trivPenalNL: Fit a Non-Linear Trivariate Joint Model for Recurrent Events...
In frailtypack: Shared, Joint (Generalized) Frailty Models; Surrogate Endpoints
trivPenalNL R Documentation
Fit a Non-Linear Trivariate Joint Model for Recurrent Events and a Terminal
Event with a Biomarker Described with an ODE Population Model
Description
Fit a non-linear trivariate joint model for a longitudinal biomarker,
recurrent events and a terminal event using a semiparametric penalized
likelihood estimation or a parametric estimation on the hazard functions.
The values yi(t) (i=1,...,N) for N subjects represent
the individual evolution of the biomarker e.g. tumor size expressed as the
sum of the longest diameters (SLD) of target lesions. The dynamics of the
biomarker are described by an ordinary differential equation (ODE) that
includes the effect of the natural net growth and the treatment effect:
The model includes the following parameters (using the interpretation of
tumor dynamics): exp(KG,0) the constant tumor growth rate,
exp(KD,0) the drug-induced tumor decline rate, \lambda
resistance effect to drug (exponential tumor decay change with time),
exp(y0) the initial level of the biomarker and di is the
treatment concentration (e.g dose). The random effects biT
= (by0,i,bG,i,bD,i,b\lambda,i)T are gaussian variables
with a diagonal covariance matrix Bi. In the trivariate model
we use the analytical solution of the equation with the population-based
approach of the non-linear mixed effects model. We can also assume a
transformation for the observations of the biomarker (one parameter Box-Cox
transformation) and we include a gaussian measurement error, for individual
i and observation k (k=1,...,ni),
\epsilonik ~ N(0,\sigma\epsilon2).
The risks of the recurrent (rij(.) the risk of the jth
event of the individual i) and terminal events (\lambdai the
risk of the event of the individual i) are represented by proportional
hazard risk models. The joint model is constructed assuming that the
processes are linked via a latent structure and includes the non-linear
mixed effects model for the longitudinal data:
where XG,i(t), XD,i(t),
XR,ij(t) and XT,i(t) are vectors of possible
time-varying fixed effects covariates and \betaG,
\betaD, \betaR and \betaT are the
associated coefficients. The random effects bi are independent
from the measurement error. The relationship between the biomarker and
recurrent events is explained via g(yi(t)) with coefficients
\etaR and between the biomarker and terminal event is
explained via h(yi(t)) with coefficients \etaT.
Currently, only one form of the functions g(.) and h(.)
is available: the random effects bi. The frailty term
vi is gaussian with mean 0 and variance \sigmav. Together with
bi constitutes the random effects of the model:
Any combination of the random effects bi, e.g.
bi=by0,i or bi =
{bG,i,bD,i,b\lambda,i} can be chosen for the model.
We consider that the longitudinal outcome can be a subject to a
quantification limit, i.e. some observations, below a level of detection
s cannot be quantified (left-censoring).
Usage
trivPenalNL(formula, formula.terminalEvent, biomarker, formula.KG,
formula.KD, dose, time.biomarker, data, data.Longi, random, id, link =
"Random-effects", BoxCox = FALSE, left.censoring = FALSE, recurrentAG =
FALSE, n.knots, kappa, maxit = 300, hazard = "Splines", init.B, init.Random,
init.Eta, init.Alpha, init.Biomarker, method.GH = "Standard", init.GH =
FALSE, n.nodes, LIMparam = 1e-3, LIMlogl = 1e-3, LIMderiv = 1e-3,
print.times = TRUE)
Arguments
formula
a formula object, with the response on the left of a
\sim operator, and the terms on the right. The response must be a
survival object as returned by the 'Surv' function like in survival package.
Interactions are possible using * or :.
formula.terminalEvent
A formula object, only requires terms on the
right to indicate which variables are modelling the terminal event.
Interactions are possible using * or :.
biomarker
Name of the variable representing the longitudinal
biomarker.
formula.KG
A formula object, only requires terms on the right to
indicate which covariates related to the biomarker growth are included in
the longitudinal sub-model. It must follow the standard form used for
linear mixed-effects models. Interactions are possible using * or :.
formula.KD
A formula object, only requires terms on the right to
indicate which covariates related to the biomarker drug-induced decline are
included in the longitudinal sub-model. It must follow the standard form
used for linear mixed-effects models. Interactions are possible using * or
:.
dose
Name of the variable representing the drug concentration
indicator.
time.biomarker
Name of the variable of times of biomarker
measurements.
data
A 'data.frame' with the variables used in formula.
data.Longi
A 'data.frame' with the variables used in
formula.KG, formula.KD, biomarker, dose,
time.biomarker and id.
random
Names of parameters for which the random effects are included
in the mixed model. The names must be chosen among "y0", "KG",
"KD" and "lambda". Any combination of the mentioned names is
allowed.
id
Name of the variable representing the individuals.
link
Type of link functions for the dependence between the biomarker
and death and between the biomarker and the recurrent events: only
"Random-effects" for the association directly via the random effects
of the biomarker is allowed for the moment (option for a future extension).
BoxCox
Should the Box-Cox transformation be used for the longitudinal
biomarker? If there is no transformation, the argument must be equal to
FALSE, otherwise the of the transformation parameter must be given,
then the transformed values are y^*=(y^{\xi}-1)/\xi, where \xi
is the Box-Cox parameter.
left.censoring
Is the biomarker left-censored below a threshold
s? If there is no left-censoring, the argument must be equal to
FALSE, otherwise the value of the threshold must be given.
recurrentAG
Logical value. Is Andersen-Gill model fitted? If so
indicates that recurrent event times with the counting process approach of
Andersen and Gill is used. This formulation can be used for dealing with
time-dependent covariates. The default is FALSE.
n.knots
Integer giving the number of knots to use. Value required in
the penalized likelihood estimation. It corresponds to the (n.knots+2)
splines functions for the approximation of the hazard or the survival
functions. We estimate I or M-splines of order 4. When the user set a
number of knots equals to k (n.knots=k) then the number of interior knots is
(k-2) and the number of splines is (k-2)+order. Number of knots must be
between 4 and 20. (See Note in frailtyPenal function)
kappa
Positive smoothing parameters in the penalized likelihood
estimation. The coefficient kappa of the integral of the squared second
derivative of hazard function in the fit (penalized log likelihood). To
obtain an initial value for kappa, a solution is to fit the
corresponding Cox model using cross validation (See cross.validation
in function frailtyPenal). We advise the user to identify several
possible tuning parameters, note their defaults and look at the sensitivity
of the results to varying them.
maxit
Maximum number of iterations for the Marquardt algorithm.
Default is 300
hazard
Type of hazard functions: "Splines" for semiparametric
hazard functions using equidistant intervals or "Splines-per" using
percentile with the penalized likelihood estimation, "Weibull" for
the parametric Weibull functions. The default is "Splines".
init.B
Vector of initial values for regression coefficients. This
vector should be of the same size as the whole vector of covariates with the
first elements for the covariates related to the recurrent events, then to
the terminal event and then to the biomarker (interactions in the end of
each component). Default is 0.5 for each.
init.Random
Initial value for variance of the elements of the matrix
of the distribution of the random effects.
init.Eta
Initial values for regression coefficients for the link
functions, first for the recurrent events (\bold{\eta}_R) and for the
terminal event (\bold{\eta}_T).
init.Alpha
Initial value for parameter alpha
init.Biomarker
Initial values for biomarker parameters: y_0,
K_{G,0}, K_{D,0} and \lambda (using this order).
method.GH
Method for the Gauss-Hermite quadrature: "Standard"
for the standard non-adaptive Gaussian quadrature and
"Pseudo-adaptive" for the pseudo-adaptive Gaussian quadrature (see
Details). The default is "Standard". When the option
"Pseudo-adaptive" is chosen, then a univariate model (non-linear
mixed model for the biomarker) is fitted in order to obtain the estimations
of the random effects \bold{b}_i.
init.GH
Only when the opiton "Pseudo-adaptive" of the argument
method.GH is chosen. If TRUE, the estimations of the biomarker
parameters (y_0, K_{G,0}, K_{D,0} and \lambda),
\sigma_{\epsilon}, \bold{\beta}_G and \bold{\beta}_D from
the univariate mixed model are used as the initial values of the parameters
related to the biomarker.
n.nodes
Number of nodes for the Gauss-Hermite quadrature (from 5 to
32). The default is 9.
LIMparam
Convergence threshold of the Marquardt algorithm for the
parameters (see Details), 10^{-3} by default.
LIMlogl
Convergence threshold of the Marquardt algorithm for the
log-likelihood (see Details), 10^{-3} by default.
LIMderiv
Convergence threshold of the Marquardt algorithm for the
gradient (see Details), 10^{-3} by default.
print.times
a logical parameter to print iteration process. Default
is TRUE.
Details
Typical usage for the joint model
trivPenalNL(Surv(time,event)~cluster(id) + var1 + var2 +
terminal(death), formula.terminalEvent =~ var1 + var3, biomarker =
"biomarker.name", dose = "dose.name", time.biomarker = "time", formula.KG ~
var1, formula.KD ~ var2, data, data.Longi, ...)
The method of the Gauss-Hermite quadrature for approximations of the
multidimensional integrals, i.e. length of random more than 2, can be
chosen among the standard (non-adaptive) and pseudo-adaptive in which the
quadrature points are transformed using the information from the fitted
mixed-effects model for the biomarker (Rizopoulos 2012) or multivariate
non-adaptive procedure proposed by Genz et al. 1996 and implemented in
FORTRAN subroutine HRMSYM. The choice of the method is important for
estimations. The standard non-adaptive Gauss-Hermite quadrature
("Standard") with a specific number of points gives accurate results
but can be time consuming. The pseudo-adaptive quadrature uses transformed
quadrature points to center and scale the integrand by utilizing estimates
of the random effects from an appropriate non-linear mixed-effects model
(this transformation does not include the frailty in the trivariate model,
for which the standard method, with 20 quadrature points, is used). This
method enables using less quadrature points while preserving the estimation
accuracy and thus lead to a better computational time.
NOTE. Data frames data and data.Longi must be consistent.
Names and types of corresponding covariates must be the same, as well as the
number and identification of individuals.
Value
The following components are included in a 'trivPenalNL' object for each
model:
b
The sequence of the corresponding estimation of the coefficients
for the hazard functions (parametric or semiparametric), the random effects
variances and the regression coefficients.
call
The code used for
the model.
formula
The formula part of the code used for the
recurrent event part of the model.
formula.terminalEvent
The formula
part of the code used for the terminal event part of the model.
formula.KG
The formula part of the code used for the longitudinal
part of the model, for the biomarker growth dynamics.
formula.KD
The
formula part of the code used for the longitudinal part of the model, for
the biomarker decline dynamics.
coef
The regression coefficients
(first for the recurrent events, then for the terminal event, then for the
biomarker growth and then for the biomarker decline.
groups
The
number of groups used in the fit.
kappa
The values of the smoothing
parameters in the penalized likelihood estimation corresponding to the
baseline hazard functions for the recurrent and terminal events.
logLikPenal
The complete marginal penalized log-likelihood in the
semiparametric case.
logLik
The marginal log-likelihood in the
parametric case.
n.measurements
The number of biomarker observations
used in the fit.
max_rep
The maximal number of repeated measurements
per individual.
n
The number of observations in 'data' (recurrent
and terminal events) used in the fit.
n.events
The number of
recurrent events observed in the fit.
n.deaths
The number of
terminal events observed in the fit.
n.iter
The number of iterations
needed to converge.
n.knots
The number of knots for estimating the
baseline hazard function in the penalized likelihood estimation.
n.strat
The number of stratum.
varH
The variance matrix of all parameters (before positivity
constraint transformation for the variance of the measurement error, for
which the delta method is used).
varHIH
The robust estimation of the
variance matrix of all parameters.
xR
The vector of times where both survival and hazard function of the
recurrent events are estimated. By default seq(0,max(time),length=99), where
time is the vector of survival times.
lamR
The array (dim=3) of
baseline hazard estimates and confidence bands (recurrent events).
survR
The array (dim=3) of baseline survival estimates and confidence
bands (recurrent events).
xD
The vector of times where both survival and hazard function of the
terminal event are estimated. By default seq(0,max(time),length=99), where
time is the vector of survival times.
lamD
The array (dim=3) of
baseline hazard estimates and confidence bands.
survD
The array
(dim=3) of baseline survival estimates and confidence bands.
medianR
The value of the median survival and its confidence bands for the recurrent event.
medianD
The value of the median survival and its confidence bands for the terminal event.
typeof
The type of the baseline hazard function (0:"Splines",
"2:Weibull").
npar
The number of parameters.
nvar
The vector
of number of explanatory variables for the recurrent events, terminal event,
biomarker growth and biomarker decline.
nvarRec
The number of
explanatory variables for the recurrent events.
nvarEnd
The number
of explanatory variables for the terminal event.
nvarKG
The number
of explanatory variables for the biomarker growth.
nvarKD
The number
of explanatory variables for the biomarker decline.
noVarRec
The
indicator of absence of the explanatory variables for the recurrent events.
noVarEnd
The indicator of absence of the explanatory variables for
the terminal event.
noVarKG
The indicator of absence of the
explanatory variables for the biomarker growth.
noVarKD
The
indicator of absence of the explanatory variables for the biomarker
decline.
LCV
The approximated likelihood cross-validation criterion
in the semiparametric case (with H minus the converged Hessian matrix, and
l(.) the full log-likelihood).
LCV=\frac{1}{n}(trace(H^{-1}_{pl}H) -
l(.))
AIC
The Akaike information Criterion for the parametric
case.
AIC=\frac{1}{n}(np - l(.))
n.knots.temp
The initial
value for the number of knots.
shape.weib
The shape parameter for
the Weibull hazard functions (the first element for the recurrences and the
second one for the terminal event).
scale.weib
The scale parameter
for the Weibull hazard functions (the first element for the recurrences and
the second one for the terminal event).
random.effects.pred
The empirical Bayes predictions of the random
effects (ie. using conditional posterior distributions).
global_chisq.testR
The binary variable equals to 0 when no
multivariate Wald is given, 1 otherwise (for the recurrent part).
global_chisq.testT
The binary variable equals to 0 when no
multivariate Wald is given, 1 otherwise (for the terminal part).
global_chisq.testKG
The binary variable equals to 0 when no
multivariate Wald is given, 1 otherwise (for the biomarker growth).
global_chisq.testKD
The binary variable equals to 0 when no
multivariate Wald is given, 1 otherwise (for the biomarker decline).
AG
The logical value. Is Andersen-Gill model fitted?
B1
The variance matrix of the random effects for the longitudinal
outcome.
sigma2
The variance of the frailty term (\sigma_v).
alpha
The coefficient \alpha associated with the frailty
parameter in the terminal hazard function.
ResidualSE
The variance
of the measurement error.
etaR
The regression coefficients for the
link function g(\cdot).
etaT
The regression coefficients for
the link function h(\cdot).
ne_re
The number of random effects
b used in the fit.
names.re
The names of variables for the random
effects \bold{b}_i.
link
The name of the type of the link
functions.
leftCensoring
The logical value. Is the longitudinal outcome
left-censored?
leftCensoring.threshold
For the left-censored
biomarker, the value of the left-censoring threshold used for the fit.
prop.censored
The fraction of observations subjected to the
left-censoring.
methodGH
The Gaussian quadrature method used in the fit.
n.nodes
The number of nodes used for the Gaussian quadrature in the
fit.
K_G0
Value of the estimate of the biomarker growth parameter.
K_D0
Value of the estimate of the biomarker decay parameter.
lambda
Value of the estimate of the biomarker resistance to drug.
y_0
Value of the estimate of the biomarker intial level.
biomarker
Name of the variable associated with the biomarker in the
data.
time.biomarker
Name of the variable associated with the time
of measurements of the biomarker in the data.
dose
Name of the
variable associated with the drug concentration in the data.
BoxCox
The logical value. Is the BoxCox transformation applied for
the biomarker?
BoxCox_parameter
The value of the BoxCox
transformation parameter.
alpha_p.value
p-value of the Wald test for the estimated coefficient
\alpha.
sigma2_p.value
p-value of the Wald test for the
estimated variance of the frailty term (\sigma_v).
etaR_p.value
p-values of the Wald test for the estimated regression
coefficients for the link function g(\cdot).
etaT_p.value
p-values of the Wald test for the estimated regression
coefficients for the link function h(\cdot).
y_0_p.value
p-value of the Wald test for the estimated biomarker
intial level.
K_G0_p.value
p-value of the Wald test for the
estimated biomarker growth parameter.
K_D0_p.value
p-value of the
Wald test for the estimated biomarker decay parameter.
lambda_p.value
p-value of the Wald test for the estimated biomarker
resistance to drug.
beta_p.value
p-values of the Wald test for the
estimated regression coefficients.
Note
It is recommended to initialize the parameter values using the results
from a corresponding reduced model (frailtyPenal for the recurrent
and terminal part). See example.
Estimations of models with more than three random effects can be very long.
References
A. Krol, C. Tournigand, S. Michiels and V. RondeauS (2018).
Multivariate joint frailty model for the analysis of nonlinear tumor
kinetics and dynamic predictions of death. Statistics in Medicine.
A. Krol, L. Ferrer, JP. Pignon, C. Proust-Lima, M. Ducreux, O. Bouche, S.
Michiels, V. Rondeau (2016). Joint Model for Left-Censored Longitudinal
Data, Recurrent Events and Terminal Event: Predictive Abilities of Tumor
Burden for Cancer Evolution with Application to the FFCD 2000-05 Trial.
Biometrics 72(3) 907-16.
D. Rizopoulos (2012). Fast fitting of joint models for longitudinal and
event time data using a pseudo-adaptive Gaussian quadrature rule.
Computational Statistics and Data Analysis 56, 491-501.
L. Claret, P. Girard, P.M. Hoff, E. Van Cutsem, K.P. Zuideveld, K. Jorga, J.
Fagerberg, R Bruno (2009). Model-based prediction of phase III overall
survival in colorectal cancer on the basis of phase II tumor dynamics.
Journal of Clinical Oncology 27(25), 4103-8.
See Also
plot.trivPenalNL,print.trivPenalNL,summary.trivPenalNL
Examples
## Not run:
###--- Non-linear trivariate joint model for longitudinal data, ---###
###--- recurrent events and a terminal event ---###
data(colorectal)
data(colorectalLongi)
# No information on dose - creation of a dummy variable
colorectalLongi$dose <- 1
# Parameters initialisation - estimation of a simplified model
# with two random effects (a frailty term and a random effect
# related to biomarker growth (KG))
initial.model <- trivPenalNL(Surv(time0, time1, new.lesions) ~ cluster(id)
+ age + treatment + terminal(state), formula.terminalEvent =~ age + treatment,
biomarker = "tumor.size", formula.KG ~ 1, formula.KD ~ treatment, dose = "dose",
time.biomarker = "year", data = colorectal, data.Longi =colorectalLongi,
random = "KG", id = "id", recurrentAG = TRUE, n.knots = 5, kappa = c(0.01, 2),
method.GH = "Pseudo-adaptive")
# Trivariate joint model with initial values for parameters
# (computation takes around 40 minutes)
model <- trivPenalNL(Surv(time0, time1, new.lesions) ~ cluster(id) + age + treatment
+ terminal(state), formula.terminalEvent =~ age + treatment, biomarker = "tumor.size",
formula.KG ~ 1, formula.KD ~ treatment, dose = "dose", time.biomarker = "year",
data = colorectal, data.Longi =colorectalLongi, random = c("y0", "KG"), id = "id",
init.B = c(-0.22, -0.16, -0.35, -0.19, 0.04, -0.41, 0.23), init.Alpha = 1.86,
init.Eta = c(0.5, 0.57, 0.5, 2.34), init.Biomarker = c(1.24, 0.81, 1.07, -1.53),
recurrentAG = TRUE, n.knots = 5, kappa = c(0.01, 2), method.GH = "Pseudo-adaptive")
## End(Not run)
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| trivPenalNL | R Documentation |
Fit a Non-Linear Trivariate Joint Model for Recurrent Events and a Terminal Event with a Biomarker Described with an ODE Population Model
Description
Fit a non-linear trivariate joint model for a longitudinal biomarker, recurrent events and a terminal event using a semiparametric penalized likelihood estimation or a parametric estimation on the hazard functions.
The values yi(t) (i=1,...,N) for N subjects represent the individual evolution of the biomarker e.g. tumor size expressed as the sum of the longest diameters (SLD) of target lesions. The dynamics of the biomarker are described by an ordinary differential equation (ODE) that includes the effect of the natural net growth and the treatment effect:
The model includes the following parameters (using the interpretation of
tumor dynamics): exp(KG,0) the constant tumor growth rate,
exp(KD,0) the drug-induced tumor decline rate, \lambda
resistance effect to drug (exponential tumor decay change with time),
exp(y0) the initial level of the biomarker and di is the
treatment concentration (e.g dose). The random effects biT
= (by0,i,bG,i,bD,i,b\lambda,i)T are gaussian variables
with a diagonal covariance matrix Bi. In the trivariate model
we use the analytical solution of the equation with the population-based
approach of the non-linear mixed effects model. We can also assume a
transformation for the observations of the biomarker (one parameter Box-Cox
transformation) and we include a gaussian measurement error, for individual
i and observation k (k=1,...,ni),
\epsilonik ~ N(0,\sigma\epsilon2).
The risks of the recurrent (rij(.) the risk of the jth
event of the individual i) and terminal events (\lambdai the
risk of the event of the individual i) are represented by proportional
hazard risk models. The joint model is constructed assuming that the
processes are linked via a latent structure and includes the non-linear
mixed effects model for the longitudinal data:
where XG,i(t), XD,i(t),
XR,ij(t) and XT,i(t) are vectors of possible
time-varying fixed effects covariates and \betaG,
\betaD, \betaR and \betaT are the
associated coefficients. The random effects bi are independent
from the measurement error. The relationship between the biomarker and
recurrent events is explained via g(yi(t)) with coefficients
\etaR and between the biomarker and terminal event is
explained via h(yi(t)) with coefficients \etaT.
Currently, only one form of the functions g(.) and h(.)
is available: the random effects bi. The frailty term
vi is gaussian with mean 0 and variance \sigmav. Together with
bi constitutes the random effects of the model:
Any combination of the random effects bi, e.g. bi=by0,i or bi = {bG,i,bD,i,b\lambda,i} can be chosen for the model.
We consider that the longitudinal outcome can be a subject to a quantification limit, i.e. some observations, below a level of detection s cannot be quantified (left-censoring).
Usage
trivPenalNL(formula, formula.terminalEvent, biomarker, formula.KG,
formula.KD, dose, time.biomarker, data, data.Longi, random, id, link =
"Random-effects", BoxCox = FALSE, left.censoring = FALSE, recurrentAG =
FALSE, n.knots, kappa, maxit = 300, hazard = "Splines", init.B, init.Random,
init.Eta, init.Alpha, init.Biomarker, method.GH = "Standard", init.GH =
FALSE, n.nodes, LIMparam = 1e-3, LIMlogl = 1e-3, LIMderiv = 1e-3,
print.times = TRUE)
Arguments
formula |
a formula object, with the response on the left of a
|
formula.terminalEvent |
A formula object, only requires terms on the right to indicate which variables are modelling the terminal event. Interactions are possible using * or :. |
biomarker |
Name of the variable representing the longitudinal biomarker. |
formula.KG |
A formula object, only requires terms on the right to indicate which covariates related to the biomarker growth are included in the longitudinal sub-model. It must follow the standard form used for linear mixed-effects models. Interactions are possible using * or :. |
formula.KD |
A formula object, only requires terms on the right to indicate which covariates related to the biomarker drug-induced decline are included in the longitudinal sub-model. It must follow the standard form used for linear mixed-effects models. Interactions are possible using * or :. |
dose |
Name of the variable representing the drug concentration indicator. |
time.biomarker |
Name of the variable of times of biomarker measurements. |
data |
A 'data.frame' with the variables used in |
data.Longi |
A 'data.frame' with the variables used in
|
random |
Names of parameters for which the random effects are included
in the mixed model. The names must be chosen among |
id |
Name of the variable representing the individuals. |
link |
Type of link functions for the dependence between the biomarker
and death and between the biomarker and the recurrent events: only
|
BoxCox |
Should the Box-Cox transformation be used for the longitudinal
biomarker? If there is no transformation, the argument must be equal to
|
left.censoring |
Is the biomarker left-censored below a threshold
|
recurrentAG |
Logical value. Is Andersen-Gill model fitted? If so indicates that recurrent event times with the counting process approach of Andersen and Gill is used. This formulation can be used for dealing with time-dependent covariates. The default is FALSE. |
n.knots |
Integer giving the number of knots to use. Value required in
the penalized likelihood estimation. It corresponds to the (n.knots+2)
splines functions for the approximation of the hazard or the survival
functions. We estimate I or M-splines of order 4. When the user set a
number of knots equals to k (n.knots=k) then the number of interior knots is
(k-2) and the number of splines is (k-2)+order. Number of knots must be
between 4 and 20. (See Note in |
kappa |
Positive smoothing parameters in the penalized likelihood
estimation. The coefficient kappa of the integral of the squared second
derivative of hazard function in the fit (penalized log likelihood). To
obtain an initial value for |
maxit |
Maximum number of iterations for the Marquardt algorithm. Default is 300 |
hazard |
Type of hazard functions: |
init.B |
Vector of initial values for regression coefficients. This vector should be of the same size as the whole vector of covariates with the first elements for the covariates related to the recurrent events, then to the terminal event and then to the biomarker (interactions in the end of each component). Default is 0.5 for each. |
init.Random |
Initial value for variance of the elements of the matrix of the distribution of the random effects. |
init.Eta |
Initial values for regression coefficients for the link
functions, first for the recurrent events ( |
init.Alpha |
Initial value for parameter alpha |
init.Biomarker |
Initial values for biomarker parameters: |
method.GH |
Method for the Gauss-Hermite quadrature: |
init.GH |
Only when the opiton |
n.nodes |
Number of nodes for the Gauss-Hermite quadrature (from 5 to 32). The default is 9. |
LIMparam |
Convergence threshold of the Marquardt algorithm for the
parameters (see Details), |
LIMlogl |
Convergence threshold of the Marquardt algorithm for the
log-likelihood (see Details), |
LIMderiv |
Convergence threshold of the Marquardt algorithm for the
gradient (see Details), |
print.times |
a logical parameter to print iteration process. Default is TRUE. |
Details
Typical usage for the joint model
trivPenalNL(Surv(time,event)~cluster(id) + var1 + var2 + terminal(death), formula.terminalEvent =~ var1 + var3, biomarker = "biomarker.name", dose = "dose.name", time.biomarker = "time", formula.KG ~ var1, formula.KD ~ var2, data, data.Longi, ...)
The method of the Gauss-Hermite quadrature for approximations of the
multidimensional integrals, i.e. length of random more than 2, can be
chosen among the standard (non-adaptive) and pseudo-adaptive in which the
quadrature points are transformed using the information from the fitted
mixed-effects model for the biomarker (Rizopoulos 2012) or multivariate
non-adaptive procedure proposed by Genz et al. 1996 and implemented in
FORTRAN subroutine HRMSYM. The choice of the method is important for
estimations. The standard non-adaptive Gauss-Hermite quadrature
("Standard") with a specific number of points gives accurate results
but can be time consuming. The pseudo-adaptive quadrature uses transformed
quadrature points to center and scale the integrand by utilizing estimates
of the random effects from an appropriate non-linear mixed-effects model
(this transformation does not include the frailty in the trivariate model,
for which the standard method, with 20 quadrature points, is used). This
method enables using less quadrature points while preserving the estimation
accuracy and thus lead to a better computational time.
NOTE. Data frames data and data.Longi must be consistent.
Names and types of corresponding covariates must be the same, as well as the
number and identification of individuals.
Value
The following components are included in a 'trivPenalNL' object for each model:
b |
The sequence of the corresponding estimation of the coefficients for the hazard functions (parametric or semiparametric), the random effects variances and the regression coefficients. |
call |
The code used for the model. |
formula |
The formula part of the code used for the recurrent event part of the model. |
formula.terminalEvent |
The formula part of the code used for the terminal event part of the model. |
formula.KG |
The formula part of the code used for the longitudinal part of the model, for the biomarker growth dynamics. |
formula.KD |
The formula part of the code used for the longitudinal part of the model, for the biomarker decline dynamics. |
coef |
The regression coefficients (first for the recurrent events, then for the terminal event, then for the biomarker growth and then for the biomarker decline. |
groups |
The number of groups used in the fit. |
kappa |
The values of the smoothing parameters in the penalized likelihood estimation corresponding to the baseline hazard functions for the recurrent and terminal events. |
logLikPenal |
The complete marginal penalized log-likelihood in the semiparametric case. |
logLik |
The marginal log-likelihood in the parametric case. |
n.measurements |
The number of biomarker observations used in the fit. |
max_rep |
The maximal number of repeated measurements per individual. |
n |
The number of observations in 'data' (recurrent and terminal events) used in the fit. |
n.events |
The number of recurrent events observed in the fit. |
n.deaths |
The number of terminal events observed in the fit. |
n.iter |
The number of iterations needed to converge. |
n.knots |
The number of knots for estimating the baseline hazard function in the penalized likelihood estimation. |
n.strat |
The number of stratum. |
varH |
The variance matrix of all parameters (before positivity constraint transformation for the variance of the measurement error, for which the delta method is used). |
varHIH |
The robust estimation of the variance matrix of all parameters. |
xR |
The vector of times where both survival and hazard function of the recurrent events are estimated. By default seq(0,max(time),length=99), where time is the vector of survival times. |
lamR |
The array (dim=3) of baseline hazard estimates and confidence bands (recurrent events). |
survR |
The array (dim=3) of baseline survival estimates and confidence bands (recurrent events). |
xD |
The vector of times where both survival and hazard function of the terminal event are estimated. By default seq(0,max(time),length=99), where time is the vector of survival times. |
lamD |
The array (dim=3) of baseline hazard estimates and confidence bands. |
survD |
The array (dim=3) of baseline survival estimates and confidence bands. |
medianR |
The value of the median survival and its confidence bands for the recurrent event. |
medianD |
The value of the median survival and its confidence bands for the terminal event. |
typeof |
The type of the baseline hazard function (0:"Splines", "2:Weibull"). |
npar |
The number of parameters. |
nvar |
The vector of number of explanatory variables for the recurrent events, terminal event, biomarker growth and biomarker decline. |
nvarRec |
The number of explanatory variables for the recurrent events. |
nvarEnd |
The number of explanatory variables for the terminal event. |
nvarKG |
The number of explanatory variables for the biomarker growth. |
nvarKD |
The number of explanatory variables for the biomarker decline. |
noVarRec |
The indicator of absence of the explanatory variables for the recurrent events. |
noVarEnd |
The indicator of absence of the explanatory variables for the terminal event. |
noVarKG |
The indicator of absence of the explanatory variables for the biomarker growth. |
noVarKD |
The indicator of absence of the explanatory variables for the biomarker decline. |
LCV |
The approximated likelihood cross-validation criterion in the semiparametric case (with H minus the converged Hessian matrix, and l(.) the full log-likelihood).
|
AIC |
The Akaike information Criterion for the parametric case.
|
n.knots.temp |
The initial value for the number of knots. |
shape.weib |
The shape parameter for the Weibull hazard functions (the first element for the recurrences and the second one for the terminal event). |
scale.weib |
The scale parameter for the Weibull hazard functions (the first element for the recurrences and the second one for the terminal event). |
random.effects.pred |
The empirical Bayes predictions of the random effects (ie. using conditional posterior distributions). |
global_chisq.testR |
The binary variable equals to 0 when no multivariate Wald is given, 1 otherwise (for the recurrent part). |
global_chisq.testT |
The binary variable equals to 0 when no multivariate Wald is given, 1 otherwise (for the terminal part). |
global_chisq.testKG |
The binary variable equals to 0 when no multivariate Wald is given, 1 otherwise (for the biomarker growth). |
global_chisq.testKD |
The binary variable equals to 0 when no multivariate Wald is given, 1 otherwise (for the biomarker decline). |
AG |
The logical value. Is Andersen-Gill model fitted? |
B1 |
The variance matrix of the random effects for the longitudinal outcome. |
sigma2 |
The variance of the frailty term ( |
alpha |
The coefficient |
ResidualSE |
The variance of the measurement error. |
etaR |
The regression coefficients for the
link function |
etaT |
The regression coefficients for
the link function |
ne_re |
The number of random effects b used in the fit. |
names.re |
The names of variables for the random
effects |
link |
The name of the type of the link functions. |
leftCensoring |
The logical value. Is the longitudinal outcome left-censored? |
leftCensoring.threshold |
For the left-censored biomarker, the value of the left-censoring threshold used for the fit. |
prop.censored |
The fraction of observations subjected to the left-censoring. |
methodGH |
The Gaussian quadrature method used in the fit. |
n.nodes |
The number of nodes used for the Gaussian quadrature in the fit. |
K_G0 |
Value of the estimate of the biomarker growth parameter. |
K_D0 |
Value of the estimate of the biomarker decay parameter. |
lambda |
Value of the estimate of the biomarker resistance to drug. |
y_0 |
Value of the estimate of the biomarker intial level. |
biomarker |
Name of the variable associated with the biomarker in the data. |
time.biomarker |
Name of the variable associated with the time of measurements of the biomarker in the data. |
dose |
Name of the variable associated with the drug concentration in the data. |
BoxCox |
The logical value. Is the BoxCox transformation applied for the biomarker? |
BoxCox_parameter |
The value of the BoxCox transformation parameter. |
alpha_p.value |
p-value of the Wald test for the estimated coefficient
|
sigma2_p.value |
p-value of the Wald test for the
estimated variance of the frailty term ( |
etaR_p.value |
p-values of the Wald test for the estimated regression
coefficients for the link function |
etaT_p.value |
p-values of the Wald test for the estimated regression
coefficients for the link function |
y_0_p.value |
p-value of the Wald test for the estimated biomarker intial level. |
K_G0_p.value |
p-value of the Wald test for the estimated biomarker growth parameter. |
K_D0_p.value |
p-value of the Wald test for the estimated biomarker decay parameter. |
lambda_p.value |
p-value of the Wald test for the estimated biomarker resistance to drug. |
beta_p.value |
p-values of the Wald test for the estimated regression coefficients. |
Note
It is recommended to initialize the parameter values using the results
from a corresponding reduced model (frailtyPenal for the recurrent
and terminal part). See example.
Estimations of models with more than three random effects can be very long.
References
A. Krol, C. Tournigand, S. Michiels and V. RondeauS (2018). Multivariate joint frailty model for the analysis of nonlinear tumor kinetics and dynamic predictions of death. Statistics in Medicine.
A. Krol, L. Ferrer, JP. Pignon, C. Proust-Lima, M. Ducreux, O. Bouche, S. Michiels, V. Rondeau (2016). Joint Model for Left-Censored Longitudinal Data, Recurrent Events and Terminal Event: Predictive Abilities of Tumor Burden for Cancer Evolution with Application to the FFCD 2000-05 Trial. Biometrics 72(3) 907-16.
D. Rizopoulos (2012). Fast fitting of joint models for longitudinal and event time data using a pseudo-adaptive Gaussian quadrature rule. Computational Statistics and Data Analysis 56, 491-501.
L. Claret, P. Girard, P.M. Hoff, E. Van Cutsem, K.P. Zuideveld, K. Jorga, J. Fagerberg, R Bruno (2009). Model-based prediction of phase III overall survival in colorectal cancer on the basis of phase II tumor dynamics. Journal of Clinical Oncology 27(25), 4103-8.
See Also
plot.trivPenalNL,print.trivPenalNL,summary.trivPenalNL
Examples
## Not run:
###--- Non-linear trivariate joint model for longitudinal data, ---###
###--- recurrent events and a terminal event ---###
data(colorectal)
data(colorectalLongi)
# No information on dose - creation of a dummy variable
colorectalLongi$dose <- 1
# Parameters initialisation - estimation of a simplified model
# with two random effects (a frailty term and a random effect
# related to biomarker growth (KG))
initial.model <- trivPenalNL(Surv(time0, time1, new.lesions) ~ cluster(id)
+ age + treatment + terminal(state), formula.terminalEvent =~ age + treatment,
biomarker = "tumor.size", formula.KG ~ 1, formula.KD ~ treatment, dose = "dose",
time.biomarker = "year", data = colorectal, data.Longi =colorectalLongi,
random = "KG", id = "id", recurrentAG = TRUE, n.knots = 5, kappa = c(0.01, 2),
method.GH = "Pseudo-adaptive")
# Trivariate joint model with initial values for parameters
# (computation takes around 40 minutes)
model <- trivPenalNL(Surv(time0, time1, new.lesions) ~ cluster(id) + age + treatment
+ terminal(state), formula.terminalEvent =~ age + treatment, biomarker = "tumor.size",
formula.KG ~ 1, formula.KD ~ treatment, dose = "dose", time.biomarker = "year",
data = colorectal, data.Longi =colorectalLongi, random = c("y0", "KG"), id = "id",
init.B = c(-0.22, -0.16, -0.35, -0.19, 0.04, -0.41, 0.23), init.Alpha = 1.86,
init.Eta = c(0.5, 0.57, 0.5, 2.34), init.Biomarker = c(1.24, 0.81, 1.07, -1.53),
recurrentAG = TRUE, n.knots = 5, kappa = c(0.01, 2), method.GH = "Pseudo-adaptive")
## End(Not run)
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