Artset1987 data

Description

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

Usage

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)

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)

Arguments

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)

Maximum likelihood estimation

Description

Estimate of the parameters.

Usage

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

Arguments

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.

See Also

optim

Examples

# 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")

Arguments

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.

See Also

optim

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

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)

Arguments

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

Randomized quantile residuals for the GTDL model

Description

Randomized quantile residuals for the GTDL model

Usage

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

Arguments

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

Tumor data

Description

Times (in days) of patients in ovarian cancer study

Usage

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)