Package {ksamplesLTRC}

Automatic test selector for survival data

Description

Performs tests for data subject to left truncation, right censoring or both.

Usage

ksample.test(
  time,
  entry = NULL,
  status = NULL,
  group,
  weights = NULL,
  tests = c("ks", "cvm", "logrank"),
  B = 500,
  plot.curves = FALSE,
  holes = c("holes", "add", "conditional"),
  p = 0,
  q = 0,
  seed = NULL,
  keep.boot = FALSE
)

Arguments

Value

A list with the following components:

• statistic: numerical value of the Kolmogorov–Smirnov and/or Cramér–von Mises statistics.

• p.value: approximated p-value of the tests.

• stat.boot: bootstrap statistics for the p-value approximation.

• logrank: a matrix with weights p and q, test statistics, and p-values.

• B: number of bootstrap replications.

• tests: tests performed, as specified in the input.

• curves: distribution function estimators of the k samples to be compared.

• groups: group indicators.

• holes: method for handling holes (if appropriate).

• stat.boot: bootstrap statistics (if keep.boot = TRUE).

Examples

if (requireNamespace("KMsurv", quietly = TRUE)) {
data("channing", package = "KMsurv")

aa <- ksample.test(time = channing$age,entry = channing$ageentry,
status = channing$death, group = channing$gender,
weights=rep(1/2,2), B = 200, holes='conditional',
tests=c('ks','cvm','logrank'),plot.curves=TRUE,
p=c(0,1,0.5,0),q=c(0,0,0.5,1),keep.boot = FALSE)

summary(aa)
}

Add estimators to an existing plot

Description

Adds the survival estimator to an existing plot.

Usage

## S3 method for class 'estimators'
lines(x, conf.int = FALSE, ...)

Arguments

Value

Invisibly returns x.

Add estimators to an existing plot

Description

Adds the survival estimator to an existing plot.

Usage

## S3 method for class 't.estimators'
lines(x, ...)

Arguments

Value

Invisibly returns x.

Cramér–von Mises-type statistic for the k-sample problem under left truncation

Description

Computes the Cramér–von Mises statistic adapted to left truncation.

Usage

lt.cvm(
  p.sample,
  weights = NULL,
  holes = c("holes", "add", "conditional"),
  plot.curves = FALSE
)

Arguments

Details

In Kiefer (1959), the k-sample version of the Cramér–von Mises test is defined as

D_{CvM} = \sum_{j=1}^k n_j \int \left( F_{jn_j}(t) - F_n(t)\right)^2 dF_n(t),

where F_{jn_j} is the cumulative distribution function estimator of calculated only with the j-th sample, and F_n is given by

F_n(t) =\sum_{j=1}^k p_j F_{jn_j}(t),

where p_1, \ldots, p_k correspond to the input weights. This function implements the previous test statistic accounting for left truncation.

Value

Numerical value of the test statistic computed from the k samples.

References

Kiefer, J. (1959). K-sample analogues of the Kolmogorov–Smirnov and Cramér–von Mises tests. The Annals of Mathematical Statistics, 30:420-447.

Examples

if (requireNamespace("mvna", quietly = TRUE)) {
data("abortion", package = "mvna")
pooled.sample <- abortion[abortion$cause==3, 2:4]
head(pooled.sample)
aa <- lt.cvm(p.sample=pooled.sample,weights=rep(1/2,2),plot.curves=TRUE)
aa
}

Survival function estimator under left truncation

Description

Computes the survival function estimator accounting for left truncation. It also returns pointwise confidence bands and includes the possibility of dealing with holes in the sample.

Usage

lt.estimator(
  o.sample,
  conf.int = FALSE,
  alpha = 0.05,
  conf.type = c("plain", "loglog"),
  holes = c("holes", "add", "conditional")
)

Arguments

Details

The estimator of the survival function when data are subject to left truncation is defined as (Lynden-Bell, 1971)

S_n(t) = \prod_{x_i \leq t} \left( 1-\frac{d_i}{r_i} \right),

where x_1, \ldots, x_m are the distinct observed event times, d_i = \# \{j : X_j = x_i\} and r_i = \#\{j : U_j \leq x_i \leq X_j\}. One should note that this definition of the risk sets is not equal to that in survival::survfit, where r_i = \#\{j : U_j < x_i \leq X_j\}, so these two implementations may not be equal when ties are present in the data.

The variance for the confidence bands follows a Greenwood's-like formula:

Var(S_n(t)) = S_n(t)^2 \sum_{x_i \leq t} \frac{d_i}{n_i(n_i-d_i)}.

A log-log transformation is also possible. The confidence band is based on the Gaussianity of the process \sqrt{n}(S_n(t) - S(t)).

The function also includes a way to deal with possible holes. A hole is an event time with associated unitary risk set, namely r_i=1 for some i \in \{1, \ldots, m-1\} . We exclude the largest event time, since it is always a hole. If a hole is present in a sample, the estimator degenerates and takes the value 1 from that time on. Different techniques are available in the literature for the treatment of holes. For instance, in Stute and Wang (2008), authors suggest to increase the risk set by 1 (holes='add'). Other references, like Klein and Moeschberger (2003), suggest conditioning on the largest hole different from the largest event time (holes='conditional'). The possibility of computing the degenerate statistic is also included (holes='holes').

Value

An object with the following content:

• fail.time: ordered distinct observed event times

• estimation: estimated survival function at each observed event time

• prob: jumps of the estimator

• If conf.int = TRUE, it also returns the confidence band lower and upper limits for both the distribution and survival function.

References

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

Lynden-Bell, D. (1971). A method of allowing for known observational selection in small samples applied to 3CR quasars. Monthly Notices of the Royal Astronomical Society, 155:95-118.

Stute, W. and Wang, J. L. (2008). The central limit theorem under random truncation. Bernoulli, 14:604–622.

Examples

if (requireNamespace("mvna", quietly = TRUE)) {
data("abortion", package = "mvna")
pooled.sample <- abortion[abortion$cause==3, 2:4]
plot( lt.estimator(pooled.sample[pooled.sample$group==0,],conf.int=TRUE,conf.type='loglog'),
xlab='Time (in weeks)',ylab='Estimated survival probability' )
lines( lt.estimator(pooled.sample[pooled.sample$group==1,],conf.int=TRUE,conf.type='loglog'),col=2 )
}

Kolmogorov–Smirnov-type statistic for the k-sample problem under left truncation

Description

Computes the Kolmogorov–Smirnov statistic adapted to left truncation.

Usage

lt.ks(
  p.sample,
  weights = NULL,
  plot.curves = FALSE,
  holes = c("holes", "add", "conditional")
)

Arguments

Details

In Kiefer (1959), the k-sample version of the Kolmogorov–Smirnov test is defined as

D_{KS} = \sup_t \sum_{j=1}^k n_j \left( F_{jn_j}(t) - F_n(t)\right)^2,

where F_{jn_j} is the cumulative distribution function estimator of calculated only with the j-th sample, and F_n is given by

F_n(t) =\sum_{j=1}^k p_j F_{jn_j}(t),

where p_1, \ldots, p_k correspond to the input weights. This function implements the previous test statistic accounting for left truncation. The two-sample version of this statistic was studied in Lago et al. (2025). Note that, when k=2, one has

D_{KS} = (n_1p_2^2 + n_2p_1^2) \left( \sup_t \mid F_{1n_1}(t) - F_{2n_2}(t) \mid \right)^2.

Value

A list with the following components:

• statistic: numerical value of the Kolmogorov–Smirnov statistic.

• time: time when the maximum is attained.

References

Kiefer, J. (1959). K-sample analogues of the Kolmogorov–Smirnov and Cramér–von Mises tests. The Annals of Mathematical Statistics, 30:420-447.

Lago, A., de Uña-Álvarez, J., & Pardo-Fernández, J. C. (2025). A Kolmogorov–Smirnov-type test for the two-sample problem with left-truncated data. Test, 34(1), 69-90.

Examples

if (requireNamespace("mvna", quietly = TRUE)) {
data("abortion", package = "mvna")
pooled.sample <- abortion[abortion$cause==3, 2:4]
head(pooled.sample)
aa <- lt.ks(p.sample=pooled.sample,weights=rep(1/2,2),plot.curves=TRUE)
aa
}

Weighted log-rank tests under left truncation

Description

Computes weighted log-rank-type tests (Fleming–Harrington family) for k groups with left truncation.

Usage

lt.logrank(
  p.sample,
  p = 0,
  q = 0,
  plot.curves = FALSE,
  holes = c("holes", "conditional")
)

Arguments

Details

The test statistic is based on the quantity

L_j = \sum_{i=1}^m \omega_i \left(d_{ji} - d_i \frac{r_{ji}}{r_i}\right),

for j \in \{1, \ldots, k\}. Denote by x_1, \ldots, x_m the distinct observed event times from the k samples. For a time x_i and a group j, d_{ji} denotes the total count of individuals that experience the event, and r_{ji} is the number of individuals at risk. In addition, d_i = \sum_{j=1}^k d_{ji} and r_i = \sum_{j=1}^k r_{ji}, which are the total number of events observed and the total number of individuals at risk at time x_i, respectively.

Finally, the weights \omega_1, \ldots, \omega_m are taken as

w_i = \hat{S}(x_i)^p (1-\hat{S}(x_i))^q,

with \hat{S}(x_i) the pooled survival just before the event time. The choice p=0=q results in the classical version of the log-rank test.

The test statistic is given by

(L_1, \ldots, L_{k-1}) \hat{\Sigma}^{-1}(L_1, \ldots, L_{k-1})^\top,

where \hat{\Sigma} is the matrix with the variance-covariance estimators. Each term in the matrix includes the correction factor (n_j-d_j)/(n_j-1) for tied event times. The test statistic follows an asymptotic \chi^2_{k-1} distribution under the null hypothesis of equal target distributions in the k populations.

Value

A matrix with columns statistic, detailing the choices of p and q, in addition to the corresponding p.value.

References

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

Examples

if (requireNamespace("mvna", quietly = TRUE)) {
data("abortion", package = "mvna")
pooled.sample <- abortion[abortion$cause==3, 2:4]
head(pooled.sample)
lt.logrank(p.sample=pooled.sample,p=c(0,1,0.5,5),q=c(0,0,0.5,1),plot.curves=TRUE)
}

Cramér–von Mises-type test for the k-sample problem under left truncation

Description

Computes the Cramér–von Mises statistic adapted to left truncation, and the corresponding p-value.

Usage

lt.pv.cvm(
  p.sample,
  weights = NULL,
  B = 500,
  plot.curves = FALSE,
  holes = c("holes", "add", "conditional"),
  seed = NULL,
  keep.boot = FALSE
)

Arguments

Details

In Kiefer (1959), the k-sample version of the Cramér–von Mises test is defined as

D_{CvM} = \sum_{j=1}^k n_j \int \left( F_{jn_j}(t) - F_n(t)\right)^2 dF_n(t),

where F_{jn_j} is the cumulative distribution function estimator of calculated only with the j-th sample, and F_n is given by

F_n(t) =\sum_{j=1}^k p_j F_{jn_j}(t),

where p_1, \ldots, p_k correspond to the input weights. This function implements the previous test statistic accounting for left truncation.

The p-value is approximated by means of the obvious bootstrap accounting for left truncation, as a particular case of the obvious bootstrap proposed for left-truncated and right-censored data in Bilker and Wang (1997). The exact algorithm for the two-sample problem can be consulted in Lago et al. (2025).

Value

A list with the following elements:

• statistic: numerical value of the Cramér–von Mises test statistic

• pvalue: p-value of the test

• curves: A list with the estimators of the k distribution functions to be compared.

• groups: Group indicator

• stat.boot: If keep.boot = TRUE, the function returns the statistic evaluated in the bootstrap resamples.

References

Bilker, W.B., and Wang, M.C. (1997). Bootstrapping left truncated and right censored data. Communications in Statistics - Simulation and Computation, 26:141-171.

Kiefer, J. (1959). K-sample analogues of the Kolmogorov–Smirnov and Cramér–von Mises tests. The Annals of Mathematical Statistics, 30:420-447.

Lago, A., de Uña-Álvarez, J., & Pardo-Fernández, J. C. (2025). A Kolmogorov–Smirnov-type test for the two-sample problem with left-truncated data. TEST, 34, 69-90.

Examples

if (requireNamespace("mvna", quietly = TRUE)) {
data("abortion", package = "mvna")
pooled.sample <- abortion[abortion$cause==3, 2:4]
head(pooled.sample)
aa <- lt.pv.cvm(p.sample=pooled.sample,weights=rep(1/2,2),plot.curves=TRUE, B = 200)
aa$pvalue
plot(aa)
}

Kolmogorov–Smirnov-type test for the k-sample problem under left truncation

Description

Computes the Kolmogorov–Smirnov test adapted to left truncation and the corresponding p-value.

Usage

lt.pv.ks(
  p.sample,
  weights = NULL,
  B = 500,
  plot.curves = FALSE,
  holes = c("holes", "add", "conditional"),
  seed = NULL,
  keep.boot = FALSE
)

Arguments

Details

In Kiefer (1959), the k-sample version of the Kolmogorov–Smirnov test is defined as

D_{KS} = \sup_t \sum_{j=1}^k n_j \left( F_{jn_j}(t) - F_n(t)\right)^2,

where F_{jn_j} is the cumulative distribution function estimator of calculated only with the j-th sample, and F_n is given by

F_n(t) =\sum_{j=1}^k p_j F_{jn_j}(t),

where p_1, \ldots, p_k correspond to the input weights. This function implements the previous test statistic accounting for left truncation. The two-sample version of this test was studied in Lago et al. (2025).

The p-value is approximated by means of the obvious bootstrap accounting for left truncation, studied in Bilker and Wang (1997). The exact algorithm for the two-sample problem can be consulted in Lago et al. (2025).

Value

A list with the following components:

• statistic: numerical value of the Kolmogorov–Smirnov statistic.

• pvalue: approximated p-value of the test.

• curves: A list with the estimators of the k distribution functions to be compared.

• groups: Group indicator

• stat.boot: statistics computed from the bootstrap resamples. Included in the output only if keep.boot = TRUE

References

Bilker, W.B., and Wang, M.C. (1997). Bootstrapping left truncated and right censored data. Communications in Statistics - Simulation and Computation, 26:141-171.

Lago, A., Pardo-Fernández, J.C., and de Uña-Álvarez, J. (2026). A Kolmogorov–Smirnov-type test for the two-sample problem with left-truncated data. TEST: 34:69-90.

Kiefer, J. (1959). K-sample analogues of the Kolmogorov–Smirnov and Cramér–von Mises tests. The Annals of Mathematical Statistics, 30:420-447.

Examples

if (requireNamespace("mvna", quietly = TRUE)) {
data("abortion", package = "mvna")
pooled.sample <- abortion[abortion$cause==3, 2:4]
head(pooled.sample)
aa <- lt.pv.ks(p.sample=pooled.sample,weights=rep(1/2,2),plot.curves=TRUE, B = 200)
aa$pvalue
plot(aa)
}

Left-truncated data simulation

Description

Simulates data under left truncation from prespecified variables for the truncation and target variables.

Usage

lt.sim(
  n,
  rtrunc,
  trunc.args = list(),
  rtarget,
  target.args = list(),
  seed = NULL
)

Arguments

Value

A data frame with n rows and two columns: the truncation times are in the first column, and the event times, in the second one.

Examples

aa <- lt.sim(1000,rnorm,trunc.args=list(4,1),rnorm,target.args = list(4,1))
head(aa)
plot(lt.estimator(aa),col=2,xlab='t',ylab='Estimated survival probiblity')
curve(1-pnorm(x,4,1),add=TRUE)

Tests for the k-sample problem under left truncation

Description

Computes k-sample tests adapted to left truncation, and the corresponding p-value.

Usage

lt.test(
  p.sample,
  weights = NULL,
  tests = c("ks", "cvm", "logrank"),
  B = 500,
  plot.curves = FALSE,
  holes = c("holes", "add", "conditional"),
  p = 0,
  q = 0,
  seed = NULL,
  keep.boot = FALSE
)

Arguments

Details

This function collects the three tests implemented in this package: Kolmogorov–Smirnov, Cramér–von Mises and weighted log-rank-type tests, (Klein and Moeschberger, 2003) adapted to left truncation. The p-values of the first two are approximated by bootstrap, and the p-value of the log-rank tests is computed by the \chi^2_{k-1} asymptotic distribution of the test statistic, where k is the number of groups.

Value

A list with the following components:

• statistic: numerical value of the Kolmogorov–Smirnov and/or Cramér–von Mises statistics.

• p.value: approximated p-value of the tests.

• stat.boot: bootstrap statistics for the p-value approximation.

• logrank: a matrix with weights p and q, test statistics, and p-values.

• B: number of bootstrap replications.

• tests: tests performed, as specified in the input.

• curves: distribution function estimators of the k samples to be compared.

• groups: group indicators.

• holes: method for handling holes.

The summary method returns a data frame with one row per test and the following columns:

Test

Name of the test. For the weighted log-rank-type tests, the weights p and q are detailed.

Statistic

Observed value of the test statistic.

p.value

Corresponding p-value.

References

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

Lago, A., de Uña-Álvarez, J., and Pardo-Fernández, J.C. (2025). A Kolmogorov–Smirnov-type test for the two-sample problem with left-truncated data. TEST: 34:69-90.

Examples

if (requireNamespace("mvna", quietly = TRUE)) {
data("abortion", package = "mvna")
head(abortion)
pooled.sample <- abortion[abortion$cause==3, 2:4]
aa <- lt.test(pooled.sample,weights=rep(1/2,2),B=200,
holes='conditional', tests=c('ks','cvm','logrank'),
p=c(0,1,0.5,0),q=c(0,0,0.5,1))
summary(aa)
plot(aa)
}

Estimator of the truncation survival function for data subject to left truncation

Description

It computes the estimator of the truncation survival function for data subject to left truncation

Usage

lt.truncation.estimator(o.sample, holes = c("holes", "add", "conditional"))

Arguments

Details

This estimator is based on the idea that if X is left-truncated by U, then -U is left-truncated by -X. In addition, note that

S_U(u) = 1-S_{_U}(-u),

where S_U is the survival function of U and S_{-U} is the survival function of -U. Thus, since lt.estimator() computes the estimator of the survival function, one can apply the Lynden-Bell estimator to the sample formed by reversing the roles of U and X by merely multiplying by -1.

The nontruncation probability is defined as

\gamma = \mathbb{P} (U \leq X) = \int G(z) dF(z),

where U \sim G and X \sim F. Then, an estimator of \gamma is given by (Woodroofe, 1985)

\gamma_n = \sum_{i=1}^m \varphi_i G_n(x_i),

where x_1, \ldots, x_m are the distinct observed event times, \varphi_1, \ldots, \varphi_m are the weights assigned by the Lynden-Bell estimator to the observed event times, and G_n(x_1), \ldots, G_n(x_m) are estimated distribution function of the observed variable evaluated at the distinct observed event times.

Value

A list with the following:

• trunc.time: observed truncation times.

• estimation: estimated survival probability.

• prob: jumps of the estimator at each observed truncation time.

• gamma: estimator of the nontruncation probability.

References

Lynden-Bell, D. (1971). A method of allowing for known observational selection in small samples applied to 3CR quasars. Monthly Notices of the Royal Astronomical Society, 155:95-118.

Woodroofe, M. (1985). Estimating a distribution function with truncated data. The Annals of Statistics, 13: 163-177.

Examples

aa <- lt.sim(500,rnorm,trunc.args=list(4,1),rnorm,target.args = list(4,1))
g.est <- lt.truncation.estimator(o.sample=aa)
plot(g.est$trunc.time,g.est$estimation,type='s',lwd=2,col=2)
curve(1-pnorm(x,4,1),add=TRUE,lwd=2,
xlab='t',ylab='Estimated distribution function')
# nontruncation probability estimator:
g.est$gamma

Cramér–von Mises-type statistic for the k-sample problem under left truncation and right censoring

Description

Computes the Cramér–von Mises statistic adapted to left truncation and right censoring.

Usage

ltrc.cvm(
  p.sample,
  weights = NULL,
  holes = c("holes", "add", "conditional"),
  plot.curves = FALSE
)

Arguments

Details

In Kiefer (1959), the k-sample version of the Cramér–von Mises test is defined as

D_{CvM} = \sum_{j=1}^k n_j \int \left( F_{jn_j}(t) - F_n(t)\right)^2 dF_n(t),

where F_{jn_j} is the cumulative distribution function estimator of calculated only with the j-th sample, and F_n is given by

F_n(t) =\sum_{j=1}^k p_j F_{jn_j}(t),

where p_1, \ldots, p_k correspond to the input weights. This function implements the previous test statistic accounting for left truncation and right censoring, as studied in Lago et al. (2026).

Value

Numerical value of the test statistic computed from the k samples.

References

Lago, A., Pardo-Fernández, J.C., and de Uña-Álvarez, J. (2026). Kolmogorov–Smirnov and Cramér–von Mises tests for the k-sample problem for left-truncated and right-censored data. Lifetime Data Analysis: 32(2).

Kiefer, J. (1959). K-sample analogues of the Kolmogorov–Smirnov and Cramér–von Mises tests. The Annals of Mathematical Statistics, 30:420-447.

Examples

if (requireNamespace("KMsurv", quietly = TRUE)) {
data("channing", package = "KMsurv")
data.channing <- data.frame(channing$ageentry, channing$age,
channing$death, channing$gender)
ltrc.cvm(data.channing,weights=rep(1/2,2), holes='conditional', plot.curves = TRUE)
}

Survival function estimator under left truncation and right censoring

Description

Computes the survival function estimator accounting for late entry times and censored observations. It also returns pointwise confidence bands.

Usage

ltrc.estimator(
  o.sample,
  conf.int = FALSE,
  alpha = 0.05,
  conf.type = c("plain", "loglog"),
  holes = c("holes", "add", "conditional")
)

Arguments

Details

The estimator is defined as

S_n(t) = \prod_{y_i \leq t} \left( 1-\frac{d_i}{n_i} \right),

where y_1, \ldots, y_m are the distinct observed event times, d_i = \# \{j : Y_j = y_i , \Delta_j=1 \} and n_i = \#\{j : U_j \leq y_i \leq Y_j\}. It reduces to the Kaplan–Meier estimator if data are not subject to truncation, and to the Lynden-Bell estimator if no censoring is present.

The variance for the confidence bands follows a Greenwood's like formula:

Var(S_n(t)) = S_n(t)^2 \sum_{y_i \leq t} \frac{d_i}{n_i(n_i-d_i)}.

The confidence intervals are based on the asymptotic Gaussianity of the process \sqrt{n} (S_n(t) - S(t)). Different transformations are also possible.

The function also includes a way to deal with possible holes. A hole is an event time with associated unitary risk set. If a hole is present in a sample, the estimator degenerates and takes the value 1 from that time on. Different techniques are available in the literature for the treatment of holes. For instance, in Stute and Wang (2008), authors suggest to increase the risk set by 1 (holes='add'). Other references, like Klein and Moeschberger (2003), suggest conditioning on the largest hole different from the largest event time (holes='conditional'). The possibility of computing the degenerate statistic is also included (holes='holes').

Value

An object with the following content:

fail.time: ordered distinct observed event times

estimation: estimated survival function at each observed event time

prob: jumps of the estimator

If conf.int is set to TRUE, it also returns the confidence band lower and upper limits

References

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

Wang, M.C. (1991). Nonparametric estimation from cross-sectional survival data. Journal of the American Statistical Association, 86:130-143.

Stute, W. and Wang, J. L. (2008). The central limit theorem under random truncation. Bernoulli, 14:604–622.

Examples

observed.sample <- ltrc.sim(n=1000,rtrunc=rnorm,rtarget=rnorm,rcens=rexp,
trunc.args=list(4,1),target.args = list(4,1),cens.args = list(1))
est <- ltrc.estimator(observed.sample,conf.int=TRUE,alpha=0.05,conf.type='loglog')
str(est)
plot(est,col='firebrick',xlab='t', ylab='Estimated survival probability')

if (requireNamespace("KMsurv", quietly = TRUE)) {
data("channing", package = "KMsurv")
o.sample <- channing[channing$gender==1,c(3,4,2)] # subgroup of men
head(o.sample)
plot( ltrc.estimator(o.sample,holes='holes'),col=2,xlab='t',
ylab='Estimated survival probability' ,
main = 'Estimated survival probability for men') # no correction
lines( ltrc.estimator(o.sample,holes='add'),col=3 ) # +1 to risk set
lines( ltrc.estimator(o.sample,holes='conditional'),col=4 )
legend('topright',legend=c('holes="holes" ','holes="add" ',
'holes="conditional"'),col=2:4,lwd=rep(1,3))
}

Kolmogorov–Smirnov-type statistic for the k-sample problem under left truncation and right censoring

Description

Computes the Kolmogorov–Smirnov statistic adapted to left truncation and right censoring.

Usage

ltrc.ks(
  p.sample,
  weights = NULL,
  plot.curves = FALSE,
  holes = c("holes", "add", "conditional")
)

Arguments

Details

In Kiefer (1959), the k-sample version of the Kolmogorov–Smirnov test is defined as

D_{KS} = \sup_t \sum_{j=1}^k n_j \left( F_{jn_j}(t) - F_n(t)\right)^2,

where F_{jn_j} is the cumulative distribution function estimator of calculated only with the j-th sample, and F_n is given by

F_n(t) =\sum_{j=1}^k p_j F_{jn_j}(t),

where p_1, \ldots, p_k correspond to the input weights. This function implements the previous test statistic accounting for left truncation and right censoring, as studied in Lago et al. (2026).

Value

A list with the following components:

• statistic: numerical value of the Kolmogorov–Smirnov statistic.

• time: time when the maximum is attained.

References

Lago, A., Pardo-Fernández, J.C., and de Uña-Álvarez, J. (2026) Kolmogorov–Smirnov and Cramér–von Mises tests for the k-sample problem for left-truncated and right-censored data. Lifetime Data Analysis: 32(2).

Examples

if (requireNamespace("KMsurv", quietly = TRUE)) {
data("channing", package = "KMsurv")

data.channing <- data.frame(channing$ageentry, channing$age,
channing$death, channing$gender)
ltrc.ks(data.channing,weights=rep(1/2,2),holes='conditional')
}

Weighted log-rank tests under left truncation and right censoring

Description

Computes weighted log-rank-type tests (Fleming–Harrington family) for k groups with left truncation and right censoring.

Usage

ltrc.logrank(
  p.sample,
  p = 0,
  q = 0,
  plot.curves = FALSE,
  holes = c("holes", "conditional")
)

Arguments

Details

The test statistic is based on the quantity

L_j = \sum_{i=1}^m \omega_i \left(d_{ji} - d_i \frac{r_{ji}}{r_i}\right),

for j \in \{1, \ldots, k\}. Denote by y_1, \ldots, y_m the distinct observed event times from the k samples. For a time y_i and a group j, d_{ji} denotes the total count of individuals that experience the event, and r_{ji} is the number of individuals at risk. In addition, d_i = \sum_{j=1}^k d_{ji} and r_i = \sum_{j=1}^k r_{ji}, which are the total number of events observed and the total number of individuals at risk at time y_i, respectively.

Finally, the weights \omega_1, \ldots, \omega_m are taken as

w_i = \hat{S}(y_i)^p (1-\hat{S}(y_i))^q,

with \hat{S}(x_i) the pooled survival just before the event time. The choice p=0=q results in the classical version of the log-rank test.

The test statistic is given by

(L_1, \ldots, L_{k-1}) \hat{\Sigma}^{-1}(L_1, \ldots, L_{k-1})^\top,

where \hat{\Sigma} is the matrix with the variance-covariance estimators. Each term in the matrix includes the correction factor (n_j-d_j)/(n_j-1) for tied event times. The test statistic follows an \chi^2_{k-1} distribution under the null hypothesis of equal target distributions in the k populations.

Value

A matrix with columns statistic and p.value.

References

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

Examples

if (requireNamespace("KMsurv", quietly = TRUE)) {
data("channing", package = "KMsurv")

data.channing <- data.frame(channing$ageentry, channing$age,
channing$death, channing$gender)
ltrc.logrank(data.channing, plot.curves=TRUE,
p = c(0,1,0.5,0),q = c(0,0,0.5,1),
holes = 'conditional')
}

Cramér–von Mises-type test for the k-sample problem under left truncation and right censoring

Description

Computes the Cramér–von Mises test adapted to left truncation and right censoring and the corresponding p-value.

Usage

ltrc.pv.cvm(
  p.sample,
  weights = NULL,
  B = 500,
  holes = c("holes", "add", "conditional"),
  plot.curves = FALSE,
  keep.boot = FALSE
)

Arguments

Details

In Kiefer (1959), the k-sample version of the Cramér–von Mises test is defined as

D_{CvM} = \sum_{j=1}^k n_j \int \left( F_{jn_j}(t) - F_n(t)\right)^2 dF_n(t),

where F_{jn_j} is the cumulative distribution function estimator of calculated only with the j-th sample, and F_n is given by

F_n(t) =\sum_{j=1}^k p_j F_{jn_j}(t),

where p_1, \ldots, p_k correspond to the input weights. This function implements the previous test statistic accounting for left truncation and right censoring, as studied in Lago et al. (2026).

The p-value is approximated by means of the obvious bootstrap accounting for left truncation and right censoring, studied in Bilker and Wang (1997). The exact algorithm for the k-sample problem can be consulted in Lago et al. (2026).

Value

A list with the following:

• statistic: Evaluation of the Cramér–von Mises statistic on the samples.

• pvalue: p-value approximated by bootstrap.

• curves: A list with the estimators of the k distribution functions to be compared.

• groups: Group indicator

• stat.boot: Evaluation of the Cramér–von Mises statistic on the bootstrap resamples. Only included in the output if keep.boot = TRUE.

References

Bilker, W.B., and Wang, M.C. (1997). Bootstrapping left truncated and right censored data. Communications in Statistics - Simulation and Computation, 26:141-171.

Lago, A., Pardo-Fernández, J.C., and de Uña-Álvarez, J. (2026). Kolmogorov–Smirnov and Cramér–von Mises tests for the k-sample problem for left-truncated and right-censored data. Lifetime Data Analysis: 32(2).

Kiefer, J. (1959). K-sample analogues of the Kolmogorov–Smirnov and Cramér–von Mises tests. The Annals of Mathematical Statistics, 30:420-447.

Examples

if (requireNamespace("KMsurv", quietly = TRUE)) {
data("channing", package = "KMsurv")

data.channing <- data.frame( channing$ageentry,channing$age,
channing$death,channing$gender )
aa <- ltrc.pv.cvm(data.channing, holes = 'conditional',
plot.curves = FALSE, B = 200)

aa$pvalue

}

Kolmogorov–Smirnov-type test for the k-sample problem under left truncation and right censoring

Description

Computes the Kolmogorov–Smirnov test adapted to left truncation and right censoring and the corresponding p-value.

Usage

ltrc.pv.ks(
  p.sample,
  weights = NULL,
  B = 500,
  plot.curves = FALSE,
  holes = c("holes", "add", "conditional"),
  keep.boot = FALSE
)

Arguments

Details

In Kiefer (1959), the k-sample version of the Kolmogorov–Smirnov test is defined as

D_{KS} = \sup_t \sum_{j=1}^k n_j \left( F_{jn_j}(t) - F_n(t)\right)^2,

where F_{jn_j} is the cumulative distribution function estimator of calculated only with the j-th sample, and F_n is given by

F_n(t) =\sum_{j=1}^k p_j F_{jn_j}(t),

where p_1, \ldots, p_k correspond to the input weights. This function implements the previous test statistic accounting for left truncation and right censoring, as studied in Lago et al. (2026).

The p-value is approximated by means of the obvious bootstrap accounting for left truncation and right censoring, studied in Bilker and Wang (1997). The exact algorithm for the k-sample problem can be consulted in Lago et al. (2026).

Value

A list with the following components:

• statistic: numerical value of the Kolmogorov–Smirnov statistic.

• p.value: approximated p-value of the tests.

• curves: A list with the k estimators of the distribution functions to be compared.

• groups: Group indicator

• stat.boot: the statistics computed from the bootstrap resamples. Included in the output only if keep.boot = TRUE.

References

Bilker, W.B., and Wang, M.C. (1997). Bootstrapping left truncated and right censored data. Communications in Statistics - Simulation and Computation, 26:141-171.

Lago, A., Pardo-Fernández, J.C., and de Uña-Álvarez, J. (2026). Kolmogorov–Smirnov and Cramér–von Mises tests for the k-sample problem for left-truncated and right-censored data. Lifetime Data Analysis: 32(2).

Kiefer, J. (1959). K-sample analogues of the Kolmogorov–Smirnov and Cramér–von Mises tests. The Annals of Mathematical Statistics, 30:420-447.

Examples

if (requireNamespace("KMsurv", quietly = TRUE)) {
data("channing", package = "KMsurv")
data.channing <- data.frame(channing$ageentry,channing$age,
channing$death,channing$gender)
aa <- ltrc.pv.ks(data.channing,weights=rep(1/2,2),
holes = 'conditional', plot.curves = FALSE, B = 200)
plot(aa)
aa$pvalue
}

Simulate left-truncated and right-censored data

Description

Simulates data under left truncation and right censoring.

Usage

ltrc.sim(
  n,
  rtrunc,
  trunc.args = list(),
  rtarget,
  target.args = list(),
  rcens,
  cens.args = list(),
  seed = NULL
)

Arguments

Value

A data frame with columns u, y, and delta.

Examples

aa <- ltrc.sim(n=500,rtrunc=rnorm,trunc.args=list(4,1),
rtarget=rnorm,target.args = list(4,1),
rcens=rexp,cens.args=list(1),seed=12)
head(aa)

Tests for the k-sample problem under left truncation and right censoring

Description

Computes k-sample tests adapted to left truncation and right censoring and the corresponding p-value.

Usage

ltrc.test(
  p.sample,
  weights = NULL,
  tests = c("ks", "cvm", "logrank"),
  B = 500,
  plot.curves = FALSE,
  holes = c("holes", "add", "conditional"),
  p = 0,
  q = 0,
  seed = NULL,
  keep.boot = FALSE
)

Arguments

Details

This function collects the three tests implemented in this package: Kolmogorov–Smirnov, Cramér–von Mises (Lago et al., 2026) and weighted log-rank-type tests, (Klein and Moeschberger, 2003) adapted to left truncation and right censoring. The p-values of the first two are approximated by bootstrap, and the p-value of the log-rank tests is computed by the \chi^2_{k-1} asymptotic distribution of the test statistic, where k is the number of groups.

Value

A list with the following components:

• statistic: numerical value of the Kolmogorov–Smirnov and/or Cramér–von Mises statistics.

• p.value: approximated p-value of the tests.

• stat.boot: bootstrap statistics for the p-value approximation.

• logrank: a matrix with weights p and q, test statistics, and p-values.

• B: number of bootstrap replications.

• tests: tests performed, as specified in the input.

• curves: distribution function estimators of the k samples to be compared.

• groups: group indicators.

• holes: method for handling holes.

The summary method returns a data frame with one row per test and the following columns:

Test

Name of the test. For the weighted log-rank-type tests, the weights p and q are detailed.

Statistic

Observed value of the test statistic.

p.value

Corresponding p-value.

References

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

Lago, A., Pardo-Fernández, J.C., and de Uña-Álvarez, J. (2026). Kolmogorov–Smirnov and Cramér–von Mises tests for the k-sample problem for left-truncated and right-censored data. Lifetime Data Analysis: 32(2).

Examples

if (requireNamespace("KMsurv", quietly = TRUE)) {
data("channing", package = "KMsurv")

data.channing <- data.frame(channing$ageentry, channing$age,
channing$death, channing$gender)
aa <- ltrc.test(data.channing,weights=rep(1/2,2),
holes='conditional',tests=c('ks','cvm','logrank'), B = 200,
p=c(0,1,0.5,0),q=c(0,0,0.5,1))
plot(aa)
summary(aa)
}

Estimator of the truncation distribution function for data subject to left truncation and right censoring

Description

It computes the estimator of the truncation distribution function for data subject to left truncation and right censoring

Usage

ltrc.truncation.estimator(o.sample, holes = c("holes", "add", "conditional"))

Arguments

Details

The estimator of the distribution function of the truncation variable is given by

G_n(t) = \sum_i \frac{S_n(u_i)^{-1}}{\sum_j S_n(u_j)^{-1} } \mathbb{I}[u_i \leq t],

where u_1, \ldots, u_m are the distinct truncation times and S_n is the survival function estimator of the target variable.

The nontruncation probability is defined as

\gamma = \mathbb{P}(U \leq X) = \int G(z)dF(z),

where F is the distribution function of the target variable. Then, it can be estimated as

\gamma_n = \sum_{i=1}^m \varphi_i G_n(y_i),

where y_1, \ldots, y_m are the distinct observed event times and \varphi_1, \ldots, \varphi_m are the weights that the estimator S_n assigns to y_1, \ldots, y_m.

In practice, it is possible to observe truncation times larger than the largest event time. This would break the formula of the estimator G_{n} defined above. In such a case, following the recommendations in Wang (1991), we reduce the distribution function of the truncation variable to a conditional distribution on the identifiable region, by trimming a small amount of data. This is analogous to conditioning to times largest than the largest hole when estimating the distribution function of the target variable.

Value

An object with the observed truncation times, the estimator at those times, the jumps of the estimator and an estimator of the nontruncation probability

References

Wang, M.C. (1991). Nonparametric Estimation from Cross-Sectional Survival Data. Journal of the American Statistical Association, 86:130-143.

Examples

aa <- ltrc.sim(n=5000,rtrunc=rnorm,trunc.args = list(4,1),rtarget=rnorm,
target.args=list(4,1),rcens=rexp,cens.args=list(1))
head(aa)
plot(ltrc.truncation.estimator(aa))
curve(pnorm(x,4,1),add=TRUE,col=2)

if (requireNamespace("KMsurv", quietly = TRUE)) {
data("channing", package = "KMsurv")

o.sample <- channing[channing$gender==1,c(3,4,2)] # subgroup of men
plot(ltrc.estimator(o.sample)) # hole in the second event time
plot(ltrc.truncation.estimator(o.sample,holes='add'))
lines(ltrc.truncation.estimator(o.sample,holes='conditional'),col=2)
}

Plot method for estimators objects

Description

Produces a step plot of the survival estimator.

Usage

## S3 method for class 'estimators'
plot(x, conf.int = FALSE, ...)

Arguments

Value

Invisibly returns x.

Plot method for k-sample test objects

Description

Plots the estimated survival curves for the groups involved in ltrc.test().

Usage

## S3 method for class 'ksamples_test'
plot(
  x,
  y = NULL,
  col = NULL,
  lwd = 2,
  lty = 1,
  xlab = "Time",
  ylab = "Survival probability",
  main = "Estimated survival curves",
  legend.pos = "topright",
  ...
)

Arguments

Value

Invisibly returns x.

Plot method for estimators objects

Description

Produces a step plot of the survival estimator.

Usage

## S3 method for class 't.estimators'
plot(x, ...)

Arguments

Value

Invisibly returns x.

Cramér–von Mises-type statistic for the k-sample problem under right censoring

Description

Computes the Cramér–von Mises statistic adapted to left truncation and right censoring.

Usage

rc.cvm(p.sample, weights = NULL, plot.curves = FALSE)

Arguments

Details

In Kiefer (1959), the k-sample version of the Cramér–von Mises test is defined as

D_{CvM} = \sum_{j=1}^k n_j \int \left( F_{jn_j}(t) - F_n(t)\right)^2 dF_n(t),

where F_{jn_j} is the cumulative distribution function estimator of calculated only with the j-th sample, and F_n is given by

F_n(t) =\sum_{j=1}^k p_j F_{jn_j}(t),

where p_1, \ldots, p_k correspond to the input weights. This function implements the previous test statistic accounting for right censoring.

Value

Numerical value of the test statistic computed from the k samples.

References

Kiefer, J. (1959). K-sample analogues of the Kolmogorov–Smirnov and Cramér–von Mises tests. The Annals of Mathematical Statistics, 30:420-447.

Schumacher, M. (1984). Two-Sample Tests of Cramér–von Mises-and Kolmogorov–Smirnov-Type for Randomly Censored Data. International Statistical Review/Revue Internationale de Statistique, 263-281.

Examples

if (requireNamespace("survival", quietly = TRUE)) {
library(survival)

data <- data.frame(y=stanford2$time,
delta=stanford2$status,
group=as.numeric(stanford2$age<=35)+as.numeric(stanford2$age<=47))
rc.cvm(data,weights=table(data$group)/nrow(data))
}

Survival function estimator under left truncation and right censoring

Description

Computes the survival function estimator accounting for censored observations. It also returns pointwise confidence bands.

Usage

rc.estimator(
  o.sample,
  conf.int = FALSE,
  alpha = 0.05,
  conf.type = c("plain", "loglog")
)

Arguments

Details

The estimator is defined as

S_n(t) = \prod_{y_i \leq t} \left( 1-\frac{d_i}{n_i} \right),

where y_1, \ldots, y_m are the distinct observed event times, d_i = \# \{j : Y_j = y_i , \Delta_j=1 \} and n_i = \#\{j : U_j \leq y_i \leq Y_j\}. It reduces to the Kaplan–Meier estimator if data are not subject to truncation, and to the Lynden-Bell estimator if no censoring is present.

The variance for the confidence bands follows a Greenwood's like formula:

Var(S_n(t)) = S_n(t)^2 \sum_{y_i \leq t} \frac{d_i}{n_i(n_i-d_i)}.

Different transformations are also possible.

Value

An object with the following content:

fail.time: ordered distinct observed event times

estimation: estimated survival function at each observed event time

prob: jumps of the estimator

If conf.int is set to TRUE, it also returns the confidence band lower and upper limits

References

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

Wang, M.C. (1991). Nonparametric estimation from cross-sectional survival data. Journal of the American Statistical Association, 86:130-143.

Stute, W. and Wang, J. L. (2008). The central limit theorem under random truncation. Bernoulli, 14:604–622.

Examples

x <- rnorm(500,4,1)
c <- rnorm(500,4,1)
o.sample <- data.frame(y = pmin(x,c),delta = as.numeric(x <= c))
plot(rc.estimator(o.sample,conf.int=TRUE,conf.type='loglog'))
curve(1-pnorm(x,4,1),add=TRUE,col=2)

Kolmogorov–Smirnov-type statistic for the k-sample problem under right censoring

Description

Computes the Kolmogorov–Smirnov statistic adapted to right censoring.

Usage

rc.ks(p.sample, weights = NULL, plot.curves = FALSE)

Arguments

Details

In Kiefer (1959), the k-sample version of the Kolmogorov–Smirnov test is defined as

D_{KS} = \sup_t \sum_{j=1}^k n_j \left( F_{jn_j}(t) - F_n(t)\right)^2,

where F_{jn_j} is the cumulative distribution function estimator of calculated only with the j-th sample, and F_n is given by

F_n(t) =\sum_{j=1}^k p_j F_{jn_j}(t),

where p_1, \ldots, p_k correspond to the input weights. This function implements the previous test statistic accounting for right censoring.

For the particular case of k=2, the test statistic reduces to

D_{KS} = (n_1p_2^2 + n_2p_1^2) \left(\sup_t \mid F_{1n_1}(t) - F_{2n_2}(t) \mid\right)^2,

thus the test indeed extends the classical two-sample Kolmogorov–Smirnov test. A two-sample Kolmogorov–Smirnov-type test for right-censored data was proposed in Schumacher (1984).

Value

A list with the following components:

• statistic: numerical value of the Kolmogorov–Smirnov statistics.

• time: time when the maximum is attained.

References

Kiefer, J. (1959). K-sample analogues of the Kolmogorov–Smirnov and Cramér–von Mises tests. The Annals of Mathematical Statistics, 30:420-447.

Schumacher, M. (1984). Two-Sample Tests of Cramér–von Mises-and Kolmogorov–Smirnov-Type for Randomly Censored Data. International Statistical Review/Revue Internationale de Statistique, 263-281.

Examples

if (requireNamespace("survival", quietly = TRUE)) {
library(survival)

data <- data.frame(y=stanford2$time,
delta=stanford2$status,
group=as.numeric(stanford2$age<=35)+as.numeric(stanford2$age<=47))
rc.ks(p.sample=data,weights=table(data$group)/nrow(data),plot.curves=TRUE)
}

Weighted log-rank tests under left truncation

Description

Computes weighted log-rank-type tests (Fleming–Harrington family) for k groups with left truncation and right censoring.

Usage

rc.logrank(p.sample, p = 0, q = 0, plot.curves = FALSE)

Arguments

Details

The test statistic is based on the quantity

L_j = \sum_{i=1}^m \omega_i \left(d_{ji} - d_i \frac{r_{ji}}{r_i}\right),

for j \in \{1, \ldots, k\}. Denote by y_1, \ldots, y_m the distinct observed event times from the k samples. For a time y_i and a group j, d_{ji} denotes the total count of individuals that experience the event, and r_{ji} is the number of individuals at risk. In addition, d_i = \sum_{j=1}^k d_{ji} and r_i = \sum_{j=1}^k r_{ji}, which are the total number of events observed and the total number of individuals at risk at time y_i, respectively.

Finally, the weights \omega_1, \ldots, \omega_m are taken as

w_i = \hat{S}(y_i)^p (1-\hat{S}(y_i))^q,

with \hat{S}(x_i) the pooled survival just before the event time. The choice p=0=q results in the classical version of the log-rank test.

The test statistic is given by

(L_1, \ldots, L_{k-1}) \hat{\Sigma}^{-1}(L_1, \ldots, L_{k-1})^\top,

where \hat{\Sigma} is the matrix with the variance-covariance estimators. Each term in the matrix includes the correction factor (n_j-d_j)/(n_j-1) for tied event times. The test statistic follows an \chi^2_{k-1} distribution under the null hypothesis of equal target distributions in the k populations.

Value

A matrix with columns statistic and p.value.

References

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

Examples

if (requireNamespace("survival", quietly = TRUE)) {
library(survival)
data <- data.frame(y=stanford2$time,
delta=stanford2$status,
group=as.numeric(stanford2$age<=35)+as.numeric(stanford2$age<=47))
rc.logrank(data, p = c(0, 1, 2), q = 0)
}

Cramér–von Mises-type test for the k-sample problem under right censoring

Description

Computes the Kolmogorov–Smirnov test adapted to left truncation and right censoring and the corresponding p-value.

Usage

rc.pv.cvm(p.sample, weights = NULL, B = 500, keep.boot = FALSE)

Arguments

Details

In Kiefer (1959), the k-sample version of the Cramér–von Mises test is defined as

D_{CvM} = \sum_{j=1}^k n_j \int \left( F_{jn_j}(t) - F_n(t)\right)^2 dF_n(t),

where F_{jn_j} is the cumulative distribution function estimator of calculated only with the j-th sample, and F_n is given by

F_n(t) =\sum_{j=1}^k p_j F_{jn_j}(t),

where p_1, \ldots, p_k correspond to the input weights. This function implements the previous test statistic accounting for right censoring.

The p-value is approximated by means of the obvious bootstrap accounting for left truncation and right censoring, studied in Efron (1981).

Value

A list with the following:

• statistic: Cramér–von Mises statistic evaluated on the the samples.

• pvalue: p-value of the test approximated via bootstrap.

• curves: A list with the estimators of the k distribution functions to be compared.

• groups: Group indicator.

• stat.boot: The Cramér–von Mises statistic evaluated from the bootstrap resamples.

References

Efron, B. (1981). Censored data and the bootstrap. Journal of the American Statistical Association, 76: 312-319.

Kiefer, J. (1959). K-sample analogues of the Kolmogorov–Smirnov and Cramér–von Mises tests. The Annals of Mathematical Statistics, 30:420-447.

Examples

if (requireNamespace("survival", quietly = TRUE)) {
library(survival)
data <- data.frame(y=stanford2$time,
delta=stanford2$status,
group=as.numeric(stanford2$age<=35)+as.numeric(stanford2$age<=47))
aa <- rc.pv.cvm(data,weights=table(data$group)/nrow(data), B = 200)
plot(aa)
aa$pvalue
}

Kolmogorov–Smirnov-type test for the k-sample problem under right censoring

Description

Computes the Kolmogorov–Smirnov test adapted to right censoring and the corresponding p-value.

Usage

rc.pv.ks(
  p.sample,
  weights = NULL,
  B = 500,
  plot.curves = FALSE,
  keep.boot = FALSE
)

Arguments

Details

In Kiefer (1959), the k-sample version of the Kolmogorov–Smirnov test is defined as

D_{KS} = \sup_t \sum_{j=1}^k n_j \left( F_{jn_j}(t) - F_n(t)\right)^2,

where F_{jn_j} is the cumulative distribution function estimator of calculated only with the j-th sample, and F_n is given by

F_n(t) =\sum_{j=1}^k p_j F_{jn_j}(t),

where p_1, \ldots, p_k correspond to the input weights. This function implements the previous test statistic accounting for right censoring.

For the particular case of k=2, the test statistic reduces to

D_{KS} = (n_1p_2^2 + n_2p_1^2) \left(\sup_t \mid F_{1n_1}(t) - F_{2n_2}(t) \mid\right)^2,

thus the test indeed extends the classical two-sample Kolmogorov–Smirnov test. A two-sample Kolmogorov–Smirnov-type test for right-censored data was proposed in Schumacher (1984).

The p-value is approximated by means of the obvious bootstrap accounting for right censoring, studied in Efron (1981).

Value

A list with the following components:

• statistic: numerical value of the Kolmogorov–Smirnov statistic.

• pvalue: approximated p-value of the test.

• curves: A list with the estimators of the k distribution functions to be compared.

• groups: Group indicator

• stat.boot: statistics computed from the bootstrap resamples.

References

Efron, B. (1981). Censored data and the bootstrap. Journal of the American Statistical Association, 76: 312-319.

Kiefer, J. (1959). K-sample analogues of the Kolmogorov–Smirnov and Cramér–von Mises tests. The Annals of Mathematical Statistics, 30:420-447.

Examples

if (requireNamespace("survival", quietly = TRUE)) {
library(survival)

data <- data.frame(y=stanford2$time,
delta=stanford2$status,
group=as.numeric(stanford2$age<=35)+as.numeric(stanford2$age<=47))
aa <- rc.pv.ks(data,weights=table(data$group)/nrow(data), B = 200)
plot(aa)
aa$pvalue
}

Right-censored data simulation

Description

Simulates right-censored survival data from prespecified event-time and censoring-time distributions.

Usage

rc.sim(
  n,
  rtarget,
  target.args = list(),
  rcens,
  cens.args = list(),
  seed = NULL
)

Arguments

Value

A data frame with n rows and two columns:

y

Observed follow-up time, i.e. \min(X, C).

delta

Event indicator, equal to 1 if the event is observed and 0 if censored.

Examples

aa <- rc.sim(n = 1000, rtarget = rweibull,
target.args = list(shape = 2, scale = 3),
rcens = rexp, cens.args = list(rate = 0.2),
seed = 123)
head(aa)
mean(aa$delta == 0)  # empirical censoring rate

Tests for the k-sample problem under right censoring

Description

Computes k-sample tests adapted to right censoring and the corresponding p-value.

Usage

rc.test(
  p.sample,
  weights = NULL,
  tests = c("ks", "cvm", "logrank"),
  B = 500,
  plot.curves = FALSE,
  p = 0,
  q = 0,
  seed = NULL,
  keep.boot = FALSE
)

Arguments

Details

This function collects the three tests implemented in this package: Kolmogorov–Smirnov, Cramér–von Mises and weighted log-rank-type tests, (Klein and Moeschberger, 2003) adapted to right censoring. The p-values of the first two are approximated by the obvious bootstrap (Efron, 1981), and the p-value of the log-rank tests is computed by the \chi^2_{k-1} asymptotic distribution of the test statistic, where k is the number of groups.

Value

A list with the following components:

• statistic: numerical value of the Kolmogorov–Smirnov and/or Cramér–von Mises statistics.

• p.value: approximated p-value of the tests.

• logrank: a matrix with weights p and q, test statistics, and p-values.

• B: number of bootstrap replications.

• tests: tests performed, as specified in the input.

• curves: distribution function estimators of the k samples to be compared.

• groups: group indicators.

• stat.boot: evaluation of the tests in the bootstrap resamples. Only when keep.boot = TRUE.

The summary method returns a data frame with one row per test and the following columns:

Test

Name of the test. For the weighted log-rank-type tests, the weights p and q are detailed.

Statistic

Observed value of the test statistic.

p.value

Corresponding p-value.

References

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

Efron, B. (1981). Censored data and the bootstrap. Journal of the American Statistical Association, 76: 312-319.

Examples

if (requireNamespace("survival", quietly = TRUE)) {
library(survival)

data <- data.frame(y=stanford2$time,delta=stanford2$status,
group=as.numeric(stanford2$age<=35)+as.numeric(stanford2$age<=47))
aa <- rc.test(p.sample=data,weights=table(data$group)/nrow(data),
plot.curves=TRUE,tests=c('ks','cvm','logrank'),
B = 200, p=c(0,1,2), q=0 )

plot(aa)
summary(aa)
}

Summary method for k-sample test objects

Description

Produces a table with the test name, test statistic and p-value for the procedures computed by ltrc.test().

Usage

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

Arguments

Value

A data frame with columns:

Test

Name of the test ("KS", "CvM", or the corresponding ⁠(p, q)⁠ values for weighted rank tests).

Statistic

Observed test statistic.

p-value

Associated p-value.