Description Usage Arguments Details Value Author(s) References See Also Examples

Generating data from a Cox model with time-dependent covariates.

1 |

`n` |
Sample size. |

`dist` |
Bivariate distribution assumed for generating the two covariates (time-fixed and time-dependent). Possible bivariate distributions are "exponential" and "weibull" (see details below). |

`corr` |
Correlation parameter. Possible values for the bivariate exponential distribution are between -1 and 1 (0 for independency). Any value between 0 (not included) and 1 (1 for independency) is accepted for the bivariate weibull distribution. |

`dist.par` |
Vector of parameters for the allowed distributions. Two (scale) parameters for the bivariate exponential distribution and four (2 shape parameters and 2 scale parameters) for the bivariate weibull distribution: (shape1, scale1, shape2, scale2). See details below. |

`model.cens` |
Model for censorship. Possible values are "uniform" and "exponential". |

`cens.par` |
Parameter for the censorship distribution. Must be greater than 0. |

`beta` |
Vector of two regression parameters for the two covariates. |

`lambda` |
Parameter for an exponential distribution. An exponential distribution is assumed for the baseline hazard function. |

The bivariate exponential distribution, also known as Farlie-Gumbel-Morgenstern distribution is given by

*F(x,y)=F_1(x)F_2(y)[1+α(1-F_1(x))(1-F_2(y))]*

for *x≥0* and *y≥0*. Where the marginal distribution functions *F_1* and *F_2* are exponential with scale parameters *θ_1* and *θ_2* and correlation parameter *α*, *-1 ≤ α ≤ 1*.

The bivariate Weibull distribution with two-parameter marginal distributions. It's survival function is given by

*S(x,y)=P(X>x,Y>y)=exp^(-[(x/θ_1)^(β_1/δ)+(y/θ_2)^(β_2/δ)]^δ)*

Where *0 < δ ≤ 1* and each marginal distribution has shape parameter *β_i* and a scale parameter *θ_i*, *i = 1, 2*.

An object with two classes, `data.frame`

and `TDCM`

.
To accommodate time-dependent effects, we used a counting process data-structure, introduced by Andersen and Gill (1982).
In this data-structure, apart the time-fixed covariates (named `covariate`

), an individual's survival data is expressed by three variables:
`start`

, `stop`

and `event`

. Individuals without change in the time-dependent covariate (named `tdcov`

) are represented by only one line of data,
whereas patients with a change in the time-dependent covariate must be represented by two lines.
For these patients, the first line represents the time period until the change in the time-dependent covariate;
the second line represents the time period that passes from that change to the end of the follow-up.
For each line of data, variables `start`

and `stop`

mark the time interval (start, stop) for the data,
while event is an indicator variable taking on value 1 if there was a death at time stop, and 0 otherwise.
More details about this data-structure can be found in papers by (Meira-Machado et al., 2009).

Artur Araújo, Luís Meira Machado and Susana Faria

Anderson, P.K., Gill, R.D. (1982). Cox's regression model for counting processes: a large sample study. *Annals of Statistics*, **10**(4), 1100-1120. doi: 10.1214/aos/1176345976

Cox, D.R. (1972). Regression models and life tables. *Journal of the Royal Statistical Society: Series B*, **34**(2), 187-202. doi: 10.1111/j.2517-6161.1972.tb00899.x

Johnson, M. E. (1987). *Multivariate Statistical Simulation*, John Wiley and Sons.

Johnson, N., Kotz, S. (1972). *Distribution in statistics: continuous multivariate distributions*, John Wiley and Sons.

Lu J., Bhattacharya G. (1990). Some new constructions of bivariate weibull models. *Annals of Institute of Statistical Mathematics*, **42**(3), 543-559. doi: 10.1007/BF00049307

Meira-Machado, L., Cadarso-Suárez, C., De Uña- Álvarez, J., Andersen, P.K. (2009). Multi-state models for the analysis of time to event data. *Statistical Methods in Medical Research*, **18**(2), 195-222. doi: 10.1177/0962280208092301

Meira-Machado L., Faria S. (2014). A simulation study comparing modeling approaches in an illness-death multi-state model. *Communications in Statistics - Simulation and Computation*, **43**(5), 929-946. doi: 10.1080/03610918.2012.718841

Meira-Machado, L., Sestelo M. (2019). Estimation in the progressive illness-death model: a nonexhaustive
review. *Biometrical Journal*, **61**(2), 245–263. doi: 10.1002/bimj.201700200

Therneau, T.M., Grambsch, P.M. (2000). *Modelling survival data: Extending the Cox Model*, New York: Springer.

1 2 3 4 5 6 7 8 9 10 11 12 | ```
tdcmdata <- genTDCM(n=1000, dist="weibull", corr=0.8, dist.par=c(2,3,2,3),
model.cens="uniform", cens.par=2.5, beta=c(-3.3,4), lambda=1)
head(tdcmdata, n=20L)
library(survival)
fit1<-coxph(Surv(start,stop,event)~tdcov+covariate,data=tdcmdata)
summary(fit1)
tdcmdata2 <- genTDCM(n=1000, dist="exponential", corr=0, dist.par=c(1,1),
model.cens="uniform", cens.par=1, beta=c(-3,2), lambda=0.5)
head(tdcmdata2, n=20L)
fit2<-coxph(Surv(start,stop,event)~tdcov+covariate,data=tdcmdata2)
summary(fit2)
``` |

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