rc_spCox_copula {CopulaCenR}

Copula regression models with Cox semiparametric margins for bivariate right-censored data

Description

Fits a copula model with Cox semiparametric margins for bivariate right-censored data.

Usage

rc_spCox_copula(
  data,
  var_list,
  copula = "Clayton",
  method = "BFGS",
  iter = 500,
  stepsize = 1e-06,
  control = list(),
  B = 100,
  seed = 1
)

Arguments

data

a data frame; must have id (subject id), ind (1,2 for two margins), obs_time, status (0 for right-censoring, 1 for event).

var_list

the list of covariates to be fitted into the model.

copula

specify the copula family.

method

optimization method (see ?optim); default is "BFGS"; also can be "Newton" (see ?nlm).

iter

number of iterations when method = "Newton"; default is 500.

stepsize

size of optimization step when method = "Newton"; default is 1e-6.

control

a list of control parameters for methods other than "Newton"; see ?optim.

B

number of bootstraps for estimating standard errors with default 100;

seed

the bootstrap seed; default is 1

Details

The input data must be a data frame with columns id (subject id), ind (1,2 for two margins; each id must have both ind = 1 and 2), obs_time, status (0 for right-censoring, 1 for event) and covariates.

The supported copula models are "Clayton", "Gumbel", "Frank", "AMH", "Joe" and "Copula2". The "Copula2" model is a two-parameter copula model that incorporates Clayton and Gumbel as special cases. The parametric generator functions of copula functions are list below:

The Clayton copula has a generator

\phi_{\eta}(t) = (1+t)^{-1/\eta},

with \eta > 0 and Kendall's \tau = \eta/(2+\eta).

The Gumbel copula has a generator

\phi_{\eta}(t) = \exp(-t^{1/\eta}),

with \eta \geq 1 and Kendall's \tau = 1 - 1/\eta.

The Frank copula has a generator

\phi_{\eta}(t) = -\eta^{-1}\log \{1+e^{-t}(e^{-\eta}-1)\},

with \eta \geq 0 and Kendall's \tau = 1+4\{D_1(\eta)-1\}/\eta, in which D_1(\eta) = \frac{1}{\eta} \int_{0}^{\eta} \frac{t}{e^t-1}dt.

The AMH copula has a generator

\phi_{\eta}(t) = (1-\eta)/(e^{t}-\eta),

with \eta \in [0,1) and Kendall's \tau = 1-2\{(1-\eta)^2 \log (1-\eta) + \eta\}/(3\eta^2).

The Joe copula has a generator

\phi_{\eta}(t) = 1-(1-e^{-t})^{1/\eta},

with \eta \geq 1 and Kendall's \tau = 1 - 4 \sum_{k=1}^{\infty} \frac{1}{k(\eta k+2)\{\eta(k-1)+2\}}.

The Two-parameter copula (Copula2) has a generator

\phi_{\eta}(t) = \{1/(1+t^{\alpha})\}^{\kappa},

with \alpha \in (0,1], \kappa > 0 and Kendall's \tau = 1-2\alpha\kappa/(2\kappa+1).

The marginal distribution is a Cox semiparametric proportional hazards model. The copula parameter and coefficient standard errors are estimated from bootstrap.

Optimization methods can be all methods (except "Brent") from optim, such as "Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN". Users can also use "Newton" (from nlm).

Value

a CopulaCenR object summarizing the model. Can be used as an input to general S3 methods including summary, print, plot, lines, coef, logLik, AIC, BIC, fitted, predict.

Source

Tao Sun, Yi Liu, Richard J. Cook, Wei Chen and Ying Ding (2019). Copula-based Score Test for Bivariate Time-to-event Data, with Application to a Genetic Study of AMD Progression. Lifetime Data Analysis 25(3), 546-568.
Tao Sun and Ying Ding (In Press). Copula-based Semiparametric Regression Model for Bivariate Data under General Interval Censoring. Biostatistics. DOI: 10.1093/biostatistics/kxz032.

Examples

# fit a Clayton-Cox model
data(DRS)
clayton_cox <- rc_spCox_copula(data = DRS, var_list = "treat",
                            copula = "Clayton", B = 2)
summary(clayton_cox)