jointSurroCopPenal: Fit the one-step Joint frailty-copula model for evaluating a...
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jointSurroCopPenal R Documentation
Fit the one-step Joint frailty-copula model for evaluating a canditate surrogate endpoint
Description
Joint Frailty-Copula model for Surrogacy definition
Fit the one-step Joint frailty-copula surrogate model for the evaluation of a canditate surrogate endpoint,
with different integration methods on the random effects, using a semiparametric penalized
likelihood estimation. This approach extends that of Burzykowski et al. (2001) by
including in the bivariate copula model the random effects treatment-by-trial interaction.
Assume Sij and Tij the failure times associated respectively
with the surrogate and the true endpoints, for subject j(j = 1,..., ni) belonging to
the trial i (i = 1,..., G).
Let vi = (ui,
vSi, vTi) be the vector of trial
level random effects; ZS,ij = (ZSij1, ..., ZSijp)'
and ZT,ij = (ZTij1, ..., ZTijp)'
be covariates associated with Sij and Tij.
The joint frailty-copula model is defined as follows:
where
and the conditional survival functions are given by
in which
ui ~ N(0,\gamma),
ui ⊥ vSi, ui ⊥ vTi;
(vSi, vTi)T ~ N(0,\Sigmav)
with
In this model, \lambda0s(x) is the baseline hazard function associated with the
surrogate endpoint and \betaS the fixed effects (or log-hazard ratio) corresponding
to the covariates ZS,ij;
\lambda0T(x) is the baseline hazard function associated with the true endpoint
and \betaT the fixed treatment effects corresponding
to the covariates ZT,ij. The copula model serves to consider dependence between
the surrogate and true endpoints at the individual level. In the copula model, \theta is the copula
parameter used to quantify the strength of association. ui is a shared frailty effect associated
with the baseline hazard function that serve to take into account the heterogeneity between trials
of the baseline hazard function, associated with the fact that we have several trials in this
meta-analytical design. The power parameter \alpha distinguishes
trial-level heterogeneity between the surrogate and the true endpoint.
vSi and vTi are two correlated random effects treatment-by-trial interactions.
ZSij1 or ZTij1 represents the treatment arm to which the patient has been randomized.
For simplicity, we focus on the Clayton and Gumbel-Hougaard copula functions. In Clayton's
model, the copula function has the form
and in Gumbel's model, the copula function has the form
Surrogacy evaluation
We propose to base validation of a candidate surrogate endpoint on Kendall's \tau at the individual level and
coefficient of determination at the trial level, as in the classical approach (Burzykowski et al., 2001).
The formulations are given below.
Individual-level surrogacy
From the proposed model, according to the copula function, it can be shown that Kendall's \tau
is defined as :
where \theta is the copula parameter. Kendall's \tau is the difference between the probability of
concordance and the probability of discordance of two realizations of Sij and Tij.
It belongs to the interval [-1,1] and assumes a zero value when Sij and Tij are
independent.
Trial-level surrogacy
The key motivation for validating a surrogate endpoint is to be able to predict the effect
of treatment on the true endpoint, based on the observed effect of treatment on the
surrogate endpoint. As shown by Buyse et al. (2000), the coefficenient of
determination obtains from the covariance matrix \Sigmav of the random effects
treatment-by-trial interaction can be used to evaluate underlined prediction, and
therefore as surrogacy evaluation measurement at trial-level. It is defined by:
The SEs of Rtrial2 and \tau are calculated using the Delta-method. We also propose
Rtrial2 and 95% CI computed using the parametric bootstrap. The use of delta-method
can lead to confidence limits violating the [0,1], as noted by
(Burzykowski et al., 2001). However, using other methods would not significantly alter
the findings of the surrogacy assessment
Usage
jointSurroCopPenal(data, maxit = 40, indicator.alpha = 1,
frail.base = 1, n.knots = 6, LIMparam = 0.001, LIMlogl = 0.001,
LIMderiv = 0.001, nb.mc = 1000, nb.gh = 20, nb.gh2 = 32,
adaptatif = 0, int.method = 0, nb.iterPGH = 5, true.init.val = 0,
thetacopula.init = 1, sigma.ss.init = 0.5, sigma.tt.init = 0.5,
sigma.st.init = 0.48, gamma.init = 0.5, alpha.init = 1,
betas.init = 0.5, betat.init = 0.5, scale = 1,
random.generator = 1, kappa.use = 4, random = 0,
random.nb.sim = 0, seed = 0, init.kappa = NULL, ckappa = c(0,0),
typecopula = 1, nb.decimal = 4, print.times = TRUE, print.iter = FALSE)
Arguments
data
A data.frame containing at least seven variables entitled:
patientID: A numeric, that represents the patient's identifier and must be unique;
trialID: A numeric, that represents the trial in which each patient was randomized;
timeS: The follow-up time associated with the surrogate endpoint;
statusS: The event indicator associated with the surrogate endpoint. Normally
0 = no event, 1 = event;
timeT: The follow-up time associated with the true endpoint;
statusT: The event indicator associated with the true endpoint. Normally
0 = no event, 1 = event;
trt: The treatment indicator for each patient, with 1 = treated, 0 = untreated.
maxit
maximum number of iterations for the Marquardt algorithm.
The default being 40.
indicator.alpha
A binary, indicating whether the power's parameter \alpha should
be estimated (1) or not (0). If 0, \alpha will be set to 1 during estimation.
The default is 1.
frail.base
A binary, indicating whether the heterogeneity between trial on the baseline
risk is considered (1) or not (0), using
the shared cluster specific frailties ui . The default is 1.
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 frailtyPenal for more details).
LIMparam
Convergence threshold of the Marquardt algorithm for the
parameters, 10-3 by default (See frailtyPenal for more details).
LIMlogl
Convergence threshold of the Marquardt algorithm for the
log-likelihood, 10-3 by default (See frailtyPenal for more details).
LIMderiv
Convergence threshold of the Marquardt algorithm for the gradient,
10-3 by default
(See frailtyPenal for more details).
nb.mc
Number of samples considered in the Monte-Carlo integration. Required in the event
int.method is equals to 0, 2 or 4. A value between 500 and 1000 most often gives
good results. The default is 1000.
nb.gh
Number of nodes for the Gaussian-Hermite quadrature. It can
be chosen among 5, 7, 9, 12, 15, 20 and 32. The default is 20. A value greater than or equals to 15
allowed good results in simulation studies
nb.gh2
Number of nodes for the Gauss-Hermite quadrature used to re-estimate the model,
in the event of non-convergence, defined as previously. The default is 32.
adaptatif
A binary, indicates whether the pseudo adaptive Gaussian-Hermite quadrature (1) or the classical
Gaussian-Hermite quadrature (0) is used. The default is 0.
int.method
A numeric, indicates the integration method: 0 for Monte carlo,
1 for Gaussian-Hermite quadrature, 3 for Laplace approximation. The default is 0.
nb.iterPGH
Number of iterations before the re-estimation of the posterior random effects,
in the event of the two-steps pseudo-adaptive Gaussian-hermite quadrature. If set to 0 there is no
re-estimation". The default is 5.
true.init.val
Numerical value. Indicates if the given initial values to parameters (0) should be considered.
If set to 2, \alpha and \gamma are initialised using two separed shared frailty model
(see frailtyPenal for more details);
\sigma2vS, \sigma2vT and
\sigmavST
are fixed by the user or the default values;
\theta, \betaS and \betaT
are initialized using a classical joint
frailty model, considering individual level random effects, with \theta the variance of individual level random effects.
If the joint frailty model is
faced to convergence issues, \betaS and \betaT
are initialized using
two shared frailty models. In all other scenarios, if the simplified model does not converge,
default given parameters values are used. Initial values for spline's associated parameters
are fixed to 0.5. The default for this argument is 0.
thetacopula.init
Initial values for the copula parameter (\theta), required if true.init.val
is set to 0 or 2. The default is 1.
sigma.ss.init
Initial values for
\sigma2vS, required if true.init.val
is set to 0 or 2. The default is 0.5.
sigma.tt.init
Initial values for
\sigma2vT, required if true.init.val
is set to 0 or 2. The default is 0.5.
sigma.st.init
Initial values for
\sigmavST, required if true.init.val
is set to 0 or 2. The default is 0.48.
gamma.init
Initial values for \gamma, required if true.init.val
is set to 0 or 2. The default is 0.5.
alpha.init
Initial values for \alpha, required if true.init.val
is set to 0 or 2. The default is 1.
betas.init
Initial values for \betaS, required if true.init.val
is set to 0 or 2. The default is 0.5.
betat.init
Initial values for \betaT, required if true.init.val
is set to 0 or 2. The default is 0.5.
scale
A numeric that allows to rescale (by multiplication) the survival times, to avoid numerical
problems in the event of some convergence issues. If no change is needed the argument is set to 1, the default value.
eg: 1/365 aims to convert days to years ".
random.generator
Random number generator used by the Fortran compiler,
1 for the intrinsec subroutine Random_number and 2 for the
subroutine uniran(). The default is 1. In the event of convergence problem
with int.method set to 0, 2 or 4, that requires
integration by Monte-Carlo, user could change the random numbers generator.
kappa.use
A numeric, that indicates how to manage the smoothing parameters
k1
and k2 in the event of convergence issues. If it is set to 1,
the given smoothing parameters or those obtained by cross-validation are used.
If it is set to 3, the associated smoothing parameters are successively divided by 10,
in the event of convergence issues until 5 times. If it is set to 4, the management of the
smoothing parameter is as in the event 1, follows by the successive division as described
in the event 3 and preceded by the changing of the number of nodes for the Gauss-Hermite quadrature.
The default is 4.
random
A binary that says if we reset the random number generation with a different environment
at each call (1) or not (0). If it is set to 1, we use the computer clock
as seed. In the last case, it is not possible to reproduce the generated datasets".
The default is 0. Required if random.generator is set to 1.
random.nb.sim
If random is set to 1, a binary that indicates the number
of generations that will be made.
seed
The seed to use for data (or samples) generation. required if random is set to 0.
The default is 0.
init.kappa
smoothing parameter used to penalized the log-likelihood. By default (init.kappa = NULL) the values used
are obtain by cross-validation.
ckappa
Vector of two fixed values to add to the smoothing parameters. By default it is set to (0,0). this argument allows
to well manage the smoothing parameters in the event of convergence issues.
typecopula
The copula function used, can be 1 for clayton or 2 for Gumbel-Hougaard. The default is 1
nb.decimal
Number of decimal required for results presentation.
print.times
a logical parameter to print estimation time. Default
is TRUE.
print.iter
a logical parameter to print iteration process. Default
is FALSE.
Details
The estimated parameter are obtained using the robust Marquardt algorithm
(Marquardt, 1963) which is a combination between a Newton-Raphson algorithm
and a steepest descent algorithm. The iterations are stopped when the
difference between two consecutive log-likelihoods was small (<
10-3 ), the estimated
coefficients were stable (consecutive
values (< 10-3 )), and the gradient
small enough (< 10-3 ), by default.
Cubic M-splines of order 4 are used for the hazard function, and I-splines
(integrated M-splines) are used for the cumulative hazard function.
The inverse of the Hessian matrix is the variance estimator and to deal
with the positivity constraint of the variance component and the spline
coefficients, a squared transformation is used and the standard errors are
computed by the \Delta-method (Knight & Xekalaki, 2000). The smooth
parameter can be chosen by maximizing a likelihood cross validation
criterion (Joly and other, 1998).
We proposed based on the joint surrogate model a new definition
of the Kendall's \tau. Moreover, distinct numerical integration methods are available to approximate the
integrals in the marginal log-likelihood.
Non-convergence case management procedure
Special attention must be given to initializing model parameters, the choice of the number of
spline knots, the smoothing parameters and the number of quadrature points to solve convergence
issues. We first initialized parameters using the user's desired strategy, as specified
by the option true.init.val. When numerical or convergence problems are encountered,
with kappa.use set to 4, the model is fitted again using a combination of the following strategies:
vary the number of quadrature point (nb.gh to nb.gh2 or nb.gh2 to nb.gh)
in the event of the use of the Gaussian Hermite quadrature integration (see int.method);
divided or multiplied the smoothing parameters ( k1,
k2) by 10 or 100 according to
their preceding values, or used parameter vectors obtained during the last iteration (with a
modification of the number of quadrature points and smoothing parameters). Using this strategy,
we usually obtained during simulation the rejection rate less than 3%. A sensitivity analysis
was conducted without this strategy, and similar results were obtained on the converged samples,
with about a 23% rejection rate.
Value
This function return an object of class jointSurroPenal with elements :
EPS
A vector containing the obtained convergence thresholds with the Marquardt algorithm,
for the parameters, the log-likelihood and for the gradient;
b
A vector containing estimates for the splines parameter's; elements of the
lower triangular matrix (L) from the Cholesky decomposition such that \Sigma = LLT, with \Sigma
the covariance of the random effects (vSi,vTi);
the coefficient \alpha (if indicator.alpha is set to 1); the satandard error
of the random effect ui; the logarithm
of the copula parameter (\theta) if the Clayton copula function is considered, or
the squared root of \theta if the Gumbel copula is considered. The last two parameters represent
the regression coefficients \betaS and \betaT;
varH
The variance matrix of all parameters in b (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 in b;
loglikPenal
The complete marginal penalized log-likelihood;
LCV
the approximated likelihood cross-validation criterion in the semiparametric case (with H
minus the converged Hessian matrix, and l(.) the full log-likelihood).
xS
vector of times for surrogate endpoint where both survival and hazard function are estimated.
By default seq(0,max(time),length=99), where time is the vector of survival times;
lamS
array (dim = 3) of hazard estimates and confidence bands, for surrogate endpoint;
survS
array (dim = 3) of baseline survival estimates and confidence bands, for surrogate endpoint;
xT
vector of times for true endpoint where both survival and hazard function are estimated.
By default seq(0, max(time), length = 99), where time is the vector of survival times;
lamT
array (dim = 3) of hazard estimates and confidence bands, for true endpoint;
survT
array (dim = 3) of baseline survival estimates and confidence bands, for true endpoint;
n.iter
number of iterations needed to converge;
theta
Estimate for \theta;
gamma
Estimate for \gamma;
alpha
Estimate for \alpha;
zeta
A value equals to 1, no really use in this function;
sigma.s
Estimate for \sigma2vS;
sigma.t
Estimate for \sigma2vT;
sigma.st
Estimate for \sigmavST;
beta.s
Estimate for \betaS;
beta.t
Estimate for \betaT;
ui
A binary, that indicates if the heterogeneity between trial on the baseline risk
has been Considered (1), using the shared cluster specific frailties
(ui),
or not (0);
ktau
The Kendall's \tau with the correspondant 95 \% CI obtained from the delta-method;
R2.boot
The
R2trial
with the correspondant 95 \% CI obtained from the parametric bootstrap;
Coefficients
The estimates with the corresponding standard errors and the 95 \% CI
kappa
Positive smoothing parameters used for convergence. These values could be different to initial
values if kappa.use is set to 3 or 4;
scale
The value used to rescale the survival times
data
The dataset used in the model
varcov.Sigma
Covariance matrix of the estimates of
the estimates of (\sigma2vS,\sigma2vT,
\sigmavST) obtained from the delta-method
parameter
List of all arguments used in the model
type.joint
A code 3 that represents the joint frailty-copula model. This output is used in other functions
Author(s)
Casimir Ledoux Sofeu casimir.sofeu@u-bordeaux.fr, scl.ledoux@gmail.com and
Virginie Rondeau virginie.rondeau@inserm.fr
References
Burzykowski, T., Molenberghs, G., Buyse, M., Geys, H., and Renard, D. (2001). Validation
of surrogate end points in multiple randomized clinical trials with failure time end points.
Journal of the Royal Statistical Society: Series C (Applied Statistics) 50, 405-422.
Buyse, M., Molenberghs, G., Burzykowski, T., Renard, D., and Geys, H. (2000). The validation
of surrogate endpoints in meta-analyses of randomized experiments. Biostatistics 1, 49-67
Sofeu, C. L., Emura, T., and Rondeau, V. (2019). One-step validation method for surrogate
endpoints using data from multiple randomized cancer clinical trials with failure-time endpoints.
Statistics in Medicine 38, 2928-2942.
R. B. Nelsen. An introduction to Copulas. Springer, 2006
Prenen, L., Braekers, R., and Duchateau, L. (2017). Extending the archimedean copula methodology
to model multivariate survival data grouped in clusters of variable size. Journal of the Royal
Statistical Society: Series B (Statistical Methodology) 79, 483-505.
Sofeu, C. L., Emura, T., and Rondeau, V. (2020). A joint frailty-copula model for meta-analytic
validation of failure time surrogate endpoints in clinical trials. Under review
See Also
jointSurrCopSimul, summary.jointSurroPenal, jointSurroPenal, jointSurroPenalSimul
Examples
## Not run:
# Data from the advanced ovarian cancer randomized clinical trials.
data(dataOvarian)
joint.surro.Gumbel <- jointSurroCopPenal(data = dataOvarian, int.method = 0,
n.knots = 8, maxit = 50, kappa.use = 4, nb.mc = 1000, typecopula = 2,
print.iter = FALSE, scale = 1/365)
print(joint.surro.Gumbel)
joint.surro.Clayton <- jointSurroCopPenal(data = dataOvarian, int.method = 0,
n.knots = 8, maxit = 50, kappa.use = 4, nb.mc = 1000, typecopula = 1,
print.iter = FALSE, scale = 1/365)
print(joint.surro.Clayton)
## End(Not run)
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| jointSurroCopPenal | R Documentation |
Fit the one-step Joint frailty-copula model for evaluating a canditate surrogate endpoint
Description
Joint Frailty-Copula model for Surrogacy definition
Fit the one-step Joint frailty-copula surrogate model for the evaluation of a canditate surrogate endpoint,
with different integration methods on the random effects, using a semiparametric penalized
likelihood estimation. This approach extends that of Burzykowski et al. (2001) by
including in the bivariate copula model the random effects treatment-by-trial interaction.
Assume Sij and Tij the failure times associated respectively
with the surrogate and the true endpoints, for subject j(j = 1,..., ni) belonging to
the trial i (i = 1,..., G).
Let vi = (ui, vSi, vTi) be the vector of trial level random effects; ZS,ij = (ZSij1, ..., ZSijp)' and ZT,ij = (ZTij1, ..., ZTijp)' be covariates associated with Sij and Tij. The joint frailty-copula model is defined as follows:
where
and the conditional survival functions are given by
in which
ui ~ N(0,\gamma),
ui ⊥ vSi, ui ⊥ vTi;
(vSi, vTi)T ~ N(0,\Sigmav)
with
In this model, \lambda0s(x) is the baseline hazard function associated with the
surrogate endpoint and \betaS the fixed effects (or log-hazard ratio) corresponding
to the covariates ZS,ij;
\lambda0T(x) is the baseline hazard function associated with the true endpoint
and \betaT the fixed treatment effects corresponding
to the covariates ZT,ij. The copula model serves to consider dependence between
the surrogate and true endpoints at the individual level. In the copula model, \theta is the copula
parameter used to quantify the strength of association. ui is a shared frailty effect associated
with the baseline hazard function that serve to take into account the heterogeneity between trials
of the baseline hazard function, associated with the fact that we have several trials in this
meta-analytical design. The power parameter \alpha distinguishes
trial-level heterogeneity between the surrogate and the true endpoint.
vSi and vTi are two correlated random effects treatment-by-trial interactions.
ZSij1 or ZTij1 represents the treatment arm to which the patient has been randomized.
For simplicity, we focus on the Clayton and Gumbel-Hougaard copula functions. In Clayton's model, the copula function has the form
and in Gumbel's model, the copula function has the form
Surrogacy evaluation
We propose to base validation of a candidate surrogate endpoint on Kendall's \tau at the individual level and
coefficient of determination at the trial level, as in the classical approach (Burzykowski et al., 2001).
The formulations are given below.
Individual-level surrogacy
From the proposed model, according to the copula function, it can be shown that Kendall's \tau
is defined as :
where \theta is the copula parameter. Kendall's \tau is the difference between the probability of
concordance and the probability of discordance of two realizations of Sij and Tij.
It belongs to the interval [-1,1] and assumes a zero value when Sij and Tij are
independent.
Trial-level surrogacy
The key motivation for validating a surrogate endpoint is to be able to predict the effect
of treatment on the true endpoint, based on the observed effect of treatment on the
surrogate endpoint. As shown by Buyse et al. (2000), the coefficenient of
determination obtains from the covariance matrix \Sigmav of the random effects
treatment-by-trial interaction can be used to evaluate underlined prediction, and
therefore as surrogacy evaluation measurement at trial-level. It is defined by:
The SEs of Rtrial2 and \tau are calculated using the Delta-method. We also propose
Rtrial2 and 95% CI computed using the parametric bootstrap. The use of delta-method
can lead to confidence limits violating the [0,1], as noted by
(Burzykowski et al., 2001). However, using other methods would not significantly alter
the findings of the surrogacy assessment
Usage
jointSurroCopPenal(data, maxit = 40, indicator.alpha = 1,
frail.base = 1, n.knots = 6, LIMparam = 0.001, LIMlogl = 0.001,
LIMderiv = 0.001, nb.mc = 1000, nb.gh = 20, nb.gh2 = 32,
adaptatif = 0, int.method = 0, nb.iterPGH = 5, true.init.val = 0,
thetacopula.init = 1, sigma.ss.init = 0.5, sigma.tt.init = 0.5,
sigma.st.init = 0.48, gamma.init = 0.5, alpha.init = 1,
betas.init = 0.5, betat.init = 0.5, scale = 1,
random.generator = 1, kappa.use = 4, random = 0,
random.nb.sim = 0, seed = 0, init.kappa = NULL, ckappa = c(0,0),
typecopula = 1, nb.decimal = 4, print.times = TRUE, print.iter = FALSE)
Arguments
data |
A
|
maxit |
maximum number of iterations for the Marquardt algorithm.
The default being |
indicator.alpha |
A binary, indicating whether the power's parameter |
frail.base |
A binary, indicating whether the heterogeneity between trial on the baseline
risk is considered ( |
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 |
LIMparam |
Convergence threshold of the Marquardt algorithm for the
parameters, 10-3 by default (See |
LIMlogl |
Convergence threshold of the Marquardt algorithm for the
log-likelihood, 10-3 by default (See |
LIMderiv |
Convergence threshold of the Marquardt algorithm for the gradient,
10-3 by default
(See |
nb.mc |
Number of samples considered in the Monte-Carlo integration. Required in the event
|
nb.gh |
Number of nodes for the Gaussian-Hermite quadrature. It can
be chosen among 5, 7, 9, 12, 15, 20 and 32. The default is |
nb.gh2 |
Number of nodes for the Gauss-Hermite quadrature used to re-estimate the model,
in the event of non-convergence, defined as previously. The default is |
adaptatif |
A binary, indicates whether the pseudo adaptive Gaussian-Hermite quadrature |
int.method |
A numeric, indicates the integration method: |
nb.iterPGH |
Number of iterations before the re-estimation of the posterior random effects,
in the event of the two-steps pseudo-adaptive Gaussian-hermite quadrature. If set to |
true.init.val |
Numerical value. Indicates if the given initial values to parameters |
thetacopula.init |
Initial values for the copula parameter ( |
sigma.ss.init |
Initial values for
|
sigma.tt.init |
Initial values for
|
sigma.st.init |
Initial values for
|
gamma.init |
Initial values for |
alpha.init |
Initial values for |
betas.init |
Initial values for |
betat.init |
Initial values for |
scale |
A numeric that allows to rescale (by multiplication) the survival times, to avoid numerical
problems in the event of some convergence issues. If no change is needed the argument is set to 1, the default value.
eg: |
random.generator |
Random number generator used by the Fortran compiler,
|
kappa.use |
A numeric, that indicates how to manage the smoothing parameters
k1
and k2 in the event of convergence issues. If it is set to |
random |
A binary that says if we reset the random number generation with a different environment
at each call |
random.nb.sim |
If |
seed |
The seed to use for data (or samples) generation. required if |
init.kappa |
smoothing parameter used to penalized the log-likelihood. By default (init.kappa = NULL) the values used are obtain by cross-validation. |
ckappa |
Vector of two fixed values to add to the smoothing parameters. By default it is set to (0,0). this argument allows to well manage the smoothing parameters in the event of convergence issues. |
typecopula |
The copula function used, can be 1 for clayton or 2 for Gumbel-Hougaard. The default is |
nb.decimal |
Number of decimal required for results presentation. |
print.times |
a logical parameter to print estimation time. Default is TRUE. |
print.iter |
a logical parameter to print iteration process. Default is FALSE. |
Details
The estimated parameter are obtained using the robust Marquardt algorithm (Marquardt, 1963) which is a combination between a Newton-Raphson algorithm and a steepest descent algorithm. The iterations are stopped when the difference between two consecutive log-likelihoods was small (< 10-3 ), the estimated coefficients were stable (consecutive values (< 10-3 )), and the gradient small enough (< 10-3 ), by default. Cubic M-splines of order 4 are used for the hazard function, and I-splines (integrated M-splines) are used for the cumulative hazard function.
The inverse of the Hessian matrix is the variance estimator and to deal
with the positivity constraint of the variance component and the spline
coefficients, a squared transformation is used and the standard errors are
computed by the \Delta-method (Knight & Xekalaki, 2000). The smooth
parameter can be chosen by maximizing a likelihood cross validation
criterion (Joly and other, 1998).
We proposed based on the joint surrogate model a new definition
of the Kendall's \tau. Moreover, distinct numerical integration methods are available to approximate the
integrals in the marginal log-likelihood.
Non-convergence case management procedure
Special attention must be given to initializing model parameters, the choice of the number of
spline knots, the smoothing parameters and the number of quadrature points to solve convergence
issues. We first initialized parameters using the user's desired strategy, as specified
by the option true.init.val. When numerical or convergence problems are encountered,
with kappa.use set to 4, the model is fitted again using a combination of the following strategies:
vary the number of quadrature point (nb.gh to nb.gh2 or nb.gh2 to nb.gh)
in the event of the use of the Gaussian Hermite quadrature integration (see int.method);
divided or multiplied the smoothing parameters ( k1,
k2) by 10 or 100 according to
their preceding values, or used parameter vectors obtained during the last iteration (with a
modification of the number of quadrature points and smoothing parameters). Using this strategy,
we usually obtained during simulation the rejection rate less than 3%. A sensitivity analysis
was conducted without this strategy, and similar results were obtained on the converged samples,
with about a 23% rejection rate.
Value
This function return an object of class jointSurroPenal with elements :
EPS |
A vector containing the obtained convergence thresholds with the Marquardt algorithm, for the parameters, the log-likelihood and for the gradient; |
b |
A vector containing estimates for the splines parameter's; elements of the
lower triangular matrix (L) from the Cholesky decomposition such that |
varH |
The variance matrix of all parameters in |
varHIH |
The robust estimation of the variance matrix of all parameters in |
loglikPenal |
The complete marginal penalized log-likelihood; |
LCV |
the approximated likelihood cross-validation criterion in the semiparametric case (with |
xS |
vector of times for surrogate endpoint where both survival and hazard function are estimated. By default seq(0,max(time),length=99), where time is the vector of survival times; |
lamS |
array (dim = 3) of hazard estimates and confidence bands, for surrogate endpoint; |
survS |
array (dim = 3) of baseline survival estimates and confidence bands, for surrogate endpoint; |
xT |
vector of times for true endpoint where both survival and hazard function are estimated. By default seq(0, max(time), length = 99), where time is the vector of survival times; |
lamT |
array (dim = 3) of hazard estimates and confidence bands, for true endpoint; |
survT |
array (dim = 3) of baseline survival estimates and confidence bands, for true endpoint; |
n.iter |
number of iterations needed to converge; |
theta |
Estimate for |
gamma |
Estimate for |
alpha |
Estimate for |
zeta |
A value equals to |
sigma.s |
Estimate for |
sigma.t |
Estimate for |
sigma.st |
Estimate for |
beta.s |
Estimate for |
beta.t |
Estimate for |
ui |
A binary, that indicates if the heterogeneity between trial on the baseline risk
has been Considered ( |
ktau |
The Kendall's |
R2.boot |
The
|
Coefficients |
The estimates with the corresponding standard errors and the 95 |
kappa |
Positive smoothing parameters used for convergence. These values could be different to initial
values if |
scale |
The value used to rescale the survival times |
data |
The dataset used in the model |
varcov.Sigma |
Covariance matrix of the estimates of
the estimates of ( |
parameter |
List of all arguments used in the model |
type.joint |
A code |
Author(s)
Casimir Ledoux Sofeu casimir.sofeu@u-bordeaux.fr, scl.ledoux@gmail.com and Virginie Rondeau virginie.rondeau@inserm.fr
References
Burzykowski, T., Molenberghs, G., Buyse, M., Geys, H., and Renard, D. (2001). Validation of surrogate end points in multiple randomized clinical trials with failure time end points. Journal of the Royal Statistical Society: Series C (Applied Statistics) 50, 405-422.
Buyse, M., Molenberghs, G., Burzykowski, T., Renard, D., and Geys, H. (2000). The validation of surrogate endpoints in meta-analyses of randomized experiments. Biostatistics 1, 49-67
Sofeu, C. L., Emura, T., and Rondeau, V. (2019). One-step validation method for surrogate endpoints using data from multiple randomized cancer clinical trials with failure-time endpoints. Statistics in Medicine 38, 2928-2942.
R. B. Nelsen. An introduction to Copulas. Springer, 2006
Prenen, L., Braekers, R., and Duchateau, L. (2017). Extending the archimedean copula methodology to model multivariate survival data grouped in clusters of variable size. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 79, 483-505.
Sofeu, C. L., Emura, T., and Rondeau, V. (2020). A joint frailty-copula model for meta-analytic
validation of failure time surrogate endpoints in clinical trials. Under review
See Also
jointSurrCopSimul, summary.jointSurroPenal, jointSurroPenal, jointSurroPenalSimul
Examples
## Not run:
# Data from the advanced ovarian cancer randomized clinical trials.
data(dataOvarian)
joint.surro.Gumbel <- jointSurroCopPenal(data = dataOvarian, int.method = 0,
n.knots = 8, maxit = 50, kappa.use = 4, nb.mc = 1000, typecopula = 2,
print.iter = FALSE, scale = 1/365)
print(joint.surro.Gumbel)
joint.surro.Clayton <- jointSurroCopPenal(data = dataOvarian, int.method = 0,
n.knots = 8, maxit = 50, kappa.use = 4, nb.mc = 1000, typecopula = 1,
print.iter = FALSE, scale = 1/365)
print(joint.surro.Clayton)
## End(Not run)
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