test between two nested joint models.

Usage

## S3 method for class 'jmcs'
anova(object, object2, digits = 4, ...)

Arguments

object

an object inheriting from class jmcs, nested in object2.

object2

an object inheriting from class jmcs.

digits

the number of significant digits to use when printing. Default is 4.

...

further arguments passed to or from other methods.

Value

A table to summarize the likelihood ratio test.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Examples


# Fit a joint model
fit <- jmcs(ydata = ydata, cdata = cdata, 
            long.formula = response ~ time + x1, 
            surv.formula = Surv(surv, failure_type) ~ x1 + x2, 
            random =  ~ time| ID)

fit2 <- jmcs(ydata = ydata, cdata = cdata, 
            long.formula = response ~ time + gender + x1 + race, 
            surv.formula = Surv(surv, failure_type) ~ x1 + gender + x2 + race, 
            random =  ~ time| ID)      

anova(fit, fit2)

Simulated competing risks data correlated with ydata

Description

The cdata data frame has 1000 rows and 7 columns.

Usage

data(cdata)

Format

This data frame contains the following columns:

ID: patient identifier.
surv: event time.
failure_type: event indicator. 0 denotes censoring, 1 risk 1, and 2 risk 2.
x1: continuous variable.
x2: treatment indicator. 0 denotes the placebo group and 1 the treatment group.
gender: gender indicator.
race: race indicator.

Simulated competing risks data where event hazards depend on within-subject variance

Description

The cdatah data frame has 200 rows and 6 columns.

Usage

data(cdatah)

Format

This data frame contains the following columns:

ID: patient identifier.
survtime: event time.
cmprsk: event indicator. 0 denotes censoring, 1 risk 1, and 2 risk 2.
var1: treatment indicator. 0 denotes the placebo group and 1 the treatment group.
var2: continuous variable.
var3: continuous variable.

Fitted values for joint models

Description

Extract fitted values for joint models.

Usage

## S3 method for class 'jmcs'
fitted(
  object,
  type = c("Marginal", "Subject"),
  process = c("Longitudinal", "Event"),
  ...
)

Arguments

object

an object inheriting from class jmcs.

type

for which type of fitted values to calculate.

process

for which sub-model to calculate the fitted values.

...

further arguments passed to or from other methods.

Value

a numeric vector of fitted values.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Examples


fit <- jmcs(ydata = ydata, cdata = cdata, 
            long.formula = response ~ time + gender + x1 + race, 
            surv.formula = Surv(surv, failure_type) ~ x1 + gender + x2 + race, 
            random =  ~ time| ID)

# fitted for the longitudinal process
head(cbind(
  "Marg" = fitted(fit, type = "Marginal", process = "Longitudinal"), 
  "Subj" = fitted(fit, type = "Subject", process = "Longitudinal")
))
# fitted for the levent process - marginal survival function
head(fitted(fit, type = "Marginal", process = "Event"))

Estimated coefficients estimates for joint models

Description

Extracts the fixed effects for a fitted joint model.

Usage

fixef(object, process = c("Longitudinal", "Event"), ...)

Arguments

object

an object inheriting from class jmcs or mvjmcs.

process

for which sub-model to extract the estimated coefficients.

...

further arguments passed to or from other methods.

Value

A numeric vector or a list of the estimated parameters for the fitted model.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Examples


# a joint model fit
fit <- jmcs(ydata = ydata, cdata = cdata, 
            long.formula = response ~ time + gender + x1 + race, 
            surv.formula = Surv(surv, failure_type) ~ x1 + gender + x2 + race, 
            random =  ~ time| ID)

# fixed effects for the longitudinal process
fixef(fit, process = "Longitudinal")
# fixed effects for the event process
fixef(fit, process = "Event")

Joint modeling of longitudinal continuous data and competing risks

Description

Joint modeling of longitudinal continuous data and competing risks

Usage

jmcs(
  ydata,
  cdata,
  long.formula,
  random = NULL,
  surv.formula,
  REML = TRUE,
  quadpoint = NULL,
  maxiter = 10000,
  print.para = FALSE,
  initial.para = NULL,
  survinitial = TRUE,
  tol = 1e-04,
  method = "pseudo-adaptive",
  opt = "nlminb"
)

Arguments

ydata

a longitudinal data frame in long format.

cdata

a survival data frame with competing risks or single failure. Each subject has one data entry.

long.formula

a formula object with the response variable and fixed effects covariates to be included in the longitudinal sub-model.

random

a one-sided formula object describing the random effects part of the longitudinal sub-model. For example, fitting a random intercept model takes the form ~ 1|ID. Alternatively. Fitting a random intercept and slope model takes the form ~ x1 + ... + xn|ID.

surv.formula

a formula object with the survival time, event indicator, and the covariates to be included in the survival sub-model.

REML

a logic object that indicates the use of REML estimator. Default is TRUE.

quadpoint

the number of pseudo-adaptive Gauss-Hermite quadrature points. to be chosen for numerical integration. Default is 6 which produces stable estimates in most dataframes.

maxiter

the maximum number of iterations of the EM algorithm that the function will perform. Default is 10000.

print.para

Print detailed information of each iteration. Default is FALSE, i.e., not to print the iteration details.

initial.para

a list of initialized parameters for EM iteration. Default is NULL.

survinitial

Fit a Cox model to obtain initial values of the parameter estimates. Default is TRUE.

tol

Tolerance parameter. Default is 0.0001.

method

Method for proceeding numerical integration in the E-step. Default is pseudo-adaptive.

opt

Optimization method to fit a linear mixed effects model, either nlminb (default) or optim.

Value

Object of class jmcs with elements

beta

the vector of fixed effects for the linear mixed effects model.

gamma1

the vector of fixed effects for type 1 failure for the survival model.

gamma2

the vector of fixed effects for type 2 failure for the survival model. Valid only if CompetingRisk = TRUE.

nu1

the vector of association parameter(s) for type 1 failure.

nu2

the vector of association parameter(s) for type 2 failure. Valid only if CompetingRisk = TRUE.

H01

H02

the matrix that collects baseline hazards evaluated at each uncensored event time for type 2 failure. The data structure is the same as H01. Valid only if CompetingRisk = TRUE.

Sig

the variance-covariance matrix of the random effects.

sigma

the variance of the measurement error for the linear mixed effects model.

iter

the total number of iterations until convergence.

convergence

convergence identifier: 1 corresponds to successful convergence, whereas 0 to a problem (i.e., when 0, usually more iterations are required).

vcov

the variance-covariance matrix of all the fixed effects for both models.

sebeta

the standard error of beta.

segamma1

the standard error of gamma1.

segamma2

the standard error of gamma2. Valid only if CompetingRisk = TRUE.

senu1

the standard error of nu1.

senu2

the standard error of nu2. Valid only if CompetingRisk = TRUE.

seSig

the vector of standard errors of covariance of random effects.

sesigma

the standard error of variance of measurement error for the linear mixed effects model.

loglike

the log-likelihood value.

fitted

a list with the fitted values:

resid: the vector of estimated residuals for the linear mixed effects model.
fitted: the vector of fitted values for the linear mixed effects model.
fittedmar: the vector of marginal fitted values for the linear mixed effects model.
residmar: the vector of estimated marginal residuals for the linear mixed effects model.

fittedSurv

the estimated survival rate evaluated at each uncensored event time.

FUNB

the estimated random effects for each subject.

CompetingRisk

logical value; TRUE if a competing event are accounted for.

quadpoint

the number of Gauss Hermite quadrature points used for numerical integration.

ydata

the input longitudinal dataset for fitting a joint model. It has been re-ordered in accordance with descending observation times in cdata.

cdata

the input survival dataset for fitting a joint model. It has been re-ordered in accordance with descending observation times.

PropEventType

a frequency table of number of events.

LongitudinalSubmodel

the component of the long.formula.

SurvivalSubmodel

the component of the surv.formula.

random

the component of the random.

tol

the convergence parameter.

call

the matched call.

Quad.method

the quadrature rule used for integration. If pseudo-adaptive quadrature rule is used, then return pseudo-adaptive. Otherwise return standard.

id

the grouping vector for the longitudinal outcome.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Examples


require(FastJM)
require(survival)
# Load a simulated longitudinal dataset
data(ydata)
# Load a simulated survival dataset with two competing events
data(cdata)

# Fit a joint model
fit <- jmcs(ydata = ydata, cdata = cdata, 
            long.formula = response ~ time + gender + x1 + race, 
            surv.formula = Surv(surv, failure_type) ~ x1 + gender + x2 + race, 
            random =  ~ time| ID)
fit
# Extract the parameter estimates of longitudinal sub-model fixed effects
fixef(fit, process = "Longitudinal")
# Extract the parameter estimates of survival sub-model fixed effects
fixef(fit, process = "Event")
# Obtain the random effects estimates for first 6 subjects 
head(ranef(fit))
# Obtain the variance-covariance matrix of all parameter estimates 
vcov(fit)
# Obtain the result summaries of the joint model fit
summary(fit, process = "Longitudinal")
summary(fit, process = "Event")
# Prediction of cumulative incidence for competing risks data
# Predict the conditional probabilities for two patients who are alive (censored)
ND <- ydata[ydata$ID %in% c(419, 218), ]
ID <- unique(ND$ID)
NDc <- cdata[cdata$ID  %in% ID, ]
survfit <- survfitjmcs(fit, 
                       ynewdata = ND, 
                       cnewdata = NDc, 
                       u = seq(3, 4.8, by = 0.2), 
                       method = "GH",
                       obs.time = "time")
survfit

res <- DynPredAccjmcs(object = fit,
                      landmark.time = 3,
                      horizon.time = c(3.6, 4, 4.4),
                      obs.time = "time",
                      method = "GH",
                      maxiter = 1000,
                      n.cv = 3,
                      metrics = c("AUC", "Cindex", "Brier", "MAE", "MAEQ"))

# Print all available evaluation metrics for the fitted joint model
summary(res, metric = "Brier")
summary(res, metric = "MAE")
summary(res, metric = "MAEQ")
summary(res, metric = "AUC")
summary(res, metric = "Cindex")

Simulated competing risks data correlated with mvydata

Description

The mvcdata data frame has 500 rows and 5 columns.

Usage

data(mvcdata)

Format

This data frame contains the following columns:

ID: patient identifier.
survtime: event time.
cmprsk: event indicator. 0 denotes censoring, 1 risk 1, and 2 risk 2.
X21: X21.
X22: X22.

Joint modeling of multivariate longitudinal and competing risks data

Description

Joint modeling of multivariate longitudinal continuous data and competing risks

Usage

mvjmcs(
  ydata,
  cdata,
  long.formula,
  random = NULL,
  surv.formula,
  maxiter = 10000,
  opt = "nlminb",
  tol = 0.005,
  print.para = TRUE,
  initial.para = NULL,
  cpu.cores = NULL
)

Arguments

ydata

A longitudinal data frame in long format.

cdata

A survival data frame with competing risks or single failure. Each subject has one data entry.

long.formula

A list of formula objects specifying fixed effects for each longitudinal outcome.

random

A formula or list of formulas describing random effects structures (e.g., ~ 1|ID).

surv.formula

A formula for the survival sub-model, including survival time and event indicator.

maxiter

Maximum number of EM iterations. Default is 10000.

opt

Optimization method for mixed model. Default is "nlminb".

tol

Convergence tolerance for EM algorithm. Default is 0.0001.

print.para

Logical; if TRUE, prints parameter values at each iteration.

initial.para

Optional list of initialized parameters. Default is NULL.

cpu.cores

Number of CPU cores for parallel computation. Default is 1.

Details

Function fits a joint model for multiple longitudinal outcomes and competing risks using a fast EM algorithm.

Value

Object of class mvjmcs with elements

beta

the vector of all biomarker-specific fixed effects for the linear mixed effects sub-models.

betaList

the list of biomarker-specific fixed effects for the linear mixed effects sub-model.

gamma1

the vector of fixed effects for type 1 failure for the survival model.

gamma2

the vector of fixed effects for type 2 failure for the survival model. Valid only if CompetingRisk = TRUE.

alpha1

the vector of association parameter(s) for type 1 failure.

alpha2

the vector of association parameter(s) for type 2 failure. Valid only if CompetingRisk = TRUE.

H01

H02

the matrix that collects baseline hazards evaluated at each uncensored event time for type 2 failure. The data structure is the same as H01. Valid only if CompetingRisk = TRUE.

Sig

the variance-covariance matrix of the random effects.

sigma

the vector of the variance of the biomarker-specific measurement error for the linear mixed effects sub-models.

iter

the total number of iterations until convergence.

convergence

convergence identifier: 1 corresponds to successful convergence, whereas 0 to a problem (i.e., when 0, usually more iterations are required).

vcov

the variance-covariance matrix of all the fixed effects for both models.

FisherInfo

the Empirical Fisher information matrix.

Score

a matrix of the score function for all subjects.

sebeta

the standard error of beta.

segamma1

the standard error of gamma1.

segamma2

the standard error of gamma2. Valid only if CompetingRisk = TRUE.

sealpha1

the standard error of nu1.

sealpha2

the standard error of nu2. Valid only if CompetingRisk = TRUE.

seSig

the vector of standard errors of covariance of random effects.

sesigma

the standard error of variance of biomarker-specific measurement error for the linear mixed effects sub-models.

pos.mode

the posterior mode of the conditional distribution of random effects.

pos.cov

the posterior covariance of the conditional distribution of random effects.

CompetingRisk

logical value; TRUE if a competing event are accounted for.

ydata

the input longitudinal dataset for fitting a joint model. It has been re-ordered in accordance with descending observation times in cdata.

cdata

the input survival dataset for fitting a joint model. It has been re-ordered in accordance with descending observation times.

PropEventType

a frequency table of number of events.

LongitudinalSubmodel

the component of the long.formula.

SurvivalSubmodel

the component of the surv.formula.

random

the component of the random.

call

the matched call.

id

the grouping vector for the longitudinal outcome.

opt

the numerical optimizer for obtaining the initial guess of the parameters in the linear mixed effects sub-models.

runtime

the total computation time.

Examples



  require(FastJM)
  require(survival)
  require(future)
  require(future.apply)
  
  data(mvcdata)
  data(mvydata)

  
  # Fit joint model with two biomarkers
  fit <- mvjmcs(ydata = mvydata, cdata = mvcdata, 
                long.formula = list(Y1 ~ X11 + X12 + time, 
                                    Y2 ~ X11 + X12 + time),
                random = list(~ time | ID,
                              ~ 1 | ID),
                surv.formula = Surv(survtime, cmprsk) ~ X21 + X22, maxiter = 1000, opt = "optim", 
                tol = 1e-3, print.para = FALSE)
  fit
  
  # Extract the parameter estimates of longitudinal sub-model fixed effects
  fixef(fit, process = "Longitudinal")
  
  # Extract the parameter estimates of survival sub-model fixed effects
  fixef(fit, process = "Event")
  
  # Obtain the random effects estimates for first 6 subjects 
  head(ranef(fit))

Simulated bivariate longitudinal data

Description

The mvydata data frame has 4060 rows and 6 columns.

Usage

data(mvydata)

Format

This data frame contains the following columns:

ID: patient identifier.
time: visit time.
Y1: response variable of biomarker 1.
Y2: response variable of biomarker 2.
X11: X11.
X12: X12.

Fitted values for joint models

Description

Plot Diagnostics for Joint Models.

Usage

## S3 method for class 'jmcs'
plot(x, add.smooth = getOption("add.smooth"), ...)

Arguments

x

x of class 'jmcs'.

add.smooth

logical; if TRUE a smooth line is superimposed in the "Residuals vs Fitted" plot.

...

further arguments passed to or from other methods.

Value

The first two plots are longitudinal sub-model diagnostics and the last two are marginal survival function and marginal cumulative hazard.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Examples


fit <- jmcs(ydata = ydata, cdata = cdata, 
            long.formula = response ~ time + gender + x1 + race, 
            surv.formula = Surv(surv, failure_type) ~ x1 + gender + x2 + race, 
            random =  ~ time| ID)

oldpar <- par(mfrow = c(2, 2))
plot(fit)
par(oldpar)

Plot conditional probabilities for new subjects

Description

Plot conditional probabilities for new subjects. If CompetingRisk = FALSE, print the survival probabilities. Otherwise, print the cumulative incidence probabilities for each failure type.

Usage

## S3 method for class 'survfitJMMLSM'
plot(
  x,
  include.y = FALSE,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim.long = NULL,
  ylim.surv = NULL,
  ...
)

Arguments

x

x of class survfitJMMLSM.

include.y

include longitudinal responses of this subject versus time. Default is FALSE.

xlab

X axis label.

ylab

Y axis label.

xlim

X axis support.

ylim.long

Y axis support for the longitudinal outcome.

ylim.surv

Y axis support for the event / survival probability.

...

further arguments passed to or from other methods.

Value

plots of conditional probabilities over different pre-specified time points for subjects. If single failure type, then survival probabilities will be returned. Otherwise, cumulative incidence probabilities for each failure type will be returned.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Print jmcs

Description

Print jmcs

Print mvjmcs

Usage

## S3 method for class 'jmcs'
print(x, digits = 4, ...)

## S3 method for class 'mvjmcs'
print(x, digits = 4, ...)

Arguments

x

Object of class 'mvjmcs'.

digits

the number of significant digits to use when printing.

...

Further arguments passed to or from other methods.

Value

a summary of data, joint model, log likelihood, and parameter estimates.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Print JMMLSM

Description

Print contents of JMMLSM object.

Usage

## S3 method for class 'JMMLSM'
print(x, digits = 4, ...)

Arguments

x

Object of class 'JMMLSM'.

digits

number of digits of decimal to be printed.

...

Further arguments passed to or from other methods.

Value

a summary of data, joint model, log likelihood, and parameter estimates.

Author(s)

Shanpeng Li

Print survfitJMMLSM

Description

Print survfitJMMLSM

Usage

## S3 method for class 'survfitJMMLSM'
print(x, ...)

Arguments

x

x of class 'survfitJMMLSM'.

...

Further arguments passed to or from other methods.

Value

a list of matrices with conditional probabilities for subjects.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Print survfitjmcs

Description

Print survfitjmcs

Usage

## S3 method for class 'survfitjmcs'
print(x, ...)

Arguments

x

x of class 'survfitjmcs'.

...

Further arguments passed to or from other methods.

Value

a list of matrices with conditional probabilities for subjects.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Print survfitmvjmcs

Description

Print survfitmvjmcs

Usage

## S3 method for class 'survfitmvjmcs'
print(x, ...)

Arguments

x

x of class 'survfitmvjmcs'.

...

Further arguments passed to or from other methods.

Value

a list of matrices with conditional probabilities for subjects.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Random effects estimates for joint models

Description

Extracts the posterior mean of the random effects for a fitted joint model.

Usage

ranef(object, ...)

Arguments

object

an object inheriting from class jmcs or mvjmcs.

...

further arguments passed to or from other methods.

Value

a matrix of random effects estimates.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Examples


# a joint model fit
fit <- jmcs(ydata = ydata, cdata = cdata, 
            long.formula = response ~ time + gender + x1 + race, 
            surv.formula = Surv(surv, failure_type) ~ x1 + gender + x2 + race, 
            random =  ~ time| ID)

# extract random effects estimates
head(ranef(fit))

Residuals for joint models

Description

Extract residuals for joint models.

Usage

## S3 method for class 'jmcs'
residuals(object, type = c("Marginal", "Subject"), ...)

Arguments

object

an object inheriting from class jmcs.

type

what type of residuals to calculate.

...

further arguments passed to or from other methods.

Value

a vector of residuals of the longitudinal sub-model.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Examples


# a joint model fit
fit <- jmcs(ydata = ydata, cdata = cdata, 
            long.formula = response ~ time + gender + x1 + race, 
            surv.formula = Surv(surv, failure_type) ~ x1 + gender + x2 + race, 
            random =  ~ time| ID)

# residuals of the longitudinal sub-model
head(cbind(
  "Marg" = residuals(fit, type = "Marginal"), 
  "Subj" = residuals(fit, type = "Subject")
))

Joint modeling of multivariate longitudinal and competing risks data

Description

Data simulation from the joint model of multivariate longitudinal biomarkers and time-to-event data

Usage

simmvJMdata(
  seed = 100,
  N = 200,
  increment = 0.7,
  beta = list(beta1 = c(5, 1.5, 2, 1), beta2 = c(10, 1, 2, 1)),
  sigma = c(0.5, 0.5),
  gamma1 = c(1, 0.5),
  gamma2 = c(-0.5, 0.5),
  alpha1 = list(alpha11 = c(0.5, 0.7), alpha12 = c(-0.5, 0.5)),
  alpha2 = list(alpha21 = c(0.5, 0.7), alpha22 = c(-0.5, 0.5)),
  lambda1 = 0.05,
  lambda2 = 0.025,
  CL = 5,
  CU = 10,
  covb = diag(rep(1, 4)),
  missprob = 0,
  CR = TRUE
)

Arguments

seed

a random seed number specified for simulating a joint model dataset.

N

an integer to specify the sample size.

increment

a scalar to specify the increment of visit time for longitudinal measurements.

beta

a list of true parameters for the linear mixed effects sub-models. Each component of the list correspond to a specific biomarker.

sigma

a vector of true error variance for all biomarkers.

gamma1

a vector of true parameters of survival fixed effects for failure 1.

gamma2

a vector of true parameters of survival fixed effects for failure 2.

alpha1

a list of true parameters for the association parameters for failure 1. Each component of the list correspond to a specific biomarker.

alpha2

a list of true parameters for the association parameters for failure 2. Each component of the list correspond to a specific biomarker.

lambda1

the baseline hazard rate of failure 1. An exponential distribution with a rate parameter of lambda1 is assumed.

lambda2

the baseline hazard rate of failure 2. An exponential distribution with a rate parameter of lambda2 is assumed.

CL

a lower limit of a uniform distribution to be specified for the censoring time.

CU

an upper limit of a uniform distribution to be specified for the censoring time.

covb

a matrix of variance-covariance matrix of random effects.

missprob

a scalar (ranging from 0 to 1) to specify the probability of missing longitudinal observations. Default is 0.

CR

logical; if TRUE, simulate competing risks time-to-event data with 2 failures. Default is TRUE.

Value

a list of datasets for both longitudinal and survival data with the elements

mvydata

a long-format data frame of longitudinal data.

mvcdata

a dataframe of survival data.

Summaries of evaluation metrics for joint models

Description

Produce result summaries of a joint model fit.

Usage

## S3 method for class 'AUCJMMLSM'
summary(object, digits = 4, ...)

## S3 method for class 'ConcordanceJMMLSM'
summary(object, digits = 4, ...)

## S3 method for class 'Concordancejmcs'
summary(object, digits = 4, ...)

## S3 method for class 'DynPredAccjmcs'
summary(
  object,
  metric = c("AUC", "Cindex", "Brier", "MAE", "MAEQ"),
  digits = 4,
  ...
)

## S3 method for class 'JMMLSM'
summary(object, process = c("longitudinal", "survival"), digits = 4, ...)

## S3 method for class 'MAEQJMMLSM'
summary(object, digits = 3, ...)

## S3 method for class 'PEJMMLSM'
summary(object, error = c("MAE", "Brier"), ...)

## S3 method for class 'jmcs'
summary(object, process = c("Longitudinal", "Event"), digits = 4, ...)

## S3 method for class 'mvjmcs'
summary(object, process = c("Longitudinal", "Event"), digits = 4, ...)

Arguments

object

an object inheriting from class mvjmcs.

digits

the number of significant digits to use when printing. Default is 4.

...

further arguments passed to or from other methods.

metric

a list to indicate what metric to summarize

process

for which model (i.e., longitudinal model or survival model) to extract the estimated coefficients.

error

a character string that specifies the loss function.

Value

a list of matrices with conditional probabilities for subjects.

a summary of the list of matrices with conditional probabilities for subjects.

A table to summarize the model results.

a list of matrices with conditional probabilities for subjects.

A table to summarize the model results.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Prediction in Joint Models

Description

This function computes the conditional probability of surviving later times than the last observed time for which a longitudinal measurement was available.

Usage

survfitJMMLSM(
  object,
  seed = 100,
  ynewdata = NULL,
  cnewdata = NULL,
  u = NULL,
  Last.time = NULL,
  obs.time = NULL,
  LOCF = FALSE,
  LOCFcovariate = NULL,
  clongdata = NULL,
  method = c("Laplace", "GH"),
  quadpoint = NULL,
  ...
)

Arguments

object

an object inheriting from class JMMLSM.

seed

a random seed number to proceed non-parametric bootstrap. Default is 100.

ynewdata

a data frame that contains the longitudinal and covariate information for the subjects for which prediction of survival probabilities is required.

cnewdata

a data frame that contains the survival and covariate information for the subjects for which prediction of survival probabilities is required.

u

a numeric vector of times for which prediction survival probabilities are to be computed.

Last.time

a numeric vector or character string. This specifies the known time at which each of the subjects in cnewdata was known to be alive. If NULL, then this is automatically taken as the survival time of each subject. If a numeric vector, then it is assumed to be greater than or equals to the last available longitudinal time point for each subject. If a character string, then it should be a variable in cnewdata.

obs.time

a character string of specifying a longitudinal time variable in ynewdata.

LOCF

LOCFcovariate

a vector of string with time-dependent survival covariates if LOCF = TRUE. Default is NULL.

clongdata

a long format data frame where time-dependent survival covariates are incorporated. Default is NULL.

method

a character string specifying the type of probability approximation; if Laplace, then a first order estimator is computed. If GH, then the standard Gauss-Hermite quadrature is used instead.

quadpoint

number of quadrature points used for estimating conditional probabilities when method = "GH". Default is NULL. If method = "GH", then 15 is used.

...

further arguments passed to or from other methods.

Value

a list of matrices with conditional probabilities for subjects.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Prediction in Joint Models

Description

This function computes the conditional probability of surviving later times than the last observed time for which a longitudinal measurement was available.

Usage

survfitjmcs(
  object,
  seed = 100,
  ynewdata = NULL,
  cnewdata = NULL,
  u = NULL,
  Last.time = NULL,
  obs.time = NULL,
  LOCF = FALSE,
  LOCFcovariate = NULL,
  clongdata = NULL,
  method = c("Laplace", "GH"),
  quadpoint = NULL,
  ...
)

Arguments

object

an object inheriting from class jmcs.

seed

a random seed number to proceed Monte Carlo simulation. Default is 100.

ynewdata

a data frame that contains the longitudinal and covariate information for the subjects for which prediction of survival probabilities is required.

cnewdata

a data frame that contains the survival and covariate information for the subjects for which prediction of survival probabilities is required.

u

a numeric vector of times for which prediction survival probabilities are to be computed.

Last.time

obs.time

a character string of specifying a longitudinal time variable in ynewdata.

LOCF

LOCFcovariate

a vector of string with time-dependent survival covariates if LOCF = TRUE. Default is NULL.

clongdata

a long format data frame where time-dependent survival covariates are incorporated. Default is NULL.

method

a character string specifying the type of probability approximation; if Laplace, then a first order estimator is computed. If GH, then the standard Gauss-Hermite quadrature is used instead.

quadpoint

number of quadrature points used for estimating conditional probabilities when method = "GH". Default is NULL. If method = "GH", then use the same amount of quadrature points obtained from object.

...

further arguments passed to or from other methods.

Value

a list of matrices with conditional probabilities for subjects.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Prediction in Joint Models

Description

This function computes the conditional probability of surviving later times than the last observed time for which a longitudinal measurement was available.

Usage

survfitmvjmcs(
  object,
  seed = 100,
  ynewdata = NULL,
  cnewdata = NULL,
  u = NULL,
  Last.time = NULL,
  obs.time = NULL,
  LOCF = FALSE,
  LOCFcovariate = NULL,
  clongdata = NULL,
  ...
)

Arguments

object

an object inheriting from class mvjmcs.

seed

a random seed number to proceed Monte Carlo simulation. Default is 100.

ynewdata

a data frame that contains the longitudinal and covariate information for the subjects for which prediction of survival probabilities is required.

cnewdata

a data frame that contains the survival and covariate information for the subjects for which prediction of survival probabilities is required.

u

a numeric vector of times for which prediction survival probabilities are to be computed.

Last.time

obs.time

a character string of specifying a longitudinal time variable in ynewdata.

LOCF

LOCFcovariate

a vector of string with time-dependent survival covariates if LOCF = TRUE. Default is NULL.

clongdata

a long format data frame where time-dependent survival covariates are incorporated. Default is NULL.

...

further arguments passed to or from other methods.

Value

a list of matrices with conditional probabilities for subjects.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Variance-covariance matrix of the estimated parameters for joint models

Description

Extract variance-covariance matrix for joint models.

Usage

## S3 method for class 'JMMLSM'
vcov(object, ...)

## S3 method for class 'jmcs'
vcov(object, ...)

Arguments

object

an object inheriting from class jmcs.

...

further arguments passed to or from other methods.

Value

a matrix of variance covariance of all parameter estimates.

Author(s)

Shanpeng Li lishanpeng0913@ucla.edu

Simulated longitudinal data

Description

The ydata data frame has 3067 rows and 6 columns.

Usage

data(ydata)

Format

This data frame contains the following columns:

ID: patient identifier.
response: response variable.
time: visit time.
x1: treatment indicator. 0 denotes the placebo group and 1 the treatment group.
gender: gender indicator.
race: race indicator.

Simulated longitudinal data with within-subject variance

Description

The ydatah data frame has 1353 rows and 6 columns.

Usage

data(ydatah)

Format

This data frame contains the following columns:

ID: patient identifier.
Y: response variable.
time: visit time.
Z1: treatment indicator. 0 denotes the placebo group and 1 the treatment group.
Z2: continuous variable..
Z3: continuous variable..

Time-dependent AUC for joint models

Description

Usage

Arguments

Value

Author(s)

See Also

Concordance for joint models

Description

Usage

Arguments

Value

Author(s)

See Also

Concordance for joint models

Description

Usage

Arguments

Value

Author(s)

See Also

Calculating evaluation metrics for joint models

Description

Usage

Arguments

Value

See Also

Joint Modeling for Continuous outcomes

Description

Usage

Arguments

Value

Examples

A metric of prediction accuracy of joint model by comparing the predicted risk with the empirical risks stratified on different predicted risk group.

Description

Usage

Arguments

Value

Author(s)

See Also

A metric of prediction accuracy of joint model by comparing the predicted risk with the counting process.

Description

Usage

Arguments

Value

Author(s)

See Also

Anova Method for Fitted Joint Models

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Simulated competing risks data correlated with ydata

Description

Usage

Format

Simulated competing risks data where event hazards depend on within-subject variance

Description

Usage

Format

Fitted values for joint models

Description

Usage

Arguments

Value

Author(s)

Examples

Estimated coefficients estimates for joint models

Description

Usage

Arguments

Value

Author(s)

Examples

Joint modeling of longitudinal continuous data and competing risks

Description

Usage