frailtyIllnessDeath: Fit a Weibull Illness-Death Model with Optional Shared...
In frailtypack: Shared, Joint (Generalized) Frailty Models; Surrogate Endpoints
frailtyIllnessDeath R Documentation
Fit a Weibull Illness-Death Model with Optional Shared Frailty
Description
Fit a three-state illness-death model (states: 0=Healthy, 1=Illness, 2=Death)
using Weibull baseline hazards for all transitions (0->1, 0->2, 1->2).
Allows for shared gamma frailty between the three transitions, acting
multiplicatively on the hazards within specified groups (clusters).
The model accommodates right-censored and left-truncated data.
The transition from the illness state to death (1->2) can be modeled using
either a Markov or a Semi-Markov assumption for the baseline hazards time scale.
Usage
frailtyIllnessDeath (formula, formula.terminalEvent, data, model = "Semi-Markov",
maxit = 300, init.B, init.Theta, init.hazard.weib,
LIMparam = 1e-3, LIMlogl = 1e-3, LIMderiv = 1e-3,
partialH, x01, x02, x12, print.info = FALSE,
print.result = TRUE, blinding = 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. Status should be 1 if event 0->1 occurred,
0 otherwise. Covariates for transition 0->1 are specified here. Shared frailty
is specified using cluster(group_variable). Left truncation can be
included via Surv(time1, time2, status).
formula.terminalEvent
A formula object. Response must be a survival::Surv object
representing the time and status for the terminal event (death). Status should be 1 if
transition to state 2 occurred (either from state 0 or state 1), 0 otherwise (censored). Covariates for
transitions 0->2 and 1->2 are specified on the RHS (currently assumes the same covariates
apply to both 0->2 and 1->2).
Note on Left Truncation: The illness-death model implemented assumes a
single entry time per subject. This entry time, specified in formula,
indicates that the subject must be in the initial state (state 0, having experienced
neither transition 0->1 nor 0->2) at that entry_time to be included
in the analysis. The entry time should not be specified again here in
formula.terminalEvent.
data
A 'data.frame' with the variables used in the formulas.
model
Character string specifying the model for the 1->2 transition
baseline hazard. Allowed values: "Semi-Markov" (default) or "Markov".
maxit
Maximum number of iterations for the Marquardt algorithm.
Default is 300.
init.B
Optional. A vector of initial values for regression coefficients.
Order is (\beta_1,\beta_2,\beta_3).
The total length must match the total number of regression coefficients.
If omitted, the regression coefficients are initialized from default values.
These defaults are obtained by fitting three independent Weibull proportional
hazards models (without frailty), one for each transition.
init.Theta
Optional. Initial value for the frailty variance \theta.
Default is 0.1. This parameter is only used if cluster() is present
in formula.
init.hazard.weib
Optional. A vector of initial values for the Weibull baseline
hazard parameters. Must be of size 6, in the order: scale(0->1),
shape(0->1), scale(0->2), shape(0->2), scale(1->2), shape(1->2).
If omitted, the baseline hazard parameters are initialized from default values.
These defaults are obtained by fitting three independent Weibull proportional
hazards models (without frailty), one for each transition.
LIMparam
Convergence threshold for the parameters based on the maximum
absolute difference between successive iterations (10^{-3} by default).
LIMlogl
Convergence threshold for the log-likelihood based on the absolute
difference between successive iterations (10^{-3} by default).
LIMderiv
Convergence threshold based on the relative distance to the optimum
(related to gradient and Hessian) (10^{-3} by default). See Details.
partialH
Optional. Integer vector specifying the indices of parameters to
exclude from the Hessian matrix when calculating the relative distance
convergence criterion (LIMderiv). This is only considered if the first
two criteria (LIMparam, LIMlogl) are met and the full Hessian
is problematic (e.g., not invertible). Default is NULL.
x01
Optional. Numeric vector of time points at which to calculate baseline
hazard and survival functions for transition 0->1. Defaults to a sequence of
99 points from 0 to the maximum observed time for transition 0->1.
x02
Optional. Numeric vector of time points at which to calculate baseline
hazard and survival functions for transition 0->2. Defaults to a sequence of
99 points from 0 to the maximum observed time for transition 0->2.
x12
Optional. Numeric vector of time points at which to calculate baseline
hazard and survival functions for transition 1->2. Defaults to a sequence of
99 points from 0 to the maximum observed time for transition 1->2.
print.info
Logical. If TRUE, prints information at each iteration
of the optimization algorithm. Default is FALSE.
print.result
Logical. If TRUE, prints a formatted summary of the
results. Default is TRUE.
blinding
Logical. If TRUE, the algorithm attempts to continue even
if the log-likelihood calculation produces non-finite values (e.g., Inf,
NaN) at some iteration. Setting to FALSE may cause the algorithm
to stop earlier in such cases. Default is TRUE.
Details
Let T_1 be the time to the non-terminal event (illness, 0->1) and
T_2 be the time to the terminal event (death, 0->2 or 1->2).
The transition intensities are defined as:
\lambda_{01}(t) = \lim_{\Delta t \to 0^+} \frac{\mathbb{P}(t \leq T_1 \leq t + \Delta t \mid T_1 \geq t, T_2 \geq t)}{\Delta t}
\lambda_{02}(t) = \lim_{\Delta t \to 0^+} \frac{\mathbb{P}(t \leq T_2 \leq t + \Delta t \mid T_1 \geq t, T_2 \geq t)}{\Delta t}
\lambda_{12}(t \mid T_1 = s) = \lim_{\Delta t \to 0^+} \frac{\mathbb{P}(t \leq T_2 \leq t + \Delta t \mid T_1 = s, T_2 \geq t)}{\Delta t} \quad (0 < s < t)
A proportional hazards model with a shared frailty term \omega_i is assumed
for each transition within group i. For the j^{th} subject
(j=1,...,n_i) in the i^{th} group (i=1,...,G) the transition intensities are
defined as follows:
\lambda_{01}^{ij}(t |\omega_i,X_{01}^{ij}) = \lambda_{0,01}(t) \omega_i \exp(\beta_1^{T} X_{01}^{ij})
\lambda_{02}^{ij}(t |\omega_i,X_{02}^{ij}) = \lambda_{0,02}(t) \omega_i \exp(\beta_2^{T} X_{02}^{ij})
\lambda_{12}^{ij}(t | T_1 = s, \omega_i, X_{12}^{ij}) = \lambda_{0,12}(t | T_1 = s) \omega_i \exp(\beta_3^{T} X_{12}^{ij}) \quad (0 < s < t)
\omega_i is the frailty term for the i^{th} group.
For subject-specific frailties, use cluster(id) where id is unique (n_i=1).
\beta_1, \beta_2 and \beta_3 are respectively the vectors
of time fixed regression coefficients for the transitions 0->1, 0->2 and 1->2.
X_{01}^{ij}, X_{02}^{ij} and X_{12}^{ij} are respectively the vectors
of time fixed covariates for the j^{th} subject in the i^{th} group for the
transitions 0->1, 0->2 and 1->2.
\lambda_{0,01}(.), \lambda_{0,02}(.) and \lambda_{0,12}(.) are respectively
the baseline hazard functions for the transitions 0->1, 0->2 and 1->2.
The baseline hazard \lambda_{0,12}(t | T_1 = s) depends on the 'model' argument:
-
Markov model: \lambda_{0,12}(t | T_1 = s) = \lambda_{0,12}(t).
The risk depends on the time since origin.This model is suitable when the risk for
the transition 1 -> 2 is primarily influenced by absolute time rather than the duration spent in state 1.
-
Semi-Markov model: \lambda_{0,12}(t | T_1 = s) = \lambda_{0,12}(t - s).
The risk depends on the time since entering state 1 (sojourn time).
This model is appropriate when the risk for the transition 1 -> 2 is
more influenced by the time spent in state 1 rather than the time elapsed
since the initial starting point.
The Weibull baseline hazard parameterization is:
\lambda(t) = \frac{\gamma}{\lambda^\gamma} \cdot t^{\gamma - 1}
where \lambda is the scale parameter and \gamma the shape parameter
Value
An object of class 'frailtyIllnessDeath ' containing:
- b
Vector of the estimated parameters. Order is:
(scale(0->1), shape(0->1), scale(0->2), shape(0->2), scale(1->2), shape(1->2),
\hat {\theta} (if frailty), \hat{\beta}_1, \hat{\beta}_2, \hat{\beta}_3).
- call
The matched function call.
- coef
Vector of estimated regression coefficients.
- loglik
The marginal log-likelihood value at the final parameter estimates.
- grad
Gradient vector of the log-likelihood at the final parameter estimates.
- n
The number of subjects (observations) used in the fit.
- n.events
Vector containing the number of observed events: count for 0->1, count for 0->2, count for 1->2 and count for censoring.
- n.iter
Number of iterations.
- vcov
Variance-covariance matrix for the parameters listed in b.
- npar
Total number of estimated parameters.
- nvar
Total number of regression coefficients.
- shape.weib
Vector of estimated Weibull baseline shape parameters (shape(0->1),shape(0->2),shape(1->2)).
- scale.weib
Vector of estimated Weibull baseline scale parameters (scale(0->1),scale(0->2),scale(1->2)).
- crit
Convergence status code: 1=converged, 2=maximum iterations reached, 3=converged using partial Hessian, 4=the algorithm encountered a problem in the loglikelihood computation.
- Frailty
Logical. TRUE if a model with shared frailty (cluster(.)) was fitted.
- beta_p.value
Vector of p-values from Wald tests for the regression coefficients in coef.
- AIC
Akaike Information Criterion, calculated as AIC=\frac{1}{n}(np - l(.)), where np is the number of parameters and l is the log-likelihood.
- x01
Vector of time points used for calculating baseline functions for transition 0->1.
- x02
Vector of time points used for calculating baseline functions for transition 0->2.
- x12
Vector of time points (or sojourn times if Semi-Markov) used for calculating baseline functions for transition 1->2.
- lam01
Matrix containing baseline hazard estimates and 95% confidence intervals for transition 0->1 calculated at x01.
- lam02
Matrix containing baseline hazard estimates and 95% confidence intervals for transition 0->2 calculated at x02.
- lam12
Matrix containing baseline hazard estimates and 95% confidence intervals for transition 1->2 calculated at x12.
- surv01
Matrix containing baseline survival estimates and 95% confidence intervals for transition 0->1 calculated at x01.
- surv02
Matrix containing baseline survival estimates and 95% confidence intervals for transition 0->2 calculated at x01.
- surv12
Matrix containing baseline survival estimates and 95% confidence intervals for transition 0->2 calculated at x12.
- median.01
Matrix containing the estimated median baseline survival time and its 95% confidence interval for transition 0->1.
- median.02
Matrix containing the estimated median baseline survival time and its 95% confidence interval for transition 0->2.
- median.12
Matrix containing the estimated median baseline survival time and its 95% confidence interval for transition 1->2.
- linear.pred01
Vector of linear predictors calculated for transition 0->1. For non-frailty models, this is \hat{\beta}_1^{t}X_{01}. For frailty models, it includes the estimated log-frailty: \hat{\beta}_1^{t}X_{01} + \log(\hat{\omega}_i).
- linear.pred02
Vector of linear predictors calculated for transition 0->2. For non-frailty models, this is \hat{\beta}_2^{t}X_{02}. For frailty models, it includes the estimated log-frailty: \hat{\beta}_2^{t}X_{02} + \log(\hat{\omega}_i).
- linear.pred12
Vector of linear predictors calculated for transition 1->2. For non-frailty models, this is \hat{\beta}_3^{t}X_{12}. For frailty models, it includes the estimated log-frailty: \hat{\beta}_3^{t}X_{12} + \log(\hat{\omega}_i).
- names.factor.01
Character vector identifying factor covariates included for transition 0->1.
- names.factor.02
Character vector identifying factor covariates included for transition 0->2.
- names.factor.12
Character vector identifying factor covariates included for transition 1->2.
- global_chisq.01
Vector containing the chi-squared statistics for global Wald tests of factor variables for transition 0->1.
- global_chisq.02
Vector containing the chi-squared statistics for global Wald tests of factor variables for transition 0->2.
- global_chisq.12
Vector containing the chi-squared statistics for global Wald tests of factor variables for transition 1->2.
- dof_chisq.01
Vector containing the degrees of freedom for the global Wald tests for transition 0->1.
- dof_chisq.02
Vector containing the degrees of freedom for the global Wald tests for transition 0->2.
- dof_chisq.12
Vector containing the degrees of freedom for the global Wald tests for transition 1->2.
- p.global_chisq.01
Vector containing the p-values for the global Wald tests for transition 0->1.
- p.global_chisq.02
Vector containing the p-values for the global Wald tests for transition 0->2.
- p.global_chisq.12
Vector containing the p-values for the global Wald tests for transition 1->2.
- global_chisq.test.01
Indicator (0/1) whether any global factor tests were performed for transition 0->1.
- global_chisq.test.02
Indicator (0/1) whether any global factor tests were performed for transition 0->2.
- global_chisq.test.12
Indicator (0/1) whether any global factor tests were performed for transition 1->2.
If Frailty is TRUE, the following components related to frailty are also included:
- groups
The number of unique groups specified by cluster(.).
- theta
The estimated variance (\hat{\theta}) of the Gamma frailty distribution.
- theta_p.value
The p-value from a Wald test for the null hypothesis H_0: \theta=0.
- VarTheta
The estimated variance of the frailty variance estimator: \hat{Var}(\hat{\theta}).
- frailty.pred
Vector containing the empirical Bayes predictions of the frailty term for each group.
- frailty.var
Vector containing the variances of the empirical Bayes frailty predictions.
- frailty.sd
Vector containing the standard errors of the empirical Bayes frailty predictions.
Note
The optimization uses the robust Marquardt algorithm (Marquardt, 1963),
combining Newton-Raphson and steepest descent steps. Iterations stop when
criteria LIMparam, LIMlogl, and LIMderiv are all met.
Confidence bands for the baseline hazard and baseline survival functions were
computed using a Monte Carlo simulation approach based on the estimated Weibull
parameters. A sample of size 1000 was drawn from the joint distribution of the
shape and scale parameters. For each sampled parameter set, the baseline functions
were evaluated over the time grid defined by the vector x0. Pointwise 95
confidence bands were then obtained by computing the 2.5th and 97.5th percentiles
of the simulated values at each time point for each baseline function.
References
Lee, C., Gilsanz, P., & Haneuse, S. (2021). Fitting a shared frailty
illness-death model to left-truncated semi-competing risks data
to examine the impact of education level on incident dementia.
BMC Medical Research Methodology, 21(1), 1-13.
Marquardt, D. W. (1963). An algorithm for least-squares estimation of nonlinear
parameters. SIAM Journal on Applied Mathematics, 11(2), 431-441.
Liquet, B., Timsit, J. F., & Rondeau, V. (2012). Investigating hospital
heterogeneity with a multi-state frailty model: application to nosocomial
pneumonia disease in intensive care units. BMC Medical Research
Methodology, 12(1), 1-14.
Examples
data(readmission2)
##-- Fitting models --##
#-- Semi-Markovian Weibull Illness-Death model with
# group frailty shared between transitions --#
ModIllnessDeath_SemiMarkov_Group <- frailtyIllnessDeath(
formula = Surv(observed_disease_time, disease_status) ~ cluster(group) + sex,
formula.terminalEvent = Surv(observed_death_time, death_status) ~ sex,
data = readmission2,
model = "Semi-Markov",
print.info = FALSE,
maxit = 100
)
#-- Markovian Weibull Illness-Death model with subject-specific
# frailty shared between transitions --#
ModIllnessDeath_Markov_Subject <- frailtyIllnessDeath(
formula = Surv(observed_disease_time, disease_status) ~ cluster(id) + sex,
formula.terminalEvent = Surv(observed_death_time, death_status) ~ sex,
data = readmission2,
model = "Markov",
print.info = FALSE,
maxit = 100
)
#--- Semi-Markovian Weibull Illness-Death model with a factor and
# no covariates for the non-terminal event ---#
ModIllnessDeath_SemiMarkov_NoCov_Factor <- frailtyIllnessDeath(
formula = Surv(observed_disease_time, disease_status) ~ 1,
formula.terminalEvent = Surv(observed_death_time, death_status) ~ factor(dukes),
data = readmission2,
model = "Semi-Markov",
print.info = FALSE,
maxit = 100
)
#--- Semi-Markovian Weibull Illness-Death model with left truncation ---#
data(Paq810)
ModIllnessDeath_SemiMarkov_LeftTrunc <- frailtyIllnessDeath(
formula = Surv(e, r, dementia) ~ gender + certif,
formula.terminalEvent = Surv(t, death) ~ certif,
data = Paq810,
model = "Semi-Markov",
print.info = FALSE,
maxit = 100
)
#--- Markovian Weibull Illness-Death model with left truncation ---#
ModIllnessDeath_Markov_LeftTrunc <- frailtyIllnessDeath(
formula = Surv(e, r, dementia) ~ gender + certif,
formula.terminalEvent = Surv(t, death) ~ certif,
data = Paq810,
model = "Markov",
print.info = FALSE,
maxit = 100
)
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| frailtyIllnessDeath | R Documentation |
Fit a Weibull Illness-Death Model with Optional Shared Frailty
Description
Fit a three-state illness-death model (states: 0=Healthy, 1=Illness, 2=Death) using Weibull baseline hazards for all transitions (0->1, 0->2, 1->2). Allows for shared gamma frailty between the three transitions, acting multiplicatively on the hazards within specified groups (clusters). The model accommodates right-censored and left-truncated data. The transition from the illness state to death (1->2) can be modeled using either a Markov or a Semi-Markov assumption for the baseline hazards time scale.
Usage
frailtyIllnessDeath (formula, formula.terminalEvent, data, model = "Semi-Markov",
maxit = 300, init.B, init.Theta, init.hazard.weib,
LIMparam = 1e-3, LIMlogl = 1e-3, LIMderiv = 1e-3,
partialH, x01, x02, x12, print.info = FALSE,
print.result = TRUE, blinding = TRUE)
Arguments
formula |
A formula object with the response on the left of a |
formula.terminalEvent |
A formula object. Response must be a Note on Left Truncation: The illness-death model implemented assumes a
single entry time per subject. This entry time, specified in |
data |
A 'data.frame' with the variables used in the formulas. |
model |
Character string specifying the model for the 1->2 transition baseline hazard. Allowed values: "Semi-Markov" (default) or "Markov". |
maxit |
Maximum number of iterations for the Marquardt algorithm. Default is 300. |
init.B |
Optional. A vector of initial values for regression coefficients.
Order is ( |
init.Theta |
Optional. Initial value for the frailty variance |
init.hazard.weib |
Optional. A vector of initial values for the Weibull baseline hazard parameters. Must be of size 6, in the order: scale(0->1), shape(0->1), scale(0->2), shape(0->2), scale(1->2), shape(1->2). If omitted, the baseline hazard parameters are initialized from default values. These defaults are obtained by fitting three independent Weibull proportional hazards models (without frailty), one for each transition. |
LIMparam |
Convergence threshold for the parameters based on the maximum
absolute difference between successive iterations ( |
LIMlogl |
Convergence threshold for the log-likelihood based on the absolute
difference between successive iterations ( |
LIMderiv |
Convergence threshold based on the relative distance to the optimum
(related to gradient and Hessian) ( |
partialH |
Optional. Integer vector specifying the indices of parameters to
exclude from the Hessian matrix when calculating the relative distance
convergence criterion ( |
x01 |
Optional. Numeric vector of time points at which to calculate baseline hazard and survival functions for transition 0->1. Defaults to a sequence of 99 points from 0 to the maximum observed time for transition 0->1. |
x02 |
Optional. Numeric vector of time points at which to calculate baseline hazard and survival functions for transition 0->2. Defaults to a sequence of 99 points from 0 to the maximum observed time for transition 0->2. |
x12 |
Optional. Numeric vector of time points at which to calculate baseline hazard and survival functions for transition 1->2. Defaults to a sequence of 99 points from 0 to the maximum observed time for transition 1->2. |
print.info |
Logical. If |
print.result |
Logical. If |
blinding |
Logical. If |
Details
Let T_1 be the time to the non-terminal event (illness, 0->1) and
T_2 be the time to the terminal event (death, 0->2 or 1->2).
The transition intensities are defined as:
\lambda_{01}(t) = \lim_{\Delta t \to 0^+} \frac{\mathbb{P}(t \leq T_1 \leq t + \Delta t \mid T_1 \geq t, T_2 \geq t)}{\Delta t}
\lambda_{02}(t) = \lim_{\Delta t \to 0^+} \frac{\mathbb{P}(t \leq T_2 \leq t + \Delta t \mid T_1 \geq t, T_2 \geq t)}{\Delta t}
\lambda_{12}(t \mid T_1 = s) = \lim_{\Delta t \to 0^+} \frac{\mathbb{P}(t \leq T_2 \leq t + \Delta t \mid T_1 = s, T_2 \geq t)}{\Delta t} \quad (0 < s < t)
A proportional hazards model with a shared frailty term \omega_i is assumed
for each transition within group i. For the j^{th} subject
(j=1,...,n_i) in the i^{th} group (i=1,...,G) the transition intensities are
defined as follows:
\lambda_{01}^{ij}(t |\omega_i,X_{01}^{ij}) = \lambda_{0,01}(t) \omega_i \exp(\beta_1^{T} X_{01}^{ij})
\lambda_{02}^{ij}(t |\omega_i,X_{02}^{ij}) = \lambda_{0,02}(t) \omega_i \exp(\beta_2^{T} X_{02}^{ij})
\lambda_{12}^{ij}(t | T_1 = s, \omega_i, X_{12}^{ij}) = \lambda_{0,12}(t | T_1 = s) \omega_i \exp(\beta_3^{T} X_{12}^{ij}) \quad (0 < s < t)
\omega_i is the frailty term for the i^{th} group.
For subject-specific frailties, use cluster(id) where id is unique (n_i=1).
\beta_1, \beta_2 and \beta_3 are respectively the vectors
of time fixed regression coefficients for the transitions 0->1, 0->2 and 1->2.
X_{01}^{ij}, X_{02}^{ij} and X_{12}^{ij} are respectively the vectors
of time fixed covariates for the j^{th} subject in the i^{th} group for the
transitions 0->1, 0->2 and 1->2.
\lambda_{0,01}(.), \lambda_{0,02}(.) and \lambda_{0,12}(.) are respectively
the baseline hazard functions for the transitions 0->1, 0->2 and 1->2.
The baseline hazard \lambda_{0,12}(t | T_1 = s) depends on the 'model' argument:
-
Markov model:
\lambda_{0,12}(t | T_1 = s) = \lambda_{0,12}(t). The risk depends on the time since origin.This model is suitable when the risk for the transition 1 -> 2 is primarily influenced by absolute time rather than the duration spent in state 1. -
Semi-Markov model:
\lambda_{0,12}(t | T_1 = s) = \lambda_{0,12}(t - s). The risk depends on the time since entering state 1 (sojourn time). This model is appropriate when the risk for the transition 1 -> 2 is more influenced by the time spent in state 1 rather than the time elapsed since the initial starting point.
The Weibull baseline hazard parameterization is:
\lambda(t) = \frac{\gamma}{\lambda^\gamma} \cdot t^{\gamma - 1}
where \lambda is the scale parameter and \gamma the shape parameter
Value
An object of class 'frailtyIllnessDeath ' containing:
- b
Vector of the estimated parameters. Order is: (scale(0->1), shape(0->1), scale(0->2), shape(0->2), scale(1->2), shape(1->2),
\hat {\theta}(if frailty),\hat{\beta}_1,\hat{\beta}_2,\hat{\beta}_3).- call
The matched function call.
- coef
Vector of estimated regression coefficients.
- loglik
The marginal log-likelihood value at the final parameter estimates.
- grad
Gradient vector of the log-likelihood at the final parameter estimates.
- n
The number of subjects (observations) used in the fit.
- n.events
Vector containing the number of observed events: count for 0->1, count for 0->2, count for 1->2 and count for censoring.
- n.iter
Number of iterations.
- vcov
Variance-covariance matrix for the parameters listed in
b.- npar
Total number of estimated parameters.
- nvar
Total number of regression coefficients.
- shape.weib
Vector of estimated Weibull baseline shape parameters (shape(0->1),shape(0->2),shape(1->2)).
- scale.weib
Vector of estimated Weibull baseline scale parameters (scale(0->1),scale(0->2),scale(1->2)).
- crit
Convergence status code: 1=converged, 2=maximum iterations reached, 3=converged using partial Hessian, 4=the algorithm encountered a problem in the loglikelihood computation.
- Frailty
Logical.
TRUEif a model with shared frailty (cluster(.)) was fitted.- beta_p.value
Vector of p-values from Wald tests for the regression coefficients in
coef.- AIC
Akaike Information Criterion, calculated as
AIC=\frac{1}{n}(np - l(.)), where np is the number of parameters and l is the log-likelihood.- x01
Vector of time points used for calculating baseline functions for transition 0->1.
- x02
Vector of time points used for calculating baseline functions for transition 0->2.
- x12
Vector of time points (or sojourn times if Semi-Markov) used for calculating baseline functions for transition 1->2.
- lam01
Matrix containing baseline hazard estimates and 95% confidence intervals for transition 0->1 calculated at
x01.- lam02
Matrix containing baseline hazard estimates and 95% confidence intervals for transition 0->2 calculated at
x02.- lam12
Matrix containing baseline hazard estimates and 95% confidence intervals for transition 1->2 calculated at
x12.- surv01
Matrix containing baseline survival estimates and 95% confidence intervals for transition 0->1 calculated at
x01.- surv02
Matrix containing baseline survival estimates and 95% confidence intervals for transition 0->2 calculated at
x01.- surv12
Matrix containing baseline survival estimates and 95% confidence intervals for transition 0->2 calculated at
x12.- median.01
Matrix containing the estimated median baseline survival time and its 95% confidence interval for transition 0->1.
- median.02
Matrix containing the estimated median baseline survival time and its 95% confidence interval for transition 0->2.
- median.12
Matrix containing the estimated median baseline survival time and its 95% confidence interval for transition 1->2.
- linear.pred01
Vector of linear predictors calculated for transition 0->1. For non-frailty models, this is
\hat{\beta}_1^{t}X_{01}. For frailty models, it includes the estimated log-frailty:\hat{\beta}_1^{t}X_{01} + \log(\hat{\omega}_i).- linear.pred02
Vector of linear predictors calculated for transition 0->2. For non-frailty models, this is
\hat{\beta}_2^{t}X_{02}. For frailty models, it includes the estimated log-frailty:\hat{\beta}_2^{t}X_{02} + \log(\hat{\omega}_i).- linear.pred12
Vector of linear predictors calculated for transition 1->2. For non-frailty models, this is
\hat{\beta}_3^{t}X_{12}. For frailty models, it includes the estimated log-frailty:\hat{\beta}_3^{t}X_{12} + \log(\hat{\omega}_i).- names.factor.01
Character vector identifying factor covariates included for transition 0->1.
- names.factor.02
Character vector identifying factor covariates included for transition 0->2.
- names.factor.12
Character vector identifying factor covariates included for transition 1->2.
- global_chisq.01
Vector containing the chi-squared statistics for global Wald tests of factor variables for transition 0->1.
- global_chisq.02
Vector containing the chi-squared statistics for global Wald tests of factor variables for transition 0->2.
- global_chisq.12
Vector containing the chi-squared statistics for global Wald tests of factor variables for transition 1->2.
- dof_chisq.01
Vector containing the degrees of freedom for the global Wald tests for transition 0->1.
- dof_chisq.02
Vector containing the degrees of freedom for the global Wald tests for transition 0->2.
- dof_chisq.12
Vector containing the degrees of freedom for the global Wald tests for transition 1->2.
- p.global_chisq.01
Vector containing the p-values for the global Wald tests for transition 0->1.
- p.global_chisq.02
Vector containing the p-values for the global Wald tests for transition 0->2.
- p.global_chisq.12
Vector containing the p-values for the global Wald tests for transition 1->2.
- global_chisq.test.01
Indicator (0/1) whether any global factor tests were performed for transition 0->1.
- global_chisq.test.02
Indicator (0/1) whether any global factor tests were performed for transition 0->2.
- global_chisq.test.12
Indicator (0/1) whether any global factor tests were performed for transition 1->2.
If Frailty is TRUE, the following components related to frailty are also included:
- groups
The number of unique groups specified by
cluster(.).- theta
The estimated variance (
\hat{\theta}) of the Gamma frailty distribution.- theta_p.value
The p-value from a Wald test for the null hypothesis
H_0: \theta=0.- VarTheta
The estimated variance of the frailty variance estimator:
\hat{Var}(\hat{\theta}).- frailty.pred
Vector containing the empirical Bayes predictions of the frailty term for each group.
- frailty.var
Vector containing the variances of the empirical Bayes frailty predictions.
- frailty.sd
Vector containing the standard errors of the empirical Bayes frailty predictions.
Note
The optimization uses the robust Marquardt algorithm (Marquardt, 1963),
combining Newton-Raphson and steepest descent steps. Iterations stop when
criteria LIMparam, LIMlogl, and LIMderiv are all met.
Confidence bands for the baseline hazard and baseline survival functions were
computed using a Monte Carlo simulation approach based on the estimated Weibull
parameters. A sample of size 1000 was drawn from the joint distribution of the
shape and scale parameters. For each sampled parameter set, the baseline functions
were evaluated over the time grid defined by the vector x0. Pointwise 95
confidence bands were then obtained by computing the 2.5th and 97.5th percentiles
of the simulated values at each time point for each baseline function.
References
Lee, C., Gilsanz, P., & Haneuse, S. (2021). Fitting a shared frailty illness-death model to left-truncated semi-competing risks data to examine the impact of education level on incident dementia. BMC Medical Research Methodology, 21(1), 1-13.
Marquardt, D. W. (1963). An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics, 11(2), 431-441.
Liquet, B., Timsit, J. F., & Rondeau, V. (2012). Investigating hospital heterogeneity with a multi-state frailty model: application to nosocomial pneumonia disease in intensive care units. BMC Medical Research Methodology, 12(1), 1-14.
Examples
data(readmission2)
##-- Fitting models --##
#-- Semi-Markovian Weibull Illness-Death model with
# group frailty shared between transitions --#
ModIllnessDeath_SemiMarkov_Group <- frailtyIllnessDeath(
formula = Surv(observed_disease_time, disease_status) ~ cluster(group) + sex,
formula.terminalEvent = Surv(observed_death_time, death_status) ~ sex,
data = readmission2,
model = "Semi-Markov",
print.info = FALSE,
maxit = 100
)
#-- Markovian Weibull Illness-Death model with subject-specific
# frailty shared between transitions --#
ModIllnessDeath_Markov_Subject <- frailtyIllnessDeath(
formula = Surv(observed_disease_time, disease_status) ~ cluster(id) + sex,
formula.terminalEvent = Surv(observed_death_time, death_status) ~ sex,
data = readmission2,
model = "Markov",
print.info = FALSE,
maxit = 100
)
#--- Semi-Markovian Weibull Illness-Death model with a factor and
# no covariates for the non-terminal event ---#
ModIllnessDeath_SemiMarkov_NoCov_Factor <- frailtyIllnessDeath(
formula = Surv(observed_disease_time, disease_status) ~ 1,
formula.terminalEvent = Surv(observed_death_time, death_status) ~ factor(dukes),
data = readmission2,
model = "Semi-Markov",
print.info = FALSE,
maxit = 100
)
#--- Semi-Markovian Weibull Illness-Death model with left truncation ---#
data(Paq810)
ModIllnessDeath_SemiMarkov_LeftTrunc <- frailtyIllnessDeath(
formula = Surv(e, r, dementia) ~ gender + certif,
formula.terminalEvent = Surv(t, death) ~ certif,
data = Paq810,
model = "Semi-Markov",
print.info = FALSE,
maxit = 100
)
#--- Markovian Weibull Illness-Death model with left truncation ---#
ModIllnessDeath_Markov_LeftTrunc <- frailtyIllnessDeath(
formula = Surv(e, r, dementia) ~ gender + certif,
formula.terminalEvent = Surv(t, death) ~ certif,
data = Paq810,
model = "Markov",
print.info = FALSE,
maxit = 100
)
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