| jointLPM {JLPM} | R Documentation |
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 effects 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 outcome-specific measurement models whose nature depends on the type of the associated outcome (linear model for Gaussian outcome, curvilinear model for non-Gaussian outcome, cumulative probit model for ordinal outcome). At the latent process level, the model estimates a standard linear mixed model. The survival submodel handles right-censored (possibly left-truncated) time-to-events with competing risks. 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 a linear predictor in the proportional hazard survival model. Parameters of the measurement models, of the latent process mixed model and of the survival model are estimated simultaneously using a maximum likelihood method, through a Marquardt-Levenberg 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",
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 |
random |
a one-sided formula for the random-effects in the
latent process mixed model. Covariates with a random-effect are separated
by |
subject |
name of the covariate representing the grouping structure. |
idiag |
optional logical for the variance-covariance structure of the
random-effects. If |
cor |
optional indicator for inclusion of an autocorrelated Gaussian
process in the linear mixed model at the latent process level. Option
|
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 |
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 |
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 ( |
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. |
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 |
subset |
optional vector giving the subset of observations in
|
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 |
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 |
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 at the latent process
level the intercept of the mixed model at 0 and the
standard error of the first random effect at 1. The nature of each measurment
model adapts to the type of the outcome it models.
1. For continuous Gaussian outcomes, linear models are used and required 2 parameters for the transformation (Y(t) - b1)/b2
2. For continuous non-Gaussian outcomes, curvilinear models use parametrized link function 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 constraint 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 discrete ordinal 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 were 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 parametrizations 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) and where 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 implement 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 appears as
time-dependent covariate in the survival model.
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 :
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
|
V |
vector containing the upper triangle matrix of variance-covariance
estimates of |
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) |
Author(s)
Viviane Philipps, Tiphaine Saulnier and Cecile Proust-Lima
References
Saulnier, Philipps, Meissner, Rascol, Pavy-Le-Traon, Foubert-Samier, Proust-Lima (2021). Joint models for the longitudinal analysis of measurement scales in the presence of informative dropout arXiv:2110.02612
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)