rc_par_copula {CopulaCenR} | R Documentation |

## 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 |

`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 |

`iter` |
number of iterations when |

`stepsize` |
size of optimization step when |

`control` |
a list of control parameters for methods other than |

### 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)
```

*CopulaCenR*version 1.2.3 Index]