Type: Package
Title: Generalized Farlie-Gumbel-Morgenstern Copula
Version: 1.0.4
Date: 2019-12-11
Author: Jia-Han Shih
Maintainer: Jia-Han Shih <tommy355097@gmail.com>
Description: Compute bivariate dependence measures and perform bivariate competing risks analysis under the generalized Farlie-Gumbel-Morgenstern (FGM) copula. See Shih and Emura (2018) <doi:10.1007/s00180-018-0804-0> and Shih and Emura (2019) <doi:10.1007/s00362-016-0865-5> for details.
Depends: cmprsk, compound.Cox, joint.Cox
License: GPL-2
Encoding: UTF-8
LazyData: true
RoxygenNote: 7.0.0
Repository: CRAN
NeedsCompilation: no
Packaged: 2019-12-11 14:12:56 UTC; Tommy
Date/Publication: 2019-12-11 16:30:25 UTC

Generalized Farlie-Gumbel-Morgenstern Copula

Description

Compute bivariate dependence measures and perform bivariate competing risks analysis under the generalized Farlie-Gumbel-Morgenstern (FGM) copula. See Shih and Emura (2016, 2018) for details.

Details

The functions in this package are based on latent failure time models with competing risks in Shih and Emura (2018). However, they can be adapted to dependent censoring models in Emura and Chen (2018). See MLE.GFGM.spline for example.

Author(s)

Jia-Han Shih

Maintainer: Jia-Han Shih <tommy355097@gmail.com>

References

Shih J-H, Emura T (2016) Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula, Statistical Papers, doi: 10.1007/s00362-016-0865-5.

Shih J-H, Emura T (2018) Likelihood-based inference for bivariate latent failure time models with competing risks udner the generalized FGM copula, Computational Statistics, doi: 10.1007/s00180-018-0804-0.

Emura T, Chen Y-H (2018) Analysis of Survival Data with Dependent Censoring, Copula-Based Approaches, JSS Research Series in Statistics, Springer, in press.

The Cramer-von Mises type statistics under the generalized FGM copula

Description

Compute the Cramer-von Mises type statistics under the generalized FGM copula.

Usage

CvM.GFGM.BurrIII(
  t.event,
  event1,
  event2,
  Alpha,
  Beta,
  Gamma,
  g1,
  g2,
  p,
  q,
  theta,
  eta = 0,
  Sdist.plot = TRUE
)

Arguments

t.event

Vector of the observed failure times.

event1

Vector of the indicators for the failure cause 1.

event2

Vector of the indicators for the failure cause 2.

Alpha

Positive shape parameter for the Burr III margin (failure cause 1).

Beta

Positive shape parameter for the Burr III margin (failure cause 2).

Gamma

Common positive shape parameter for the Burr III margins.

g1

Splines coefficients for the failure cause 1.

g2

Splines coefficients for the failure cause 2.

p

Copula parameter that greater than or equal to 1.

q

Copula parameter that greater than 1 (integer).

theta

Copula parameter with restricted range.

eta

Location parameter with default value 0.

Sdist.plot

Plot sub-distribution functions if TRUE.

Details

The copula parameter q is restricted to be a integer due to the binominal theorem. The admissible range of theta is given in Dependence.GFGM.

Value

S.overall

Cramer-von Mises type statistic based on parametric and non-parametric estimators of sub-distribution functions for testing overall model.

S.GFGM

Cramer-von Mises type statistic based on semi-parametric and non-parametric estimators of sub-distribution functions for testing the generalized FGM copula.

References

Shih J-H, Emura T (2018) Likelihood-based inference for bivariate latent failure time models with competing risks udner the generalized FGM copula, Computational Statistics, 33:1293-1323.

See Also

Dependence.GFGM, MLE.GFGM.BurrIII, MLE.GFGM.spline

Examples

con   = c(16,224,16,80,128,168,144,176,176,568,392,576,128,56,112,160,384,600,40,416,
          408,384,256,246,184,440,64,104,168,408,304,16,72,8,88,160,48,168,80,512,
          208,194,136,224,32,504,40,120,320,48,256,216,168,184,144,224,488,304,40,160,
          488,120,208,32,112,288,336,256,40,296,60,208,440,104,528,384,264,360,80,96,
          360,232,40,112,120,32,56,280,104,168,56,72,64,40,480,152,48,56,328,192,
          168,168,114,280,128,416,392,160,144,208,96,536,400,80,40,112,160,104,224,336,
          616,224,40,32,192,126,392,288,248,120,328,464,448,616,168,112,448,296,328,56,
          80,72,56,608,144,408,16,560,144,612,80,16,424,264,256,528,56,256,112,544,
          552,72,184,240,128,40,600,96,24,184,272,152,328,480,96,296,592,400,8,280,
          72,168,40,152,488,480,40,576,392,552,112,288,168,352,160,272,320,80,296,248,
          184,264,96,224,592,176,256,344,360,184,152,208,160,176,72,584,144,176)
uncon = c(368,136,512,136,472,96,144,112,104,104,344,246,72,80,312,24,128,304,16,320,
          560,168,120,616,24,176,16,24,32,232,32,112,56,184,40,256,160,456,48,24,
          200,72,168,288,112,80,584,368,272,208,144,208,114,480,114,392,120,48,104,272,
          64,112,96,64,360,136,168,176,256,112,104,272,320,8,440,224,280,8,56,216,
          120,256,104,104,8,304,240,88,248,472,304,88,200,392,168,72,40,88,176,216,
          152,184,400,424,88,152,184)
cen   = rep(630,44)

t.event = c(con,uncon,cen)
event1  = c(rep(1,length(con)),rep(0,length(uncon)),rep(0,length(cen)))
event2  = c(rep(0,length(con)),rep(1,length(uncon)),rep(0,length(cen)))

library(GFGM.copula)
#res.BurrIII = MLE.GFGM.BurrIII(t.event,event1,event2,5000,3,2,0.75,eta = -71)
#Alpha = res.BurrIII$Alpha[1]
#Beta = res.BurrIII$Beta[1]
#Gamma = res.BurrIII$Gamma[1]
#res.spline = MLE.GFGM.spline(t.event,event1,event2,3,2,0.75)
#g1 = res.spline$g1
#g2 = res.spline$g2
#CvM.GFGM.BurrIII(t.event,event1,event2,Alpha,Beta,Gamma,g1,g2,3,2,0.75,eta = -71)

Bivariate dependence measures under the generalized FGM copula

Description

Compute Kendall's tau and Spearman's rho with their boundaries under the generalized FGM copula.

Usage

Dependence.GFGM(p, q, theta)

Arguments

p

Copula parameter that greater than or equal to 1.

q

Copula parameter that greater than 1.

theta

Copula parameter with restricted range.

Details

The admissible range of theta (\theta) is

-\min\bigg\{1,\frac{1}{p^{2q}}\bigg(\frac{1+pq}{q-1}\bigg)^{2q-2}\bigg\} \leq \theta \leq \frac{1}{p^{q}}\bigg(\frac{1+pq}{q-1}\bigg)^{q-1}.

See also Shih and Emura (2019) for details.

Value

theta

Dependence parameter.

tau

Kendall's tau.

rho

Spearman's rho.

References

Shih J-H, Emura T (2019) Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula, Statistical Papers, 60:1101-1118.

Examples

library(GFGM.copula)
Dependence.GFGM(3,2,0.75)

Generate samples from the generalized FGM copula with the Burr III margins

Description

Generate samples from the generalized FGM copula with the Burr III margins.

Usage

GFGM.BurrIII(n, p, q, theta, Alpha, Beta, Gamma)

Arguments

n

Sample size.

p

Copula parameter that greater than or equal to 1.

q

Copula parameter that greater than 1.

theta

Copula parameter with restricted range.

Alpha

Positive shape parameter for the Burr III margin.

Beta

Positive shape parameter for the Burr III margin.

Gamma

Common positive shape parameter for the Burr III margins.

Details

The admissible range of theta is given in Dependence.GFGM.

Value

X

X is asscoiated with the parameter Alpha.

Y

Y is asscoiated with the parameter Beta.

References

Shih J-H, Emura T (2018) Likelihood-based inference for bivariate latent failure time models with competing risks udner the generalized FGM copula, Computational Statistics, 33:1293-1323.

Shih J-H, Emura T (2019) Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula, Statistical Papers, 60:1101-1118.

See Also

Dependence.GFGM

Examples

library(GFGM.copula)
GFGM.BurrIII(5,3,2,0.75,1,1,1)

Maximum likelihood estimation for bivariate dependent competing risks data under the generalized FGM copula with the Burr III margins

Description

Maximum likelihood estimation for bivariate dependent competing risks data under the generalized FGM copula with the Burr III margins.

Usage

MLE.GFGM.BurrIII(
  t.event,
  event1,
  event2,
  D,
  p,
  q,
  theta,
  eta = 0,
  Gamma.0 = 1,
  epsilon.0 = 1e-05,
  epsilon.1 = 1e-10,
  epsilon.2 = 1e-300,
  r.1 = 1,
  r.2 = 1,
  r.3 = 1
)

Arguments

t.event

Vector of the observed failure times.

event1

Vector of the indicators for the failure cause 1.

event2

Vector of the indicators for the failure cause 2.

D

Positive tunning parameter in the NR algorithm.

p

Copula parameter that greater than or equal to 1.

q

Copula parameter that greater than 1 (integer).

theta

Copula parameter with restricted range.

eta

Location parameter with default value 0.

Gamma.0

Initial guess for the common shape parameter gamma with default value 1.

epsilon.0

Positive tunning parameter in the NR algorithm with default value 1e-5.

epsilon.1

Positive tunning parameter in the NR algorithm with default value 1e-10.

epsilon.2

Positive tunning parameter in the NR algorithm with default value 1e-300.

r.1

Positive tunning parameter in the NR algorithm with default value 1.

r.2

Positive tunning parameter in the NR algorithm with default value 1.

r.3

Positive tunning parameter in the NR algorithm with default value 1.

Details

The copula parameter q is restricted to be a integer due to the binominal theorem. The admissible range of theta is given in Dependence.GFGM.

Value

n

Sample size.

count

Iteration number.

random

Randomization number.

Alpha

Positive shape parameter for the Burr III margin (failure cause 1).

Beta

Positive shape parameter for the Burr III margin (failure cause 2).

Gamma

Common shape parameter for the Burr III margins.

MeanX

Mean lifetime due to failure cause 1.

MeanY

Mean lifetime due to failure cause 2.

logL

Log-likelihood value under the fitted model.

References

Shih J-H, Emura T (2018) Likelihood-based inference for bivariate latent failure time models with competing risks udner the generalized FGM copula, Computational Statistics, 33:1293-1323.

Shih J-H, Emura T (2019) Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula, Statistical Papers, 60:1101-1118.

See Also

Dependence.GFGM

Examples

con   = c(16,224,16,80,128,168,144,176,176,568,392,576,128,56,112,160,384,600,40,416,
          408,384,256,246,184,440,64,104,168,408,304,16,72,8,88,160,48,168,80,512,
          208,194,136,224,32,504,40,120,320,48,256,216,168,184,144,224,488,304,40,160,
          488,120,208,32,112,288,336,256,40,296,60,208,440,104,528,384,264,360,80,96,
          360,232,40,112,120,32,56,280,104,168,56,72,64,40,480,152,48,56,328,192,
          168,168,114,280,128,416,392,160,144,208,96,536,400,80,40,112,160,104,224,336,
          616,224,40,32,192,126,392,288,248,120,328,464,448,616,168,112,448,296,328,56,
          80,72,56,608,144,408,16,560,144,612,80,16,424,264,256,528,56,256,112,544,
          552,72,184,240,128,40,600,96,24,184,272,152,328,480,96,296,592,400,8,280,
          72,168,40,152,488,480,40,576,392,552,112,288,168,352,160,272,320,80,296,248,
          184,264,96,224,592,176,256,344,360,184,152,208,160,176,72,584,144,176)
uncon = c(368,136,512,136,472,96,144,112,104,104,344,246,72,80,312,24,128,304,16,320,
          560,168,120,616,24,176,16,24,32,232,32,112,56,184,40,256,160,456,48,24,
          200,72,168,288,112,80,584,368,272,208,144,208,114,480,114,392,120,48,104,272,
          64,112,96,64,360,136,168,176,256,112,104,272,320,8,440,224,280,8,56,216,
          120,256,104,104,8,304,240,88,248,472,304,88,200,392,168,72,40,88,176,216,
          152,184,400,424,88,152,184)
cen   = rep(630,44)

t.event = c(con,uncon,cen)
event1  = c(rep(1,length(con)),rep(0,length(uncon)),rep(0,length(cen)))
event2  = c(rep(0,length(con)),rep(1,length(uncon)),rep(0,length(cen)))

library(GFGM.copula)
MLE.GFGM.BurrIII(t.event,event1,event2,5000,3,2,0.75,eta = -71)

Maximum likelihood estimation for bivariate dependent competing risks data under the generalized FGM copula with the marginal distributions approximated by splines

Description

Maximum likelihood estimation for bivariate dependent competing risks data under the generalized FGM copula with the marginal distributions approximated by splines.

Usage

MLE.GFGM.spline(t.event, event1, event2, p, q, theta, h.plot = TRUE)

Arguments

t.event

Vector of the observed failure times.

event1

Vector of the indicators for the failure cause 1.

event2

Vector of the indicators for the failure cause 2.

p

Copula parameter that greater than or equal to 1.

q

Copula parameter that greater than 1 (integer).

theta

Copula parameter with restricted range.

h.plot

Plot hazard functions if TRUE.

Details

The copula parameter q is restricted to be a integer due to the binominal theorem. The admissible range of theta is given in Dependence.GFGM.

To adapt our functions to dependent censoring models in Emura and Chen (2018), one can simply set event2 = 1-event1.

Value

n

Sample size.

g1

Maximum likelihood estimator of the splines coefficients for the failure cause 1.

g2

Maximum likelihood estimator of the splines coefficients for the failure cause 2.

g1.var

Covariance matrix of splines coefficients estimates for the failure cause 1.

g2.var

Covariance matrix of splines coefficients estimates for the failure cause 2.

References

Emura T, Chen Y-H (2018) Analysis of Survival Data with Dependent Censoring, Copula-Based Approaches, JSS Research Series in Statistics, Springer, Singapore.

Shih J-H, Emura T (2018) Likelihood-based inference for bivariate latent failure time models with competing risks udner the generalized FGM copula, Computational Statistics, 33:1293-1323.

Shih J-H, Emura T (2019) Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula, Statistical Papers, 60:1101-1118.

See Also

Dependence.GFGM

Examples

con   = c(16,224,16,80,128,168,144,176,176,568,392,576,128,56,112,160,384,600,40,416,
          408,384,256,246,184,440,64,104,168,408,304,16,72,8,88,160,48,168,80,512,
          208,194,136,224,32,504,40,120,320,48,256,216,168,184,144,224,488,304,40,160,
          488,120,208,32,112,288,336,256,40,296,60,208,440,104,528,384,264,360,80,96,
          360,232,40,112,120,32,56,280,104,168,56,72,64,40,480,152,48,56,328,192,
          168,168,114,280,128,416,392,160,144,208,96,536,400,80,40,112,160,104,224,336,
          616,224,40,32,192,126,392,288,248,120,328,464,448,616,168,112,448,296,328,56,
          80,72,56,608,144,408,16,560,144,612,80,16,424,264,256,528,56,256,112,544,
          552,72,184,240,128,40,600,96,24,184,272,152,328,480,96,296,592,400,8,280,
          72,168,40,152,488,480,40,576,392,552,112,288,168,352,160,272,320,80,296,248,
          184,264,96,224,592,176,256,344,360,184,152,208,160,176,72,584,144,176)
uncon = c(368,136,512,136,472,96,144,112,104,104,344,246,72,80,312,24,128,304,16,320,
          560,168,120,616,24,176,16,24,32,232,32,112,56,184,40,256,160,456,48,24,
          200,72,168,288,112,80,584,368,272,208,144,208,114,480,114,392,120,48,104,272,
          64,112,96,64,360,136,168,176,256,112,104,272,320,8,440,224,280,8,56,216,
          120,256,104,104,8,304,240,88,248,472,304,88,200,392,168,72,40,88,176,216,
          152,184,400,424,88,152,184)
cen   = rep(630,44)

t.event = c(con,uncon,cen)
event1  = c(rep(1,length(con)),rep(0,length(uncon)),rep(0,length(cen)))
event2  = c(rep(0,length(con)),rep(1,length(uncon)),rep(0,length(cen)))

library(GFGM.copula)
MLE.GFGM.spline(t.event,event1,event2,3,2,0.75)

Sub-distribution functions under the generalized FGM copula with the Burr III margins

Description

Sub-distribution functions under the generalized FGM copula with the Burr III margins.

Usage

Sdist.GFGM.BurrIII(time, Alpha, Beta, Gamma, p, q, theta, eta = 0)

Arguments

time

Vector of times.

Alpha

Positive shape parameter for the Burr III margin (failure cause 1).

Beta

Positive shape parameter for the Burr III margin (failure cause 2).

Gamma

Common positive shape parameter for the Burr III margins.

p

Copula parameter that greater than or equal to 1.

q

Copula parameter that greater than 1 (integer).

theta

Copula parameter with restricted range.

eta

Location parameter with default value 0.

Details

The copula parameter q is restricted to be a integer due to the binominal theorem. The admissible range of theta is given in Dependence.GFGM.

Value

time

Failure times

Sdist.1

Probability of an object fails due to the failure cause 1.

Sdist.2

Probability of an object fails due to the failure cause 2.

References

Shih J-H, Emura T (2018) Likelihood-based inference for bivariate latent failure time models with competing risks udner the generalized FGM copula, Computational Statistics, 33:1293-1323.

Shih J-H, Emura T (2019) Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula, Statistical Papers, 60:1101-1118.

See Also

MLE.GFGM.BurrIII, Dependence.GFGM

Examples

library(GFGM.copula)
Sdist.GFGM.BurrIII(c(1:5),1,1,1,3,2,0.75,eta = 1)

Sub-distribution functions under the generalized FGM copula with the marginal distributions approximated by splines

Description

Sub-distribution functions under the generalized FGM copula with the marginal distributions approximated by splines.

Usage

Sdist.GFGM.spline(time, g1, g2, tmin, tmax, p, q, theta)

Arguments

time

Vector of times.

g1

Splines coefficients for the failure cause 1.

g2

Splines coefficients for the failure cause 2.

tmin

Lower bound of times.

tmax

upper bound of times.

p

Copula parameter that greater than or equal to 1.

q

Copula parameter that greater than 1 (integer).

theta

Copula parameter with restricted range.

Details

The splines coefficients g1 and g2 are usually computed by MLE.GFGM.spline. The copula parameter q is restricted to be a integer due to the binominal theorem. The admissible range of theta is given in Dependence.GFGM.

Value

time

Failure times

Sdist.1

Probability of an object fails due to the failure cause 1.

Sdist.2

Probability of an object fails due to the failure cause 2.

References

Shih J-H, Emura T (2018) Likelihood-based inference for bivariate latent failure time models with competing risks udner the generalized FGM copula, Computational Statistics, 33:1293-1323.

Shih J-H, Emura T (2019) Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula, Statistical Papers, 60:1101-1118.

See Also

MLE.GFGM.spline, Dependence.GFGM

Examples

library(GFGM.copula)
Sdist.GFGM.spline(seq(1,5,1),rep(0.1,5),rep(0.1,5),1,5,3,2,0.75)