ic_spTran_copula {CopulaCenR}

Copula regression models with semiparametric margins for bivariate interval-censored data

Description

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

Usage

ic_spTran_copula(
  data,
  var_list,
  l = 0,
  u,
  copula = "Copula2",
  m = 3,
  r = 3,
  method = "BFGS",
  iter = 300,
  stepsize = 1e-06,
  hes = TRUE,
  control = list()
)

Arguments

data

a data frame; must have id (subject id), ind (1,2 for two units in each subject), Left (0 if left-censoring), Right (Inf if right-censoring), status (0 for right-censoring, 1 for interval-censoring or left-censoring), and covariates by column.

var_list

the list of covariates to be fitted into the copula model.

l

the left bound for all Left and Right endpoints of observed finite intervals; default is 0.

u

the right bound for all Left and Right endpoints of observed finite intervals; has to be a finite value

copula

Types of copula model.

m

integer, degree of Berstein polynomials for both margins; default is 3

r

postive transformation parameter for the semiparametric transformation marginal model.

method

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

iter

number of iterations when method = "Newton"; default is 300.

stepsize

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

hes

default is TRUE for hessian calculation; if LRT is desired, can set hes = FALSE to save time.

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 (sample id), ind (1,2 for the two units from the same id), Left (0 if left-censoring), Right (Inf if right-censoring), status (0 for right-censoring, 1 for interval-censoring or left-censoring), and covariates. The function does not allow Left == Right.

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 semiparametric transformation models are built based on Bernstein polynomials, which is formulated below:

S(t|Z) = \exp[-G\{\Lambda(t) e^{Z^{\top}\beta}\}],

where t is time, Z is covariate, \beta is coefficient and \Lambda(t) is an unspecified function with infinite dimensions. We approximate \Lambda(t) in a sieve space constructed by Bernstein polynomials with degree m. By default, m=3. In the end, all model parameters are estimated by the sieve estimators (Sun and Ding, In Press).

The G(\cdot) function is the transformation function with a parameter r > 0, which has a form of G(x) = \frac{(1+x)^r - 1}{r}, when 0 < r \leq 2 and G(x) = \frac{\log\{1 + (r-2)x\}}{r - 2} when r > 2. When r = 1, the marginal model becomes a proportional hazards model; when r = 3, the marginal model becomes a proportional odds model. In practice, m and r can be selected based on the AIC value.

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 Copula2-Semiparametric model
data(AREDS)
copula2_sp <- ic_spTran_copula(data = AREDS, copula = "Copula2",
              l = 0, u = 15, m = 3, r = 3,
              var_list = c("ENROLLAGE","rs2284665","SevScaleBL"))
summary(copula2_sp)