# corrTP: Correlation between two gap times In TPmsm: Estimation of Transition Probabilities in Multistate Models

 corrTP R Documentation

## Correlation between two gap times

### Description

Provides the correlation between the bivariate times for some copula distributions.

### Usage

`corrTP(dist, corr, dist.par)`

### Arguments

 `dist` The distribution. Possible bivariate distributions are “exponential” and “weibull”. `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 location parameters and 2 scale parameters) for the bivariate Weibull distribution. See details below.

### Details

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.

### Author(s)

Artur Araújo, Javier Roca-Pardiñas and Luís Meira-Machado

### References

Araújo A, Meira-Machado L, Roca-Pardiñas J (2014). TPmsm: Estimation of the Transition Probabilities in 3-State Models. Journal of Statistical Software, 62(4), 1-29. doi: 10.18637/jss.v062.i04

Johnson N., Kotz S. (1972). Distributions 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

`dgpTP`.

### Examples

```# Example for the bivariate Weibull distribution
corrTP(dist = "weibull", corr = 0.5, dist.par = c(2, 7, 2, 7))

# Example for the bivariate Exponential distribution
corrTP(dist = "exponential", corr = 1, dist.par = c(1, 1))
```

TPmsm documentation built on Jan. 14, 2023, 1:17 a.m.