rc_par_copula {CopulaCenR}

Copula regression models with parametric margins for bivariate right-censored data

Description

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

Usage

rc_par_copula(
  data,
  var_list,
  copula = "Clayton",
  m.dist = "Weibull",
  method = "BFGS",
  iter = 500,
  stepsize = 1e-06,
  control = list()
)

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.

m.dist

specify the marginal baseline distribution.

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.

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 supported marginal distributions are "Weibull" (proportional hazards), "Gompertz" (proportional hazards) and "Loglogistic" (proportional odds). These marginal distributions are listed below and we also assume the same baseline parameters between two margins.

The Weibull (PH) survival distribution is

\exp \{-(t/\lambda)^k e^{Z^{\top}\beta}\},

with \lambda > 0 as scale and k > 0 as shape.

The Gompertz (PH) survival distribution is

\exp \{-\frac{b}{a}(e^{at}-1) e^{Z^{\top}\beta}\},

with a > 0 as shape and b > 0 as rate.

The Loglogistic (PO) survival distribution is

\{1+(t/\lambda)^{k} e^{Z^{\top}\beta} \}^{-1},

with \lambda > 0 as scale and k > 0 as shape.

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-Weibull model
data(DRS)
clayton_wb <- rc_par_copula(data = DRS, var_list = "treat",
                            copula = "Clayton",
                            m.dist = "Weibull")
summary(clayton_wb)