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

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

fixed: a two-sided linear formula object for specifying the fixed-effects in the linear mixed model at the latent process level. The response outcomes are separated by + on the left of ~ and the covariates are separated by + on the right of the ~. For identifiability purposes, the intercept specified by default should not be removed by a -1. Variables on which a contrast above the different outcomes should also be estimated are included with contrast().
random: a one-sided formula for the random-effects in the latent process mixed model. Covariates with a random-effect are separated by +. An intercept should always be included for identifiability.
subject: name of the covariate representing the grouping structure.
idiag: optional logical for the variance-covariance structure of the random-effects. If FALSE, a non structured matrix of variance-covariance is considered (by default). If TRUE a diagonal matrix of variance-covariance is considered.
cor: optional indicator for inclusion of an autocorrelated Gaussian process in the linear mixed model at the latent process level. Option BM(time) indicates a brownian motion with parameterized variance while option AR(time) specifies an autoregressive process in continuous time with parameterized variance and correlation intensity. In both cases, time is the variable representing the measurement times. By default, no autocorrelated Gaussian process is added.
link: optional vector of families of parameterized link functions defining the measurement models (one by outcome). Option "linear" (by default) specifies a linear link function. Other possibilities include "beta" for estimating a link function from the family of Beta cumulative distribution functions, "Splines" for approximating the link function by I-splines and "thresholds" for ordinal outcomes modelled by cumulative probit models. For splines case, the number of nodes and the nodes location should be also specified. The number of nodes is first entered followed by -, then the location is specified with "equi", "quant" or "manual" for respectively equidistant nodes, nodes at quantiles of the marker distribution or interior nodes entered manually in argument intnodes. It is followed by -splines. For example, "7-equi-splines" means I-splines with 7 equidistant nodes, "6-quant-splines" means I-splines with 6 nodes located at the quantiles of the marker distribution and "9-manual-splines" means I-splines with 9 nodes, the vector of 7 interior nodes being entered in the argument intnodes.
intnodes: optional vector of interior nodes. This argument is only required for a I-splines link function with nodes entered manually.
epsY: optional positive real used to rescale the marker in (0,1) when the beta link function is used. By default, epsY=0.5.
randomY: optional logical for including an outcome-specific random intercept. If FALSE no outcome-specific random intercept is added (default). If TRUE independent outcome-specific random intercepts with parameterized variance are included.
var.time: name of the variable representing the measurement times.
survival: two-sided formula object. The left side of the formula corresponds to a Surv() object for right-censored (Surv(EntryTime,Time,Indicator)) and possibly left-truncated (Surv(EntryTime,Time,Indicator)). Multiple causes of event can be considered in the Indicator (0 for censored, k for event of cause k). The right side of the formula specifies the covariates to include in the survival model. Cause-specific covariate effect are specified with cause().For example, Surv(Time,Indicator) ~ X1 + cause(X2) indicates a common effect of X1 and a cause-specific effect of X2.
hazard: optional family of hazard function assumed for the survival model. By default, "Weibull" specifies a Weibull baseline risk function. Other possibilities are "piecewise" for a piecewise constant risk function or "splines" for a cubic M-splines baseline risk function. For these two latter families, the number of nodes and the location of the nodes should be specified as well, separated by -. The number of nodes is entered first followed by -, then the location is specified with "equi", "quant" or "manual" for respectively equidistant nodes, nodes at quantiles of the times of event distribution or interior nodes entered manually in argument hazardnodes. It is followed by - and finally "piecewise" or "splines" indicates the family of baseline risk function considered. Examples include "5-equi-splines" for M-splines with 5 equidistant nodes, "6-quant-piecewise" for piecewise constant risk over 5 intervals and nodes defined at the quantiles of the times of events distribution and "9-manual-splines" for M-splines risk function with 9 nodes, the vector of 7 interior nodes being entered in the argument hazardnodes. In the presence of competing events, a vector of hazards should be provided such as hazard=c("Weibull","5-quant-splines") with 2 causes of event, the first one modelled by a Weibull baseline cause-specific risk function and the second one by splines.
hazardrange: optional vector indicating the range of the survival times (that is the minimum and maximum). By default, the range is defined according to the minimum and maximum observed values of the survival times. The option should be used only for piecewise constant and Splines hazard functions.
hazardnodes: optional vector containing interior nodes if splines or piecewise is specified for the baseline hazard function in hazard.
TimeDepVar: optional vector containing an intermediate time corresponding to a change in the risk of event. This time-dependent covariate can only take the form of a time variable with the assumption that there is no effect on the risk before this time and a constant effect on the risk of event after this time (example: initiation of a treatment to account for).
logscale: optional boolean indicating whether an exponential (logscale=TRUE) or a square (logscale=FALSE -by default) transformation is used to ensure positivity of parameters in the baseline risk functions. See details section
startWeibull: optional numeric with Weibull hazard functions only. Indicates the shift in the Weibull distribution.
sharedtype: indicator of shared random function type. 'RE' indicates an association through the random effects included in the linear mixed model. 'CL' defines a association through the predicted current level of the latent process.
centerpoly: for polynomial associations only, value(s) to center degree 2 and 3 polynomial terms.
methInteg: character indicating the type of integration to compute the log-likelihood. 'MCO' for ordinary Monte Carlo, 'MCA' for antithetic Monte Carlo, 'QMC' for quasi Monte Carlo. Default to "QMC".
nMC: integer, number of Monte Carlo simulations. Default to 1000.
data: data frame containing all variables named in fixed, random, cor, survival and subject.
subset: optional vector giving the subset of observations in data to use. By default, all lines.
na.action: Integer indicating how NAs are managed. The default is 1 for 'na.omit'. The alternative is 2 for 'na.fail'. Other options such as 'na.pass' or 'na.exclude' are not implemented in the current version.
B: optional specification for the initial values for the parameters. Initial values should be entered in the order detailed in details section.
posfix: Optional vector giving the indices in vector B of the parameters that should not be estimated. Default to NULL, all parameters are estimated.
maxiter: optional maximum number of iterations for the Marquardt iterative algorithm. By default, maxiter=100.
convB: optional threshold for the convergence criterion based on the parameter stability. By default, convB=0.0001.
convL: optional threshold for the convergence criterion based on the log-likelihood stability. By default, convL=0.0001.
convG: optional threshold for the convergence criterion based on the derivatives. By default, convG=0.0001.
partialH: optional vector giving the indices in vector B of parameters that can be dropped from the Hessian matrix to define convergence criteria.
nsim: number of points used to plot the estimated link functions. By default, nsim=100.
range: optional vector indicating the range of the outcomes (that is the minimum and maximum). By default, the range is defined according to the minimum and maximum observed values of the outcome. The option should be used only for Beta and Splines transformations.
verbose: logical indicating if information about computation should be reported. Default to TRUE.
returndata: logical indicating if data used for computation should be returned. Default to FALSE, data are not returned.
nproc: number of cores for parallel computation.
clustertype: the type of cluster that should internally be created. See parallel::makeCluster for possible values.

Value

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

ns: number of grouping units in the dataset
loglik: log-likelihood of the model
best: vector of parameter estimates in the same order as specified in B and detailed in section details
V: vector containing the upper triangle matrix of variance-covariance estimates of best with exception for variance-covariance parameters of the random-effects for which V contains the variance-covariance estimates of the Cholesky transformed parameters displayed in cholesky
gconv: vector of convergence criteria: 1. on the parameters, 2. on the likelihood, 3. on the derivatives
conv: status of convergence: =1 if the convergence criteria were satisfied, =2 if the maximum number of iterations was reached, =4 or 5 if a problem occured during optimisation
call: the matched call
niter: number of Marquardt iterations
nevent: number of occured event
pred: table of individual predictions and residuals in the underlying latent process scale; it includes marginal predictions (pred_m), marginal residuals (resid_m), subject-specific predictions (pred_ss) and subject-specific residuals (resid_ss) and finally the transformed observations in the latent process scale (obs).
predRE: table containing individual predictions of the random-effects : a column per random-effect, a line per subject.
predRE_Y: table containing individual predictions of the outcome-specific random intercept
predSurv: table containing the predicted baseline risk function and the predicted cumulative baseline risk function
cholesky: vector containing the estimates of the Cholesky transformed parameters of the variance-covariance matrix of the random-effects
estimlink: table containing the simulated values of each outcome and the corresponding estimated link function
epsY: definite positive reals used to rescale the markers in (0,1) when the beta link function is used. By default, epsY=0.5.
AIC: the Akaike's information criterion
BIC: the Bayesian information criterion
CPUtime: the runtime in seconds
data: the original data set (if returndata is TRUE)

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.

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.

Author

Viviane Philipps, Tiphaine Saulnier and Cecile Proust-Lima

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
#> event1 +/-sqrt(Weibull1) event1 +/-sqrt(Weibull2)                     male 
#>                  1.00000                  1.00000                  0.00000 
#>               event1 RE1               event1 RE2                    age65 
#>                  0.00000                  0.00000                  0.00000 
#>                     male                      CEP               age65:male 
#>                  0.00000                  0.00000                  0.00000 
#>                age65:CEP                 varcov 1                 varcov 2 
#>                  0.00000                  0.00000                  1.00000 
#>                 Linear 1                 Linear 2                std.err 1 
#>                 26.52485                  1.00000                  1.00000 

# \donttest{ 
## 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)
#> ------------------ iteration  0 ------------------
#> Function value  -6453.469 
#> Convergence criteria: parameters stability= 1.0001 
#>                     : function stability= 1.0001 
#>                     : relative distance to maximum(RDM)= 0.01591934 
#>                   coef
#> parameter1   0.1039000
#> parameter2   5.3060000
#> parameter3  -0.1887000
#> parameter4  -1.0355000
#> parameter5  -4.3817000
#> parameter6  -1.0543000
#> parameter7  -0.1161000
#> parameter8   0.8588000
#> parameter9   0.0538000
#> parameter10 -0.1722000
#> parameter11 -0.2224000
#> parameter12  0.5292809
#> parameter13 30.7768000
#> parameter14  4.6169000
#> parameter15  0.7396000
#> 
#> ------------------ iteration  1 ------------------
#> Function value  -6453.407 
#> Convergence criteria: parameters stability= 9.175e-05 
#>                     : function stability= 0.06247235 
#>                     : relative distance to maximum(RDM)= 0.00731849 
#>                    coef
#> parameter1   0.10386857
#> parameter2   5.30394308
#> parameter3  -0.18748354
#> parameter4  -1.03785060
#> parameter5  -4.37931555
#> parameter6  -1.05150809
#> parameter7  -0.11467558
#> parameter8   0.85610214
#> parameter9   0.05201593
#> parameter10 -0.17438490
#> parameter11 -0.22255581
#> parameter12  0.52235956
#> parameter13 30.77764754
#> parameter14  4.61601298
#> parameter15  0.74018106
#> 
#> ------------------ iteration  2 ------------------
#> Function value  -6453.393 
#> Convergence criteria: parameters stability= 0.00059179 
#>                     : function stability= 0.01411669 
#>                     : relative distance to maximum(RDM)= 0.0053065 
#>                    coef
#> parameter1   0.10385797
#> parameter2   5.29104424
#> parameter3  -0.17913610
#> parameter4  -1.04401250
#> parameter5  -4.36517003
#> parameter6  -1.04860593
#> parameter7  -0.10973426
#> parameter8   0.85088839
#> parameter9   0.04824421
#> parameter10 -0.17377508
#> parameter11 -0.22231010
#> parameter12  0.52079302
#> parameter13 30.78197871
#> parameter14  4.62062144
#> parameter15  0.73945238
#> 
#> ------------------ iteration  3 ------------------
#> Function value  -6453.367 
#> Convergence criteria: parameters stability= 0.00673365 
#>                     : function stability= 0.02561852 
#>                     : relative distance to maximum(RDM)= 0.00163754 
#>                    coef
#> parameter1   0.10384250
#> parameter2   5.25410826
#> parameter3  -0.16170611
#> parameter4  -1.04523276
#> parameter5  -4.29865215
#> parameter6  -1.04610474
#> parameter7  -0.10579579
#> parameter8   0.84253869
#> parameter9   0.04402949
#> parameter10 -0.17061547
#> parameter11 -0.22403884
#> parameter12  0.51968939
#> parameter13 30.79965596
#> parameter14  4.63465623
#> parameter15  0.73689152
#> 
#> ------------------ iteration  4 ------------------
#> Function value  -6453.356 
#> Convergence criteria: parameters stability= 0.00977786 
#>                     : function stability= 0.01161208 
#>                     : relative distance to maximum(RDM)= 3.689e-05 
#>                    coef
#> parameter1   0.10383639
#> parameter2   5.21631330
#> parameter3  -0.15650882
#> parameter4  -1.03486659
#> parameter5  -4.21019917
#> parameter6  -1.04813378
#> parameter7  -0.10513702
#> parameter8   0.83842698
#> parameter9   0.04279221
#> parameter10 -0.16839464
#> parameter11 -0.22497477
#> parameter12  0.52072977
#> parameter13 30.81858515
#> parameter14  4.63298088
#> parameter15  0.73694547
#> 
#> ------------------ iteration  5 ------------------
#> Function value  -6453.355 
#> Convergence criteria: parameters stability= 0.00030411 
#>                     : function stability= 0.00027544 
#>                     : relative distance to maximum(RDM)= 2e-08 
#>                    coef
#> parameter1   0.10383609
#> parameter2   5.21017374
#> parameter3  -0.15661830
#> parameter4  -1.03218977
#> parameter5  -4.19458272
#> parameter6  -1.04896199
#> parameter7  -0.10516274
#> parameter8   0.83813919
#> parameter9   0.04274243
#> parameter10 -0.16808829
#> parameter11 -0.22497073
#> parameter12  0.52111390
#> parameter13 30.82159457
#> parameter14  4.63070562
#> parameter15  0.73729071
#> 
#> ------------------ iteration  6 ------------------
#> Function value  -6453.355 
#> Convergence criteria: parameters stability= 2e-07 
#>                     : function stability= 1.7e-07 
#>                     : relative distance to maximum(RDM)= 0 
#>                    coef
#> parameter1   0.10383607
#> parameter2   5.21001639
#> parameter3  -0.15661941
#> parameter4  -1.03212040
#> parameter5  -4.19418524
#> parameter6  -1.04898353
#> parameter7  -0.10516464
#> parameter8   0.83813440
#> parameter9   0.04274189
#> parameter10 -0.16808109
#> parameter11 -0.22497084
#> parameter12  0.52112452
#> parameter13 30.82166408
#> parameter14  4.63064416
#> parameter15  0.73730022
#> 
## Optimized the parameters to be interpreted :
summary(M1)
#> Joint latent process model with shared random effects 
#>      fitted by maximum likelihood method 
#>  
#> jointLPM(fixed = IST ~ age65 * (male + CEP), random = ~age65, 
#>     subject = "ID", idiag = FALSE, link = "linear", var.time = "age65", 
#>     survival = Surv(age_init, agedem, dem) ~ male, hazard = "Weibull", 
#>     sharedtype = "RE", data = paq)
#>  
#> Statistical Model: 
#>      Dataset: paq 
#>      Number of subjects: 494 
#>      Number of observations: 2052 
#>      Number of parameters: 15  
#>      Event 1 : 
#>         Number of events: 128
#>         Weibull baseline risk function 
#>      Link functions: Linear for IST  
#>  
#> Iteration process: 
#>      Convergence criteria satisfied 
#>      Number of iterations:  6 
#>      Convergence criteria: parameters= 2e-07 
#>                          : likelihood= 1.7e-07 
#>                          : second derivatives= 7e-13 
#>  
#> Goodness-of-fit statistics: 
#>      maximum log-likelihood: -6453.36  
#>      AIC: 12936.71  
#>      BIC: 12999.75  
#>  
#> Maximum Likelihood Estimates: 
#>  
#> 
#> Parameters in the proportional hazard model:
#> 
#> 
#>                              coef      Se    Wald p-value
#> event1 +/-sqrt(Weibull1)  0.10384 0.00041 253.773 0.00000
#> event1 +/-sqrt(Weibull2)  5.21002 0.33779  15.424 0.00000
#> male                     -0.15662 0.35502  -0.441 0.65910
#> event1 RE1               -1.03212 0.22406  -4.606 0.00000
#> event1 RE2               -4.19419 0.70641  -5.937 0.00000
#> 
#> Fixed effects in the longitudinal model:
#> 
#>                               coef      Se   Wald p-value
#> intercept (not estimated)  0.00000                       
#> age65                     -1.04898 0.11719 -8.951 0.00000
#> male                      -0.10516 0.15137 -0.695 0.48722
#> CEP                        0.83813 0.18578  4.511 0.00001
#> age65:male                 0.04274 0.09836  0.435 0.66390
#> age65:CEP                 -0.16808 0.09295 -1.808 0.07055
#> 
#> 
#> Variance-covariance matrix of the random-effects:
#> (the variance of the first random effect is not estimated)
#>           intercept   age65
#> intercept   1.00000        
#> age65      -0.22497 0.32218
#> 
#>                                  
#> Residual standard error:  0.73730
#> 
#> Parameters of the link functions:
#> 
#>                  coef      Se    Wald p-value
#> IST-Linear 1 30.82166 0.71831  42.909 0.00000
#> IST-Linear 2  4.63064 0.36127  12.818 0.00000
#> 


#### 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)
#> ------------------ iteration  0 ------------------
#> Function value  -2563.295 
#> Convergence criteria: parameters stability= 1.0001 
#>                     : function stability= 1.0001 
#>                     : relative distance to maximum(RDM)= 0.220159 
#>                    coef
#> parameter1   0.08210000
#> parameter2   2.44920000
#> parameter3   0.12230000
#> parameter4   1.78640000
#> parameter5   0.07990000
#> parameter6  -0.28640000
#> parameter7   0.00550000
#> parameter8  -0.03270000
#> parameter9   0.02511394
#> parameter10  0.33130000
#> parameter11  0.97630000
#> parameter12  0.99180000
#> parameter13 -0.44020000
#> 
#> ------------------ iteration  1 ------------------
#> Function value  -2562.612 
#> Convergence criteria: parameters stability= 0.00728533 
#>                     : function stability= 0.6824102 
#>                     : relative distance to maximum(RDM)= 0.03597139 
#>                    coef
#> parameter1   0.08036165
#> parameter2   2.37994811
#> parameter3   0.12344651
#> parameter4   1.78402472
#> parameter5   0.08392983
#> parameter6  -0.28287296
#> parameter7   0.00550449
#> parameter8  -0.03264766
#> parameter9   0.02650268
#> parameter10  0.37219575
#> parameter11  0.99072111
#> parameter12  1.01063126
#> parameter13 -0.45482427
#> 
#> ------------------ iteration  2 ------------------
#> Function value  -2562.377 
#> Convergence criteria: parameters stability= 0.01069505 
#>                     : function stability= 0.2350088 
#>                     : relative distance to maximum(RDM)= 0.0052143 
#>                    coef
#> parameter1   0.07846394
#> parameter2   2.28594657
#> parameter3   0.11971715
#> parameter4   1.76099386
#> parameter5   0.08698839
#> parameter6  -0.27188858
#> parameter7   0.00484851
#> parameter8  -0.03212819
#> parameter9   0.02720867
#> parameter10  0.39875322
#> parameter11  1.00259760
#> parameter12  1.02408407
#> parameter13 -0.46716481
#> 
#> ------------------ iteration  3 ------------------
#> Function value  -2562.338 
#> Convergence criteria: parameters stability= 0.00348402 
#>                     : function stability= 0.03901784 
#>                     : relative distance to maximum(RDM)= 0.00067168 
#>                    coef
#> parameter1   0.07721460
#> parameter2   2.22996311
#> parameter3   0.11567884
#> parameter4   1.77902667
#> parameter5   0.08728371
#> parameter6  -0.27292554
#> parameter7   0.00485636
#> parameter8  -0.03204439
#> parameter9   0.02730191
#> parameter10  0.39891311
#> parameter11  1.00381398
#> parameter12  1.02541891
#> parameter13 -0.46870123
#> 
#> ------------------ iteration  4 ------------------
#> Function value  -2562.333 
#> Convergence criteria: parameters stability= 0.00059098 
#>                     : function stability= 0.00482363 
#>                     : relative distance to maximum(RDM)= 2.061e-05 
#>                    coef
#> parameter1   0.07667626
#> parameter2   2.20839070
#> parameter3   0.11467443
#> parameter4   1.79013742
#> parameter5   0.08727103
#> parameter6  -0.27338459
#> parameter7   0.00487914
#> parameter8  -0.03203803
#> parameter9   0.02730992
#> parameter10  0.39811585
#> parameter11  1.00373249
#> parameter12  1.02531163
#> parameter13 -0.46869267
#> 
#> ------------------ iteration  5 ------------------
#> Function value  -2562.333 
#> Convergence criteria: parameters stability= 2.207e-05 
#>                     : function stability= 0.00013953 
#>                     : relative distance to maximum(RDM)= 6e-08 
#>                    coef
#> parameter1   0.07657185
#> parameter2   2.20415912
#> parameter3   0.11444806
#> parameter4   1.79215223
#> parameter5   0.08726965
#> parameter6  -0.27350682
#> parameter7   0.00488456
#> parameter8  -0.03203648
#> parameter9   0.02731215
#> parameter10  0.39794908
#> parameter11  1.00372746
#> parameter12  1.02530336
#> parameter13 -0.46870157
#> 
#> ------------------ iteration  6 ------------------
#> Function value  -2562.333 
#> Convergence criteria: parameters stability= 8e-08 
#>                     : function stability= 4e-07 
#>                     : relative distance to maximum(RDM)= 0 
#>                    coef
#> parameter1   0.07656592
#> parameter2   2.20390363
#> parameter3   0.11442777
#> parameter4   1.79226097
#> parameter5   0.08726930
#> parameter6  -0.27351836
#> parameter7   0.00488511
#> parameter8  -0.03203640
#> parameter9   0.02731225
#> parameter10  0.39793428
#> parameter11  1.00372675
#> parameter12  1.02530250
#> parameter13 -0.46870171
#> 
summary(M2)
#> Joint latent process model with shared random effects 
#>      fitted by maximum likelihood method 
#>  
#> jointLPM(fixed = HIER ~ I(age - 65) * male, random = ~I(age - 
#>     65), subject = "ID", link = "thresholds", var.time = "age", 
#>     survival = Surv(age_init, agedem, dem) ~ male, sharedtype = "CL", 
#>     methInteg = "QMC", nMC = 1000, data = paq, nproc = 2)
#>  
#> Statistical Model: 
#>      Dataset: paq 
#>      Number of subjects: 499 
#>      Number of observations: 2203 
#>      Number of parameters: 13  
#>      Event 1 : 
#>         Number of events: 128
#>         Weibull baseline risk function 
#>      Link functions: Thresholds for HIER 
#>  
#> Iteration process: 
#>      Convergence criteria satisfied 
#>      Number of iterations:  6 
#>      Convergence criteria: parameters= 7.8e-08 
#>                          : likelihood= 4e-07 
#>                          : second derivatives= 6.8e-11 
#>  
#> Goodness-of-fit statistics: 
#>      maximum log-likelihood: -2562.33  
#>      AIC: 5150.67  
#>      BIC: 5205.43  
#>  
#> Maximum Likelihood Estimates: 
#>  
#> 
#> Parameters in the proportional hazard model:
#> 
#> 
#>                              coef      Se    Wald p-value
#> event1 +/-sqrt(Weibull1)  0.07657 0.00738  10.375 0.00000
#> event1 +/-sqrt(Weibull2)  2.20390 0.31606   6.973 0.00000
#> male                      0.11443 0.21966   0.521 0.60241
#> event1 CL                 1.79226 0.27586   6.497 0.00000
#> 
#> Fixed effects in the longitudinal model:
#> 
#>                               coef      Se   Wald p-value
#> intercept (not estimated)  0.00000                       
#> I(age - 65)                0.08727 0.00886  9.847 0.00000
#> male                      -0.27352 0.14624 -1.870 0.06144
#> I(age - 65):male           0.00489 0.00765  0.638 0.52319
#> 
#> 
#> Variance-covariance matrix of the random-effects:
#> (the variance of the first random effect is not estimated)
#>             intercept I(age - 65)
#> intercept     1.00000            
#> I(age - 65)  -0.03204     0.00177
#> 
#>                                  
#> Residual standard error:  0.46870
#> 
#> Parameters of the link functions:
#> 
#>                  coef      Se    Wald p-value
#> HIER-Thresh1  0.39793 0.10672   3.729 0.00019
#> HIER-Thresh2  1.00373 0.04079  24.605 0.00000
#> HIER-Thresh3  1.02530 0.04410  23.251 0.00000
#> 

# }