data_sim_copula {CopulaCenR} | R Documentation |

## Simulate bivariate time-to-event times based on specific copula and marginal distributions

### Description

To generate a sample of subjects with two correlated event times based on specific copula and marginal models

### Usage

```
data_sim_copula(n, copula, eta, dist, baseline, var_list, COV_beta, x1, x2)
```

### Arguments

`n` |
sample size |

`copula` |
types of copula, including |

`eta` |
copula parameter |

`dist` |
marginal distributions, including |

`baseline` |
marginal distribution parameters.
For |

`var_list` |
a vector of covariate names; assume the same covariates for two margins |

`COV_beta` |
a vector of regression coefficients corresponding to |

`x1` |
a data frame of covariates for margin 1; it shall have n rows,
with columns corresponding to the |

`x2` |
a data frame of covariates for margin 2 |

### Details

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 marginal survival distributions are listed below:

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.

### Value

a data frame of bivariate time-to-event data with covariates

### Examples

```
library(CopulaCenR)
set.seed(1)
dat <- data_sim_copula(n = 500, copula = "Clayton", eta = 3,
dist = "Weibull", baseline = c(0.1,2),
var_list = c("var1", "var2"),
COV_beta = c(0.1, 0.1),
x1 = cbind(rnorm(500, 6, 2),
rbinom(500, 1, 0.5)),
x2 = cbind(rnorm(500, 6, 2),
rbinom(500, 1, 0.5)))
plot(x = dat$time[dat$ind == 1], y = dat$time[dat$ind == 2],
xlab = expression(t[1]), ylab = expression(t[2]),
cex.axis = 1, cex.lab = 1.3)
```

*CopulaCenR*version 1.2.3 Index]