Package 'depCensoring'

Title: Statistical Methods for Survival Data with Dependent Censoring
Description: Several statistical methods for analyzing survival data under various forms of dependent censoring are implemented in the package. In addition to accounting for dependent censoring, it offers tools to adjust for unmeasured confounding factors. The implemented approaches allow users to estimate the dependency between survival time and dependent censoring time, based solely on observed survival data. For more details on the methods, refer to Deresa and Van Keilegom (2021) <doi:10.1093/biomet/asaa095>, Czado and Van Keilegom (2023) <doi:10.1093/biomet/asac067>, Crommen et al. (2024) <doi:10.1007/s11749-023-00903-9> and Willems et al. (2024+) <doi:10.48550/arXiv.2403.11860>.
Authors: Ilias Willems [aut] , Gilles Crommen [aut] , Negera Wakgari Deresa [aut, cre] , Ingrid Van Keilegom [aut] , Claudia Czado [aut]
Maintainer: Negera Wakgari Deresa <[email protected]>
License: GPL-3
Version: 0.1.3
Built: 2024-10-17 13:36:47 UTC
Source: https://github.com/nago2020/depcensoring

Help Index

Nonparametric bootstrap approach for a Semiparametric transformation model under dependent censpring

Description

This function estimates the bootstrap standard errors for the finite-dimensional model parameters and for the non-parametric transformation function. Parallel computing using foreach has been used to speed up the estimation of standard errors.

Usage

boot.nonparTrans(init, resData, X, W, n.boot, n.iter, eps)

Arguments

init

Initial values for the finite dimensional parameters obtained from the fit of NonParTrans
resData

Data matrix with three columns; Z = the observed survival time, d1 = the censoring indicator of T and d2 = the censoring indicator of C.
X

Data matrix with covariates related to T
W

Data matrix with covariates related to C.
n.boot

Number of bootstraps to use in the estimation of bootstrap standard errors.
n.iter

Number of iterations; the default is n.iter = 15. The larger the number of iterations, the longer the computational time.
eps

Convergence error. This is set by the user in such away that the desired convergence is met; the default is eps = 1e-3

Value

Bootstrap standard errors for parameter estimates and for estimated cumulative hazard function.

Compute bivariate survival probability

Description

This function calculates a bivariate survival probability based on multivariate normal distribution.

Usage

Bvprob(lx, ly, rho)

Arguments

lx

The first lower bound of integration
ly

The second lower bound
rho

Association parameter

Transform Cholesky decomposition to covariance matrix

Description

This function transforms the parameters of the Cholesky de- composition to the covariance matrix, represented as a the row-wise con- catenation of the upper-triangular elements.

Usage

chol2par(par.chol1)

Arguments

par.chol1

The vector of Cholesky parameters.

Value

Covariance matrix corresponding to the provided Cholesky decomposition.

Transform Cholesky decomposition to covariance matrix parameter element.

Description

This function transforms the parameters of the Cholesky de- composition to a covariance matrix element. This function is used in chol2par.R.

Usage

chol2par.elem(a, b, par.chol1)

Arguments

a

The row index of the covariance matrix element to be computed.
b

The column index of the covariance matrix element to be computed.
par.chol1

The vector of Cholesky parameters.

Value

Specified element of the covariance matrix resulting from the provided Cholesky decomposition.

Competing risk likelihood function.

Description

This function implements the second step likelihood function of the competing risk model defined in Willems et al. (2024+).

Usage

cr.lik(
  n,
  s,
  Y,
  admin,
  cens.inds,
  M,
  Sigma,
  beta.mat,
  sigma.vct,
  rho.mat,
  theta.vct
)

Arguments

n

The sample size.
s

The number of competing risks.
Y

The observed times.
admin

Boolean value indicating whether or not administrative censoring should be taken into account.
cens.inds

matrix of censoring indicators (each row corresponds to a single observation).
M

Design matrix, consisting of [intercept, exo.cov, Z, cf]. Note that cf represents the multiple ways of 'handling' the endogenous covariate Z, see also the documentation of 'estimate.cmprsk.R'. When there is no confounding, M will be [intercept, exo.cov].
Sigma

The covariance matrix.
beta.mat

Matrix containing all of the covariate effects.
sigma.vct

Vector of standard deviations. Should be equal to sqrt(diag(Sigma)).
rho.mat

The correlation matrix.
theta.vct

Vector containing the parameters of the Yeo-Johnsontrans- formations.

Value

Evaluation of the log-likelihood function

References

Willems et al. (2024+). Flexible control function approach under competing risks (in preparation).

Data generation function for competing risks data

Description

This function generates competing risk data that can be used in simulation studies.

Usage

dat.sim.reg.comp.risks(
  n,
  par,
  iseed,
  s,
  conf,
  Zbin,
  Wbin,
  type.cov,
  A.upper = 15
)

Arguments

n

The sample size of the generated data set.
par

List of parameter vectors for each component of the transformation model.
iseed

Random seed.
s

The number of competing risks. Note that the given parameter vector could specify the parameters for the model with more than s competing risks, but in this case only the first s sets of parameters will be considered.
conf

Boolean value indicating whether the data set should contain confounding.
Zbin

Indicator whether the confounded variable is binary Zbin = 1 or not Zbin = 0. If conf = FALSE, this variable is ignored.
Wbin

Indicator whether the instrument is binary (Zbin = 1) or not Zbin = 0.
type.cov

Vector of characters "c" and "b", indicating which exogenous covariates should be continuous "c" or binary "b".
A.upper

The upper bound on the support of the administrative censoring distribution. This can be used to control for the amount of administrative censoring in the data. Default is A.upper = 15. A.upper = NULL will return a data set without administrative censoring.

Value

A generated data set

Derivative of transform Cholesky decomposition to covariance matrix.

Description

This function defines the derivative of the transformation function that maps Cholesky parameters to the full covariance matrix.

Usage

dchol2par(par.chol1)

Arguments

par.chol1

The vector of Cholesky parameters.

Value

Derivative of the function that transforms the cholesky parameters to the full covariance matrix.

Derivative of transform Cholesky decomposition to covariance matrix element.

Description

This function defines the derivative of the transformation function that maps Cholesky parameters to a covariance matrix element. This function is used in dchol2par.R.

Usage

dchol2par.elem(k, q, a, b, par.chol1)

Arguments

k

The row index of the parameter with respect to which to take the derivative.
q

the column index of the parameter with respect to which to take the derivative.
a

The row index of the covariance matrix element to be computed.
b

The column index of the covariance matrix element to be computed.
par.chol1

The vector of Cholesky parameters.

Value

Derivative of the function that transforms the cholesky parameters to the specified element of the covariance matrix, evaluated at the specified arguments.

Distance between vectors

Description

This function computes distance between two vectors based on L2-norm

Usage

Distance(b, a)

Arguments

b

Second vector
a

First vector

Derivative of the Yeo-Johnson transformation function

Description

Evaluates the derivative of the Yeo-Johnson transformation at the provided argument.

Usage

DYJtrans(y, theta)

Arguments

y

The argument to be supplied to the derivative of the Yeo-Johnson transformation.
theta

The parameter of the Yeo-Johnson transformation. This should be a number in the range [0,2].

Value

The transformed value of y.

Estimate the control function

Description

This function estimates the control function for the endogenous variable based on the provided covariates. This function is called inside estimate.cmprsk.R.

Usage

estimate.cf(XandW, Z, Zbin, gammaest = NULL)

Arguments

XandW

Design matrix of exogenous covariates.
Z

Endogenous covariate.
Zbin

Boolean value indicating whether endogenous covariate is binary.
gammaest

Vector of pre-estimated parameter vector. If NULL, this function will first estimate gammaest. Default value is gammaest = NULL.

Value

List containing the vector of values for the control function and the regression parameters of the first step.

Estimate the competing risks model of Rutten, Willems et al. (20XX).

Description

This function estimates the parameters in the competing risks model described in Willems et al. (2024+). Note that this model extends the model of Crommen, Beyhum and Van Keilegom (2024) and as such, this function also implements their methodology.

Usage

estimate.cmprsk(
  data,
  admin,
  conf,
  eoi.indicator.names = NULL,
  Zbin = NULL,
  inst = "cf",
  realV = NULL,
  eps = 0.001
)

Arguments

data

A data frame, adhering to the following formatting rules:

  • The first column, named "y", contains the observed times.

  • The next columns, named "delta1", delta2, etc. contain the indicators for each of the competing risks.

  • The next column, named da, contains the censoring indicator (independent censoring).

  • The next column should be a column of all ones (representing the intercept), names x0.

  • The subsequent columns should contain the values of the covariates, named x1, x2, etc.

  • When applicable, the next column should contain the values of the endogenous variable. This column should be named z.

  • When z is provided and an instrument for z is available, the next column, named w, should contain the values for the instrument.

admin

Boolean value indicating whether the data contains administrative censoring.
conf

Boolean value indicating whether the data contains confounding and hence indicating the presence of z and, possibly, w.
eoi.indicator.names

Vector of names of the censoring indicator columns pertaining to events of interest. Events of interest will be modeled allowing dependence between them, whereas all censoring events (corresponding to indicator columns not listed in eoi.indicator.names) will be treated as independent of every other event. If eoi.indicator.names == NULL, all events will be modeled dependently.
Zbin

Indicator value indicating whether (Zbin = TRUE) or not Zbin = FALSE the endogenous covariate is binary. Default is Zbin = NULL, corresponding to the case when conf == FALSE.
inst

Variable encoding which approach should be used for dealing with the confounding. inst = "cf" indicates that the control function approach should be used. inst = "W" indicates that the instrumental variable should be used 'as is'. inst = "None" indicates that Z will be treated as an exogenous covariate. Finally, when inst = "oracle", this function will access the argument realV and use it as the values for the control function. Default is inst = "cf".
realV

Vector of numerics with length equal to the number of rows in data. Used to provide the true values of the instrumental function to the estimation procedure.
eps

Value that will be added to the diagonal of the covariance matrix during estimation in order to ensure strictly positive variances.

Value

A list of parameter estimates in the second stage of the estimation algorithm (hence omitting the estimates for the control function), as well as an estimate of their variance and confidence intervals.

References

Willems et al. (2024+). Flexible control function approach under competing risks (in preparation).

Crommen, G., Beyhum, J., and Van Keilegom, I. (2024). An instrumental variable approach under dependent censoring. Test, 33(2), 473-495.

Examples

n <- 200

# Set parameters
gamma <- c(1, 2, 1.5, -1)
theta <- c(0.5, 1.5)
eta1 <- c(1, -1, 2, -1.5, 0.5)
eta2 <- c(0.5, 1, 1, 3, 0)

# Generate exogenous covariates
x0 <- rep(1, n)
x1 <- rnorm(n)
x2 <- rbinom(n, 1, 0.5)

# Generate confounder and instrument
w <- rnorm(n)
V <- rnorm(n, 0, 2)
z <- cbind(x0, x1, x2, w) %*% gamma + V
realV <- z - (cbind(x0, x1, x2, w) %*% gamma)

# Generate event times
err <- MASS::mvrnorm(n, mu = c(0, 0), Sigma =
matrix(c(3, 1, 1, 2), nrow = 2, byrow = TRUE))
bn <- cbind(x0, x1, x2, z, realV) %*% cbind(eta1, eta2) + err
Lambda_T1 <- bn[,1]; Lambda_T2 <- bn[,2]
x.ind = (Lambda_T1>0)
y.ind <- (Lambda_T2>0)
T1 <- rep(0,length(Lambda_T1))
T2 <- rep(0,length(Lambda_T2))
T1[x.ind] = ((theta[1]*Lambda_T1[x.ind]+1)^(1/theta[1])-1)
T1[!x.ind] = 1-(1-(2-theta[1])*Lambda_T1[!x.ind])^(1/(2-theta[1]))
T2[y.ind] = ((theta[2]*Lambda_T2[y.ind]+1)^(1/theta[2])-1)
T2[!y.ind] = 1-(1-(2-theta[2])*Lambda_T2[!y.ind])^(1/(2-theta[2]))
# Generate adminstrative censoring time
C <- runif(n, 0, 40)

# Create observed data set
y <- pmin(T1, T2, C)
delta1 <- as.numeric(T1 == y)
delta2 <- as.numeric(T2 == y)
da <- as.numeric(C == y)
data <- data.frame(cbind(y, delta1, delta2, da, x0, x1, x2, z, w))
colnames(data) <- c("y", "delta1", "delta2", "da", "x0", "x1", "x2", "z", "w")

# Estimate the model
admin <- TRUE                # There is administrative censoring in the data.
conf <- TRUE                 # There is confounding in the data (z)
eoi.indicator.names <- NULL  # We will not impose that T1 and T2 are independent
Zbin <- FALSE                # The confounding variable z is not binary
inst <- "cf"                 # Use the control function approach
# Since we don't use the oracle estimator, this argument is ignored anyway
realV <- NULL
estimate.cmprsk(data, admin, conf, eoi.indicator.names, Zbin, inst, realV)

Inverse Yeo-Johnson transformation function

Description

Computes the inverse Yeo-Johnson transformation of the provided argument.

Usage

IYJtrans(y, theta)

Arguments

y

The argument to be supplied to the inverse Yeo-Johnson transformation.
theta

The parameter of the inverted Yeo-Johnson transformation. This should be a number in the range [0,2].

Value

The transformed value of y.

Second step log-likelihood function.

Description

This function defines the log-likelihood used to estimate the second step in the competing risks extension of the model described in Willems et al. (2024+).

Usage

LikF.cmprsk(par, data, admin, conf, cf)

Arguments

par

Vector of all second step model parameters, consisting of the regression parameters, variance-covariance matrix elements and transformation parameters.
data

Data frame resulting from the 'uniformize.data.R' function.
admin

Boolean value indicating whether the data contains administrative censoring.
conf

Boolean value indicating whether the data contains confounding and hence indicating the presence of Z and W.
cf

"Control function" to be used. This can either be the (i) estimated control function, (ii) the true control function, (iii) the instrumental variable, or (iv) nothing (cf = NULL). Option (ii) is used when comparing the two-step estimator to the oracle estimator, and option (iii) is used to compare the two-step estimator with the naive estimator.

Value

Log-likelihood evaluation of the second step.

References

Willems et al. (2024+). Flexible control function approach under competing risks (in preparation).

Wrapper implementing likelihood function using Cholesky factorization.

Description

This function parametrizes the covariance matrix using its Cholesky decomposition, so that optimization of the likelihood can be done based on this parametrization, and positive-definiteness of the covariance matrix is guaranteed at each step of the optimization algorithm.

Usage

likF.cmprsk.Cholesky(par.chol, data, admin, conf, cf, eps = 0.001)

Arguments

par.chol

Vector of all second step model parameters, consisting of the regression parameters, Cholesky decomposition of the variance-covariance matrix elements and transformation parameters.
data

Data frame resulting from the 'uniformize.data.R' function.
admin

Boolean value indicating whether the data contains administrative censoring.
conf

Boolean value indicating whether the data contains confounding and hence indicating the presence of Z and W.
cf

"Control function" to be used. This can either be the (i) estimated control function, (ii) the true control function, (iii) the instrumental variable, or (iv) nothing (cf = NULL). Option (ii) is used when comparing the two-step estimator to the oracle estimator, and option (iii) is used to compare the two-step estimator with the naive estimator.
eps

Minimum value for the diagonal elements in the covariance matrix. Default is eps = 0.001.

Value

Log-likelihood evaluation of the second step.

First step log-likelihood function for Z continuous

Description

This function defines the maximum likelihood used to estimate the control function in the case of a continuous endogenous variable.

Usage

LikGamma1(gamma, Z, M)

Arguments

gamma

Vector of coefficients in the linear model used to estimate the control function.
Z

Endogenous covariate.
M

Design matrix, containing an intercept, the exogenous covariates and the instrumental variable.

Value

Returns the log-likelihood function corresponding to the data, evaluated at the point gamma.

First step log-likelihood function for Z binary.

Description

This function defines the maximum likelihood used to estimate the control function in the case of a binary endogenous variable.

Usage

LikGamma2(gamma, Z, M)

Arguments

gamma

Vector of coefficients in the logistic model used to estimate the control function.
Z

Endogenous covariate.
M

Design matrix, containing an intercept, the exogenous covariates and the instrumental variable.

Value

Returns the log-likelihood function corresponding to the data, evaluated at the point gamma.

Second likelihood function needed to fit the independence model in the second step of the estimation procedure.

Description

This function defines the log-likelihood used in estimating the second step in the competing risks extension of the model described in Willems et al. (2024+). The results of this function will serve as starting values for subsequent optimizations (LikI.comprsk.R and LikF.cmprsk.R)

Usage

LikI.bis(par, data, admin, conf, cf)

Arguments

par

Vector of all second step model parameters, consisting of the regression parameters, variance-covariance matrix elements and transformation parameters.
data

Data frame resulting from the 'uniformize.data.R' function.
admin

Boolean value indicating whether the data contains administrative censoring.
conf

Boolean value indicating whether the data contains confounding and hence indicating the presence of Z and W
cf

"Control function" to be used. This can either be the (i) estimated control function, (ii) the true control function, (iii) the instrumental variable, or (iv) nothing (cf = NULL). Option (ii) is used when comparing the two-step estimator to the oracle estimator, and option (iii) is used to compare the two-step estimator with the naive estimator.

Value

Starting values for subsequent optimization function used in the second step of the estimation procedure.

References

Willems et al. (2024+). Flexible control function approach under competing risks (in preparation).

Second step log-likelihood function under independence assumption.

Description

This function defines the log-likelihood used to estimate the second step in the competing risks extension assuming independence of some of the competing risks in the model described in Willems et al. (2024+).

Usage

LikI.cmprsk(par, data, eoi.indicator.names, admin, conf, cf)

Arguments

par

Vector of all second step model parameters, consisting of the regression parameters, variance-covariance matrix elements and transformation parameters.
data

Data frame resulting from the 'uniformize.data.R' function.
eoi.indicator.names

Vector of names of the censoring indicator columns pertaining to events of interest. Events of interest will be modeled allowing dependence between them, whereas all censoring events (corresponding to indicator columns not listed in eoi.indicator.names) will be treated as independent of every other event.
admin

Boolean value indicating whether the data contains administrative censoring.
conf

Boolean value indicating whether the data contains confounding and hence indicating the presence of Z and W
cf

"Control function" to be used. This can either be the (i) estimated control function, (ii) the true control function, (iii) the instrumental variable, or (iv) nothing (cf = NULL). Option (ii) is used when comparing the two-step estimator to the oracle estimator, and option (iii) is used to compare the two-step estimator with the naive estimator.

Value

Log-likelihood evaluation for the second step in the esimation procedure.

References

Willems et al. (2024+). Flexible control function approach under competing risks (in preparation).

Wrapper implementing likelihood function assuming independence between competing risks and censoring using Cholesky factorization.

Description

This function does the same as LikI.cmprsk (in fact, it even calls said function), but it parametrizes the covariance matrix using its Cholesky decomposition in order to guarantee positive definiteness. This function is never used, might not work and could be deleted.

Usage

LikI.cmprsk.Cholesky(
  par.chol,
  data,
  eoi.indicator.names,
  admin,
  conf,
  cf,
  eps = 0.001
)

Arguments

par.chol

Vector of all second step model parameters, consisting of the regression parameters, Cholesky decomposition of the variance-covariance matrix elements and transformation parameters.
data

Data frame resulting from the 'uniformize.data.R' function.
eoi.indicator.names

Vector of names of the censoring indicator columns pertaining to events of interest. Events of interest will be modeled allowing dependence between them, whereas all censoring events (corresponding to indicator columns not listed in eoi.indicator.names) will be treated as independent of every other event.
admin

Boolean value indicating whether the data contains administrative censoring.
conf

Boolean value indicating whether the data contains confounding and hence indicating the presence of Z and W.
cf

"Control function" to be used. This can either be the (i) estimated control function, (ii) the true control function, (iii) the instrumental variable, or (iv) nothing (cf = NULL). Option (ii) is used when comparing the two-step estimator to the oracle estimator, and option (iii) is used to compare the two-step estimator with the naive estimator.
eps

Minimum value for the diagonal elements in the covariance matrix. Default is eps = 0.001.

Value

Log-likelihood evaluation for the second step in the estimation procedure.

Full likelihood (including estimation of control function).

Description

This function defines the 'full' likelihood of the model. Specifically, it includes the estimation of the control function in the computation of the likelihood. This function is used in the estimation of the variance of the estimates (variance.cmprsk.R).

Usage

likIFG.cmprsk.Cholesky(
  parhatG,
  data,
  eoi.indicator.names,
  admin,
  conf,
  Zbin,
  inst
)

Arguments

parhatG

The full parameter vector.
data

Data frame.
eoi.indicator.names

Vector of names of the censoring indicator columns pertaining to events of interest. Events of interest will be modeled allowing dependence between them, whereas all censoring events (corresponding to indicator columns not listed in eoi.indicator.names) will be treated as independent of every other event. If eoi.indicator.names == NULL, all events will be modelled dependently.
admin

Boolean value indicating whether the data contains administrative censoring.
conf

Boolean value indicating whether the data contains confounding and hence indicating the presence of Z and W.
Zbin

Boolean value indicating whether the confounding variable is binary.
inst

Type of instrumental function to be used.

Value

Full model log-likelihood evaluation.

Logarithmic transformation function.

Description

Computes the logarithm of a number.

Usage

log_transform(y)

Arguments

y

Numerical value of which the logarithm is computed.

Value

This function returns the logarithm of the provided argument y if it is greater than zero. If y is smaller than zero, it will return 0.

Log-likelihood function for the Clayton copula.

Description

This likelihood function is maximized to estimate the model parameters under the Clayton copula.

Usage

loglike.clayton.unconstrained(para, Y, Delta, Dist.T, Dist.C)

Arguments

para

Estimated parameter values/initial values.
Y

Follow-up time.
Delta

Censoring indicator.
Dist.T

The distribution to be used for the survival time T. This argument can take one of the values from c("lnorm", "weibull") and has to be the same as Dist.C.
Dist.C

The distribution to be used for the censoring time C. This argument can take one of the values from c("lnorm", "weibull") and has to be the same as Dist.T.

Value

Maximized log-likelihood value.

Log-likelihood function for the Frank copula.

Description

This likelihood function is maximized to estimate the model parameters under the Frank copula.

Usage

loglike.frank.unconstrained(para, Y, Delta, Dist.T, Dist.C)

Arguments

para

Estimated parameter values/initial values.
Y

Follow-up time.
Delta

Censoring indicator.
Dist.T

The distribution to be used for the survival time T. This argument can take one of the values from c("lnorm", "weibull", "llogis").
Dist.C

The distribution to be used for the censoring time C. This argument can take one of the values from c("lnorm", "weibull", "llogis").

Value

Maximized log-likelihood value.

Log-likelihood function for the Gaussian copula.

Description

This likelihood function is maximized to estimate the model parameters under the Gaussian copula.

Usage

loglike.gaussian.unconstrained(para, Y, Delta, Dist.T, Dist.C)

Arguments

para

Estimated parameter values/initial values.
Y

Follow-up time.
Delta

Censoring indicator.
Dist.T

The distribution to be used for the survival time T. This argument can only the value "lnorm".
Dist.C

The distribution to be used for the censoring time C. This argument can only the value "lnorm".

Value

Maximized log-likelihood value.

Log-likelihood function for the Gumbel copula.

Description

This likelihood function is maximized to estimate the model parameters under the Gumbel copula.

Usage

loglike.gumbel.unconstrained(para, Y, Delta, Dist.T, Dist.C)

Arguments

para

Estimated parameter values/initial values.
Y

Follow-up time.
Delta

Censoring indicator.
Dist.T

The distribution to be used for the survival time T. This argument can take one of the values from c("lnorm", "weibull") and has to be the same as Dist.C.
Dist.C

The distribution to be used for the censoring time C. This argument can take one of the values from c("lnorm", "weibull") and has to be the same as Dist.T.

Value

Maximized log-likelihood value.

Log-likelihood function for the independence copula.

Description

This likelihood function is maximized to estimate the model parameters under the independence copula.

Usage

loglike.indep.unconstrained(para, Y, Delta, Dist.T, Dist.C)

Arguments

para

Estimated parameter values/initial values.
Y

Follow-up time.
Delta

Censoring indicator.
Dist.T

The distribution to be used for the survival time T. This argument can take one of the values from c("lnorm", "weibull", "llogis").
Dist.C

The distribution to be used for the censoring time C. This argument can take one of the values from c("lnorm", "weibull", "llogis").

Value

Maximized log-likelihood value.

Change H to long format

Description

Change a nonparametric transformation function to long format

Usage

Longfun(Z, T1, H)

Arguments

Z

Observed survival time, which is the minimum of T, C and A, where A is the administrative censoring time.
T1

Distinct observed survival time
H

Nonparametric transformation function estimate

Fit a semiparametric transformation model for dependent censoring

Description

This function allows to estimate the dependency parameter along all other model parameters. First, estimates a non-parametric transformation function, and then at the second stage it estimates other model parameters assuming that the non-parametric function is known. The details for implementing the dependent censoring methodology can be found in Deresa and Van Keilegom (2021).

Usage

NonParTrans(
  resData,
  X,
  W,
  start = NULL,
  n.iter = 15,
  bootstrap = FALSE,
  n.boot = 50,
  eps = 0.001
)

Arguments

resData

Data matrix with three columns; Z = the observed survival time, d1 = the censoring indicator of T and d2 = the censoring indicator of C.
X

Data matrix with covariates related to T
W

Data matrix with covariates related to C
start

Initial values for the finite dimensional parameters. If start is NULL, the initial values will be obtained by fitting an Accelerated failure time models.
n.iter

Number of iterations; the default is n.iter = 20. The larger the number of iterations, the longer the computational time.
bootstrap

A boolean indicating whether to compute bootstrap standard errors for making inferences.
n.boot

Number of bootstrap samples to use in the estimation of bootstrap standard errors if bootstrap = TRUE. The default is n.boot = 50. But, higher values of n.boot are recommended for obtaining good estimates of bootstrap standard errors.
eps

Convergence error. This is set by the user in such away that the desired convergence is met; the default is eps = 1e-3.

Value

This function returns a fit of a semiparametric transformation model; parameter estimates, estimate of the non-parametric transformation function, bootstrap standard errors for finite-dimensional parameters, the nonparametric cumulative hazard function, etc.

References

Deresa, N. and Van Keilegom, I. (2021). On semiparametric modelling, estimation and inference for survival data subject to dependent censoring, Biometrika, 108, 965–979.

Examples

# Toy data example to illustrate implementation
n = 300
beta = c(0.5, 1); eta = c(1,1.5); rho = 0.70
sigma = matrix(c(1,rho,rho,1),ncol=2)
err = MASS::mvrnorm(n, mu = c(0,0) , Sigma=sigma)
err1 = err[,1]; err2 = err[,2]
x1 = rbinom(n,1,0.5); x2 = runif(n,-1,1)
X = matrix(c(x1,x2),ncol=2,nrow=n); W = X   # data matrix
T1 = X%*%beta+err1
C =  W%*%eta+err2
T1 = exp(T1); C = exp(C)
A = runif(n,0,8); Y = pmin(T1,C,A)
d1 = as.numeric(Y==T1)
d2 = as.numeric(Y==C)
resData = data.frame("Z" = Y,"d1" = d1, "d2" = d2)   # should be data frame
colnames(X) = c("X1", "X2")
colnames(W) = c("W1","W2")

#  Bootstrap is false by default
output = NonParTrans(resData = resData, X = X, W = W, n.iter = 2)
output$parameterEstimates

Fit the dependent censoring models.

Description

Estimates the model parameters by maximizing the log-likelihood.

Usage

optimlikelihood(Y, Delta, Copula, Dist.T, Dist.C, start)

Arguments

Y

Follow-up time.
Delta

Censoring indicator.
Copula

The copula family. This argument can take values from c("frank","gumbel","clayton","gaussian","indep").
Dist.T

The distribution to be used for the survival time T. This argument can take one of the values from c("lnorm", "weibull", "llogis").
Dist.C

The distribution to be used for the censoring time C. This argument can take one of the values from c("lnorm", "weibull", "llogis").
start

Starting values

Value

A list containing the minimized negative log-likelihood using the independence copula model, the estimated parameter values for the model with the independence copula, the minimized negative log-likelihood using the specified copula model and the estimated parameter values for the model with the specified copula.

Estimation of a parametric dependent censoring model without covariates.

Description

Note that it is not assumed that the association parameter of the copula function is known, unlike most other papers in the literature. The details for implementing the methodology can be found in Czado and Van Keilegom (2023).

Usage

ParamCop(Y, Delta, Copula, Dist.T, Dist.C, start = c(1, 1, 1, 1))

Arguments

Y

Follow-up time.
Delta

Censoring indicator.
Copula

The copula family. This argument can take values from c("frank","gumbel","clayton","gaussian","indep").
Dist.T

The distribution to be used for the survival time T. This argument can take one of the values from c("lnorm", "weibull", "llogis").
Dist.C

The distribution to be used for the censoring time C. This argument can take one of the values from c("lnorm", "weibull", "llogis").
start

Starting values

Value

A table containing the minimized negative log-likelihood using the independence copula model, the estimated parameter values for the model with the independence copula, the minimized negative log-likelihood using the specified copula model and the estimated parameter values for the model with the specified copula.

References

Czado and Van Keilegom (2023). Dependent censoring based on parametric copulas. Biometrika, 110(3), 721-738.

Examples

tau = 0.75
Copula = "frank"
Dist.T = "weibull"
Dist.C = "lnorm"
par.T = c(2,1)
par.C = c(1,2)
n=1000

simdata<-TCsim(tau,Copula,Dist.T,Dist.C,par.T,par.C,n)

Y = simdata[[1]]
Delta = simdata[[2]]

ParamCop(Y,Delta,Copula,Dist.T,Dist.C)

Power transformation function.

Description

Computes a given power of a number.

Usage

power_transform(y, pw)

Arguments

y

The number which one wants to raise to a certain power pw.
pw

The power to which to raise y.

Value

This function returns the result of raising y to the power pw when y > 0. Otherwise, it will return 1.

Score equations of finite parameters

Description

This function computes the score vectors and the Jacobean matrix for finite model parameters.

Usage

ScoreEqn(theta, resData, X, W, H)

Arguments

theta

Vector of parameters in the semiparametric transformation model.
resData

Data matrix with three columns; Z = the observed survival time, d1 = the censoring indicator of T and d2 = the censoring indicator of C.
X

Data matrix with covariates related to T.
W

Data matrix with covariates related to C.
H

The estimated non-parametric transformation function for a given value of theta

Search function

Description

Function to indicate position of t in observed survival time

Usage

SearchIndicate(t, T1)

Arguments

t

fixed time t
T1

distinct observed survival time

Estimate a nonparametric transformation function

Description

This function estimates the nonparametric transformation function H when the survival time and censoring time are dependent given covariates. The estimating equation of H was derived based on the martingale ideas. More details about the derivation of a nonparmaetric estimator of H and its estimation algorithm can be found in Deresa and Van Keilegom (2021).

Usage

SolveH(theta, resData, X, W)

Arguments

theta

Vector of parameters in the semiparametric transformation model.
resData

Data matrix with three columns; Z = the observed survival time, d1 = the censoring indicator of T and d2 = the censoring indicator of C.
X

Data matrix with covariates related to T.
W

Data matrix with covariates related to C.

Value

Returns the estimated transformation function H for a fixed value of parameters theta.

References

Deresa, N. and Van Keilegom, I. (2021). On semiparametric modelling, estimation and inference for survival data subject to dependent censoring, Biometrika, 108, 965–979.

Estimating equation for Ht1

Description

This function obtains an estimating equation of H at the first observed survival time t1.

Usage

SolveHt1(Ht1, Z, nu, t, X, W, theta)

Arguments

Ht1

The solver solves for an optimal value of Ht1 by equating the estimating equation to zero.
Z

The observed survival time, which is the minimum of T, C and A.
nu

The censoring indicator for T or C
t

A fixed time point
X

Data matrix with covariates related to T.
W

Data matrix with covariates related to C.
theta

Vector of parameters

Estimate finite parameters based on score equations

Description

This function estimates the model parameters

Usage

SolveScore(theta, resData, X, W, H, eps = 0.001)

Arguments

theta

Vector of parameters in the semiparametric transformation model.
resData

Data matrix with three columns; Z = the observed survival time, d1 = the censoring indicator of T and d2 = the censoring indicator of C.
X

Data matrix with covariates related to T.
W

Data matrix with covariates related to C.
H

The estimated non-parametric transformation function for a given value of theta.
eps

Convergence error.

Function to simulate (Y,Delta) from the copula based model for (T,C).

Description

Generates the follow-up time and censoring indicator according to the specified model.

Usage

TCsim(
  tau = 0,
  Copula = "frank",
  Dist.T = "lnorm",
  Dist.C = "lnorm",
  par.T = c(0, 1),
  par.C = c(0, 1),
  n = 10000
)

Arguments

tau

Value of Kendall's tau for (T,C). The default value is 0.
Copula

The copula family. This argument can take values from c("frank","gumbel","clayton","gaussian","indep"). The default copula model is "frank".
Dist.T

Distribution of the survival time T. This argument can take one of the values from c("lnorm", "weibull", "llogis"). The default distribution is "lnorm".
Dist.C

Distribution of the censoring time C. This argument can take one of the values from c("lnorm", "weibull", "llogis"). The default distribution is "lnorm".
par.T

Parameter values for the distribution of T.
par.C

Parameter values for the distribution of C.
n

Sample size.

Value

A list containing the generated follow-up times and censoring indicators.

Examples

tau = 0.5
Copula = "gaussian"
Dist.T = "lnorm"
Dist.C = "lnorm"
par.T = c(1,1)
par.C = c(2,2)
n=1000

simdata <- TCsim(tau,Copula,Dist.T,Dist.C,par.T,par.C,n)
Y = simdata[[1]]
Delta = simdata[[2]]
hist(Y)
mean(Delta)

Standardize data format

Description

Checks the required preconditions of the data and possibly restructures the data.

Usage

uniformize.data(
  data,
  admin = FALSE,
  conf = FALSE,
  comp.risks = FALSE,
  Zbin = NULL,
  Wbin = NULL
)

Arguments

data

A data frame that should contain columns named Y and delta (unless comp.risks = TRUE, see later).
admin

Boolean value indicating whether the provided data frame contains administrative (i.e. independent) censoring on top of the dependent censoring (in the column named delta). The default is admin = FALSE.
conf

Boolean value indicating whether the provided data frame contains a confounded variable and a corresponding instrument. If cond = TRUE, the provided data frame should contain columns named Z and W, corresponding to the confounded variable and instrument, respectively. Moreover, Zbin and Wbin should be specified. The default value is conf = FALSE.
comp.risks

Boolean value indicating whether the provided data frame contains competing risks. If comp.risks = TRUE, the given data frame should contain the columns delta1, delta2, etc., corresponding to the indicators I(Y = T1), I(Y = T2), etc. respectively. The default is comp.risks = FALSE.
Zbin

Boolean or integer value (0, 1) indicating whether the confounded variable is binary. Zbin = TRUE or Zbin = 1 means that Z is binary. Zbin = FALSE or Zbin = 0 means that Z is continuous.
Wbin

Boolean or integer value (0, 1) indicating whether the instrument is binary. Wbin = TRUE or Wbin = 1 means that W is binary. Wbin = FALSE or Wbin = 0 means that W is continuous.

Value

Returns the uniformized data set.

Compute the variance of the estimates.

Description

This function computes the variance of the estimates computed by the 'estimate.cmprsk.R' function.

Usage

variance.cmprsk(
  parhatc,
  gammaest,
  data,
  admin,
  conf,
  inst,
  cf,
  eoi.indicator.names,
  Zbin,
  use.chol,
  n.trans,
  totparl
)

Arguments

parhatc

Vector of estimated parameters, computed in the first part of estimate.cmprsk.R.
gammaest

Vector of estimated parameters in the regression model for the control function.
data

A data frame.
admin

Boolean value indicating whether the data contains administrative censoring.
conf

Boolean value indicating whether the data contains confounding and hence indicating the presence of z and, possibly, w.
inst

Variable encoding which approach should be used for dealing with the confounding. inst = "cf" indicates that the control function approach should be used. inst = "W" indicates that the instrumental variable should be used 'as is'. inst = "None" indicates that Z will be treated as an exogenous covariate. Finally, when inst = "oracle", this function will access the argument realV and use it as the values for the control function. Default is inst = "cf".
cf

The control function used to estimate the second step.
eoi.indicator.names

Vector of names of the censoring indicator columns pertaining to events of interest. Events of interest will be modeled allowing dependence between them, whereas all censoring events (corresponding to indicator columns not listed in eoi.indicator.names) will be treated as independent of every other event. If eoi.indicator.names == NULL, all events will be modeled dependently.
Zbin

Indicator value indicating whether (Zbin = TRUE) or not Zbin = FALSE the endogenous covariate is binary. Default is Zbin = NULL, corresponding to the case when conf == FALSE.
use.chol

Boolean value indicating whether the cholesky decomposition was used in estimating the covariance matrix.
n.trans

Number of competing risks in the model (and hence, number of transformation models).
totparl

Total number of covariate effects (including intercepts) in all of the transformation models combined.

Value

Variance estimates of the provided vector of estimated parameters.

Yeo-Johnson transformation function

Description

Computes the Yeo-Johnson transformation of the provided argument.

Usage

YJtrans(y, theta)

Arguments

y

The argument to be supplied to the Yeo-Johnson transformation.
theta

The parameter of the Yeo-Johnson transformation. This should be a number in the range [0,2].

Value

The transformed value of y.