Semi-parametric bivariate correlated frailty model.

Description

Fit a semiparametric Bivariate correlated frailty model with Proportional Hazard structure.

Usage

bcfrailph(
  formula,
  data,
  initfrailp = NULL,
  frailty = c("gamma", "lognormal"),
  weights = NULL,
  control = bcfrailph.control(),
  ...
)

Arguments

Value

An object of that contains the following components.

• coefficients - A vector of estimated Covariate coefficients.

• frailparest - A vector of estimated Frailty parameters i.e. frailty variance and correlation.

• stderr-A vector containing the Standard error of the Estimated parameters both covariate coefficients and frailty parameters.

• loglilk0- Log likelihood of without frailty model or loglik of coxph fit.

• loglilk-Log likelihood of Cox PH model with frailty.

• Iloglilk- Log likelihood of with frailty. For gamma fit it is I-likelihood or the likelihood after integrating out the frailty term.For lognormal fit it is the approximate likelihood.

• bhaz- an array containing unique event times and estimated baseline hazard.

• X-Matrix of observed covariates.

• time-the observed survival time.

• censor-censoring indicator.

• resid-the martingale residuals.

• lin.prid-the vector of linear predictors.

• frail-estimated Frailty values.

• iteration-Number of outer iterations.

• e.time-the vector of unique event times.

• n.event- the number of events at each of the unique event times.

• convergence-an indicator, 1 if converge and 0 otherwise.

• history-an array containing records of estimates and other information on each iterations.

Note

Parameters of Bivariate correlated gamma frailty model was estimated using a modified EM approach given in Kifle et al (2022). Parameters of Bivariate correlated lognormal frailty model is based on the penalized partial likelihood approach by Rippatti and Palmgren (2000).

References

Kifle YG, Chen DG, Haileyesus MT (2022). Multivariate Frailty Models using Survey Weights with Applications to Twins Infant Mortality in Ethiopia. Statistics and Its Interface,106(4), 1\-10.

Rippatti, S. and Palmgren, J (2000). Estimation of multivariate frailty models using penalized partial likelihood. Biometrics, 56: 1016-1022.

See Also

bcfrailph.control,simbcfrailph

Examples

set.seed(4)
simdata<-simbcfrailph(psize=300, cenr= c(0.3),beta=c(2),frailty=c("gamma"),
frailpar=c(0.5,0.5),bhaz=c("weibull"),
bhazpar=list(shape =c(5), scale = c(0.1)),
covartype= c("B"),covarpar=list(fargs=c(1),sargs=c(0.5)))
dataa<-simdata$data

fitbcfrgam=bcfrailph(Surv(time,censor)~ X1+frailty(PID) ,data=dataa,frailty="gamma")
fitbcfrgam


# for lognormal

set.seed(18)
simdata<-simbcfrailph(psize=100, cenr= c(0.2),beta=c(1,-0.7,0.5),
frailty=c("lognormal"),frailpar=c(0.5,-0.25),bhaz=c("exponential"),
bhazpar=list(scale = c(0.1)),covartype= c("N","N","B"),
covarpar=list(fargs=c(0,0,1),sargs=c(1,1,0.5)),comncovar=2)
dataa<-simdata$data

#fit
fitbcfrlogn=bcfrailph(Surv(time,censor)~ X1+X2+X3+frailty(PID) ,data=dataa,frailty="lognormal")
fitbcfrlogn

## one can set the initial parameter for the frailty parameters
fitbcfrailph=bcfrailph(Surv(time,censor)~ X1+frailty(PID),data=dataa,initfrailp = c(0.1,0.5),
frailty="lognormal")
fitbcfrailph

# Not run

#if covariates are not included
fitmoe=try(bcfrailph(Surv(time,censor)~0+frailty(PID),data=dataa,
frailty="lognormal"),silent = TRUE)

fitmoe=try(bcfrailph(Surv(time,censor)~1+frailty(PID),data=dataa),silent = TRUE)

# if control is not specified correctly.
# if one needs to change only max.iter to be 100,

fitmoe=try(bcfrailph(Surv(time,censor)~ X1+frailty(PID),data=dataa,
control=c(max.iter=100)),silent = TRUE)

#the correct way is
fitmoe=bcfrailph(Surv(time,censor)~ X1+frailty(PID),data=dataa,
control=bcfrailph.control(max.iter=100))
fitmoe

#if initial frailty parameters are in the boundary of parameter space
fitmoe=try(bcfrailph(Surv(time,censor)~ X1+frailty(PID),data=dataa,
initfrailp=c(0.2,1)),silent = TRUE)

fitmoe=try(bcfrailph(Surv(time,censor)~ X1+frailty(PID),data=dataa,
initfrailp=c(0,0.1)),silent = TRUE)

#if a frailty distribution other than gamma and lognormal are specified

fitmoe=try(bcfrailph(Surv(time,censor)~ X1,data=dataa,,frailty="exp"),silent = TRUE)

# End Not run

Arguments for controlling bcfrailph fits.

Description

This is used to set various numeric parameters controlling a bcfrailph model fits.

Usage

bcfrailph.control(
  max.iter = 400,
  tol = 1e-04,
  eval.max = 500,
  iter.max = 500,
  trace = 0,
  abs.tol = 1e-20,
  rel.tol = 1e-10,
  x.tol = 1.5e-08,
  xf.tol = 2.2e-14,
  step.min = 1,
  step.max = 1
)

Arguments

Value

A list of control parameters.

See Also

bcfrailph

Bivariate correlated gamma frailty model fitting function.

Description

Semi-parametric Bivariate correlated gamma frailty model fitting function.

Usage

fitbccv.gammasp(
  X,
  Y,
  initfrailp,
  weights = NULL,
  control = bcfrailph.control(),
  SE = TRUE
)

Arguments

Value

An object of that contains the following components.

• coefficients - A vector of estimated Covariate coefficients.

• frailparest - A vector of estimated Frailty parameters i.e. frailty variance and correlation.

• stderr-A vector containing the Standard error of the Estimated parameters both covariate coefficients and frailty parameters.

• loglilk0- Log likelihood of without frailty model or loglik of coxph fit.

• loglilk-Log likelihood of Cox PH model with frailty.

• Iloglilk- Log likelihood of with frailty. For gamma fit it is I-likelihood or the likelihood after integrating out the frailty term.For lognormal fit it is the approximate likelihood.

• bhaz- an array containing unique event times and estimated baseline hazard.

• X-Matrix of observed covariates.

• time-the observed survival time.

• censor-censoring indicator.

• resid-the martingale residuals.

• lin.prid-the vector of linear predictors.

• frail-estimated Frailty values.

• iteration-Number of outer iterations.

• e.time-the vector of unique event times.

• n.event- the number of events at each of the unique event times.

• convergence-an indicator, 1 if converge and 0 otherwise.

• history-an array containing records of estimates and other information on each iterations.

Note

This function is important especially for simulation studies as it reduced checking time. Parameters of Bivariate correlated gamma frailty model was estimated using a modified EM approach given in Kifle et al (2022).

References

Kifle YG, Chen DG, Haileyesus MT (2022). Multivariate Frailty Models using Survey Weights with Applications to Twins Infant Mortality in Ethiopia. Statistics and Its Interface,106(4), 1\-10.

See Also

bcfrailph

Examples

set.seed(4)
simdata<-simbcfrailph(psize=300, cenr= c(0.3),beta=c(2),frailty=c("gamma"),
frailpar=c(0.5,0.5),bhaz=c("weibull"),
bhazpar=list(shape =c(5), scale = c(0.1)),
covartype= c("B"),covarpar=list(fargs=c(1),sargs=c(0.5)))
Y<-simdata$Y;X<-simdata$X

bcspfit<-fitbccv.gammasp(X=X,Y=Y,initfrailp=NULL)
bcspfit$coef
bcspfit$frailpar

Bivariate correlated lognormal frailty model fitting function.

Description

Semi-parametric Bivariate correlated lognormal frailty model fitting function.

Usage

fitbccv.lognsp(X, Y, initfrailp, control = bcfrailph.control())

Arguments

Value

An object of that contains the following components.

• coefficients - A vector of estimated Covariate coefficients.

• frailparest - A vector of estimated Frailty parameters i.e. frailty variance and correlation.

• stderr-A vector containing the Standard error of the Estimated parameters both covariate coefficients and frailty parameters.

• loglilk0- Log likelihood of without frailty model or loglik of coxph fit.

• loglilk-Log likelihood of Cox PH model with frailty.

• Iloglilk- Log likelihood of with frailty. For gamma fit it is I-likelihood or the likelihood after integrating out the frailty term.For lognormal fit it is the approximate likelihood.

• bhaz- an array containing unique event times and estimated baseline hazard.

• X-Matrix of observed covariates.

• time-the observed survival time.

• censor-censoring indicator.

• resid-the martingale residuals.

• lin.prid-the vector of linear predictors.

• frail-estimated Frailty values.

• iteration-Number of outer iterations.

• e.time-the vector of unique event times.

• n.event- the number of events at each of the unique event times.

• convergence-an indicator, 1 if converge and 0 otherwise.

• history-an array containing records of estimates and other information on each iterations.

Note

This function is important especially for simulation studies as it reduced checking time. Parameters of Bivariate correlated lognormal frailty model is based on the penalized partial likelihood approach by Rippatti and Palmgren (2000).

References

Rippatti, S. and Palmgren, J (2000). Estimation of multivariate frailty models using penalized partial likelihood. Biometrics, 56: 1016-1022.

See Also

bcfrailph

Examples

set.seed(18)
simdata<-simbcfrailph(psize=100, cenr= c(0.2),beta=c(1,-0.7,0.5),frailty=c("lognormal"),
frailpar=c(0.5,-0.25),bhaz=c("exponential"),
bhazpar=list(scale = c(0.1)),covartype= c("N","N","B"),
covarpar=list(fargs=c(0,0,1),sargs=c(1,1,0.5)),comncovar=2)
Y<-simdata$Y;X<-simdata$X

bcspfit<-fitbccv.lognsp(X=X,Y=Y,initfrailp=NULL)
bcspfit$coef 
bcspfit$frailpar

Plot bcfrailph

Description

Generics to print the S3 class bcfrailph.

Usage

## S3 method for class 'bcfrailph'
plot(
  x,
  lty = 1,
  col = 1,
  type = "l",
  xlim = NULL,
  ylim = NULL,
  xlab = NULL,
  main = NULL,
  conf.int = FALSE,
  ...
)

Arguments

Details

Calls plot.bcfrailph().

Value

An plot of plot.bcfrailph object.

Note

The plot of cumulative baseline hazard function.

See Also

bcfrailph

Examples

set.seed(24)
simdata<-simbcfrailph(psize=100, cenr= c(0),beta=c(-1),frailty=c("gamma"),
frailpar=c(0.4,0.5),bhaz=c("weibull"),
bhazpar=list(shape =c(0.9), scale = c(2)),
covartype= c("B"),covarpar=list(fargs=c(1),sargs=c(0.5)))
dataa<-simdata$data ## the generated data set.

#fit
bcfit=bcfrailph(Surv(time, censor) ~ X1+frailty(PID),data=dataa)
plot(bcfit)

Print bcfrailph

Description

Generics to print the S3 class bcfrailph.

Usage

## S3 method for class 'bcfrailph'
print(x, ...)

Arguments

Details

Calls print.bcfrailph().

Value

An object of print.bcfrailph, with some more human-readable results from bcfrailph object.

Note

The summary function is currently identical to the print function.

See Also

bcfrailph

Examples

set.seed(4)
simdata<-simbcfrailph(psize=300, cenr= c(0.3),beta=c(2),frailty=c("gamma"),
frailpar=c(0.5,0.5),bhaz=c("weibull"),
bhazpar=list(shape =c(5), scale = c(0.1)),
covartype= c("B"),covarpar=list(fargs=c(1),sargs=c(0.5)))
dataa<-simdata$data

fitbcfrailph=bcfrailph(Surv(time,censor)~ X1+frailty(PID) ,data=dataa,frail_distrn=c("gamma"))
fitbcfrailph

Print shrgamsp

Description

Generics to print the S3 class shrgamsp.

Usage

## S3 method for class 'shrgamsp'
print(x, ...)

Arguments

Details

Calls print.shrgamsp().

Value

An object of print.shrgamsp, with some more human-readable results from shrgamsp object.

Note

The summary function is currently identical to the print function.

See Also

bcfrailph

Cox PH model with univariate and bivariate shared gamma frailty model.

Description

Fit Cox PH model with univariate and bivariate shared gamma frailty model.

Usage

shrgamsp(
  formula,
  data,
  weights = NULL,
  initfrailp = NULL,
  control = bcfrailph.control(),
  ...
)

Arguments

Value

An object of shrgamsp contains the following components.

• coefficients - A vector of estimated Covariate coefficients.

• frailparest - A vector of estimated Frailty parameters i.e. frailty variance and correlation.

• vcov- Variance Covariance matrix of the Estimated Covariate coefficients obtained from the observed information matrix.

• stderr-A vector containing the Standard error of the Estimated parameters both covariate coefficients and frailty parameter.

• loglik0- Log likelihood of without frailty model.

• loglik-Log likelihood of Cox PH model with frailty.

• Iloglilk- Log likelihood of with frailty model after integrating out the frailty term.

• bhaz- an array containing unique event times and estimated baseline hazard.

• X-Matrix of observed covariates.

• time-the observed survival time.

• censor-censoring indicator.

• resid-the martingale residuals.

• lin.prid-the vector of linear predictors.

• frail-estimated Frailty values.

• iteration-Number of outer iterations.

• e.time-the vector of unique event times.

• n.event- the number of events at each of the unique event times.

• convergence-an indicator of convergence . see nlminb.

Note

This is just a coxph model with gamma frailty and the differences between coxph with gamma frailty fit and shrgamsp fit is on the standard errors of the covariates cofficients. Here, the standard errors of the estimated covariate coefficients and the frailty variance parameter are obtained using the standard errors estimation approach given in Klein and Moeschberger (2003).

References

Duchateau, L., Janssen, P. (2008) The Frailty Model. Springer, New York.

Klein, J. P., and Moeschberger, M. L. (2003), Survival analysis: techniques for censored and truncated data, New York: Springer.

See Also

bcfrailph

Examples

set.seed(2)
n1=500;IID=array(1:n1)
X1<-runif(n1,  min=0, max=1)
z=rgamma(n1,shape=2,scale=0.5)
u1<-runif(n1,  min=0, max=1)
time<- 1/0.1*log(1-0.1*log(u1)/(0.0001*exp(3*X1)*z))
censor=rep(1,n1)
dataa <- data.frame(time=time, X1=X1,censor=censor,IID=IID)

fitcoxfr=shrgamsp(Surv(time,censor)~ X1+frailty(IID) ,data=dataa)
fitcoxfr

Simulate data from bivariate correlated frailty models.

Description

Simulate data from bivariate correlated gamma or lognormal frailty models with or without covariates.

Usage

simbcfrailph(
  psize,
  cenr = c(0),
  beta = c(0.5),
  frailty,
  frailpar = c(0.5, 0.25),
  bhaz = c("weibull"),
  bhazpar = list(shape = c(0.5), scale = c(0.01)),
  covartype = c("B"),
  covarpar = list(fargs = c(1), sargs = c(0.5)),
  inpcovar = NULL,
  inpcen = NULL,
  comncovar = NULL
)

Arguments

Value

An object of class simbcfrailph that contain the following:

• data A data frame i.e, the simulated data set. IID is individual Id, PID is pair ID, time is the simulated survival time, censor is censoring indicator and X1 denote the simulated covariate.

• X Covariates in Matrix form.

• Y A matrix contains generated survival time and censoring.

• numberofpair The specified number of pairs.

• censoredrate The specified censored rate.

• fraildist The specified frailty distribution.

• frailpar The specified frailty parameters.

See Also

bcfrailph

Examples

set.seed(4)
simdata<-simbcfrailph(psize=300, cenr= c(0.3),beta=c(2),frailty=c("gamma"),
frailpar=c(0.5,0.5),bhaz=c("weibull"),
bhazpar=list(shape =c(5), scale = c(0.1)),
covartype= c("B"),covarpar=list(fargs=c(1),sargs=c(0.5)))
dataa<-simdata$data
head(dataa)


# If data generation is from bivariate correlated lognormal frailty model,
set.seed(18)
simdata<-simbcfrailph(psize=100, cenr= c(0.2),beta=c(1,-0.7,0.5),frailty=c("lognormal"),
frailpar=c(0.5,-0.25),bhaz=c("exponential"),
bhazpar=list(scale = c(0.1)),covartype= c("N","N","B"),
covarpar=list(fargs=c(0,0,1),sargs=c(1,1,0.5)))
dataa<-simdata$data
head(dataa)

# If common covariate is desired, i.e., here out of
#the three covariates covariate 2 is common for the pair.
set.seed(18)
simdata<-simbcfrailph(psize=100, cenr= c(0.2),beta=c(1,-0.7,0.5),frailty=c("lognormal"),
frailpar=c(0.5,-0.25),bhaz=c("exponential"),
bhazpar=list(scale = c(0.1)),covartype= c("N","N","B"),
covarpar=list(fargs=c(0,0,1),sargs=c(1,1,0.5)),comncovar=2)
dataa<-simdata$data
head(dataa)

# If the data generation is from bivariate correlated gamma
# frailty model, weibull baseline and without covariate,
set.seed(4)
simdata<-simbcfrailph(psize=300, cenr= c(0.3),beta=NULL,frailty=c("gamma"),
frailpar=c(0.5,0.5),bhaz=c("weibull"),bhazpar=list(shape =c(5), scale = c(0.1)))
dataa<-simdata$data
head(dataa)

Simulation study for bivariate correlated frailty models.

Description

Simulation study for bivariate correlated gamma and lognormal frailty models with and without covariates.

Usage

simstdybcf(
  Rep,
  mfit = NULL,
  psize,
  cenr = c(0),
  beta = c(0.5),
  frailty,
  frailpar = c(0.5, 0.25),
  bhaz = c("weibull"),
  bhazpar = list(shape = c(0.5), scale = c(0.01)),
  covartype = c("B"),
  covarpar = list(fargs = c(1), sargs = c(0.5)),
  inpcovar = NULL,
  inpcen = NULL,
  comncovar = NULL
)

Arguments

Value

An object of class simstdybcf that contain the following:

• Result a summary result containing true parameter, mean of estimates, mean of the standard errors of the estimates, standard deviation of estimates, and 95% CI coverage probability.

• estimates a matrix containing estimates of parameters at each replications.

• estimateSE a matrix containing standard error of estimates at each replications.

• coverage a matrix containing an indicator whether the true parameter lies within a 95% CI at each replications or not.

• TMAT a matrix containing the generated artificial unique event times at each replications for gamma model.

• h0MAT a matrix containing the estimated baseline hazards at each replications for gamma model.

• h0SEMAT a matrix containing SE of the estimated baseline hazards at each replications for gamma model.

See Also

simbcfrailph

Examples

set.seed(2)
sim<-simstdybcf(Rep=5,psize=100, cenr= c(0.2),beta=c(1,-0.7,0.5),
frailty=c("lognormal"),frailpar=c(0.5,-0.25),bhaz=c("exponential"),
bhazpar=list(scale = c(0.1)),covartype= c("N","N","B"),
covarpar=list(fargs=c(0,0,1),sargs=c(1,1,0.5)),comncovar=2)
Res<-sim$Result
Res


# In addition to bcfrailph fit, if coxph with univariate lognormal frailty model is desired to run,

sim<-simstdybcf(Rep=5,mfit="cox",psize=100, cenr= c(0.2),beta=c(1,-0.7,0.5),
frailty=c("lognormal"),frailpar=c(0.5,-0.25),bhaz=c("exponential"),
bhazpar=list(scale = c(0.1)),covartype= c("N","N","B"),
covarpar=list(fargs=c(0,0,1),sargs=c(1,1,0.5)),comncovar=2)
Res<-sim$Result # bcfrailph fit result
Res
Resc<-sim$Resultc # coxph with univariate lognormal frailty model fit result
Resc

Print bcfrailph

Description

Generics to print the S3 class bcfrailph.

Usage

## S3 method for class 'bcfrailph'
summary(object, ...)

Arguments

Details

Calls print.bcfrailph().

Value

An object of summary.bcfrailph, with some more human-readable results from bcfrailph object.

Note

The summary function is currently identical to the print function.

See Also

bcfrailph

Examples

set.seed(4)
simdata<-simbcfrailph(psize=300, cenr= c(0.3),beta=c(2),frailty=c("gamma"),
frailpar=c(0.5,0.5),bhaz=c("weibull"),
bhazpar=list(shape =c(5), scale = c(0.1)),
covartype= c("B"),covarpar=list(fargs=c(1),sargs=c(0.5)))
dataa<-simdata$data

fitbcfrailph=bcfrailph(Surv(time,censor)~ X1+frailty(PID) ,data=dataa,frail_distrn=c("gamma"))
fitbcfrailph
summary(fitbcfrailph)

Print shrgamsp

Description

Generics to print the S3 class shrgamsp.

Usage

## S3 method for class 'shrgamsp'
summary(object, ...)

Arguments

Details

Calls print.shrgamsp().

Value

An object of summary.shrgamsp, with some more human-readable results from shrgamsp object.

Note

The summary function is currently identical to the print function.

See Also

bcfrailph