lsjm: lsjm : Estimation of joint model for longitudinal data with a...

In FlexVarJM: Estimate Joint Models with Subject-Specific Variance

lsjm : Estimation of joint model for longitudinal data with a subject-specific time-dependent variability and time-to-event data.

Description

This function fits complex joint models with shared random effects. The longitudinal submodel estimates longitudinal data with a mixed-effects model in which we suppose that the variance of the residual error is time-dependent and subject-specific. The survival submodel handles right-censored and left-truncated time-to-event data and competing risks. The dependence structure between the longitudinal and the survival data can be the random effects from the mixed model or the current value of the marker and/or the slope of the marker. We can also adjust on the current variance of the marker. (See below) Parameters are estimated simultaneously through a maximum likelihood method, using a Marquardt-Levenberg algorithm.

Usage

lsjm(
  formFixed,
  formRandom,
  formGroup,
  formSurv,
  timeVar,
  data.long,
  variability_hetero = TRUE,
  formFixedVar,
  formRandomVar,
  correlated_re = FALSE,
  sharedtype = c("current value", "variability"),
  hazard_baseline = "Exponential",
  formSlopeFixed = NULL,
  formSlopeRandom = NULL,
  indices_beta_slope = NULL,
  nb_pointsGK = 15,
  ord.splines = 3,
  competing_risk = FALSE,
  formSurv_CR = NULL,
  hazard_baseline_CR = "Exponential",
  sharedtype_CR = c("current value", "variability"),
  left_trunc = FALSE,
  Time.0 = NULL,
  S1 = 1000,
  S2 = 5000,
  nproc = 1,
  clustertype = "SOCK",
  maxiter = 100,
  print.info = FALSE,
  file = NULL,
  epsa = 0.001,
  epsb = 0.001,
  epsd = 0.001,
  binit = NULL,
  Comp.Rcpp = TRUE
)

Arguments

formFixed	A formula for the fixed effects of the longitudinal submodel
formRandom	A formula for the random effects of the longitudinal submodel
formGroup	A formula which indicates the group variable
formSurv	A formula which indicates the variables used in the survival submodel
timeVar	The name of the column of time in data.long. This variable must appears in data.long
data.long	A dataframe with the longitudinal data
variability_hetero	A logical to indicate if we suppose a subject_specific variability
formFixedVar	A formula for the fixed effects of the variance predictor
formRandomVar	A formula for the random effects of the variance predictor
correlated_re	A logical to indicate if the random effects of the marker and the variance predictors are correlated (By default there are supposed to be independent)
sharedtype	char : dependence structure for survival model : "RE" (random effects) or "CV" (current value) or "CVS" (current value and slope) or "S" (slope)
hazard_baseline	char : baseline hazard function : "Exponential" or "Weibull" or "Splines"
formSlopeFixed	A formula for the fixed effects of the slope of the longitudinal submodel : the derivative of the formFixed
formSlopeRandom	A formula for the random effects of the slope of the longitudinal submodel : the derivative of the formRandom
indices_beta_slope	A vector of index indicating which beta of the formFixed formula is used in the formSlopeFixed formula
nb_pointsGK	the number of points for Gauss-Kronrod approximation : choice between 7 and 15. 15 by default.
ord.splines	A numeric, the order of splines for the baseline risk function (3 by default)
competing_risk	A logical indicating if the model handles with competing risks
formSurv_CR	In case of competing risk A formula which indicates the variables used in the survival submodel for the second event
hazard_baseline_CR	In case of competing risk : a character for the baseline hazard function of the second event
sharedtype_CR	In case of competing risk ; a character for the dependence structure
left_trunc	A logical indicating if the model handles with left truncated data
Time.0	In case of left truncation : a vector of entry times
S1	An integer : the number of QMC draws for the first step
S2	An integer : the number of QMC draws for the second step
nproc	An integer : the number of processors for parallel computing
clustertype	one of the supported types from makeCluster function
maxiter	optional maximum number of iterations for the marqLevAlg iterative algorithm.
print.info	logical indicating if the outputs of each iteration should be written
file	optional character giving the name of the file where the outputs of each iteration should be written (if print.info=TRUE)
epsa	optional threshold for the convergence criterion based on the parameter stability.
epsb	optional threshold for the convergence criterion based on the objective function stability.
epsd	optional threshold for the relative distance to maximum. This criterion has the nice interpretation of estimating the ratio of the approximation error over the statistical error, thus it can be used for stopping the iterative process whatever the problem.
binit	optional initials parameters.
Comp.Rcpp	boolean to indicate if the computation is performed with RCPP program or R program. True by default.

Details

A. LONGITUDINAL SUBMODEL

The longitudinal submodel is defined by a linear mixed effects model with the residual variance which could be supposed to be time-dependent and subject-specific : \quad\left\{\begin{array}{ll} Y_{ij} = Y_{i}(t_{ij}) = \widetilde{Y}_i(t_{ij}) + \epsilon_{ij} = X_{ij}^{\top} \beta+Z_{ij}^{\top} b_{i}+\epsilon_{ij}, \\ \epsilon_{ij}(t_{ij}) \sim \mathcal{N}(0,\sigma_i^2(t_{ij})) \hspace{3mm} \text{with} \hspace{3mm} \log(\sigma_i(t_{ij})) = O_{ij}^{\top} \mu+M_{ij}^{\top} \tau_{i} \end{array} \right.

with X_{ij}, O_{ij}, Z_{ij} and M_{ij} four vectors of explanatory variables for subject i at visit j, respectively associated with the fixed-effect vectors \beta and \mu, and the subject-specific random-effect vector b_i and \tau_i, such as \quad\left(\begin{array}{c} b_{i} \\ \tau_i \end{array}\right) \sim N\left(\left(\begin{array}{c} 0 \\ 0 \end{array}\right),\left(\begin{array}{cc} \Sigma_{b} & \Sigma_{\tau b} \\ \Sigma_{\tau b}' & \Sigma_{\tau} \end{array}\right)\right)

Y_{i}(t_{ij}) = \tilde{Y}_i(t_{ij}) + \epsilon_{ij} = X_{ij}^{\top} \beta+Z_{ij}^{\top} b_{i}+\epsilon_{ij}

with X_{ij} and Z_{ij} two covariate vectors for subject i at visit j, respectively associated with the vector of fixed effects \beta and the vector of subject-specific individual random effects b_i. The vector b_i is assumed to be normally distributed and a specific-subject random effect on the variance of the measure error can be added: \epsilon_{ij} \sim \mathcal{N}(0,\sigma_i^2) and

\quad\left(\begin{array}{c} b_{i} \\ \log \sigma_{i} \end{array}\right) \sim \mathcal{N}\left(\left(\begin{array}{c} 0 \\ \mu_{\sigma} \end{array}\right),\left(\begin{array}{cc} \Sigma_{b} & 0 \\ 0 & \tau_{\sigma}^{2} \end{array}\right)\right)

B. SURVIVAL SUBMODEL

The risk function for the event $k = {1,2}$ is defined by: \lambda_{ik}(t)=\lambda_{0k}(t) \exp \left(W_{i}^{\top} \gamma_{k}+\alpha_{1k}\tilde{y}_i(t)+\\ \alpha_{2k}\tilde{y}'_i(t)+ \alpha_{\sigma k} \sigma_i(t) \right)

with \lambda_{0k}(t) the baseline risk function, W_{i} a vector of baseline covariates associated with the regression coefficient \gamma_k, and \alpha_{1k}, \alpha_{2k} and \alpha_{\sigma k} the regression coefficients associated with the current value \tilde{y}_i(t), the current slope \tilde{y}'_i(t) and the current variability \sigma_i(t) of the marker, respectively. Different parametric forms for the baseline risk function can be considered, such as exponential, Weibull, or, for more flexibility, a B-splines base.

Value

A FlexVarJoint object which contains the following elements :

result

A marqLevAlg object with the results of the estimation.

table.res

The table of results : Estimation and SE

time.compute

Computation time

control

A list of control elements

Examples

#fit a joint model with competing risks and subject-specific variability
example <- lsjm(formFixed = y~visit,
formRandom = ~ visit,
formGroup = ~ID,
formSurv = Surv(time, event ==1 ) ~ 1,
timeVar = "visit",
data.long = Data_toy,
variability_hetero = TRUE,
formFixedVar =~visit,
formRandomVar =~visit,
correlated_re = TRUE,
sharedtype = c("current value", "variability"),
hazard_baseline = "Weibull",
formSlopeFixed =~1,
formSlopeRandom = ~1,
indices_beta_slope = c(2), 
competing_risk = TRUE,
formSurv_CR = Surv(time, event ==2 ) ~ 1,
hazard_baseline_CR = "Weibull",
sharedtype_CR = c("current value", "variability"),
S1 = 100,
S2 = 1000,
nproc = 1,
maxiter = 100,
Comp.Rcpp = TRUE
)
 
summary(example)

FlexVarJM documentation built on Nov. 21, 2023, 1:06 a.m.