Delta matrix

Description

Deltamatrix computes the Delta matrix, along with maximal intersection intervals, for a set of intervals.

Usage

Deltamatrix(LR)

Arguments

Details

An intersection interval is a nonempty intersection of any combination of the given intervals, and a maximal intersection interval is an intersection interval that contains no other intersection interval.

The Delta matrix is a matrix of indicators (TRUE or FALSE). The rows correspond to the given interval-censored observations, and the columns the maximal intersection intervals. A TRUE value of the (i,j)-th element means that the i-th observation covers the j-th maximal intersection interval, and a FALSE value means the opposite.

Value

A list with components:

Author(s)

Yong Wang <yongwang@auckland.ac.nz>

References

Wang, Y. (2008). Dimension-reduced nonparametric maximum likelihood computation for interval-censored data. Computational Statistics & Data Analysis, 52, 2388-2402.

See Also

icendata, idf.

Examples

(x = cbind(1:5,1:5*3-2))
Deltamatrix(x)

U-shaped Hazard Function Estimation

Description

Uhaz computes the nonparametric maximum likelihood esimate (NPMLE) of a U-shaped hazard function from exact or interval-censored data, or a mix of the two types of data.

Usage

Uhaz(data, w = 1, deg = 1, maxit = 100, tol = 1e-06, verb = 0)

Arguments

Details

If data is a vector, it contains only exact observations, with weights given in w.

If data is a matrix with two columns, it contains interval-censored observations, with the two columns storing their left and right end-points, respectively. If the left and right end-points are equal, then the observation is exact. Weights are provided by w.

If data is a matrix with three columns, it contains interval-censored observations, with the first two columns storing their left and right end-points, respectively. The weight of each observation is the third-column value multiplied by the corresponding weight value in w.

The algorithm used for the computing the NPMLE of a hazard function under the U-shape restriction is is proposed by Wang and Fani (2015). Such a hazard function is given by

A U-shaped hazard function is given by

h(t) = \alpha + \sum_{j = 1}^k \nu_j(\tau_j - t)_+^p + \sum_{j = 1}^{m} \mu_j (t-\eta_j)_+^p,

where \alpha,\nu_j,\mu_j \ge 0, \tau_1 < \cdots < \tau_k \le \eta_1 < \cdots < \eta_m, and p \ge 0 is the the spline degree which determines the smoothness of the U-shaped hazard. As p increases, the family of hazard functions becomes increasingly smoother, but at the time, smaller. When p = 0, the hazard function is U-shaped, as studied by Bray et al. (1967). When p = 1, the hazard function is convex, as studied by Jankowski and Wellner (2009a,b).

Note that deg (i.e., p in the above mathematical display) can take on any nonnegative real value.

Value

An object of class Uhaz, which is a list with components:

Author(s)

Yong Wang <yongwang@auckland.ac.nz>

References

Bray, T. A., Crawford, G. B., and Proschan, F. (1967). Maximum Likelihood Estimation of a U-shaped Failure Rate Function. Defense Technical Information Center.

Jankowski, H. K. and Wellner, J. A. (2009a). Computation of nonparametric convex hazard estimators via profile methods. Journal of Nonparametric Statistics, 21, 505-518.

Jankowski, H. K. and Wellner, J. A. (2009b). Nonparametric estimation of a convex bathtub-shaped hazard function. Bernoulli, 15, 1010-1035.

Wang, Y. and Fani, S. (2018). Nonparametric maximum likelihood computation of a U-shaped hazard function. Statistics and Computing, 28, 187-200.

See Also

icendata, nzmort.

Examples

## Interval-censored observations
data(ap)
(r = Uhaz(ap, deg=0))
plot(r, ylim=c(0,.3), col=1)
for(i in 1:6) plot(Uhaz(ap, deg=i/2), add=TRUE, col=i+1)
legend(15, 0.01, paste0("deg = ", 0:6/2), lwd=2, col=1:7, xjust=1, yjust=0)

## Exact observations
data(nzmort)
x = with(nzmort, nzmort[ethnic=="maori",])[,1:2]   # Maori mortality
(h0 = Uhaz(x[,1]+0.5, x[,2], deg=0)$h)     # U-shaped hazard
(h1 = Uhaz(x[,1]+0.5, x[,2], deg=1)$h)     # convex hazard
(h2 <- Uhaz(x[,1]+0.5, x[,2], deg=2)$h)    # smooth U-shaped hazard

plot(h0, pch=2)                            # plot hazard functions
plot(h1, add=TRUE, col="green3", pch=1)
plot(h2, add=TRUE, col="red3", pch=19)

age = 0:max(x[,1])                         # plot densities
count = integer(length(age))
count[x[,"age"]+1] = x[,"deaths"]
barplot(count/sum(count), space=0, col="lightgrey")
axis(1, pos=NA, at=0:10*10)
plot(h0, fn="d", add=TRUE, pch=2)
plot(h1, fn="d", add=TRUE, col="green3", pch=1)
plot(h2, fn="d", add=TRUE, col="red3", pch=19)

plot(h0, fn="s", pch=2)                    # plot survival functions
plot(h1, fn="s", add=TRUE, col="green3", pch=1)
plot(h2, fn="s", add=TRUE, col="red3", pch=19)

## Exact and right-censored observations
data(gastric)
plot(h0<-Uhaz(gastric, deg=0)$h)           # plot hazard functions
plot(h1<-Uhaz(gastric, deg=1)$h, add=TRUE, col="green3")
plot(h2<-Uhaz(gastric, deg=2)$h, add=TRUE, col="red3")

plot(npsurv(gastric), fn="s", col="grey")  # plot survival functions
plot(h0, fn="s", add=TRUE)
plot(h1, fn="s", add=TRUE, col="green3")
plot(h2, fn="s", add=TRUE, col="red3")

Air Conditioner Failure Data

Description

Contains the number of operating hours between successive failure times of the air conditioning systems in Boeing airplanes

Format

A numeric vector storing the failure times.

Source

Proschan (1963)

References

Proschan, F. (1963). Theoretical explanation of observed decreasing failure rate. Technometrics, 5, 375-383.

See Also

Uhaz.

Examples

data(acfail)
r = Uhaz(acfail, deg=2)
plot(r$h, fn="h")
plot(r$h, fn="d")

Angina Pectoris Survival Data

Description

Contains the survival times in years from the time of diagnosis for 2418 male patients with angina pectoris. Some patients are lost to follow-up, hence giving right-censored observations. Each integer-valued survival time is treated as being censored within a one-year interval.

Format

A data frame with 30 observations and 3 variables:

L: left-end point of an interval-censored retraction time;

R: right-end point of an interval-censored retraction time;

count: number of patients in the interval.

Source

Lee and Wang (2003), page 92.

References

Lee, E. T. and Wang, J. W. (2003). Statistical Methods for Survival Data Analysis. Wiley.

See Also

npsurv.

Examples

data(ap)
r = Uhaz(ap, deg=2)           # smooth U-shaped hazard
plot(r$h, fn="h")             # hazard
plot(r$h, fn="d")             # density

# NPMLE and shape-restricted estimation
plot(npsurv(ap), fn="s")      # survival under no shape restriction
plot(r$h, fn="s", add=TRUE)   # survival with smooth U-shaped hazard

Breast Retraction Times after Beast Cancer Treatments.

Description

Contains the breast retraction times in months for 94 breast cancer patients who received either radiation therapy or radiation therapy plus adjuvant chemotherapy.

Format

A data frame with 94 observations and 3 variables:

L: left-end points of the interval-censored retraction times;

R: right-end points of the interval-censored retraction times;

group: either RT (radiation therapy) or RCT (radiation therapy plus adjuvant chemotherapy).

Source

Finkelstein and Wolfe (1985).

References

Finkelstein, D. M. and R. A. Wolfe (1985). A semiparametric model for regression analysis of interval-censored failure time data. Biometrics, 41, pp.933-945.

See Also

npsurv.

Examples

data(cancer)
i = cancer$group == "RT"
plot(npsurv(cancer[i,1:2]), xlim=c(0,60))
plot(npsurv(cancer[!i,1:2]), add=TRUE, col="green3")

Gastric Cancer Survival Data

Description

Contains the survival times of 45 gastrointestinal tumor patients who were treated with both chemotherapy and radiotherapy. It has both exact and right-censored observations.

Format

A data frame with 30 observations and 3 variables:

L: left-end points of the interval-censored survival times;

R: right-end points of the interval-censored survival times.

Source

Klein and Moeschberger (2003), page 224.

References

Klein, J. P. and Moeschberger, M. L. (2003). Survival Analysis: Techniques for Censored and Truncated Data (2nd ed.). Springer.

See Also

npsurv, Uhaz.

Examples

data(gastric)
plot(npsurv(gastric), col="grey")      # survival function
plot(h0<-Uhaz(gastric, deg=0)$h, fn="s", add=TRUE, col="green3")
plot(h1<-Uhaz(gastric, deg=1)$h, fn="s", add=TRUE)
plot(h2<-Uhaz(gastric, deg=2)$h, fn="s", add=TRUE, col="red3")

plot(h0, fn="h", col="green3")         # hazard function
plot(h1, fn="h", add=TRUE)
plot(h2, fn="h", add=TRUE, col="red3")

plot(h0, fn="d", col="green3")         # density function
plot(h1, fn="d", add=TRUE)
plot(h2, fn="d", add=TRUE, col="red3") 

Distributional Functions given a U-shaped Hazard Function

Description

Given an object of class uh:

Usage

hazuh(t, h)
chazuh(t, h)
survuh(t, h)
denuh(t, h)

Arguments

Details

hazuh computes the hazard values;

chazuh computes the cumulative hazard values;

survuh computes the survival function values;

denuh computes the density function values.

Value

A numeric vector of the function values.

Author(s)

Yong Wang <yongwang@auckland.ac.nz>

References

Wang, Y. and Fani, S. (2018). Nonparametric maximum likelihood computation of a U-shaped hazard function. Statistics and Computing, 28, 187-200.

See Also

Uhaz, icendata, plot.uh

Examples

data(ap)
h = Uhaz(icendata(ap), deg=2)$h
hazuh(0:15, h)     # hazard
chazuh(0:15, h)    # cumulative hazard
survuh(0:15, h)    # survival probability
denuh(0:15, h)     # density

Class of Interval-censored Data

Description

Class icendata can be used to store general interval-censored data, which may possibly contain exact observations.There are several functions associated with the class.

Usage

icendata(x, w=1)
is.icendata(x)

Arguments

Details

Function icendata creates an object of class 'icendata', which can be used to save both interval-censored and exact observations.

Function is.icendata simply checks if an object is of class 'icendata'.

If x is a vector, it contains only exact observations, with weights given in w.

If x is a two-column matrix, it contains interval-censored observations and stores their left and right endpoints in the first and second column, respectively. If the left and right endpoints are equal, then the observation is exact. Weights are provided by w.

If x is a three-column matrix, it contains interval-censored observations and stores their left and right endpoints in the first and second column, respectively. The weight of each observation is the third-column value multiplied by the corresponding weight value in w.

It is useful to turn interval-censored (and exact) observations into the format imposed by icendata so that they can be processed in a standardized format by other functions. Also, exact and interval-censored observations are stored separately in this format and can hence be dealt with more easily. Most functions in the package npsurv first ensure that the data has this format before processing.

Observations of zero weights are removed. Identical observations are aggregated.

An interval-valued observation is either (L_i, R_i] if L_i < R_i, or [L_i, R_i] if L_i = R_i.

Value

Author(s)

Yong Wang <yongwang@auckland.ac.nz>

References

Wang, Y. (2008). Dimension-reduced nonparametric maximum likelihood computation for interval-censored data. Computational Statistics & Data Analysis, 52, 2388-2402.

Wang, Y. and Fani, S. (2017). Nonparametric maximum likelihood computation of a U-shaped hazard function. Statistics and Computing, (in print).

See Also

npsurv, Uhaz.

Examples

data(ap)
(x = icendata(ap))
is.icendata(x)

data(gastric)
icendata(gastric)

data(leukemia)
i = leukemia[,"group"] == "6-MP"
icendata(leukemia[i,1:2])

Interval Distribution Function

Description

Class idf can be used to store a distribution function defined on a set of intervals. There are several functions associated with the class.

Usage

idf(left, right, p)
## S3 method for class 'idf'
print(x, ...)

Arguments

Details

idf creates an object of class idf. An idf object stores a distribution function defined on a set of intervals.

When left and right endpoints are identical, the intervals just represent exact points.

print.idf prints an object of class idf as a three-coumn matrix.

Value

Author(s)

Yong Wang <yongwang@auckland.ac.nz>

See Also

icendata, Deltamatrix, npsurv.

Examples

idf(1:5, 1:5*3-2, c(1,1,2,2,4))
npsurv(cbind(1:5, 1:5*3-2))$f    # NPMLE 

Kaplan-Meier Estimation

Description

km computes the nonparametric maximum likelihood esimate (NPMLE) of a survival function for right-censored data.

Usage

km(data, w = 1)

Arguments

Details

For details about the arguments, see icendata.

Value

A list with components:

Author(s)

Yong Wang <yongwang@auckland.ac.nz>

References

Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53, 457-481.

See Also

icendata, npsurv, idf.

Examples

x = cbind(1:5, c(1,Inf,3,4,Inf))
(f = km(x)$f)
plot(f)

data(leukemia)
i = leukemia[,"group"] == "Placebo"
plot(km(leukemia[i,1:2])$f, xlim=c(0,40), col="green3") # placebo
plot(km(leukemia[!i,1:2])$f, add=TRUE)                  # 6-MP

Remission Times for Acute Leukemia Patients

Description

Contains remission times in weeks of 42 acute leukemia patients, who received either the treatment of drug 6-mercaptopurine or the placebo treatment. Each remission time is either exactly observed or right-censored.

Format

A data frame with 42 observations and 3 variables:

L: left-end points of the interval-censored remission times in weeks;

R: right-end points of the interval-censored remission times;

group: either 6-MP (6-mercaptopurine) or Placebo.

Source

Freireich et al. (1963).

References

Freireich, E. O. et al. (1963). The effect of 6-mercaptopmine on the duration of steroid induced remission in acute leukemia. Blood, 21, 699-716.

See Also

npsurv.

Examples

data(leukemia)
i = leukemia[,"group"] == "Placebo"
plot(npsurv(leukemia[i,1:2]), xlim=c(0,40), col="green3") # placebo
plot(npsurv(leukemia[!i,1:2]), add=TRUE)                  # 6-MP

## Treat each remission time as interval-censored:
x = leukemia
ii = x[,1] == x[,2]
x[ii,2] = x[ii,1] + 1
plot(npsurv(x[i,1:2]), xlim=c(0,40), col="green3")        # placebo
plot(npsurv(x[!i,1:2]), add=TRUE)                         # 6-MP

Computes the Log-likelihood Value of a U-shaped Hazard Function

Description

logLikuh returns the log-likelihood value of a U-shaped hazard function, given a data set.

Usage

logLikuh(h, data)

Arguments

Value

Log-likelihood value evaluated at h, given data.

Author(s)

Yong Wang <yongwang@auckland.ac.nz>

References

Wang, Y. and Fani, S. (2018). Nonparametric maximum likelihood computation of a U-shaped hazard function. Statistics and Computing, 28, 187-200.

See Also

Uhaz, icendata, plot.uh

Examples

data(ap)
(h0 = uh(.2, NULL, NULL, NULL, NULL, 15, 1))   # Uniform hazard
plot(h0, ylim=c(0,.3))
logLikuh(h0, ap)

r = Uhaz(ap, deg=2)
r$ll
logLikuh(r$h, ap)
plot(r$h, add=TRUE, col="red3")

Angina Pectoris Survival Data

Description

Contains the answers of 191 California high school students to the question: "When did you first use marijuana?". An answer can be an exact age, or "I have never used it", which gives rise to a right-censored observation, or "I have used it but cannot recall just when the first time was", which gives rise to a left-censored observation.

Format

A data frame with 21 observations and 3 variables:

L: left-end point of an interval-censored time;

R: right-end point of an interval-censored time;

count: number of students in the interval.

Source

Turnbull and Weiss (1978). See also Klein and Moeschberger (1997), page 17.

References

Turnbull and Weiss (1978). A likelihood ratio statistic fortesting goodness of fit with randomly censored data. Biometrics, 34, 367-375.

Klein and Moeschberger (2003). Survival Analysis: Techniques for Censored and Truncated Data (2nd ed.). Springer

See Also

npsurv.

Examples

data(marijuana)
r = Uhaz(marijuana, deg=2)
plot(r$h, fn="h")
plot(r$h, fn="s")

Nonparametric Survival Function Estimation

Description

npsurv computes the nonparametric maximum likelihood esimate (NPMLE) of a survival function for general interval-censored data.

Usage

npsurv(data, w = 1, maxit = 100, tol = 1e-06, verb = 0)

Arguments

Details

If data is a vector, it contains only exact observations, with weights given in w.

If data is a matrix with two columns, it contains interval-censored observations, with the two columns storing their left and right end-points, respectively. If the left and right end-points are equal, then the observation is exact. Weights are provided by w.

If data is a matrix with three columns, it contains interval-censored observations, with the first two columns storing their left and right end-points, respectively. The weight of each observation is the third-column value multiplied by the corresponding weight value in w.

The algorithm used for computing the NPMLE is either the constrained Newton method (CNM) (Wang, 2008), or the hierachical constrained Newton method (HCNM) (Wang and Taylor, 2013) when there are a large number of maximal intersection intervals.

Inside the function, it examines if data has only right censoring, and if so, the Kaplan-Meier estimate is computed directly by function km.

An interval-valued observation is either (L_i, R_i] if L_i < R_i, or [L_i, R_i] if L_i = R_i.

Value

An object of class npsurv, which is a list with components:

Author(s)

Yong Wang <yongwang@auckland.ac.nz>

References

Wang, Y. (2008). Dimension-reduced nonparametric maximum likelihood computation for interval-censored data. Computational Statistics & Data Analysis, 52, 2388-2402.

Wang, Y. and Taylor, S. M. (2013). Efficient computation of nonparametric survival functions via a hierarchical mixture formulation. Statistics and Computing, 23, 713-725.

See Also

icendata, Deltamatrix, idf, km.

Examples

## all exact observations
data(acfail)
plot(npsurv(acfail))

## right-censored (and exact) observations
data(gastric)
plot(npsurv(gastric))

data(leukemia)
i = leukemia[,"group"] == "Placebo"
plot(npsurv(leukemia[i,1:2]), xlim=c(0,40), col="blue") # placebo
plot(npsurv(leukemia[!i,1:2]), add=TRUE, col="red")     # 6-MP

## purely interval-censored data
data(ap)
plot(npsurv(ap))

data(cancer)
cancerRT = with(cancer, cancer[group=="RT",1:2])
plot(npsurv(cancerRT), xlim=c(0,60))                  # survival of RT 
cancerRCT = with(cancer, cancer[group=="RCT",1:2])
plot(npsurv(cancerRCT), add=TRUE, col="green")        # survival of RCT 

New Zealand Mortality in 2000

Description

Contains the number of deaths of Maori and Non-Maori people at each age in New Zealand in 2000.

Format

A data frame with 210 observations and 3 variables:

age: at which age the deaths occurred;

deaths: number of people died at the age;

ethnic: either Maori or Non-Maori.

Details

Data contains no age with zero death.

Source

https://www.mortality.org/

See Also

Uhaz.

Examples

data(nzmort)
x = with(nzmort, nzmort[ethnic=="maori",])[,1:2]      # Maori mortality
# x = with(nzmort, nzmort[ethnic!="maori",])[,1:2]    # Non-Maori mortality

## As exact observations
# Plot hazard functions
h0 = Uhaz(x[,1]+0.5, x[,2], deg=0)$h    # U-shaped hazard
plot(h0, fn="h", col="green3", pch=2)
h1 = Uhaz(x[,1]+0.5, x[,2], deg=1)$h    # convex hazard
plot(h1, fn="h", add=TRUE, pch=1)
h2 = Uhaz(x[,1]+0.5, x[,2], deg=2)$h    # smooth U-shaped hazard
plot(h2, fn="h", add=TRUE, col="red3")

# Plot densities
age = 0:max(x[,1])
count = integer(length(age))
count[x[,"age"]+1] = x[,"deaths"]
barplot(count/sum(count), space=0, col="lightgrey", ylab="Density")
axis(1, pos=NA, at=0:10*10)
plot(h0, fn="d", add=TRUE, col="green3", pch=2)
plot(h1, fn="d", add=TRUE, col="blue3", pch=1)
plot(h2, fn="d", add=TRUE, col="red3", pch=19)

## As interval-censored observations
# Plot hazard functions
x2 = cbind(x[,1], x[,1]+1, x[,2])
h0 = Uhaz(x2, deg=0)$h      # U-shaped hazard
plot(h0, fn="h", col="green3", pch=2)
h1 = Uhaz(x2, deg=1)$h      # convex hazard
plot(h1, fn="h", add=TRUE, pch=1)
h2 = Uhaz(x2, deg=2)$h      # smooth U-shaped hazard
plot(h2, fn="h", add=TRUE, col="red3", pch=1)

# Plot densities
barplot(count/sum(count), space=0, col="lightgrey")
axis(1, pos=NA, at=0:10*10)
plot(h0, fn="d", add=TRUE, col="green3", pch=2)
plot(h1, fn="d", add=TRUE, col="blue3", pch=1)
plot(h2, fn="d", add=TRUE, col="red3", pch=19)

Plot Functions for U-shaped Hazard Estimation

Description

Functions for plotting various functions in U-shaped hazard estimation

Usage

## S3 method for class 'Uhaz'
plot(x, ...)
## S3 method for class 'uh'
plot(x, data, fn=c("haz","grad","surv","den","chaz"), ...)
plothazuh(h, add=FALSE, col="darkblue", lty=1, xlim, ylim,
         lwd=2, pch=19, len=500, vert=FALSE, add.knots=TRUE,
         xlab="Time", ylab="Hazard", ...)
plotchazuh(h, add=FALSE, lwd=2, len=500, col="darkblue",
          pch=19, add.knots=TRUE, vert=FALSE, xlim, ylim, ...)
plotdenuh(h, add=FALSE, lty=1, lwd=2, col="darkblue",
         add.knots=TRUE, pch=19, ylim, len=500, vert=FALSE, ...)
plotsurvuh(h, add=FALSE, lty=1, lwd=2, len=500, vert=FALSE,
          col="darkblue", pch=19, add.knots=TRUE, xlim, ylim, ...)
plotgraduh(h, data, w=1, len=500, xlim, ylim, vert=TRUE,
          add=FALSE, xlab="Time", ylab="Gradient",
          col0="red3", col1="blue3", col2="green3", order=0, ...)

Arguments

Details

plot.Uhaz and plot.uh are wrapper functions that can be used to invoke plot.hazuh, plot.chazuh, plot.survuh, plot.denuh or plot.graduh.

plothazuh plots a U-shaped hazard function.

plotchazuh plots a cumulative hazard function that has a U-shaped hazard function.

plotsurvuh plots the survival function that has a U-shaped hazard function.

plotdenuh plots the density function that has a U-shaped hazard function.

plotgraduh plots the gradient function that has a U-shaped hazard function.

A U-shaped hazard function is given by

h(t) = \alpha + \sum_{j = 1}^k \nu_j(\tau_j - t)_+^p + \sum_{j = 1}^{m} \mu_j (t-\eta_j)_+^p,

where \alpha,\nu_j,\mu_j \ge 0, \tau_1 < \cdots < \tau_k \le \eta_1 < \cdots < \eta_m, and p \ge 0.

Author(s)

Yong Wang <yongwang@auckland.ac.nz>

References

Wang, Y. and Fani, S. (2018). Nonparametric maximum likelihood computation of a U-shaped hazard function. Statistics and Computing, 28, 187-200.

See Also

icendata, uh, npsurv.

Examples

## Angina Pectoris Survival Data
data(ap)
plot(r<-Uhaz(ap))              # hazard function for a convex hazard
plot(r, fn="c")                # cumulative hazard function
plot(r, fn="s")                # survival function
plot(r, fn="d")                # density function
plot(r, ap, fn="g")            # gradient functions
plot(r, ap, fn="g", order=1)   # first derivatives of gradient functions
plot(r, ap, fn="g", order=2)   # second derivatives of gradient functions

## New Zealand Mortality in 2000
data(nzmort)
i = nzmort$ethnic == "maori"
x = nzmort[i,1:2]                            # Maori mortality
h = Uhaz(x[,1]+0.5, x[,2], deg=2)$h          # smooth U-shaped hazard
plot(h)                        # hazard function
plot(h, fn="d")                # density function
plot(h, fn="s")                # survival function

x2 = nzmort[!i,1:2]                          # Non-Maori mortality
h2 = Uhaz(x2[,1]+0.5, x2[,2], deg=2)$h
plot(h2, fn="s", add=TRUE, col="green3")

Plot Functions for Nonparametric Survival Estimation

Description

Functions for plotting nonparametric survival functions and related ones.

Usage

## S3 method for class 'npsurv'
plot(x, ...)
## S3 method for class 'idf'
plot(x, data, fn=c("surv","grad"), ...)
plotsurvidf(f, style=c("box","uniform","left","right","midpoint"),
           xlab="Time", ylab="Survival Probability", col="blue3", fill=0,  
           add=FALSE, lty=1, lty.inf=2, xlim, ...)
plotgradidf(f, data, w=1, col1="red3", col2="blue3", 
           xlab="Survival Time", ylab="Gradient", xlim, ...)

Arguments

Details

plot.npsurv and plot.idf are wrapper functions that call either plotsurvidf or plotgradidf.

plotsurvidf plots the survival function of the nonparametric maximum likelihood estimate (NPMLE).

plotgradidf plots the gradient function of the NPMLE.

plotsurvidf by default chooses a less saturated color for fill than col.

plotgradidf plots gradient values as vertical lines located as the left endpoints of the maximal intersection intervals. Each maximal intersection interval is plotted as a wider line on the horizontal zero-gradient line, with a circle to represent the open left endpoint of the interval and a solid point the closed right endpoint of the interval. The maximal intersection intervals allocated with positive probabilities have zero gradients, and hence no vertical lines are drawn for them.

Author(s)

Yong Wang <yongwang@auckland.ac.nz>

References

Wang, Y. (2008). Dimension-reduced nonparametric maximum likelihood computation for interval-censored data. Computational Statistics & Data Analysis, 52, 2388-2402.

See Also

icendata, idf, npsurv.

Examples

data(ap)
plot(r<-npsurv(ap))              # survival function
plot(r$f, ap, fn="g")            # all gradients virtually zeros.

data(cancer)
cancerRT = with(cancer, cancer[group=="RT",1:2])
plot(rt<-npsurv(cancerRT), xlim=c(0,60))                  # survival of RT 
cancerRCT = with(cancer, cancer[group=="RCT",1:2])
plot(rct<-npsurv(cancerRCT), add=TRUE, col="green3") # survival of RCT 
## as uniform dististrbutions.
plot(rt, add=TRUE, style="uniform", col="blue3")
plot(rct, add=TRUE, style="uniform", col="green3")

## plot gradients; must supply data
plot(rt, cancerRT, fn="g")        # for group RT
plotgradidf(rct$f, cancerRCT)   # or, for group RCT

U-shaped Hazard Function

Description

Class uh can be used to store U-shaped hazard functions. There are a couple of functions associated with the class.

Usage

uh(alpha, tau, nu, eta, mu, upper=Inf, deg=1, collapse=TRUE)
## S3 method for class 'uh'
print(x, ...)

Arguments

Details

uh creates an object of class uh, which stores a U-shaped hazard function.

print.uh prints an object of class uh.

A U-shape hazard function, as generalized by Wang and Fani (2018), is given by

h(t) = \alpha + \sum_{j = 1}^k \nu_j(\tau_j - t)_+^p + \sum_{j = 1}^{m} \mu_j (t-\eta_j)_+^p,

where \alpha,\nu_j,\mu_j \ge 0, \tau_1 < \cdots < \tau_k \le \eta_1 < \cdots < \eta_m, and p \ge 0 is the the spline degree which determines the smoothness of the U-shaped hazard. As p increases, the family of hazard functions becomes increasingly smoother, but at the same time, smaller. When p = 0, the hazard function is U-shaped, as studied by Bray et al. (1967). When p = 1, the hazard function is convex, as studied by Jankowski and Wellner (2009a,b).

print.uh prints an object of class uh. While alpha, upper and deg are printed as they are, tau and nu are printed as a two-column matrix, and so are eta and mu.

Value

uh returns an object of class uh. It is a list with components alpha, tau, nu, eta, mu, upper and deg, which store their corresponding values as described above.

Author(s)

Yong Wang <yongwang@auckland.ac.nz>

References

Bray, T. A., Crawford, G. B., and Proschan, F. (1967). Maximum Likelihood Estimation of a U-shaped Failure Rate Function. Defense Technical Information Center.

Jankowski, H. K. and Wellner, J. A. (2009a). Computation of nonparametric convex hazard estimators via profile methods. Journal of Nonparametric Statistics, 21, 505-518.

Jankowski, H. K. and Wellner, J. A. (2009b). Nonparametric estimation of a convex bathtub-shaped hazard function. Bernoulli, 15, 1010-1035.

Wang, Y. and Fani, S. (2018). Nonparametric maximum likelihood computation of a U-shaped hazard function. Statistics and Computing, 28, 187-200.

See Also

Uhaz, icendata, plot.uh

Examples

(h0 = uh(3, 2, 3, 4, 5, 7, deg=0))              # deg = 0
plot(h0, ylim=c(0,20))
(h1 = uh(4, 2, 3, 5, 6, 7, deg=1))              # deg = 1
plot(h1, add=TRUE, col="green3")
(h2 = uh(1, 1:2, 3:4, 5:6, 7:8, 9, deg=2))      # deg = 2
plot(h2, add=TRUE, col="red3")