| Version: | 0.1-5 |
| Date: | 2020-02-28 |
| Title: | Nonparametric Estimation in Mixture Cure Models |
| Author: | Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut] |
| Maintainer: | Ignacio López-de-Ullibarri <ignacio.lopezdeullibarri@udc.es> |
| Depends: | R (≥ 3.5.0) |
| Suggests: | KMsurv |
| Description: | Performs nonparametric estimation in mixture cure models, and significance tests for the cure probability. For details, see López-Cheda et al. (2017a) <doi:10.1016/j.csda.2016.08.002> and López-Cheda et al. (2017b) <doi:10.1007/s11749-016-0515-1>. |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| Imports: | permute, stats, utils, zoo |
| NeedsCompilation: | yes |
| Encoding: | UTF-8 |
| Packaged: | 2020-02-28 17:11:35 UTC; ilu |
| Repository: | CRAN |
| Date/Publication: | 2020-02-29 10:10:02 UTC |
Nonparametric Estimation in Mixture Cure Models
Description
Performs nonparametric estimation in mixture cure models, and significance tests for the cure probability. For details, see López-Cheda et al. (2017a) <doi:10.1016/j.csda.2016.08.002> and López-Cheda et al. (2017b) <doi:10.1007/s11749-016-0515-1>.
Details
Index of help topics:
beran Compute Beran's Estimator of the Conditional
Survival
berancv Compute the Cross-Validation Bandwidth for
Beran's Estimator of the Conditional Survival
controlpars Control Values for the Bootstrap or
Cross-validation
hpilot Compute the Pilot Bandwidth for the
Nonparametric Estimators of Cure Probability
and Latency
latency Compute Nonparametric Estimator of the
Conditional Latency
latencyhboot Compute the Bootstrap Bandwidth for the
Nonparametric Estimator of the Latency
npcure-package Nonparametric Estimation in Mixture Cure Models
print.npcure Print Method for Objects of Class 'npcure'
probcure Compute Nonparametric Estimator of the
Conditional Probability of Cure
probcurehboot Compute the Bootstrap Bandwidth for the
Nonparametric Estimator of the Cure Probability
summary.npcure Summary Method for Objects of Class 'npcure'
testcov Covariate Significance Test of the Cure
Probability
testmz Test of Maller-Zhou
Author(s)
Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut]
References
López-Cheda, A., Cao, R., Jácome, M. A., Van Keilegom, I. (2017). Nonparametric incidence estimation and bootstrap bandwidth selection in mixture cure models. Computational Statistics & Data Analysis 105: 144–165. https://doi.org/10.1016/j.csda.2016.08.002.
López-Cheda, A., Jácome, M. A., Cao, R. (2017). Nonparametric latency estimation for mixture cure models. Test, 26: 353–376. https://doi.org/10.1007/s11749-016-0515-1.
Compute Beran's Estimator of the Conditional Survival
Description
This function computes the Beran nonparametric estimator of the conditional survival function.
Usage
beran(x, t, d, dataset, x0, h, local = TRUE, testimate = NULL,
conflevel = 0L, cvbootpars = if (conflevel == 0 && !missing(h)) NULL
else controlpars())
Arguments
x |
If |
t |
If |
d |
If |
dataset |
An optional data frame in which the variables named in
|
x0 |
A numeric vector of covariate values where the survival estimates will be computed. |
h |
A numeric vector of bandwidths. If it is missing the default
is to use the cross-validation bandwidth computed by the
|
local |
A logical value, |
testimate |
A numeric vector specifying the times at which the
survival is estimated. By default it is |
conflevel |
A value controlling whether bootstrap confidence intervals (CI) of the survival are to be computed. With the default value, 0L, the CIs are not computed. If a numeric value between 0 and 1 is passed, it specifies the confidence level of the CIs. |
cvbootpars |
A list of parameters controlling the bootstrap when
computing the CIs of the survival: |
Details
This function computes the kernel type product-limit estimator
of the conditional survival function S(t | x) = P(Y > t | X = x)
under censoring, using the Nadaraya-Watson weights. The kernel used is
the Epanechnikov. If the smoothing parameter h is not provided,
then the cross-validation bandwidth selector in Geerdens et al. (2018)
is used. The function is available only for one continuous covariate
X.
Value
An object of S3 class 'npcure'. Formally, a list of components:
type |
The constant string "survival". |
local |
The value of the |
h |
The value of the |
x0 |
The value of the |
testim |
The numeric vector of time values where the survival function is estimated. |
S |
A list whose components are the estimates of the survival function
for each one of the covariate values, i.e., those specified by the
|
Author(s)
Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut]
References
Beran, R. (1981). Nonparametric regression with randomly censored survival data. Technical report, University of California, Berkeley.
Geerdens, C., Acar, E. F., Janssen, P. (2018). Conditional copula models for right-censored clustered event time data. Biostatistics, 19(2): 247-262. https://doi.org/10.1093/biostatistics/kxx034.
See Also
Examples
## Some artificial data
set.seed(123)
n <- 50
x <- runif(n, -2, 2) ## Covariate values
y <- rweibull(n, shape = .5*(x + 4)) ## True lifetimes
c <- rexp(n) ## Censoring values
p <- exp(2*x)/(1 + exp(2*x)) ## Probability of being susceptible
u <- runif(n)
t <- ifelse(u < p, pmin(y, c), c) ## Observed times
d <- ifelse(u < p, ifelse(y < c, 1, 0), 0) ## Uncensoring indicator
data <- data.frame(x = x, t = t, d = d)
## Survival estimates for covariate values 0, 0.5 using...
## ... (a) global bandwidths 0.3, 0.5, 1.
## By default, the estimates are computed at the observed times
x0 <- c(0, .5)
S1 <- beran(x, t, d, data, x0 = x0, h = c(.3, .5, 1), local = FALSE)
## Plot predicted survival curves for covariate value 0.5
plot(S1$testim, S1$S$h0.3$x0.5, type = "s", xlab = "Time", ylab =
"Survival", ylim = c(0, 1))
lines(S1$testim, S1$S$h0.5$x0.5, type = "s", lty = 2)
lines(S1$testim, S1$S$h1$x0.5, type = "s", lty = 3)
## The true survival curve is plotted for reference
p0 <- exp(2*x0[2])/(1 + exp(2*x0[2]))
lines(S1$testim, 1 - p0 + p0*pweibull(S1$testim, shape = .5*(x0[2] + 4),
lower.tail = FALSE), col = 2)
legend("topright", c("Estimate, h = 0.3", "Estimate, h = 0.5",
"Estimate, h = 1", "True"), lty = c(1:3, 1), col = c(rep(1, 3), 2))
## As before, but with estimates computed at fixed times 0.1, 0.2,...,1
S2 <- beran(x, t, d, data, x0 = x0, h = c(.3, .5, 1), local = FALSE,
testimate = .1*(1:10))
## ... (b) local bandwidths 0.3, 0.5.
## Note that the length of the covariate vector x0 and the bandwidth h
## must be the same.
S3 <- beran(x, t, d, data, x0 = x0, h = c(.3, .5), local = TRUE)
## ... (c) the cross-validation (CV) bandwidth selector (the default
## when the bandwidth argument is not provided).
## The CV bandwidth is searched in a grid of 150 bandwidths (hl = 150)
## between 0.2 and 2 times the standardized interquartile range
## of the covariate values (hbound = c(.2, 2)).
## 95% confidence intervals are also given.
S4 <- beran(x, t, d, data, x0 = x0, conflevel = .95, cvbootpars =
controlpars(hl = 150, hbound = c(.2, 2)))
## Plot of predicted survival curve and confidence intervals for
## covariate value 0.5
plot(S4$testim, S4$S$x0.5, type = "s", xlab = "Time", ylab = "Survival",
ylim = c(0, 1))
lines(S4$testim, S4$conf$x0.5$lower, type = "s", lty = 2)
lines(S4$testim, S4$conf$x0.5$upper, type = "s", lty = 2)
lines(S4$testim, 1 - p0 + p0 * pweibull(S4$testim, shape = .5*(x0[2] +
4), lower.tail = FALSE), col = 2)
legend("topright", c("Estimate with CV bandwidth", "95% CI limits",
"True"), lty = c(1, 2, 1), col = c(1, 1, 2))
## Example with the dataset 'bmt' in the 'KMsurv' package
## to study the survival of patients aged 25 and 40.
data("bmt", package = "KMsurv")
x0 <- c(25, 40)
S <- beran(z1, t2, d3, bmt, x0 = x0, conflevel = .95)
## Plot of predicted survival curves and confidence intervals
plot(S$testim, S$S$x25, type = "s", xlab = "Time", ylab = "Survival",
ylim = c(0, 1))
lines(S$testim, S$conf$x25$lower, type = "s", lty = 2)
lines(S$testim, S$conf$x25$upper, type = "s", lty = 2)
lines(S$testim, S$S$x40, type = "s", lty = 1, col = 2)
lines(S$testim, S$conf$x40$lower, type = "s", lty = 2, col = 2)
lines(S$testim, S$conf$x40$upper, type = "s", lty = 2, col = 2)
legend("topright", c("Age 25: Estimate", "Age 25: 95% CI limits",
"Age 40: Estimate", "Age 40: 95% CI limits"), lty = 1:2,
col = c(1, 1, 2, 2))
Compute the Cross-Validation Bandwidth for Beran's Estimator of the Conditional Survival
Description
This function computes the cross-validation bandwidth for Beran's estimator of the conditional survival function.
Usage
berancv(x, t, d, dataset, x0, cvpars = controlpars())
Arguments
x |
If |
t |
If |
d |
If |
dataset |
An optional data frame in which the variables named in
|
x0 |
A numeric vector of covariate values where the local cross-validation bandwidth will be computed. |
cvpars |
A list of parameters controlling the process of bandwidth
selection. The default is the value returned by the |
Details
The cross-validation (CV) bandwidth is taken as the largest
local minimizer of the leave-one-out cross-validated criterion in
Geerdens et al. (2018). Let F^{(-i)}(t | x_i), i = 1,
\ldots, n be the Beran estimator obtained using the
data points (x_j, t_j, d_j), j = 1, \ldots, i-1, i+1,
\ldots, n. For the CV criterion,
the differences I(t_i \leq t_j)-F^{(-i)}(t_j|x_i) are computed only for the so-called 'useful
pairs' of observed times (t_i, t_j). A pair (T_i, T_j) is
useful if the value of the indicator I(T_i \leq T_j) gives an unambiguous correct value for the indicator I(Y_i
\leq Y_j) which contains the corresponding true
(possibly unknown) event times, see Geerdens et al. (2018) for
details. Gannoun et al. (2007) apply a similar criterion to perform
bandwidth selection for the Beran estimator, but they consider only
the pairs of true (uncensored) event times. Note that the inclusion of
useful pairs of observed times would be especially advantageous if the
censoring rate is high.
Value
An object of S3 class 'npcure'. Formally, a list of components:
type |
The constant character string c("Cross-validation bandwidth", "survival"). |
x0 |
Grid of covariate values. |
h |
Selected local cross-validation bandwidths. |
hgrid |
Grid of bandwidths used (optional). |
Author(s)
Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut]
References
Gannoun, A., Saracco, J., Yu, K. (2007). Comparison of kernel estimators of conditional distribution function and quantile regression under censoring. Statistical Modeling, 7: 329-344. https://doi.org/10.1177/1471082X0700700404.
Geerdens, C., Acar, E. F., Janssen, P. (2018). Conditional copula models for right-censored clustered event time data. Biostatistics, 19(2): 247-262. https://doi.org/10.1093/biostatistics/kxx034.
See Also
Examples
## Some artificial data
set.seed(123)
n <- 50
x <- runif(n, -2, 2) ## Covariate values
y <- rweibull(n, shape = .5*(x + 4)) ## True lifetimes
c <- rexp(n) ## Censoring values
p <- exp(2*x)/(1 + exp(2*x)) ## Probability of being susceptible
u <- runif(n)
t <- ifelse(u < p, pmin(y, c), c) ## Observed times
d <- ifelse(u < p, ifelse(y < c, 1, 0), 0) ## Uncensoring indicator
data <- data.frame(x = x, t = t, d = d)
## Computation of cross-validation (CV) local bandwidth for Beran's
## estimator of survival for covariate values 0, 1, ...
#### ... with the default control parameters (passed through 'cvpars')
x0 <- c(0, 1)
hcv <- berancv(x, t, d, data, x0 = x0)
#### ... changing the default 'cvpars' by calling the 'controlpars()'
#### function:
#### (a) the CV local bandwidth is searched in a grid of 150 bandwidths
#### ('hl = 150') between 0.2 and 4 times the standardized interquartile
#### range of the covariate values of x ('hbound = c(.2, 4'))
#### (b) all the grid bandwidths are saved ('hsave = TRUE')
hcv <- berancv(x, t, d, data, x0 = x0, cvpars = controlpars(hbound =
c(.2, 4), hl = 150, hsave = TRUE))
## Survival estimates for covariate values 0, 1, with CV local bandwidth
S1 <- beran(x, t, d, data, x0 = x0, h = hcv$h)
## Plot predicted survival curves for covariate values 0, 1
plot (S1$testim, S1$S$x0, type = "s", xlab = "Time", ylab = "Survival",
ylim = c(0, 1))
lines(S1$testim, S1$S$x1, type = "s", lty = 2)
## The survival curves are displayed for reference
p0 <- exp(2*x0)/(1 + exp(2*x0))
lines(S1$testim, 1 - p0[1] + p0[1]*pweibull(S1$testim, shape = .5*(x0[1]
+ 4), lower.tail = FALSE), col = 2)
lines(S1$testim, 1 - p0[2] + p0[2]*pweibull(S1$testim, shape = .5*(x0[2]
+ 4), lower.tail = FALSE), lty = 2, col = 2)
legend("topright", c("Estimate, x = 0", "True, x = 0",
"Estimate, x = 1", "True, x = 1"), lty = c(1, 1, 2, 2), col = 1:2)
## Example with the dataset 'bmt' of the 'KMsurv' package to study the
## survival of patients aged 25 and 40.
data("bmt", package = "KMsurv")
x0 <- c(25, 40)
hcv <- berancv(z1, t2, d3, bmt, x0 = x0, cvpars = controlpars(hbound =
c(.2, 4), hl = 150, hsave = TRUE))
S <- beran(z1, t2, d3, bmt, x0 = x0, h = hcv$h, conflevel = .95)
## Plot of predicted survival curves and confidence intervals
plot(S$testim, S$S$x25, type = "s", xlab = "Time", ylab = "Survival",
ylim = c(0, 1))
lines(S$testim, S$conf$x25$lower, type = "s", lty = 2)
lines(S$testim, S$conf$x25$upper, type = "s", lty = 2)
lines(S$testim, S$S$x40, type = "s", lty = 1, col = 2)
lines(S$testim, S$conf$x40$lower, type = "s", lty = 2, col = 2)
lines(S$testim, S$conf$x40$upper, type = "s", lty = 2, col = 2)
legend("topright", c("Age 25: Estimate", "Age 25: 95% CI limits",
"Age 40: Estimate", "Age 40: 95% CI limits"), lty = 1:2,
col = c(1, 1, 2, 2))
Control Values for the Bootstrap or Cross-validation
Description
This function returns a list of values for the control parameters of the functions of the package that use the bootstrap or cross-validation.
Usage
controlpars(B = 999L, hbound = c(0.1, 3), hl = 100L, hsave =
FALSE, nnfrac = 0.25, fpilot = NULL, qt = 0.75, hsmooth = 1L, ...)
Arguments
B |
An integer giving the number of bootstrap resamples. |
hbound |
A numeric vector of length 2 specifying the minimum (default, 0.1) and maximum (default, 3), respectively, of the initial grid of bandwidths as a multiple of the standardized interquartile range of the covariate values. |
hl |
A numeric value giving the length of the initial grid of bandwidths. The default is 100. |
hsave |
A logical value specifying if the grids of bandwidths
must be saved as a component of the list returned by the
|
nnfrac |
A numeric value giving the fraction of the sample size that determines the order of the nearest neighbor used when choosing the pilot bandwidth. The default is 0.25. |
fpilot |
A function name or |
qt |
In bandwidth selection for the latency estimator (see
|
hsmooth |
An integer. Its value controls whether the bandwidths
selected by the |
... |
Arguments of |
Details
The output of controlpars is a list of control
parameters required by the package functions which use the bootstrap
or cross-validation. This is mainly the case of the berancv
function, which computes a cross-validation bandwidth for Beran's
estimator of survival, and of the latencyhboot and
probcurehboot functions, which compute the bootstrap bandwidth
selectors of the estimators of the latency and the probability of
cure, respectively. Since these functions are indirectly called by,
respectively, the beran, latency and probcure
functions when their h argument is missing, the output of
controlpars is also the expected (and default) way of passing
to them the parameters for bandwidth selection.
Additionally, controlpars is used by beran,
latency and probcure to set the number of bootstrap
resamples and the value of nnfrac (see above) when confidence
intervals are computed. The testcov function also uses it for
setting the number of bootstrap resamples.
Value
A list whose components are the arguments of the function, their defaults being replaced with the values the function was called with.
Author(s)
Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut]
See Also
beran,berancv,
hpilot, latency,
latencyhboot, probcure,
probcurehboot,testcov
Compute the Pilot Bandwidth for the Nonparametric Estimators of Cure Probability and Latency
Description
This function computes local pilot bandwidths for the nonparametric estimators of the probability of cure and the latency function.
Usage
hpilot(x, x0, nnfrac = 0.25)
Arguments
x |
A numeric vector of observed covariate values. |
x0 |
A numeric vector specifying a grid of covariate values. |
nnfrac |
A numeric value giving the fraction of the sample size
that determines the order of the nearest neighbor. This is taken as
|
Details
The function computes a data-driven local pilot bandwidth, required for the bootstrap bandwidth selector of the nonparametric estimators of the cure rate and latency functions. Simulations in López-Cheda et al. (2017) show that the choice of pilot bandwidth has small effect on the bootstrap bandwidth. This pilot bandwidth only depends on the sample size and the distribution of the covariate x (see López-Cheda, 2018):
g(x0) = 0.5(d_k^{+}(x0) + d_k^{-}(x0))(100/n)^{1/9}
where d_k^{+}(x0) and d_k^{-}(x0) are the distances from
x0 to the k-th nearest neighbor on the right and the left,
respectively, and k is a suitable integer depending on the
sample size n. If there are not at least k neighbors on
the right or on the left, we use d_k^{+}(x0) = d_k^{-}(x0). The
default value of k is n/4. The order
n^{-1/9} satisfies the conditions in Theorem 1 of Li
and Datta (2001) and coincides with the order obtained by Cao and
González-Manteiga (1993) for the uncensored case.
Value
A numeric vector of local pilot bandwidths corresponding to each
one of the values of the grid of covariate values given by x0.
Author(s)
Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut]
References
Cao R., González-Manteiga W. (1993). Bootstrap methods in regression smoothing. Journal of Nonparametric Statistics, 2: 379-388. https://doi.org/10.1080/10485259308832566.
Li, G., Datta, S. (2001). A bootstrap approach to nonparametric regression for right censored data. Annals of the Institute of Statistical Mathematics, 53(4): 708-729. https://doi.org/10.1023/A:1014644700806.
López-Cheda A. (2018). Nonparametric Inference in Mixture Cure Models. PhD dissertation, Universidade da Coruña. Spain.
López-Cheda, A., Cao, R., Jácome, M. A., Van Keilegom, I. (2017). Nonparametric incidence estimation and bootstrap bandwidth selection in mixture cure models. Computational Statistics & Data Analysis, 105: 144–165. https://doi.org/10.1016/j.csda.2016.08.002.
López-Cheda, A., Jácome, M. A., Cao, R. (2017). Nonparametric latency estimation for mixture cure models. TEST, 26: 353–376. https://doi.org/10.1007/s11749-016-0515-1.
See Also
controlpars, latencyhboot,
probcurehboot
Examples
## Some artificial data
set.seed(123)
n <- 50
x <- runif(n, -2, 2) ## Covariate values
y <- rweibull(n, shape = .5*(x + 4)) ## True lifetimes
c <- rexp(n) ## Censoring values
p <- exp(2*x)/(1 + exp(2*x)) ## Probability of being susceptible
u <- runif(n)
t <- ifelse(u < p, pmin(y, c), c) ## Observed times
d <- ifelse(u < p, ifelse(y < c, 1, 0), 0) ## Uncensoring indicator
data <- data.frame(x = x, t = t, d = d)
## Computing pilot bandwidths for covariate values -1, -0.8, ..., 1
## by taking the 5-th nearest neighbor
hpilot(data$x, x0 = seq(-1, 1, by = .2), nnfrac = .05)
Compute Nonparametric Estimator of the Conditional Latency
Description
This function computes the nonparametric estimator of the conditional latency function proposed by López-Cheda et al. (2017).
Usage
latency(x, t, d, dataset, x0, h, local = TRUE, testimate = NULL,
conflevel = 0L, bootpars = if (conflevel == 0 && !missing(h)) NULL else
controlpars())
Arguments
x |
If |
t |
If |
d |
If |
dataset |
An optional data frame in which the variables named in
|
x0 |
A numeric vector of covariate values where the latency estimates will be computed. |
h |
A numeric vector of bandwidths. If it is missing the default
is to use the local bootstrap bandwidth computed by the
|
local |
A logical value, |
testimate |
A numeric vector specifying the times at which the
latency is estimated. By default it is |
conflevel |
A value controlling whether bootstrap confidence intervals (CI) of the latency are to be computed. With the default value, 0L, the CIs are not computed. If a numeric value between 0 and 1 is passed, it specifies the confidence level of the CIs. |
bootpars |
A list of parameters controlling the bootstrap when
computing the CIs of the latency: |
Details
The function computes the nonparametric estimator of the
conditional latency S_0(t | X = x_0) = P(Y>t | Y<\infty,
X=x_0)
proposed by López-Cheda et al. (2017). It is only available for a
continuous covariate X.
Value
An object of S3 class 'npcure'. Formally, a list of components:
type |
The constant string "latency". |
local |
The value of the |
h |
The value of the |
x0 |
The value of the |
testim |
The numeric vector of time values where the latency function is estimated. |
S |
A list whose components are the estimates of the latency
function for each one of the covariate values, i.e., those specified
by the |
conf |
A list of components |
Author(s)
Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut]
References
López-Cheda, A., Jácome, M. A., Cao, R. (2017). Nonparametric latency estimation for mixture cure models. Test, 26: 353–376. https://doi.org/10.1007/s11749-016-0515-1.
See Also
Examples
## Some artificial data
set.seed(123)
n <- 50
x <- runif(n, -2, 2) ## Covariate values
y <- rweibull(n, shape = .5*(x + 4)) ## True lifetimes
c <- rexp(n) ## Censoring values
p <- exp(2*x)/(1 + exp(2*x)) ## Probability of being susceptible
u <- runif(n)
t <- ifelse(u < p, pmin(y, c), c) ## Observed times
d <- ifelse(u < p, ifelse(y < c, 1, 0), 0) ## Uncensoring indicator
data <- data.frame(x = x, t = t, d = d)
## Latency estimates for covariate value 0.5...
x0 <- .5
## ... (a) with global bandwidths 0.5, 1, 2.
## By default, estimates are computed at the time values of 't'
S1 <- latency(x, t, d, data, x0 = x0, h = c(.5, 1, 2), local = FALSE)
plot(S1$testim, S1$S$h0.5$x0.5, type = "s", xlab = "Time", ylab =
"Latency", ylim = c(0, 1))
lines(S1$testim, S1$S$h1$x0.5, type = "s", lty = 2)
lines(S1$testim, S1$S$h2$x0.5, type = "s", lty = 3)
## The true latency curve is plotted as reference
lines(S1$testim, pweibull(S1$testim, shape = .5*(x0 + 4), lower.tail =
FALSE), col = 2)
legend("topright", c(paste("Estimate, ", c("h = 0.5", "h = 1",
"h = 2")), "True"), lty = c(1:3, 1), col = c(rep(1, 3), 2))
## As before, but with estimates computed at times 0.1, 0.2,..., 1
S2 <- latency(x, t, d, data, x0 = x0, h = c(.5, 1, 2), local = FALSE,
testimate = .1*(1:10))
## ... (b) with local bandwidth 2.
S3 <- latency(x, t, d, data, x0 = x0, h = 2, local = TRUE)
#### Note that with only one covariate value the results with
#### 'local = FALSE' and 'local = TRUE' coincide, but the output formats
#### differ slightly. Compare with
S3 <- latency(x, t, d, data, x0 = x0, h = 2, local = FALSE)
## ... (c) with local bootstrap bandwidth
b <- latencyhboot(x, t, d, data, x0 = x0)
S4 <- latency(x, t, d, data, x0 = x0, h = b$h)
## ... (d) when the bandwidth is not specified, the bootstrap bandwidth
#### selector given by the 'latencyhboot' function is used by default.
#### The computation of 95% confidence intervals based on 1999 bootstrap
#### resamples is also illustrated
S5 <- latency(x, t, d, data, x0 = x0, conflevel = .95, bootpars =
controlpars(B = 1999))
plot(S5$testim, S5$S$x0, type = "s", xlab = "Time", ylab = "Latency",
ylim = c(0, 1))
lines(S5$testim, S5$conf$x0$lower, type = "s", lty = 2)
lines(S5$testim, S5$conf$x0$upper, type = "s", lty = 2)
lines(S5$testim, pweibull(S5$testim, shape = .5*(x0 + 4), lower.tail =
FALSE), col = 2)
legend("topright", c("Estimate", "95% CI limits", "True"), lty = c(1,
2, 1), col = c(1, 1, 2))
## Example with the dataset 'bmt' of the 'KMsurv' package
## to study the survival of the uncured patients aged 25 and 40
data("bmt", package = "KMsurv")
x0 <- c(25, 40)
S <- latency(z1, t2, d3, bmt, x0 = x0, conflevel = .95)
## Plot of predicted latency curves and confidence intervals
plot(S$testim, S$S$x25, type = "s", xlab = "Time (days)",
ylab = "Latency", ylim = c(0,1))
lines(S$testim, S$conf$x25$lower, type = "s", lty = 2)
lines(S$testim, S$conf$x25$upper, type = "s", lty = 2)
lines(S$testim, S$S$x40, type = "s", lty = 1, col = 2)
lines(S$testim, S$conf$x40$lower, type = "s", lty = 2, col = 2)
lines(S$testim, S$conf$x40$upper, type = "s", lty = 2, col = 2)
legend("topright", c("Age 25: Estimate", "Age 25: 95% CI limits",
"Age 40: Estimate","Age 40: 95% CI limits"), lty = 1:2,
col = c(1, 1, 2, 2))
Compute the Bootstrap Bandwidth for the Nonparametric Estimator of the Latency
Description
This function computes the bootstrap bandwidth for the nonparametric estimator of the conditional latency function.
Usage
latencyhboot(x, t, d, dataset, x0, bootpars = controlpars())
Arguments
x |
If |
t |
If |
d |
If |
dataset |
An optional data frame in which the variables named in
|
x0 |
A numeric vector of covariate values where the local bootstrap bandwidth will be computed. |
bootpars |
A list of parameters controlling the process of
bandwidth selection. The default is the value returned by the
|
Details
The function computes the bootstrap bandwidth selector for the
nonparametric estimator of the conditional latency function at the
covariate values given by x0. The bootstrap bandwidth is the
minimizer of a bootstrap version of the Mean Integrated Squared Error
(MISE) of the latency estimator, which is approximated by Monte Carlo
by simulating a large number of bootstrap resamples, B. For
each value of x0, the bootstrap MISE is the bootstrap
expectation of the integrated difference between the value of the
latency estimator computed with the bootstrap sample in a grid of
bandwidths and its value computed with the original sample and a pilot
bandwidth. The bootstrap resamples are generated by using the simple
weighted bootstrap resampling method, fixing the covariate. This
method is equivalent to the simple weighted bootstrap of Li and Datta
(2001). All the parameters typically involved in the bootstrap
bandwidth selection process (number of bootstrap resamples, grid of
bandwidths, pilot bandwidth, and right boundary of the integration
interval for computing the MISE) are typically set through the
controlpars function, whose output is passed to the
bootpars argument. Also, the bootstrap bandwidths can be
smoothed, and, if so, the smoothed bandwidths are returned as a
separate component of the output. See the help of controlpars
for details.
Value
An object of S3 class 'npcure'. Formally, a list of components:
type |
The constant character string c("Bootstrap bandwidth", "latency"). |
x0 |
Grid of covariate values. |
h |
Selected local bootstrap bandwidths. |
hsmooth |
Smoothed selected local bootstrap bandwidths (optional) |
hgrid |
Grid of bandwidths used (optional). |
Author(s)
Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut]
References
Li, G., Datta, S. (2001). A bootstrap approach to nonparametric regression for right censored data. Annals of the Institute of Statistical Mathematics, 53: 708–729. https://doi.org/10.1023/A:1014644700806.
López-Cheda, A., Jácome, M. A., Cao, R. (2017). Nonparametric latency estimation for mixture cure models. TEST, 26: 353–376. https://doi.org/10.1007/s11749-016-0515-1.
See Also
Examples
## Some artificial data
set.seed(123)
n <- 50
x <- runif(n, -2, 2) ## Covariate values
y <- rweibull(n, shape = .5*(x + 4)) ## True lifetimes
c <- rexp(n) ## Censoring values
p <- exp(2*x)/(1 + exp(2*x)) ## Probability of being susceptible
u <- runif(n)
t <- ifelse(u < p, pmin(y, c), c) ## Observed times
d <- ifelse(u < p, ifelse(y < c, 1, 0), 0) ## Uncensoring indicator
data <- data.frame(x = x, t = t, d = d)
## A vector of covariate values
vecx0 <- seq(-1.5, 1.5, by = .1)
## Computation of bootstrap local bandwidths at the values of 'vecx0'...
#### ... with the default control parameters
hb1 <- latencyhboot(x, t, d, data, x0 = vecx0)
#### ... changing the default 'bootpars' with 'controlpars()':
#### (a) 'B = 1999' (1999 bootstrap resamples are generated),
#### (b) 'hbound = c(.2, 4)' and 'hl = 50' (a grid of 50 bandwidths
#### between 0.2 and 4 times the standardized interquartile range of
#### the covariate values is built), and
#### (c) 'hsave = TRUE' (the grid bandwidths are saved), and
hb2 <- latencyhboot(x, t, d, data, x0 = vecx0, bootpars =
controlpars(B = 1999, hbound = c(.2, 4), hl = 50, hsave = TRUE))
## Estimates of the conditional latency at the covariate value x0 = 0
## with the selected bootstrap bandwidths
S1 <- latency(x, t, d, data, x0 = 0, h = hb1$h[hb1$x0 == 0])
S2 <- latency(x, t, d, data, x0 = 0, h = hb2$h[hb2$x0 == 0])
## A plot comparing the estimates with bootstrap bandwidths obtained
## with default and non-default 'bootpars'
plot(S1$testim, S1$S$x0, type = "s", xlab = "Time", ylab = "Latency",
ylim = c(0, 1))
lines(S2$testim, S2$S$x0, type = "s", lty = 2)
lines(S1$testim, pweibull(S1$testim, shape = .5*(0 + 4), lower.tail =
FALSE), col = 2)
legend("topright", c("Estimate with 'hb1'", "Estimate with 'hb2'",
"True"), lty = c(1, 2, 1), col = c(1, 1, 2))
## Example with the dataset 'bmt' of the 'KMsurv' package
## to study the survival of the uncured patients aged 25 and 40
data("bmt", package = "KMsurv")
x0 <- c(25, 40)
hb <- latencyhboot(z1, t2, d3, bmt, x0 = x0, bootpars = controlpars(B =
1999, hbound = c(.2, 4), hl = 150, hsave = TRUE))
S0 <- latency(z1, t2, d3, bmt, x0 = x0, hb$h, conflevel = .95)
## Plot of predicted latency curves and confidence bands
plot(S0$testim, S0$S$x25, type = "s", xlab = "Time (days)",
ylab = "Survival", ylim = c(0,1))
lines(S0$testim, S0$conf$x25$lower, type = "s", lty = 2)
lines(S0$testim, S0$conf$x25$upper, type = "s", lty = 2)
lines(S0$testim, S0$S$x40, type = "s", lty = 1, col = 2)
lines(S0$testim, S0$conf$x40$lower, type = "s", lty = 2, col = 2)
lines(S0$testim, S0$conf$x40$upper, type = "s", lty = 2, col = 2)
legend("topright", c("Age 25: Estimate", "Age 25: 95% CI limits",
"Age 40: Estimate", "Age 40: 95% CI limits"), lty = 1:2,
col = c(1, 1, 2, 2))
Print Method for Objects of Class 'npcure'
Description
This function implements a print method for 'npcure' objects.
Usage
## S3 method for class 'npcure'
print(x, how, head = FALSE, ...)
Arguments
x |
An object of class 'npcure'. |
how |
A character string with values "wide" or "long". If missing, the function itself chooses a convenient default. |
head |
A logical value that controls whether the function's
output must be abbreviated ( |
... |
Further optional arguments. Excepting for |
Value
A formatted output.
Author(s)
Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut]
See Also
Examples
## Some artificial data
set.seed(123)
n <- 50
x <- runif(n, -2, 2) ## Covariate values
y <- rweibull(n, shape = .5*(x + 4)) ## True lifetimes
c <- rexp(n) ## Censoring values
p <- exp(2*x)/(1 + exp(2*x)) ## Probability of being susceptible
u <- runif(n)
t <- ifelse(u < p, pmin(y, c), c) ## Observed times
d <- ifelse(u < p, ifelse(y < c, 1, 0), 0) ## Uncensoring indicator
data <- data.frame(x = x, t = t, d = d)
## Calling 'print()' with an object of class 'npcure' created by
## 'latency()'
S1 <- latency(x, t, d, data, x0 = c(0, .5), h = c(1, 1.5))
## In this case (latency estimation with local bandwidths and without
## confidence bands), the 'wide' format is used by default
S1
print(S1, how = "wide")
print(S1, how = "long")
## How to control the number of significant digits of the output, and
## how to abbreviate the output
print(S1, digits = 5, head = TRUE, n = 4)
## Calling 'print()' with a 'npcure' object created by 'probcure()'
q1 <- probcure(x, t, d, data, x0 = c(0, .5), h = c(.5, 1, 1.5), local =
FALSE, conflevel = .95)
## Only the 'long' format is available when confidence bands are
## computed
q1
print(q1, how = "long")
print(q1, how = "wide")
Compute Nonparametric Estimator of the Conditional Probability of Cure
Description
This function computes the nonparametric estimator of the conditional probability of cure proposed by Xu and Peng (2014).
Usage
probcure(x, t, d, dataset, x0, h, local = TRUE, conflevel = 0L,
bootpars = if (conflevel == 0 && !missing(h)) NULL else controlpars())
Arguments
x |
If |
t |
If |
d |
If |
dataset |
An optional data frame in which the variables named in
|
x0 |
A numeric vector of covariate values where the estimates of cure probability will be computed. |
h |
A numeric vector of bandwidths. If it is missing the default
is to use the local bootstrap bandwidth computed by the
|
local |
A logical value, |
conflevel |
A value controlling whether bootstrap confidence intervals (CI) of the cure probability are to be computed. With the default value, 0L, the CIs are not computed. If a numeric value between 0 and 1 is passed, it specifies the confidence level of the CIs. |
bootpars |
A list of parameters controlling the bootstrap when
computing either the CIs of the cure probability or the bootstrap
bandwidth (if |
Details
The function computes the nonparametric estimator of the
conditional cure probability q(X = x_0) \equiv 1 - p(X =
x_0)= P(Y=\infty | X=x_0) proposed by Xu and Peng (2014), and also studied
by López-Cheda et al (2017). It is only available for a continuous
covariate X.
Value
An object of S3 class 'npcure'. Formally, a list of components:
estimate |
The constant string "cure". |
local |
The value of the |
h |
The value of the |
x0 |
The value of the |
q |
A list with the estimates of the probability of cure. |
conf |
A list of components |
Author(s)
Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut]
References
López-Cheda, A., Cao, R., Jácome, M. A., Van Keilegom, I. (2017). Nonparametric incidence estimation and bootstrap bandwidth selection in mixture cure models. Computational Statistics & Data Analysis, 105: 144–165. https://doi.org/10.1016/j.csda.2016.08.002.
Xu, J., Peng, Y. (2014). Nonparametric cure rate estimation with covariates. The Canadian Journal of Statistics 42: 1-17. https://doi.org/10.1002/cjs.11197.
See Also
Examples
## Some artificial data
set.seed(123)
n <- 50
x <- runif(n, -2, 2) ## Covariate values
y <- rweibull(n, shape = 0.5 * (x + 4)) ## True lifetimes
c <- rexp(n) ## Censoring values
p <- exp(2*x)/(1 + exp(2*x)) ## Probability of being susceptible
u <- runif(n)
t <- ifelse(u < p, pmin(y, c), c) ## Observed times
d <- ifelse(u < p, ifelse(y < c, 1, 0), 0) ## Uncensoring indicator
data <- data.frame(x = x, t = t, d = d)
## Covariate values where cure probability is estimated
x0 <- seq(-1.5, 1.5, by = 0.1)
## Nonparametric estimates of cure probability at 'x0'...
## ... (a) with global bandwidths 1, 1.5, 2
q1 <- probcure(x, t, d, data, x0 = x0, h = c(1, 1.5, 2), local = FALSE)
#### Plot predicted cure probabilities at 'x0' for each bandwidth in 'h'
#### (the true cure probability is displayed for reference)
plot(q1$x0, q1$q$h1, type = "l", xlab = "Covariate", ylab =
"Cure probability", ylim = c(0, 1))
lines(q1$x0, q1$q$h1.5, lty = 2)
lines(q1$x0, q1$q$h2, lty = 3)
lines(q1$x0, 1 - exp(2*q1$x0)/(1 + exp(2*q1$x0)), col = 2)
legend("topright", c(paste("Estimate, ", c("h = 1", "h = 1.5",
"h = 2")), "True"), lty = c(1, 2, 3, 1), col = c(1, 1, 1, 2))
## ... (b) with local bandwidths (default)
#### (the vectors passed to 'x0' and 'h' must have the same length)
q2 <- probcure(x, t, d, data, x0 = x0, h = seq(1, 2.5, along = x0))
## ... (c) with local bootstrap bandwidths (based on 1999 booostrap
#### resamples). Besides, 95% confidence intervals are computed and
#### smoothed (with a 15-th order moving average)
set.seed(1) ## Not needed, just for reproducibility.
hb <- probcurehboot(x, t, d, data, x0 = x0, bootpars = controlpars(B =
1999, hsmooth = 15))
q3 <- probcure(x, t, d, data, x0 = x0, h = hb$hsmooth, conflevel = .95,
bootpars = controlpars(B = 1999))
## ... (d) If the bandwidth is not specified, the local bootstrap
#### bandwidth is used (same results as in (c))
set.seed(1) ## Not needed, just for reproducibility.
q4 <- probcure(x, t, d, data, x0 = x0, conflevel = .95, bootpars =
controlpars(B = 1999, hsmooth = 15))
#### Plot of the estimated cure probabilities evaluated at 'x0'
#### (true cure rate displayed as reference)
plot (q4$x0, q4$q, type = "l", ylim = c(0, 1), xlab = "Covariate X",
ylab = "Cure probability")
lines(q4$x0, q4$conf$lower, lty = 2)
lines(q4$x0, q4$conf$upper, lty = 2)
lines(q4$x0, 1-exp(2 * q4$x0)/(1 + exp(2 * q4$x0)), col = 2)
legend("topright", c("Estimate", "95% CI limits", "True"),
lty = c(1, 2, 1), col = c(1, 1, 2))
## Example with the dataset 'bmt' in the 'KMsurv' package
## to study the probability of cure as a function of the age (z1).
data("bmt", package = "KMsurv")
x0 <- seq(quantile(bmt$z1, .05), quantile(bmt$z1, .95), length.out = 100)
q.age <- probcure(z1, t2, d3, bmt, x0 = x0, conflevel = .95, bootpars =
controlpars(B = 1999, hsmooth = 10))
## Plot of estimated cure probability and confidence intervals
par(mar = c(5, 4, 4, 5) + .1)
plot(q.age$x0, q.age$q, type = "l", ylim = c(0, 1), xlab =
"Patient age (years)", ylab = "Cure probability")
lines(q.age$x0, q.age$conf$lower, lty = 2)
lines(q.age$x0, q.age$conf$upper, lty = 2)
## The estimated density of age (z1) is added for reference
par(new = TRUE)
d.age <- density(bmt$z1)
plot(d.age, xaxt = "n", yaxt = "n", xlab = "", ylab = "", col = 2,
main = "", zero.line = FALSE)
mtext("Density", side = 4, col = 2, line = 3)
axis(4, ylim = c(0, max(d.age$y)), col = 2, col.axis = 2)
legend("topright", c("Estimate", "95% CI limits", "Covariate density"),
lty = c(1, 2, 1), col = c(1, 1, 2))
Compute the Bootstrap Bandwidth for the Nonparametric Estimator of the Cure Probability
Description
This function computes the bootstrap bandwidth for the nonparametric estimator of the conditional probability of cure.
Usage
probcurehboot(x, t, d, dataset, x0, bootpars = controlpars())
Arguments
x |
If |
t |
If |
d |
If |
dataset |
An optional data frame in which the variables named in
|
x0 |
A numeric vector of covariate values where the local bootstrap bandwidth will be computed. |
bootpars |
A list of parameters controlling the process of
bandwidth selection. The default is the value returned by the
|
Details
The function computes the bootstrap bandwidth selector for the
nonparametric estimator of the cure probability at the covariate
values given by x0. The bootstrap bandwidth is the minimizer of
a bootstrap version of the Mean Squared Error (MSE) of the cure rate
estimator, which is approximated by Monte Carlo by simulating a large
number, B, of bootstrap resamples. The bootstrap MSE is the
bootstrap expectation of the difference between the value of the cure
rate estimator computed with the bootstrap sample in a grid of
bandwidths and its value computed with the original sample and a pilot
bandwidth. The bootstrap resamples are generated by using the simple
weighted bootstrap resampling method, fixing the covariate. This
method is equivalent to the simple weighted bootstrap of Li and Datta
(2001). All the parameters involved in the bootstrap bandwidth
selection process (number of bootstrap resamples, grid of bandwidths,
and pilot bandwidth) are typically set through the controlpars
function, whose output is passed to the bootpars
argument. See the help of controlpars for details.
Given the local nature of bootstrap bandwidth selection, estimates
obtained from sets of bootstrap bandwidths may sometimes look
wiggly. To counter this behavior, the selected vector of bootstrap
bandwidths can be smoothed by computing a moving average (its order
being set by controlpars). Then, the smoothed bandwidths are
contained in the hsmooth component of the returned value.
Value
An object of S3 class 'npcure'. Formally, a list of components:
type |
The constant character string c("Bootstrap bandwidth", "cure"). |
x0 |
Grid of covariate values. |
h |
Selected local bootstrap bandwidths. |
hsmooth |
Smoothed selected local bootstrap bandwidths (optional) |
hgrid |
Grid of bandwidths used (optional). |
Author(s)
Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut]
References
Li, G., Datta, S. (2001). A bootstrap approach to nonparametric regression for right censored data. Annals of the Institute of Statistical Mathematics, 53: 708-729. https://doi.org/10.1023/A:1014644700806.
López-Cheda, A., Cao, R., Jácome, M. A., Van Keilegom, I. (2017). Nonparametric incidence estimation and bootstrap bandwidth selection in mixture cure models. Computational Statistics & Data Analysis, 105: 144–165. https://doi.org/10.1016/j.csda.2016.08.002.
See Also
Examples
## Some artificial data
set.seed(123)
n <- 50
x <- runif(n, -2, 2) ## Covariate values
y <- rweibull(n, shape = .5*(x + 4)) ## True lifetimes
c <- rexp(n) ## Censoring values
p <- exp(2*x)/(1 + exp(2*x)) ## Probability of being susceptible
u <- runif(n)
t <- ifelse(u < p, pmin(y, c), c) ## Observed times
d <- ifelse(u < p, ifelse(y < c, 1, 0), 0) ## Uncensoring indicator
data <- data.frame(x = x, t = t, d = d)
## A vector of covariate values
vecx0 <- seq(-1.5, 1.5, by = .1)
## Computation of bootstrap local bandwidth at the values of 'vecx0'...
#### ... with the default control parameters
set.seed(1) ## Not needed, just for reproducibility.
hb1 <- probcurehboot(x, t, d, data, x0 = vecx0)
#### ... changing the default 'bootpars' through 'controlpars()', with
#### arguments:
#### (a) 'B = 1999' (1999 bootstrap resamples are generated),
#### (b) 'hbound = c(.2, 4)' and 'hl = 50' (a grid of 50 bandwidths
#### between 0.2 and 4 times the standardized interquartilic range of
#### the covariate values is built),
#### (c) 'hsave = TRUE' (the grid bandwidths are saved), and
#### (d) 'hsmooth = 7' (the bootstrap bandwidths are smoothed by a
#### moving average of 7-th order)
set.seed(1) ## Not needed, just for reproducibility.
hb2 <- probcurehboot(x, t, d, data, x0 = vecx0, bootpars =
controlpars(B = 1999, hbound = c(.2, 4), hl = 50, hsave = TRUE, hsmooth
= 7))
## Estimates of the conditional probability of cure at the covariate
## values of 'vecx0' with the selected bootstrap bandwidths
q1 <- probcure(x, t, d, data, x0 = vecx0, h = hb1$h)
q2 <- probcure(x, t, d, data, x0 = vecx0, h = hb2$h)
q2sm <- probcure(x, t, d, data, x0 = vecx0, h = hb2$hsmooth)
## A plot comparing the estimates obtained with the bootstrap bandwidths
plot(q1$x0, q1$q, type = "l", xlab = "Covariate", ylab =
"Cure probability", ylim = c(0,1))
lines(q2$x0, q2$q, type = "l", lty = 2)
lines(q2sm$x0, q2sm$q, type = "l", lty = 3)
lines(q1$x0, 1 - exp(2*q1$x0)/(1 + exp(2*q1$x0)), col = 2)
legend("topright", c("Estimate with 'hb1'", "Estimate with 'hb2'",
"Estimate with 'hb2' smoothed", "True"), lty = c(1, 2, 3, 1), col = c(1,
1, 1, 2))
## Example with the dataset 'bmt' of the 'KMsurv' package
## to study the probability of cure as a function of the age (z1).
data("bmt", package = "KMsurv")
x0 <- seq(quantile(bmt$z1, .05), quantile(bmt$z1, .95), length.out =
100)
## This might take a while
hb <- probcurehboot(z1, t2, d3, bmt, x0 = x0, bootpars =
controlpars(B = 1999, hbound = c(.2, 2), hl = 50, hsave = TRUE, hsmooth
= 10))
q.age <- probcure(z1, t2, d3, bmt, x0 = x0, h = hb$h)
q.age.smooth <- probcure(z1, t2, d3, bmt, x0 = x0, h = hb$hsmooth)
## Plot of estimated cure probability
plot(q.age$x0, q.age$q, type = "l", ylim = c(0, 1), xlab =
"Patient age (years)", ylab = "Cure probability")
lines(q.age.smooth$x0, q.age.smooth$q, col = 2)
legend("topright", c("Estimate with h bootstrap",
"Estimate with smoothed h bootstrap"), lty = 1, col = 1:2)
Summary Method for Objects of Class 'npcure'
Description
This function prints a summary of a 'npcure' object.
Usage
## S3 method for class 'npcure'
summary(object, ...)
Arguments
object |
An object of class 'npcure'. |
... |
Further optional arguments for the default method
(i.e., |
Value
A compact summary showing the components of the object.
Author(s)
Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut]
See Also
Examples
## Some artificial data
set.seed(123)
n <- 50
x <- runif(n, -2, 2) ## Covariate values
y <- rweibull(n, shape = .5*(x + 4)) ## True lifetimes
c <- rexp(n) ## Censoring values
p <- exp(2*x)/(1 + exp(2*x)) ## Probability of being susceptible
u <- runif(n)
t <- ifelse(u < p, pmin(y, c), c) ## Observed times
d <- ifelse(u < p, ifelse(y < c, 1, 0), 0) ## Uncensoring indicator
data <- data.frame(x = x, t = t, d = d)
## Calling 'summary()' with an object of class 'npcure' created by
## 'latency()'
S1 <- latency(x, t, d, data, x0 = c(0, .5), h = c(1, 1.5))
summary(S1)
## If needed, the number of significant digits of the output can be set
summary(S1, digits = 5)
## Calling 'summary()' with an object created by 'probcure()'
q1 <- probcure(x, t, d, data, x0 = c(0, .5), h = c(.5, 1, 1.5), local =
FALSE, conflevel = .95)
summary(q1)
Covariate Significance Test of the Cure Probability
Description
This function carries out a significance test of covariate effect on the probability of cure.
Usage
testcov(x, t, d, dataset, bootpars = controlpars())
Arguments
x |
If |
t |
If |
d |
If |
dataset |
An optional data frame in which the variables named in
|
bootpars |
A list of parameters controlling the test. Currently,
the only accepted component is |
Details
The function computes a statistic, based on the method by
Delgado and González-Manteiga (2004), to test whether a covariate X
has an effect on the cure probability (H_1: cure probability =
q(x)) or not (H_0: cure probability = q). Since the cure rate
can be written as the regression function E(\nu | X = x) =
q(x), where \nu is the cure
indicator, the procedure is carried out as a significance test in
nonparametric regression. The challenge of the test is that the cure
indicator \nu is only partially observed due to censoring:
for the uncensored observations \nu = 0, but it is
unknown if a censored individual will be eventually cured or not
(\nu unknown). The approach consists in expressing the cure
rate as a regression function with another response, not observable
but estimable, say \eta. The estimated values of
\eta depend on suitable estimates of the conditional
distribution of the censoring variable and \tau(x), an
unknown time beyond which a subject could be considered as cured; see
López-Cheda (2018). For the computation of the values of
\eta, the censoring distribution is estimated
unconditionally using the Kaplan-Meier product-limit estimator. The
time \tau is estimated by the largest uncensored time.
The test statistic is a weighted mean of the difference between the
observations of \eta_i and the values of the conditional
mean of \eta under the null hypothesis:
T_{n(x)} = 1/n \sum (\eta_i - \bar{\eta}) I(x_i \leq x)
.
For a qualitative covariate there is no natural way to order the values
x_i. In principle, this makes impossible to compute the indicator
function in the test statistic. The problem is solved by considering all
the possible combinations of the covariate values, and by computing the
test statistic for each 'ordered' combination. T_{n(x)} is taken
as the largest value of these test statistics.
The distribution of the test statistic under the null hypothesis is approximated by bootstrap, using independent naive resampling.
Value
An object of S3 class 'npcure'. Formally, a list of components:
type |
The constant character string c("test", "Covariate"). |
x |
The name of the covariate. |
CM |
The result of the Cramer-von Mises test: a list with
components |
KS |
The result of the Kolmogorov-Smirnov test: a list with
components |
Author(s)
Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut]
References
Delgado M. A., González-Manteiga W. (2001). Significance testing in nonparametric regression based on the bootstrap. Annals of Statistics, 29: 1469-1507.
López-Cheda A. (2018). Nonparametric Inference in Mixture Cure Models. PhD dissertation, Universidade da Coruña. Spain.
See Also
Examples
## Some artificial data
set.seed(123)
n <- 50
x <- runif(n, -2, 2) ## Covariate values
y <- rweibull(n, shape = .5*(x + 4)) ## True lifetimes
c <- rexp(n) ## Censoring values
p <- exp(2*x)/(1 + exp(2*x)) ## Probability of being susceptible
u <- runif(n)
t <- ifelse(u < p, pmin(y, c), c) ## Observed times
d <- ifelse(u < p, ifelse(y < c, 1, 0), 0) ## Uncensoring indicator
data <- data.frame(x = x, t = t, d = d)
## Test of the significance of the covariate 'x'
testcov(x, t, d, data)
## Test carried out with 1999 bootstrap resamples (the default is 999)
testcov(x, t, d, data, bootpars = controlpars(B = 1999))
## How to apply the test repeatedly when there is more than one
## covariate...
## ... 'y' is another continuous covariate of the data frame 'data'
data$y <- runif(n, -1, 1)
namecovar <- c("x", "y")
## ... testcov is called from a 'for' loop
for (i in 1:length(namecovar)) {
result <- testcov(data[, namecovar[i]], data$t, data$d)
print(result)
}
## In the previous example, testcov() was called without using the
## argument 'dataset'. To use it, the 'for' must be avoided
testcov(x, t, d, data)
testcov(y, t, d, data)
## Non-numeric covariates can also be tested...
## ... 'z' is a nominal covariate of the data frame 'data'
data$z <- rep(factor(letters[1:5]), each = n/5)
testcov(z, t, d, data)
## Example with the dataset 'bmt' in the 'KMsurv' package
## to study the effect on the probability of cure of...
## ... (a) a continuous covariate (z1 = age of the patient)
data("bmt", package = "KMsurv")
set.seed(1) ## Not needed, just for reproducibility.
testcov(z1, t2, d3, bmt, bootpars = controlpars(B = 4999))
## ... (b) a qualitative covariate (z3 = patient gender)
set.seed(1) ## Not needed, just for reproducibility.
testcov(z3, t2, d3, bmt, bootpars = controlpars(B = 4999))
Test of Maller-Zhou
Description
This function carries out the nonparametric test of Maller and Zhou (1992).
Usage
testmz(t, d, dataset)
Arguments
t |
If |
d |
If |
dataset |
An optional data frame in which the variables named in
|
Details
The function implements Maller and Zhou's (1992) method to
test the null hypothesis H_{0} : \tau_{F_{0}} > \tau_{G} vs. H_{1} : \tau_{F_{0}} \leq \tau_{G}, where \tau_{F_{0}} and
\tau_{G} are the supports of, respectively, the
distribution function of the survival time of the uncured and the
distribution function of the censoring time.
Value
An object of S3 class 'npcure'. Formally, a list of components:
type |
The constant character string c("test", "Maller-Zhou"). |
pvalue |
The p-value of the test. |
aux |
A list of components: |
Author(s)
Ignacio López-de-Ullibarri [aut, cre], Ana López-Cheda [aut], Maria Amalia Jácome [aut]
References
Maller R. A., Zhou S. (1992). Estimating the proportion of immunes in a censored sample. Biometrika, 79: 731-739. https://doi.org/10.1093/biomet/79.4.731.
See Also
Examples
## Some artificial data
set.seed(123)
n <- 50
y <- qweibull(runif(n)*pweibull(2, shape = 2), shape = 2) ## True lifetimes
c <- qexp(runif(n)*pexp(2.5)) ## Censoring values
u <- runif(n)
## Probability of being susceptible is constantly equal to .5
t <- ifelse(u < .5, pmin(y, c), c) ## Observed times
d <- ifelse(u < .5, ifelse(y < c, 1, 0), 0) ## Uncensoring indicator
data <- data.frame(t = t, d = d)
## Maller-Zhou test
testmz(t, d, data)