Package 'GTDL' reference manual

Title:	The Generalized Time-Dependent Logistic Family
Description:	Computes the probability density, survival function, the hazard rate functions and generates random samples from the GTDL distribution given by Mackenzie, G. (1996) <doi:10.2307/2348408>. The likelihood estimates, the randomized quantile (Louzada, F., et al. (2020) <doi:10.1109/ACCESS.2020.3040525>) residuals and the normally transformed randomized survival probability (Li,L., et al. (2021) <doi:10.1002/sim.8852>) residuals are obtained for the GTDL model.
Authors:	Jalmar Carrasco [aut, cre], Luciano Santana [aut], Lizandra Fabio [aut]
Maintainer:	Jalmar Carrasco <[email protected]>
License:	GPL (>=3)
Version:	1.0.0
Built:	2025-03-05 03:15:49 UTC
Source:	https://github.com/carrascojalmar/gtdl

Artset1987 data

Description

Times to failure of 50 devices put on life test at time 0.

Usage

data(artset1987)
data(artset1987)

Format

This data frame contains the following columns:

t: Times to failure

References

Aarset, M. V. (1987). How to Identify a Bathtub Hazard Rate. IEEE Transactions on Reliability, 36, 106–108.

Examples


data(artset1987)
head(artset1987)

data(artset1987)
head(artset1987)

The GTDL distribution

Description

Density function, survival function, failure function and random generation for the GTDL distribution.

Usage

dGTDL(t, param, log = FALSE)

hGTDL(t, param)

sGTDL(t, param)

rGTDL(n, param)
dGTDL(t, param, log = FALSE)

hGTDL(t, param)

sGTDL(t, param)

rGTDL(n, param)

Arguments

`t`	vector of integer positive quantile.
`param`	parameters (alpha and gamma are scalars, lambda non-negative).
`log`	logical; if TRUE, probabilities p are given as log(p).
`n`	number of observations.

Details

Density function

$f(t\mid \boldsymbol{\theta})=\lambda\left(\frac{\exp\{\alpha{t}+\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}{1+\exp\{\alpha{t}+\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}\right)\times\left(\frac{1+\exp\{\alpha{t}+\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}{1+\exp\{\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}\right)^{-\lambda/\alpha}$
Survival function

$S(t \mid \boldsymbol{\theta})=\left(\frac{1+\exp\{\alpha{t}+\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}{1+\exp\{\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}\right)^{-\lambda/\alpha}$
Failure function

$h(t\mid\boldsymbol{\theta})=\lambda\left(\frac{\exp\{\alpha{t}+\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}{1+\exp\{\alpha{t}+\boldsymbol{X}^{\top}\boldsymbol{\beta}\}}\right)$

Value

dGTDL gives the density function, hGTDL gives the failure function, sGTDL gives the survival function and rGTDL generates random samples.

Invalid arguments will return an error message.

Source

[d-p-q-r]GTDL are calculated directly from the definitions.

References

Mackenzie, G. (1996). Regression Models for Survival Data: The Generalized Time-Dependent Logistic Family. Journal of the Royal Statistical Society. Series D (The Statistician). 45. 21-34.

Examples


library(GTDL)
t <- seq(0,20,by = 0.1)
lambda <- 1.00
alpha <- -0.05
gamma <- -1.00
param <- c(lambda,alpha,gamma)
y1 <- hGTDL(t,param)
y2 <- sGTDL(t,param)
y3 <- dGTDL(t,param,log = FALSE)
tt <- as.matrix(cbind(t,t,t))
yy <- as.matrix(cbind(y1,y2,y3))
matplot(tt,yy,type="l",xlab="time",ylab="",lty = 1:3,col=1:3,lwd=2)


y1 <- hGTDL(t,c(1,0.5,-1.0))
y2 <- hGTDL(t,c(1,0.25,-1.0))
y3 <- hGTDL(t,c(1,-0.25,1.0))
y4 <- hGTDL(t,c(1,-0.50,1.0))
y5 <- hGTDL(t,c(1,-0.06,-1.6))
tt <- as.matrix(cbind(t,t,t,t,t))
yy <- as.matrix(cbind(y1,y2,y3,y4,y5))
matplot(tt,yy,type="l",xlab="time",ylab="Hazard function",lty = 1:3,col=1:3,lwd=2)



library(GTDL)
t <- seq(0,20,by = 0.1)
lambda <- 1.00
alpha <- -0.05
gamma <- -1.00
param <- c(lambda,alpha,gamma)
y1 <- hGTDL(t,param)
y2 <- sGTDL(t,param)
y3 <- dGTDL(t,param,log = FALSE)
tt <- as.matrix(cbind(t,t,t))
yy <- as.matrix(cbind(y1,y2,y3))
matplot(tt,yy,type="l",xlab="time",ylab="",lty = 1:3,col=1:3,lwd=2)


y1 <- hGTDL(t,c(1,0.5,-1.0))
y2 <- hGTDL(t,c(1,0.25,-1.0))
y3 <- hGTDL(t,c(1,-0.25,1.0))
y4 <- hGTDL(t,c(1,-0.50,1.0))
y5 <- hGTDL(t,c(1,-0.06,-1.6))
tt <- as.matrix(cbind(t,t,t,t,t))
yy <- as.matrix(cbind(y1,y2,y3,y4,y5))
matplot(tt,yy,type="l",xlab="time",ylab="Hazard function",lty = 1:3,col=1:3,lwd=2)

Maximum likelihood estimation

Description

Estimate of the parameters.

Usage

mle1.GTDL(start, t, method = "BFGS")
mle1.GTDL(start, t, method = "BFGS")

Arguments

`start`	Initial values for the parameters to be optimized over.
`t`	non-negative random variable representing the failure time and leave the snapshot failure rate, or danger.
`method`	The method to be used.

Value

Returns a list of summary statistics of the fitted GTDL distribution.

References

Aarset, M. V. (1987). How to Identify a Bathtub Hazard Rate. IEEE Transactions on Reliability, 36, 106–108.
Mackenzie, G. (1996) Regression Models for Survival Data: The Generalized Time-Dependent Logistic Family. Journal of the Royal Statistical Society. Series D (The Statistician). 45. 21-34.

Examples


# times data (from Aarset, 1987))
data(artset1987)
mod <- mle1.GTDL(c(1,-0.05,-1),t = artset1987)

# times data (from Aarset, 1987))
data(artset1987)
mod <- mle1.GTDL(c(1,-0.05,-1),t = artset1987)

Maximum likelihood estimates of the GTDL model

Description

Maximum likelihood estimates of the GTDL model

Usage

mle2.GTDL(t, start, formula, censur, method = "BFGS")
mle2.GTDL(t, start, formula, censur, method = "BFGS")

Arguments

`t`	non-negative random variable representing the failure time and leave the snapshot failure rate, or danger.
`start`	Initial values for the parameters to be optimized over.
`formula`	The structure matrix of covariates of dimension n x p.
`censur`	censoring status 0=censored, a=fail.
`method`	The method to be used.

Value

Returns a list of summary statistics of the fitted GTDL model.

References

Mackenzie, G. (1996) Regression Models for Survival Data: The Generalized Time-Dependent Logistic Family. Journal of the Royal Statistical Society. Series D (The Statistician). (45). 21-34.

Examples


### Example 1

require(survival)
data(lung)

lung <- lung[-14,]
lung$sex <- ifelse(lung$sex==2, 1, 0)
lung$ph.ecog[lung$ph.ecog==3]<-2
t1 <- lung$time
start1 <- c(0.03,0.05,-1,0.7,2,-0.1)
formula1 <- ~lung$sex+factor(lung$ph.ecog)+lung$age
censur1 <- ifelse(lung$status==1,0,1)
fit.model1 <- mle2.GTDL(t = t1,start = start1,
                     formula = formula1,
                     censur = censur1)
fit.model1

### Example 2

data(tumor)
t2 <- tumor$time
start2 <- c(1,-0.05,1.7)
formula2 <- ~tumor$group
censur2 <- tumor$censured
fit.model2 <- mle2.GTDL(t = t2,start = start2,
                       formula = formula2,
                       censur = censur2)
fit.model2

### Example 1

require(survival)
data(lung)

lung <- lung[-14,]
lung$sex <- ifelse(lung$sex==2, 1, 0)
lung$ph.ecog[lung$ph.ecog==3]<-2
t1 <- lung$time
start1 <- c(0.03,0.05,-1,0.7,2,-0.1)
formula1 <- ~lung$sex+factor(lung$ph.ecog)+lung$age
censur1 <- ifelse(lung$status==1,0,1)
fit.model1 <- mle2.GTDL(t = t1,start = start1,
                     formula = formula1,
                     censur = censur1)
fit.model1

### Example 2

data(tumor)
t2 <- tumor$time
start2 <- c(1,-0.05,1.7)
formula2 <- ~tumor$group
censur2 <- tumor$censured
fit.model2 <- mle2.GTDL(t = t2,start = start2,
                       formula = formula2,
                       censur = censur2)
fit.model2

Normally-transformed randomized survival probability residuals for the GTDL model

Description

Normally-transformed randomized survival probability residuals for the GTDL model

Usage

nrsp.GTDL(t, formula, pHat, censur)
nrsp.GTDL(t, formula, pHat, censur)

Arguments

`t`	non-negative random variable representing the failure time and leave the snapshot failure rate, or danger.
`formula`	The structure matrix of covariates of dimension n x p.
`pHat`	Estimate of the parameters from the GTDL model.
`censur`	Censoring status 0=censored, a=fail.

Value

Normally-transformed randomized survival probability residuals

References

Li, L., Wu, T., e Cindy, F. (2021). Model diagnostics for censored regression via randomized survival probabilities. Statistics in Medicine, 40, 1482–1497.
de Oliveira, L. E. F., dos Santos L. S., da Silva, P. H. F., Fabio, L. C., Carrasco, J. M. F.(2022). Análise de resíduos para o modelo logístico generalizado dependente do tempo (GTDL). Submitted.

Examples


### Example 1

require(survival)
data(lung)
lung <- lung[-14,]
lung$sex <- ifelse(lung$sex==2, 1, 0)
lung$ph.ecog[lung$ph.ecog==3]<-2
t1 <- lung$time
formula1 <- ~lung$sex+factor(lung$ph.ecog)+lung$age
censur1 <- ifelse(lung$status==1,0,1)
start1 <- c(0.03,0.05,-1,0.7,2,-0.1)
fit.model1 <- mle2.GTDL(t = t1,start = start1,
           formula = formula1,
           censur = censur1)
r1 <- nrsp.GTDL(t = t1,formula = formula1 ,pHat = fit.model1$Coefficients[,1],
             censur = censur1)
r1

### Example 2

data(tumor)
t2 <- tumor$time
formula2 <- ~tumor$group
censur2 <- tumor$censured
start2 <- c(1,-0.05,1.7)
fit.model2 <- mle2.GTDL(t = t2,start = start2,
                       formula = formula2,
                       censur = censur2)
r2 <- nrsp.GTDL(t = t2,formula = formula2, pHat = fit.model2$Coefficients[,1],
            censur = censur2)
r2
### Example 1

require(survival)
data(lung)
lung <- lung[-14,]
lung$sex <- ifelse(lung$sex==2, 1, 0)
lung$ph.ecog[lung$ph.ecog==3]<-2
t1 <- lung$time
formula1 <- ~lung$sex+factor(lung$ph.ecog)+lung$age
censur1 <- ifelse(lung$status==1,0,1)
start1 <- c(0.03,0.05,-1,0.7,2,-0.1)
fit.model1 <- mle2.GTDL(t = t1,start = start1,
           formula = formula1,
           censur = censur1)
r1 <- nrsp.GTDL(t = t1,formula = formula1 ,pHat = fit.model1$Coefficients[,1],
             censur = censur1)
r1

### Example 2

data(tumor)
t2 <- tumor$time
formula2 <- ~tumor$group
censur2 <- tumor$censured
start2 <- c(1,-0.05,1.7)
fit.model2 <- mle2.GTDL(t = t2,start = start2,
                       formula = formula2,
                       censur = censur2)
r2 <- nrsp.GTDL(t = t2,formula = formula2, pHat = fit.model2$Coefficients[,1],
            censur = censur2)
r2

Randomized quantile residuals for the GTDL model

Description

Randomized quantile residuals for the GTDL model

Usage

random.quantile.GTDL(t, formula, pHat, censur)
random.quantile.GTDL(t, formula, pHat, censur)

Arguments

`t`	non-negative random variable representing the failure time and leave the snapshot failure rate, or danger.
`formula`	The structure matrix of covariates of dimension n x p.
`pHat`	Estimate of the parameters from the GTDL model.
`censur`	censoring status 0=censored, a=fail.

Details

The randomized quantile residual (Dunn and Smyth, 1996), which follow a standard normal distribution is used to assess departures from the GTDL model.

Value

Randomized quantile residuals

References

Dunn, P. K. e Smyth, G. K. (1996). Randomized quantile residuals. Journal of Computational and Graphical Statistics, 5, 236–244.
Louzada, F., Cuminato, J. A., Rodriguez, O. M. H., Tomazella, V. L. D., Milani, E. A., Ferreira, P. H., Ramos, P. L., Bochio, G., Perissini, I. C., Junior, O. A. G., Mota, A. L., Alegr´ıa, L. F. A., Colombo, D., Oliveira, P. G. O., Santos, H. F. L., e Magalh˜aes, M. V. C. (2020). Incorporation of frailties into a non-proportional hazard regression model and its diagnostics for reliability modeling of downhole safety valves. IEEE Access, 8, 219757 – 219774.
de Oliveira, L. E. F., dos Santos L. S., da Silva, P. H. F., Fabio, L. C., Carrasco, J. M. F.(2022). Análise de resíduos para o modelo logístico generalizado dependente do tempo (GTDL). Submitted.

Examples


### Example 1

require(survival)
data(lung)
lung <- lung[-14,]
lung$sex <- ifelse(lung$sex==2, 1, 0)
lung$ph.ecog[lung$ph.ecog==3]<-2
t1 <- lung$time
formula1 <- ~lung$sex+factor(lung$ph.ecog)+lung$age
censur1 <- ifelse(lung$status==1,0,1)
start1 <- c(0.03,0.05,-1,0.7,2,-0.1)
fit.model1 <- mle2.GTDL(t = t1,start = start1,
           formula = formula1,
           censur = censur1)
r1 <- random.quantile.GTDL(t = t1,formula = formula1 ,pHat = fit.model1$Coefficients[,1],
             censur = censur1)
r1

### Example 2

data(tumor)
t2 <- tumor$time
formula2 <- ~tumor$group
censur2 <- tumor$censured
start2 <- c(1,-0.05,1.7)
fit.model2 <- mle2.GTDL(t = t2,start = start2,
                       formula = formula2,
                       censur = censur2)
r2 <- random.quantile.GTDL(t = t2,formula = formula2, pHat = fit.model2$Coefficients[,1],
            censur = censur2)
r2
### Example 1

require(survival)
data(lung)
lung <- lung[-14,]
lung$sex <- ifelse(lung$sex==2, 1, 0)
lung$ph.ecog[lung$ph.ecog==3]<-2
t1 <- lung$time
formula1 <- ~lung$sex+factor(lung$ph.ecog)+lung$age
censur1 <- ifelse(lung$status==1,0,1)
start1 <- c(0.03,0.05,-1,0.7,2,-0.1)
fit.model1 <- mle2.GTDL(t = t1,start = start1,
           formula = formula1,
           censur = censur1)
r1 <- random.quantile.GTDL(t = t1,formula = formula1 ,pHat = fit.model1$Coefficients[,1],
             censur = censur1)
r1

### Example 2

data(tumor)
t2 <- tumor$time
formula2 <- ~tumor$group
censur2 <- tumor$censured
start2 <- c(1,-0.05,1.7)
fit.model2 <- mle2.GTDL(t = t2,start = start2,
                       formula = formula2,
                       censur = censur2)
r2 <- random.quantile.GTDL(t = t2,formula = formula2, pHat = fit.model2$Coefficients[,1],
            censur = censur2)
r2

Tumor data

Description

Times (in days) of patients in ovarian cancer study

Usage

data(tumor)
data(tumor)

Format

This data frame contains the following columns:

time: survival time in days
censured: censored = 0, dead = 1
group: large tumor = 0, small tumor = 1

References

Colosimo, E. A and Giolo, S. R. Análise de Sobrevivência Aplicada. Edgard Blucher: São Paulo. 2006.

Examples



data(tumor)
head(tumor)

data(tumor)
head(tumor)

Package 'GTDL'

Help Index

Artset1987 data

Description

Usage

Format

References

Examples

The GTDL distribution

Description

Usage

Arguments

Details

Value

Source

References

Examples

Maximum likelihood estimation

Description

Usage

Arguments

Value

References

See Also

Examples

Maximum likelihood estimates of the GTDL model

Description

Usage

Arguments

Value

References

See Also

Examples

Normally-transformed randomized survival probability residuals for the GTDL model

Description

Usage

Arguments

Value

References

Examples

Randomized quantile residuals for the GTDL model

Description

Usage

Arguments

Details

Value

References

Examples

Tumor data

Description

Usage

Format

References

Examples