ic_spTran_copula {CopulaCenR} | R Documentation |

## 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 |

`var_list` |
the list of covariates to be fitted into the copula model. |

`l` |
the left bound for all |

`u` |
the right bound for all |

`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 |

`iter` |
number of iterations when |

`stepsize` |
size of optimization step when method is |

`hes` |
default is |

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

### 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)
```

*CopulaCenR*version 1.2.3 Index]