Estimation of joint latent process models

Description

Functions for the estimation of joint latent process models (JLPM). Continuous and ordinal outcomes are handled for the longitudinal part, whereas the survival part considers multiple competing events. The likelihood is computed using Monte-Carlo integration. Estimation is achieved by maximizing the log-likelihood using a robust iterative algorithm.

Details

Please report to the JLPM-team any question or suggestion regarding the package via github only (https://github.com/VivianePhilipps/JLPM/issues).

Author(s)

Viviane Philipps, Tiphaine Saulnier and Cecile Proust-Lima

References

Saulnier, Philipps, Meissner, Rascol, Pavy-Le-Traon, Foubert-Samier, Proust-Lima (2022). Joint models for the longitudinal analysis of measurement scales in the presence of informative dropout, Methods; 203:142-51.

Philipps, Hejblum, Prague, Commenges, Proust-Lima (2021). Robust and efficient optimization using a Marquardt-Levenberg algorithm with R package marqLevAlg, The R Journal 13:2.

For internal use only ...

Description

For internal use only ...

Fisher information function, used in Step 4 _ Selecting from the 4Smethod

Description

This function computes the contribution if each item during stages, based on the Fisher information.

Usage

IRT4Sselecting(D, proj)

Arguments

Value

a list of two matrices

###@examples

Author(s)

Tiphaine Saulnier, Viviane Philipps and Cecile Proust-Lima

References

preprint : arXiv:2407.08278

Projection function, used in Step 3 _ Staging from the 4Smethod

Description

This function determines the stage thresholds in the latent scale of the dimension process. Specifically, this function predicts the dimension scores corresponding to stage changes and deducts the corresponding thresholds in the dimension scale based on a correspondence between the predicted scores and the expected sum of the items. The output is a vector of the estimated stage transition thresholds.

Usage

IRT4Sstaging(D, Dss, nsim = 1000, bounds = c(-10, 10))

Arguments

Value

a vector of the estimated stage transition thresholds

###@examples

Author(s)

Tiphaine Saulnier, Viviane Philipps and Cecile Proust-Lima

References

preprint : arXiv:2407.08278

REconddendity

Description

Wrapper to the Fortran subroutine computing the conditional density of the random effects

Usage

REconddensity(
  ui0,
  Y0,
  X0,
  Tentr0,
  Tevt0,
  Devt0,
  ind_survint0,
  idea0,
  idg0,
  idcor0,
  idcontr0,
  idsurv0,
  idtdv0,
  typrisq0,
  nz0,
  zi0,
  nbevt0,
  idtrunc0,
  logspecif0,
  ny0,
  nv0,
  nobs0,
  nmes0,
  idiag0,
  ncor0,
  nalea0,
  npm0,
  b0,
  epsY0,
  idlink0,
  nbzitr0,
  zitr0,
  uniqueY0,
  indiceY0,
  nvalSPLORD0,
  idst0,
  nXcl0,
  Xcl_Ti0,
  Xcl_GK0,
  Xcs_Ti0,
  Xcs_GK0,
  nonlin0,
  centerpoly0
)

Arguments

Value

the log-density

Standard methods for estimated models

Description

coef and vcov methods for estimated jointLPM models.

Usage

## S3 method for class 'jointLPM'
coef(object, ...)

Arguments

Value

For coef, the vector of the estimated parameters.

For vcov, the variance-covariance matrix of the estimated parameters.

Author(s)

Viviane Philipps

WaldMult Performs multivariate Wald tests.

Description

Given a vector of estimated parameters (coef = (theta_1, ... theta_n)') and their associated variance matrix (vcov = Var(coef)), the WaldMult function performs either the test of the null hypothesis :

Usage

WaldMult(
  Mod,
  coef,
  vcov,
  pos = NULL,
  contrasts = NULL,
  A = NULL,
  name = NULL,
  value = 0
)

Arguments

Details

1. H0 : theta_m = (theta_j1, ..., theta_jm)' = 0 ie, tests if a subset of m parameters are simultaneously equal to zero. The Wald statistic is w = theta_m' * Var(theta_m)^(-1) * theta_m and the associated p-value is obtained by p = P(chi2_(dof=m) > w)

2. H0 : theta = c1*theta_j1 + ... + cm*theta_jm = 0 ie, tests if a combination of m parameters is equal to zero. The Wald statistic is w = theta^2 / Var(theta) = (c1*theta_j1 + ... + cm*theta_jm)^2 / Var(c1*theta_j1 + ... + cm*theta_jm) and the associated p-value is obtained by p = P(chi2_(dof=1) > w)

3. H0 : Theta = A*coef = 0 with A a matrix with m lines and n columns. ie, tests if multiple combinations of parameters are simultaneously equal to zero. The Wald statistic is w = Theta * Var(Theta)^(-1) * Theta' = (A*coef)' * (A*vcov*A')^(-1) * (A*coef) and the associated p-value is obtained by p = P(chi2_(dof=m) > w)

The function can be applied to any object having a coef

Value

If contrasts is NULL, the function returns a matrix with 1 row and 2 columns containing the value of the Wald test's statistic and the associated p-value.

If contrasts is not NULL, the function returns a matrix with 1 row and 4 columns containing the value of the coefficient (dot product of pos and contrasts), his standard deviation, the value of the Wald test's statistic and the associated p-value.

If A is specified, the function returns a matrix with m rows and 2 columns containing the value of the Wald test's statistic and the associated p-value.

Examples

require(lcmm)
m <- hlme(Y ~ Time + X2, random = ~ 1, subject = "ID", data = data_hlme)
summary(m)

mm <- hlme(Y ~ -1 + Time + factor(X2), random = ~ 1, subject = "ID", data = data_hlme)
summary(mm)

## Retrieve from model mm the difference between X2 levels as in model m :
WaldMult(Mod = mm, pos = c(2, 3), contrasts = c(-1, 1))
## or
WaldMult(coef = coef(mm), vcov = vcov(mm), A = matrix(c(0, -1, 1, 0, 0), 1, 5))

Conversion

Description

This function converts a jointLPM object to an object of type multlcmm or Jointlcmm from lcmm package. The conversion to multlcmm unable the use of the dedicated postfit and predictions functions implemented in the lcmm package (plot, predictL, predictY, predictlink, predictYcond, fitY, VarExpl). The conversion to Jointlcmm permits the use of functions cuminc and plot (with options "baselinerisk" or "survival").

Usage

convert(object, to)

Arguments

Value

an object of class multlcmm or Jointlcmm.

Likelihood arguments

Description

Create the arguments to be passed to the loglik function

Usage

createFargs(x, data)

Arguments

Value

a named list

createX0

Description

Create the matrix containing all covariates appearing in the formula

Usage

createX0(form, data)

Arguments

Details

Caution 1 : 'data' should not contain any missing values. There is no check for that within the function. Caution 2 : if interaction terms are present, the names are sorted. For example x:age becomes age:x in X0. Caution 3 : intercept is not systematically in the first column. It will only if form[[1]] contains an intercept.

Value

a list containing

Difference of sojourn times

Description

This function computes the difference between two sojourn times.

Usage

diffSojournTime(x1, x2)

Arguments

Value

returns the median, the 2.5% and 97.5% quantiles, the mean, the standard deviation of the difference between the two sojourn times and the number of removed draws (eventually due to computational issues)

Estimation of latent process joint models for multivariate longitudinal outcomes and time-to-event data.

Description

This function fits extended joint models with shared random effects. The longitudinal submodel handles multiple continuous longitudinal outcomes (Gaussian or non-Gaussian, curvilinear) as well as ordinal longitudinal outcomes in a mixed effect framework. The model assumes that all the outcomes measure the same underlying latent process defined as their common factor, and each outcome is related to this latent common factor by a outcome-specific measurement model whose nature depends on the type of the outcome (linear model for Gaussian outcome, curvilinear model for non-Gaussian outcome, probit model for binary outcome, cumulative probit model for ordinal outcome). At the latent process level, the model assumes a linear mixed model. The survival submodel handles right-censored (possibly left-truncated) time-to-event with competing causes The association between the longitudinal and the survival data is captured by including the random effect from the mixed model or the predicted current level of the underlying process as linear predictors in the cause-specific proportional hazard survival model. Parameters of the measurement models, of the latent process mixed model and of the survival model are simultaneously estimated using a maximum likelihood method, through a Marquardt-Levenberg optimization algorithm.

Usage

jointLPM(
  fixed,
  random,
  subject,
  idiag = FALSE,
  cor = NULL,
  link = "linear",
  intnodes = NULL,
  epsY = 0.5,
  randomY = FALSE,
  var.time,
  survival = NULL,
  hazard = "Weibull",
  hazardrange = NULL,
  hazardnodes = NULL,
  TimeDepVar = NULL,
  logscale = FALSE,
  startWeibull = 0,
  sharedtype = "RE",
  centerpoly = 0,
  methInteg = "QMC",
  nMC = 1000,
  data,
  subset = NULL,
  na.action = 1,
  B,
  posfix = NULL,
  maxiter = 100,
  convB = 1e-04,
  convL = 1e-04,
  convG = 1e-04,
  partialH = NULL,
  nsim = 100,
  range = NULL,
  verbose = TRUE,
  returndata = FALSE,
  nproc = 1,
  clustertype = NULL
)

Arguments

Details

A. THE MEASUREMENT MODELS

jointLPM function estimates one measurement model per outcome to link each outcome Y_k(t) with the underlying latent common factor L(t) they measure. To fix the latent process dimension, we chose to constrain the latent process dimension with the intercept of the mixed model fixed at 0 and the standard error of the first random effect fixed at 1. The nature of each measurement model adapts to the type of the outcome it models.

1. For continuous Gaussian outcomes, linear models are used and require 2 parameters for the transformation (Y(t) - b1)/b2

2. For continuous non-Gaussian outcomes, curvilinear models use parameterized link functions to link outcomes to the latent process. With the "beta" link function, 4 parameters are required for the following transformation: [ h(Y(t)',b1,b2) - b3]/b4 where h is the Beta CDF with canonical parameters c1 and c2 that can be derived from b1 and b2 as c1=exp(b1)/[exp(b2)*(1+exp(b1))] and c2=1/[exp(b2)*(1+exp(b1))], and Y(t)' is the rescaled outcome i.e. Y(t)'= [ Y(t) - min(Y(t)) + epsY ] / [ max(Y(t)) - min(Y(t)) +2*epsY ]. With the "splines" link function, n+2 parameters are required for the following transformation b_1 + b_2*I_1(Y(t)) + ... + b_(n+2)*I_(n+1)(Y(t)), where I_1,...,I_(n+1) is the basis of quadratic I-splines. To constrain the parameters to be positive, except for b_1, the program estimates b_k^* (for k=2,...,n+2) so that b_k=(b_k^*)^2.

3. For ordinal outcomes (and binary outcomes), cumulative probit models are used. For a (n+1)-level outcome, the model consist of determining n thresholds t_k in the latent process scale which correspond to the outcome level changes. Then, Y(t) = n' <=> t_n' < L(t) + e <= t_(n'+1) with e the standard error of the outcome. To ensure that t_1 < t_2 < ... < t_n, the program estimates t'_1, t'_2, ..., t'_n such that t_1=t'_1, t_2=t_1+(t'_2)^2, t_3=t_2+(t'_3)^2, ...

B. THE SURVIVAL MODEL

a. BASELINE RISK FUNCTIONS

For the baseline risk functions, the following parameterizations are considered.

1. With the "Weibull" function: 2 parameters are necessary w_1 and w_2 so that the baseline risk function a_0(t) = w_1^2*w_2^2*(w_1^2*t)^(w_2^2-1) if logscale=FALSE and
a_0(t) = exp(w_1)*exp(w_2)(t)^(exp(w_2)-1) if logscale=TRUE.

2. with the "piecewise" step function and nz nodes (y_1,...y_nz), nz-1 parameters are necesssary p_1,...p_nz-1 so that the baseline risk function a_0(t) = p_j^2 for y_j < t =< y_j+1 if logscale=FALSE and a_0(t) = exp(p_j) for y_j < t =< y_j+1 if logscale=TRUE.

3. with the "splines" function and nz nodes (y_1,...y_nz), nz+2 parameters are necessary s_1,...s_nz+2 so that the baseline risk function a_0(t) = sum_j s_j^2 M_j(t) if logscale=FALSE and a_0(t) = sum_j exp(s_j) M_j(t) if logscale=TRUE where M_j is the basis of cubic M-splines. Two parameterizations of the baseline risk function are proposed (logscale=TRUE or FALSE) because in some cases, especially when the instantaneous risks are very close to 0, some convergence problems may appear with one parameterization or the other. As a consequence, we recommend to try the alternative parameterization (changing logscale option) when a model does not converge (maximum number of iterations reached) although convergence criteria based on the parameters and likelihood are small.

b. ASSOCIATION BETWEEN LONGITUDINAL AND SURVIVAL DATA

The association between the longitudinal and the survival data is captured by including a function of the elements from the latent process mixed model as a predictor in the survival model. We implemented so far two association structures, that should be specified through sharedtype argument.

1. the random effect from the latent process linear mixed model (sharedtype='RE'): the q random effects modeling the individual deviation in the longitudinal model are also included in the survival model, so that a q-vector of parameters measures the association between the risk of event and the longitudinal outcome(s).

2. the predicted current level of the underlying process (sharedtype='CL'): the predicted latent process defined by the mixed model at time t is included as time-dependent covariate in the risk of event model for time t. The association between the longitudinal process and the risk of event is then quantified by a unique parameter.

C. THE VECTOR OF PARAMETERS B

The parameters in the vector of initial values B or in the vector of maximum likelihood estimates best are included in the following order: (1) parameters for the baseline risk function: 2 parameters for each Weibull, nz-1 for each piecewise constant risk and nz+2 for each splines risk. In the presence of competing events, the number of parameters should be adapted to the number of causes of event; (2) for all covariates in survival, one parameter is required. Covariates parameters should be included in the same order as in survival. In the presence of cause-specific effects, the number of parameters should be multiplied by the number of causes; (3) parameter(s) of association between the longitudinal and the survival process: for sharedtype='RE', one parameter per random effect and per cause of event is required; for sharedtype='CL', one parameter per cause of event is required; (4) for all covariates in fixed, one parameter is required. Parameters should be included in the same order as in fixed; (5)for all covariates included with contrast() in fixed, one supplementary parameter per outcome is required excepted for the last outcome for which the parameter is not estimated but deduced from the others; (6) the variance of each random-effect specified in random (excepted the intercept which is constrained to 1) if idiag=TRUE and the inferior triangular variance-covariance matrix of all the random-effects if idiag=FALSE; (7) if cor is specified, the standard error of the Brownian motion or the standard error and the correlation parameter of the autoregressive process; (8) parameters of each measurement model: 2 for "linear", 4 for "beta", n+2 for "splines" with n nodes, n for "thresholds" for a (n+1)-level outcome; (9) if randomY=TRUE, the standard error of the outcome-specific random intercept (one per outcome); (10) the outcome-specific standard errors (one per outcome)

C. CAUTIONS REGARDING THE USE OF THE PROGRAM

Some caution should be made when using the program. Convergence criteria are very strict as they are based on the derivatives of the log-likelihood in addition to the parameter and log-likelihood stability. In some cases, the program may not converge and reach the maximum number of iterations fixed at 100 by default. In this case, the user should check that parameter estimates at the last iteration are not on the boundaries of the parameter space.

If the parameters are on the boundaries of the parameter space, the identifiability of the model is critical. This may happen especially with splines parameters that may be too close to 0 (lower boundary). When identifiability of some parameters is suspected, the program can be run again from the former estimates by fixing the suspected parameters to their value with option posfix. This usually solves the problem. An alternative is to remove the parameters of the Beta of Splines link function from the inverse of the Hessian with option partialH. If not, the program should be run again with other initial values, with a higher maximum number of iterations or less strict convergence tolerances.

To reduce the computation time, this program can be carried out in parallel mode, ie. using multiple cores which number can be specified with argument nproc.

Value

An object of class "jointLPM" is returned containing some internal information used in related functions. Users may investigate the following elements :

Author(s)

Viviane Philipps, Tiphaine Saulnier and Cecile Proust-Lima

References

Saulnier, Philipps, Meissner, Rascol, Pavy-Le-Traon, Foubert-Samier, Proust-Lima (2022). Joint models for the longitudinal analysis of measurement scales in the presence of informative dropout, Methods; 203:142-51.

Philipps, Hejblum, Prague, Commenges, Proust-Lima (2021). Robust and efficient optimization using a Marquardt-Levenberg algorithm with R package marqLevAlg, The R Journal 13:2.

Examples

#### Examples with paquid data from R-package lcmm
library(lcmm)
paq <- paquid[which(paquid$age_init<paquid$agedem),]
paq$age65 <- (paq$age-65)/10

#### Example with one Gaussian marker :
## We model the cognitive test IST according to age, sexe and eduction level. We assume
## a Weibull distribution for the time to dementia and link the longitudinal and survival
## data using the random effects.
## We provide here the call to the jointLPM function without optimization (maxiter=0). The
## results should therefore not be interpreted.
M0 <- jointLPM(fixed = IST~age65*(male+CEP),
                random=~age65,
                idiag=FALSE,
                subject="ID",
                link="linear",
                survival=Surv(age_init,agedem,dem)~male,
                sharedtype='RE',
                hazard="Weibull",
                data=paq,
                var.time="age65",
                maxiter=0)
M0$best ## these are the initial values of each of the 15 parameters

 
## Estimation with one Gaussian marker
## We remove the maxiter=0 option to estimate the model. We specify initial values
## to reduce the runtime, but this can take several minutes.
binit1 <- c(0.1039, 5.306, -0.1887, -1.0355, -4.3817, -1.0543, -0.1161, 0.8588,
0.0538, -0.1722, -0.2224, 0.3296, 30.7768, 4.6169, 0.7396)
M1 <- jointLPM(fixed = IST~age65*(male+CEP),
                random=~age65,
                idiag=FALSE,
                subject="ID",
                link="linear",
                survival=Surv(age_init,agedem,dem)~male,
                sharedtype='RE',
                hazard="Weibull",
                data=paq,
                var.time="age65",
                B=binit1)
## Optimized the parameters to be interpreted :
summary(M1)


#### Estimation with one ordinal marker :
## We consider here the 4-level hierarchical scale of dependence HIER and use "thresholds"
## to model it as an ordinal outcome. We assume an association between the current level
## of dependency and the risk of dementia through the option sharedtype="CL".
## We use a parallel optimization on 2 cores to reduce computation time.
binit2 <- c(0.0821, 2.4492, 0.1223, 1.7864, 0.0799, -0.2864, 0.0055, -0.0327, 0.0017,
0.3313, 0.9763, 0.9918, -0.4402)
M2 <- jointLPM(fixed = HIER~I(age-65)*male,
                random = ~I(age-65),
                subject = "ID", 
                link = "thresholds",
                survival = Surv(age_init,agedem,dem)~male,
                sharedtype = 'CL',
                var.time = "age",
                data = paq, 
                methInteg = "QMC", 
                nMC = 1000,
                B=binit2,
                nproc=2)
summary(M2)

Wrapper to the Fortran subroutine that computes the log-likelihood

Description

Computes the log-likelihood of a jointLPM model

Usage

loglik(
  b0,
  Y0,
  X0,
  Tentr0,
  Tevt0,
  Devt0,
  ind_survint0,
  idea0,
  idg0,
  idcor0,
  idcontr0,
  idsurv0,
  idtdv0,
  typrisq0,
  nz0,
  zi0,
  nbevt0,
  idtrunc0,
  logspecif0,
  ny0,
  ns0,
  nv0,
  nobs0,
  nmes0,
  idiag0,
  ncor0,
  nalea0,
  npm0,
  nfix0,
  bfix0,
  epsY0,
  idlink0,
  nbzitr0,
  zitr0,
  uniqueY0,
  indiceY0,
  indiceYinf0,
  nvalSPLORD0,
  fix0,
  methInteg0,
  nMC0,
  dimMC0,
  seqMC0,
  idst0,
  nXcl0,
  Xcl_Ti0,
  Xcl_GK0,
  Xcs_Ti0,
  Xcs_GK0,
  nonlin0,
  centerpoly0,
  expectancy0
)

Arguments

Value

the log-likelihood

Random effects prediction

Description

The random effects of a jointLPM model are predicted as the mode of the conditional distribution of the random effects given the subject's longitudinal and survival observations.

Usage

predictRE(x, data, control = NULL, variance = FALSE)

Arguments

Value

a matrix containing the predicted random effects (in columns) for each subject (in rows). If variance = TRUE, returns a list containing the previous matrix and a vector containing the upper part of each variance matrix.

Brief summary of a joint latent process model

Description

This function provides a brief summary of model estimated with the jointLPM function.

Usage

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

Arguments

Value

the function is used for its side effects, no value is returned.

Author(s)

Viviane Philipps, Tiphaine Saulnier and Cecile Proust-Lima

removeNA

Description

Remove missing values

Usage

removeNA(form, data)

Arguments

Details

Caution : if several response variables are included in 'form', then the resulting data frame will contain one column (named outcome) grouping all these response variables. The irst lines will refer to the first response variables, a second block to the second, etc. All covariates will be repeated for each response variable.

Value

a list containing

Examples

 d <- data.frame(y1=c(3,6,4,2), y2=c(6.5,NA,9.5,4.5), x=c(0,0,1,0))
 removeNA(form=list(y1+y2~x), data=d)
## 1      3.0 0   | first block referring to y1
## 2      6.0 0   |
## 3      4.0 1   |
## 4      2.0 0   |
## 11     6.5 0     | second block referring to y2
## 31     9.5 1     |
## 41     4.5 0     |

Computation of sojourn time under specific condition

Description

This function computes posterior computations from a joint shared random-effect model with ordinal longitudinal outcomes, aka a joint item response theory model. Specifically, the function computes the expected time spent before reaching a given level of impairment specified by one or multiple items for a covariate profile.

Usage

sojournTime(
  x,
  maxState,
  condState = NULL,
  newdata,
  var.time,
  startTime = 0,
  nMC = 1000,
  upperTime = 150,
  subdivisions = 100L,
  rel.tol = .Machine$double.eps^0.25,
  draws = FALSE,
  ndraws = 2000,
  returndraws = FALSE,
  cl = NULL,
  seed = NULL
)

Arguments

Details

1. Lifetime expected sojourn time: the expected time before reaching level k at item(s) Y is the integral over time t, from 0 to infinity, of P(Y(t) <= k, T > t), i.e., int_0^infty P(Y(t) <= k, T > t) dt.

2. Residual expected sojourn time from a time s: conditionally on being below m at item(s) Y at time s and being alive at time s, the sojourn time below level k at item(s) Y is computed as: int_s^infty P(Y(t) <= k, T > t | Y(s) <= m, T > s) dt = int_s^infty P(Y(t) <= k, T > t, Y(s) <= m) dt / P(Y(s) <= m, T > s)

Value

if draws = FALSE, returns a single value. If draws = TRUE and returndraws = FALSE, returns the median, the 2.5% and 97.5% quantiles, the mean, the standard deviation and the number of removed draws (eventually due to computational issues). If draws = TRUE and returndraws = TRUE, returns the ndraws values.

Author(s)

Viviane Philipps and Cecile Proust-Lima

Examples

library(lcmm)
paq <- paquid[which(paquid$age_init < paquid$agedem), ]
paq$age65 <- (paq$age - 65) / 10
paq$ageinit65 <- (paq$age_init - 65) / 10
paq$agedem65 <- (paq$agedem - 65) / 10

#### Estimation of the joint model with one ordinal longitudinal item
## Not run: 
M2 <- jointLPM(fixed = HIER ~ age65 * male,
                random = ~ age65,
                subject = "ID", 
                link = "thresholds",
                survival = Surv(ageinit65, agedem65, dem) ~ male,
                sharedtype = 'RE',
                var.time = "age65",
                data = paq, 
                methInteg = "QMC", 
                nMC = 1000,
                B = c(0.6, 2.399, -0.409, 2.076, 6.338, 0.994, -0.223, -0.005,
                     -0.299, 0.174, 0.523, 1.044, 1.064, 0.506))

#### Computation of the expected lifetime sojourn time with HIER impairment
up to 2 (HIER = 2)
#### (=int_0^150 P(HIER(t) <= 2, T > t) dt)
sojournTime(M2, list(HIER = 2), newdata = data.frame(male = 0), 
var.time = "age65")

#### Computation of the expected residual time with maximum HIER impairment
 of 2 (HIER = 2), given the impairment was 1 at time 0.5
#### (=int_0.5^150 P(HIER(t) <= 2, T > t | HIER(0.5) <= 1, T > 0.5) dt)
sojournTime(M2, list(HIER = 2), condState = list(HIER = 1), startTime = 0.5,
newdata = data.frame(male = 0), var.time = "age65")

## End(Not run)

Summary of a joint latent process model

Description

This function provides a summary of model estimated with the jointLPM function.

Usage

## S3 method for class 'jointLPM'
summary(object, ...)

Arguments

Value

The function is mainly used for its side effects. It returns invisibily a list of two matrices containing the estimates, their standard errors, Wald statistics and associated p-values for the survival submodel (first element of the list) and for the mixed model's fixed effects (second element).

Author(s)

Viviane Philipps, Tiphaine Saulnier and Cecile Proust-Lima