Package {BMeRL}

Type:	Package
Title:	Bivariate Median Residual Life Estimation
Version:	1.0.0
Author:	Ruhul Ali Khan [aut, cre], Musie Ghebremichael [aut] (Project PI)
Maintainer:	Ruhul Ali Khan <ruhulali.khan@gmail.com>
Description:	Nonparametric estimation of the bivariate median residual life function of Kayid et al. (2020) for complete and right-censored paired lifetimes. Order-restricted estimation for two groups is also provided, following Khan and Ghebremichael (2026). For censored data the bivariate survival function is estimated by the method of Lin and Ying (1993). Simultaneous confidence intervals are obtained by the percentile bootstrap approach of Mandel and Betensky (2008), and the estimated functions can be drawn as surface plots. See the accompanying vignette 'BMeRL' for a detailed introduction. Kayid, M., Shafaei Noughabi, M., and Abouammoh, A. M. (2020) <doi:10.1007/s00357-018-9300-z>; Khan, R. A., and Ghebremichael, M. (2026) <doi:10.1002/bimj.70112>; Lin, D. Y., and Ying, Z. (1993) <doi:10.1093/biomet/80.3.573>; Mandel, M., and Betensky, R. A. (2008) <doi:10.1016/j.csda.2007.07.005>.
License:	GPL-3
Encoding:	UTF-8
LazyData:	true
Depends:	R (≥ 3.6.0)
Imports:	stats, utils, graphics, grDevices
Suggests:	knitr, rmarkdown
VignetteBuilder:	knitr
RoxygenNote:	7.3.3
NeedsCompilation:	no
Packaged:	2026-10-05 20:40:59 UTC; rk128
Repository:	CRAN
Date/Publication:	2026-10-15 17:40:09 UTC

BMeRL: Bivariate Median Residual Life

Description

Nonparametric estimation of the bivariate median residual life function of Kayid et al. (2020) for complete and right-censored paired lifetimes. Order-restricted estimation for two groups is also provided, following Khan and Ghebremichael (2026). For censored data, the bivariate survival function is estimated by the method of Lin and Ying (1993). Simultaneous confidence intervals are obtained by the percentile bootstrap approach of Mandel and Betensky (2008), and the estimated functions can be drawn as surface plots.

Estimation

• BMeRL_Complete: bivariate median residual life for complete data.

• BMeRL_Censored: bivariate median residual life for right-censored data.

• BMeRL_Complete_ordered: bivariate median residual life for complete data in two groups, under an order restriction between the groups.

• BMeRL_Censored_ordered: bivariate median residual life for right-censored data in two groups, under an order restriction between the groups.

Surface plots

Each estimator has a matching surface-plot function: BMeRL_Surface_complete, BMeRL_Surface_censored, BMeRL_Surface_complete_ordered and BMeRL_Surface_censored_ordered.

Data

Two artificial data sets, complete and censored, simulated to illustrate all four estimators. The column disease is used only by the order-restricted functions. It must contain exactly two groups; otherwise these functions stop with an error.

Author(s)

Maintainer: Ruhul Ali Khan ruhulali.khan@gmail.com (ORCID)

Authors:

• Musie Ghebremichael musie_ghebremichael@dfci.harvard.edu (Project PI)

References

Kayid, M., Shafaei Noughabi, M., and Abouammoh, A. M. (2020). A nonparametric estimator of bivariate quantile residual life model with application to tumor recurrence data set. Journal of Classification, 37, 237–253. doi:10.1007/s00357-018-9300-z

Khan, R. A., and Ghebremichael, M. (2026). Bivariate quantile residual life functions for estimating immunological and virological recovery in HIV patients. Biometrical Journal, 68(2), e70112. doi:10.1002/bimj.70112

Lin, D. Y., and Ying, Z. (1993). A simple nonparametric estimator of the bivariate survival function under univariate censoring. Biometrika, 80(3), 573–581. doi:10.1093/biomet/80.3.573

Mandel, M., and Betensky, R. A. (2008). Simultaneous confidence intervals based on the percentile bootstrap approach. Computational Statistics and Data Analysis, 52(4), 2158–2165. doi:10.1016/j.csda.2007.07.005

Bivariate Median Residual Life for Right-Censored Data

Description

Estimates the bivariate median residual life function at one or more points from right-censored paired lifetimes and, optionally, computes bootstrap simultaneous confidence intervals (SCI).

Usage

BMeRL_Censored(
  x1,
  x2,
  dataset,
  time1,
  time2,
  status1,
  status2,
  SCI = FALSE,
  B = 1000,
  level = 0.95,
  seed = NULL,
  verbose = TRUE
)

Arguments

Details

Censoring model. Each subject has a common censoring time C, independent of the lifetimes (X_1, X_2). The observed data are T_j = \min(X_j, C) and \delta_j = I(X_j \le C), j = 1, 2.

Survival function. Following Lin and Ying (1993), the bivariate survival function is estimated by

\hat R(x_1, x_2) = \frac{n^{-1}\sum_i I(T_{1i} > x_1,\ T_{2i} > x_2)} {\hat G(\max(x_1, x_2))},

where \hat G is the Kaplan-Meier estimator of P(C > t), computed from \max(T_{1i}, T_{2i}) with censoring indicators 1 - \delta_{1i}\delta_{2i}.

Median residual life. At a point (a, b), let p = \hat R(a, b)/2. Along the first coordinate, let t_1^{\le} be the first time t \ge a such that \hat R(t, b) \le p, and let t_1^{<} be the first time such that \hat R(t, b) < p. The estimated median location is the midpoint of t_1^{\le} and t_1^{<}, with t_1^{\le} used alone if the strict crossing does not occur. Then \hat q_1(a, b) is this estimated median location minus a. The estimator \hat q_2(a, b) is defined analogously along the second coordinate. In the absence of censoring, this construction reduces to the conventional sample median and agrees with BMeRL_Complete.

If no paired observation lies beyond a point, both estimates are set to 0. An estimate is NA when \hat G(\max(a, b)) = 0 or when the estimated survival function does not reach the target level within the observed data.

SCI. The paired observations, including both observed times and their corresponding event indicators, are resampled with replacement, and \hat G is re-estimated in every bootstrap sample. The SCI are based on the relative-rank method of Mandel and Betensky (2008), as implemented in BMeRL_Complete, and are simultaneous for q_1 and q_2 across all specified points. Bootstrap samples containing an undefined estimate are dropped with a warning.

Value

If SCI = FALSE, a data frame with columns x1, x2, q1, and q2, where q1 and q2 are the estimated first and second components of the bivariate median residual life at each specified point.

If SCI = TRUE, the data frame additionally contains q1_lower and q1_upper, the lower and upper simultaneous confidence limits for q1, and q2_lower and q2_upper, the corresponding simultaneous confidence limits for q2. The intervals are simultaneous across both components and all specified points. Additional bootstrap diagnostics are stored as attributes of the returned data frame.

References

Kayid, M., Shafaei Noughabi, M., and Abouammoh, A. M. (2019). A nonparametric estimator of bivariate quantile residual life model with application to tumor recurrence data set. Journal of Classification. doi:10.1007/s00357-018-9300-z

Lin, D. Y. and Ying, Z. (1993). A simple nonparametric estimator of the bivariate survival function under univariate censoring. Biometrika, 80(3), 573–581.

Mandel, M. and Betensky, R. A. (2008). Simultaneous confidence intervals based on the percentile bootstrap approach. Computational Statistics and Data Analysis, 52(4), 2158–2165.

See Also

BMeRL_Censored_ordered for two ordered groups, BMeRL_Complete for complete data, and BMeRL_Surface_censored for surface plots.

Examples

data(censored)

## Point estimates only
BMeRL_Censored(
  c(0, 5, 10), c(0, 5, 5), censored,
  time1, time2, status1, status2
)

## Point estimates with 95% simultaneous confidence intervals
out <- BMeRL_Censored(
  c(0, 5), c(0, 5), censored,
  time1, time2, status1, status2,
  SCI = TRUE, B = 200, seed = 1691, verbose = FALSE
)
out

Order-Restricted Bivariate Median Residual Life for Right-Censored Data

Description

Estimates the bivariate median residual life functions for two groups under an order restriction for right-censored data and, optionally, computes bootstrap simultaneous confidence intervals (SCI).

Usage

BMeRL_Censored_ordered(
  x1,
  x2,
  dataset,
  time1,
  time2,
  status1,
  status2,
  group,
  smaller,
  SCI = TRUE,
  B = 1000,
  level = 0.95,
  seed = NULL,
  verbose = TRUE
)

Arguments

Details

Let \mathbf{q}_F(x) = (q_{F1}(x), q_{F2}(x)) denote the bivariate median residual life for the group specified by smaller, and let \mathbf{q}_G(x) = (q_{G1}(x), q_{G2}(x)) denote that for the other group, with sample sizes n_F and n_G, respectively. The componentwise order restriction q_{Fj}(x) \le q_{Gj}(x), j = 1, 2, is imposed at each specified point x.

Let \hat{\mathbf{q}}_F and \hat{\mathbf{q}}_G denote the unrestricted bivariate median residual life estimates for groups F and G, respectively, obtained as in BMeRL_Censored. Define the pooled estimate by

\hat{\mathbf{q}}_P = \frac{n_F}{n_F + n_G}\hat{\mathbf{q}}_F + \frac{n_G}{n_F + n_G}\hat{\mathbf{q}}_G.

The order-restricted estimates are then defined componentwise as

\hat{\mathbf{q}}_F^* = \min(\hat{\mathbf{q}}_F, \hat{\mathbf{q}}_P), \qquad \hat{\mathbf{q}}_G^* = \max(\hat{\mathbf{q}}_G, \hat{\mathbf{q}}_P).

If the unrestricted estimates already satisfy \hat q_{Fj} \le \hat q_{Gj}, they remain unchanged; otherwise, the corresponding components are replaced by the pooled estimate. Thus, \hat q_{Fj}^* \le \hat q_{Gj}^*, j = 1, 2, at every specified point.

The censoring distribution is estimated separately within each group and re-estimated in every bootstrap sample. The paired observations, including both observed times and their corresponding event indicators, are resampled with replacement separately within groups. The pooled and order-restricted estimates are then recomputed for each bootstrap sample.

The SCI are based on the relative-rank method of Mandel and Betensky (2008) and are simultaneous for both components of groups F and G across all specified points. Within each coordinate, the bootstrap estimates are ranked, and the largest absolute deviation of the coordinatewise ranks from the central rank is used to construct the simultaneous confidence limits. Near the boundary \mathbf{q}_F = \mathbf{q}_G, the order-restricted estimators are nonsmooth, and the percentile bootstrap may have coverage below the nominal level.

Value

A data frame with one row for each specified point and group, with columns x1, x2, group, q1, and q2, where q1 and q2 are the order-restricted estimates of the first and second components of the bivariate median residual life.

If SCI = TRUE, the data frame additionally contains q1_lower and q1_upper, the lower and upper simultaneous confidence limits for q1, and q2_lower and q2_upper, the corresponding simultaneous confidence limits for q2. The intervals are simultaneous across both components, both groups, and all specified points.

The unrestricted estimates are stored in the unrestricted attribute. Additional bootstrap diagnostics are stored as attributes of the returned data frame when SCI = TRUE.

References

Kayid, M., Shafaei Noughabi, M., and Abouammoh, A. M. (2019). A nonparametric estimator of bivariate quantile residual life model with application to tumor recurrence data set. Journal of Classification. doi:10.1007/s00357-018-9300-z

Lin, D. Y. and Ying, Z. (1993). A simple nonparametric estimator of the bivariate survival function under univariate censoring. Biometrika, 80(3), 573–581.

Mandel, M., and Betensky, R. A. (2008). Simultaneous confidence intervals based on the percentile bootstrap approach. Computational Statistics and Data Analysis, 52(4), 2158–2165.

See Also

BMeRL_Censored for unrestricted censored-data estimation, BMeRL_Complete_ordered for complete data, and BMeRL_Surface_censored_ordered for surface plots.

Examples

data(censored)

## Point estimates, with group "Yes" denoted by F and assumed to have
## the smaller bivariate median residual life
BMeRL_Censored_ordered(
  c(0, 5, 10), c(0, 5, 5), censored,
  time1, time2, status1, status2, disease,
  smaller = "Yes", SCI = FALSE
)

## With 95% simultaneous confidence intervals
BMeRL_Censored_ordered(
  c(0, 5), c(0, 5), censored,
  time1, time2, status1, status2, disease,
  smaller = "Yes", B = 200, seed = 1691,
  verbose = FALSE
)

Bivariate Median Residual Life for Complete Data

Description

Estimates the bivariate median residual life function at one or more points from complete (uncensored) paired lifetimes and, optionally, computes bootstrap simultaneous confidence intervals (SCI).

Usage

BMeRL_Complete(
  x1,
  x2,
  dataset,
  time1,
  time2,
  SCI = TRUE,
  B = 1000,
  level = 0.95,
  seed = NULL,
  verbose = TRUE
)

Arguments

Details

At a point x = (x_1, x_2), the median residual lives are

\hat q_1(x_1, x_2) = \mathrm{median}\{X_{1i} : X_{1i} > x_1,\ X_{2i} > x_2\} - x_1,

\hat q_2(x_1, x_2) = \mathrm{median}\{X_{2i} : X_{1i} > x_1,\ X_{2i} > x_2\} - x_2,

where the conventional sample median is used, averaging the two middle values when the number of paired observations beyond the point is even. Together, \hat q_1 and \hat q_2 define the bivariate median residual life at x. If no paired observation lies beyond a point, both estimates are set to 0.

The SCI are based on the relative-rank method of Mandel and Betensky (2008) and are simultaneous for q_1 and q_2 across all specified points. The paired observations are resampled with replacement B times. Within each coordinate, the bootstrap estimates are ranked: tied values below the median receive the minimum rank of their group, tied values above the median receive the maximum rank, and values equal to the median are treated as central. For each bootstrap sample, the largest absolute deviation of the coordinatewise ranks from the central rank is computed. The central level proportion of bootstrap samples is retained, and the minimum and maximum retained bootstrap estimates in each coordinate define the simultaneous confidence limits.

Value

If SCI = FALSE, a data frame with columns x1, x2, q1, and q2, where q1 and q2 are the estimated first and second components of the bivariate median residual life at each specified point.

If SCI = TRUE, the data frame additionally contains q1_lower and q1_upper, the lower and upper simultaneous confidence limits for q1, and q2_lower and q2_upper, the corresponding simultaneous confidence limits for q2. The intervals are simultaneous across both components and all specified points. Additional bootstrap diagnostics are stored as attributes of the returned data frame.

References

Kayid, M., Shafaei Noughabi, M., and Abouammoh, A. M. (2020). A nonparametric estimator of bivariate quantile residual life model with application to tumor recurrence data set. Journal of Classification, 37, 237–253. doi:10.1007/s00357-018-9300-z

Mandel, M., and Betensky, R. A. (2008). Simultaneous confidence intervals based on the percentile bootstrap approach. Computational Statistics and Data Analysis, 52(4), 2158–2165. doi:10.1016/j.csda.2007.07.005

See Also

BMeRL_Complete_ordered for two ordered groups, BMeRL_Censored for censored data, and BMeRL_Surface_complete for surface plots.

Examples

data(complete)

## Point estimates only
BMeRL_Complete(c(0, 5, 10), c(0, 5, 5), complete, time1, time2,
               SCI = FALSE)

## Point estimates with 95% simultaneous confidence intervals
out <- BMeRL_Complete(c(0, 5), c(0, 5), complete, time1, time2,
                      B = 200, seed = 1691, verbose = FALSE)
out

Order-Restricted Bivariate Median Residual Life for Complete Data

Description

Estimates the bivariate median residual life functions for two groups under an order restriction for complete (uncensored) data and, optionally, computes bootstrap simultaneous confidence intervals (SCI).

Usage

BMeRL_Complete_ordered(
  x1,
  x2,
  dataset,
  time1,
  time2,
  group,
  smaller,
  SCI = TRUE,
  B = 1000,
  level = 0.95,
  seed = NULL,
  verbose = TRUE
)

Arguments

Details

Let \mathbf{q}_F(x) = (q_{F1}(x), q_{F2}(x)) denote the bivariate median residual life for the group specified by smaller, and let \mathbf{q}_G(x) = (q_{G1}(x), q_{G2}(x)) denote that for the other group, with sample sizes n_F and n_G, respectively. The componentwise order restriction q_{Fj}(x) \le q_{Gj}(x), j = 1, 2, is imposed at each specified point x.

Let \hat q_F and \hat q_G denote the unrestricted bivariate median residual life estimates for groups F and G, respectively, obtained as in BMeRL_Complete. Define the pooled estimate by

\hat q_P = \frac{n_F}{n_F + n_G}\hat q_F + \frac{n_G}{n_F + n_G}\hat q_G.

The order-restricted estimates are then defined componentwise as

\hat q_F^* = \min(\hat q_F, \hat q_P), \qquad \hat q_G^* = \max(\hat q_G, \hat q_P).

If the unrestricted estimates already satisfy \hat q_F \le \hat q_G, they remain unchanged; otherwise, both are replaced by the pooled estimate. Thus, \hat q_F^* \le \hat q_G^* at every specified point.

The SCI are based on the relative-rank method of Mandel and Betensky (2008) and are simultaneous for both components of groups F and G across all specified points. The paired observations are resampled with replacement separately within each group, and the pooled and order-restricted estimates are recomputed in each bootstrap sample. Within each coordinate, the bootstrap estimates are ranked, and the largest absolute deviation of the coordinatewise ranks from the central rank is used to construct the simultaneous confidence limits.

Value

A data frame with one row for each specified point and group, with columns x1, x2, group, q1, and q2, where q1 and q2 are the order-restricted estimates of the first and second components of the bivariate median residual life.

If SCI = TRUE, the data frame additionally contains q1_lower and q1_upper, the lower and upper simultaneous confidence limits for q1, and q2_lower and q2_upper, the corresponding simultaneous confidence limits for q2. The intervals are simultaneous across both components, both groups, and all specified points.

The unrestricted estimates are stored in the unrestricted attribute. Additional bootstrap diagnostics are stored as attributes of the returned data frame when SCI = TRUE.

References

Kayid, M., Shafaei Noughabi, M., and Abouammoh, A. M. (2019). A nonparametric estimator of bivariate quantile residual life model with application to tumor recurrence data set. Journal of Classification. doi:10.1007/s00357-018-9300-z

Mandel, M., and Betensky, R. A. (2008). Simultaneous confidence intervals based on the percentile bootstrap approach. Computational Statistics and Data Analysis, 52(4), 2158–2165.

See Also

BMeRL_Complete, BMeRL_Censored_ordered, and BMeRL_Surface_complete_ordered.

Examples

data(complete)

## Point estimates, with group "Yes" denoted by F and assumed to have
## the smaller bivariate median residual life
est <- BMeRL_Complete_ordered(
  c(0, 5, 10), c(0, 5, 5), complete,
  time1, time2, disease, smaller = "Yes",
  SCI = FALSE
)
est

## With 95% simultaneous confidence intervals
BMeRL_Complete_ordered(
  c(0, 5), c(0, 5), complete,
  time1, time2, disease, smaller = "Yes",
  B = 200, seed = 1691, verbose = FALSE
)

Surface Plots of the Bivariate Median Residual Life

Description

Evaluates a bivariate median residual life estimator over a grid of points and displays the two components \hat q_1 and \hat q_2 as surface plots.

Usage

BMeRL_Surface_complete(
  dataset,
  time1,
  time2,
  x1.grid = NULL,
  x2.grid = NULL,
  n.grid = 30,
  which = c("both", "q1", "q2"),
  plot = TRUE,
  theta = 35,
  phi = 25,
  col.palette = 50,
  main = NULL,
  ...
)

BMeRL_Surface_censored(
  dataset,
  time1,
  time2,
  status1,
  status2,
  x1.grid = NULL,
  x2.grid = NULL,
  n.grid = 30,
  which = c("both", "q1", "q2"),
  plot = TRUE,
  theta = 35,
  phi = 25,
  col.palette = 50,
  main = NULL,
  ...
)

BMeRL_Surface_complete_ordered(
  dataset,
  time1,
  time2,
  group,
  smaller,
  x1.grid = NULL,
  x2.grid = NULL,
  n.grid = 30,
  which = c("both", "q1", "q2"),
  plot = TRUE,
  theta = 35,
  phi = 25,
  col.palette = 50,
  main = NULL,
  ...
)

BMeRL_Surface_censored_ordered(
  dataset,
  time1,
  time2,
  status1,
  status2,
  group,
  smaller,
  x1.grid = NULL,
  x2.grid = NULL,
  n.grid = 30,
  which = c("both", "q1", "q2"),
  plot = TRUE,
  theta = 35,
  phi = 25,
  col.palette = 50,
  main = NULL,
  ...
)

Arguments

Details

• BMeRL_Surface_complete uses BMeRL_Complete;

• BMeRL_Surface_censored uses BMeRL_Censored;

• BMeRL_Surface_complete_ordered uses BMeRL_Complete_ordered;

• BMeRL_Surface_censored_ordered uses BMeRL_Censored_ordered.

The estimator is evaluated with SCI = FALSE at each grid point. The surfaces of \hat q_1 and \hat q_2 are displayed side by side. For the order-restricted estimators, one row is shown for each group, with group F first and group G second. The two groups share the same vertical and colour scales for each component, allowing their surfaces to be compared directly. Surface facets are coloured according to their height. Grid points at which an estimate is NA are left blank. Graphical parameters modified for the layout are restored on exit.

Value

Invisibly, a list with components x1 and x2, containing the grid values, and q1 and q2, containing the corresponding bivariate median residual life estimates. For the unrestricted estimators, q1 and q2 are matrices with rows corresponding to x1 and columns corresponding to x2. For the order-restricted estimators, each is a list containing one such matrix for each group.

References

Kayid, M., Shafaei Noughabi, M., and Abouammoh, A. M. (2019). A nonparametric estimator of bivariate quantile residual life model with application to tumor recurrence data set. Journal of Classification. doi:10.1007/s00357-018-9300-z

Examples

data(complete)
data(censored)

BMeRL_Surface_complete(
  complete, time1, time2,
  x1.grid = 0:30, x2.grid = 0:30
)

BMeRL_Surface_censored(
  censored, time1, time2, status1, status2,
  x1.grid = 0:30, x2.grid = 0:30
)

BMeRL_Surface_complete_ordered(
  complete, time1, time2, disease, "Yes",
  x1.grid = 0:30, x2.grid = 0:30
)

BMeRL_Surface_censored_ordered(
  censored, time1, time2, status1, status2,
  disease, "Yes",
  x1.grid = 0:30, x2.grid = 0:30,
  which = "q1"
)

## Return the grid of estimates without drawing the surfaces
s <- BMeRL_Surface_complete(
  complete, time1, time2,
  n.grid = 10, plot = FALSE
)
round(s$q1, 1)

Simulated Right-Censored Bivariate Lifetimes

Description

A simulated data set containing the subjects in complete observed under right censoring. The data were generated to illustrate the censored-data methods in the package and do not represent real subjects. The variables time1, time2, status1, and status2 are used by BMeRL_Censored and BMeRL_Surface_censored. The order-restricted functions BMeRL_Censored_ordered and BMeRL_Surface_censored_ordered additionally use the group variable disease.

Usage

censored

Format

A data frame with 120 rows and 6 variables:

id

Subject identifier.

time1

Observed time for the first component, in months, defined as the minimum of the lifetime and censoring time.

time2

Observed time for the second component, in months, defined as the minimum of the lifetime and censoring time.

status1

Event indicator for time1: 1 if the event time was observed and 0 if it was censored.

status2

Event indicator for time2: 1 if the event time was observed and 0 if it was censored.

disease

Group indicator used by the order-restricted functions: "Yes" for 50 subjects and "No" for 70 subjects.

Details

Each subject has a single censoring time drawn from a uniform distribution on 0 to 60 months and applied to both component lifetimes in complete. Approximately 42% of the subjects have at least one censored component.

Source

Simulated data generated for this package.

See Also

complete, which contains the same subjects without censoring.

Examples

data(censored)
head(censored)
mean(censored$status1 == 0)

Simulated Complete Bivariate Lifetimes

Description

A simulated/artificial data set of paired lifetimes (in months) for 120 subjects, generated to illustrate the complete-data methods in the package. The data are uncensored and do not represent real subjects. The variables time1 and time2 are used by BMeRL_Complete and BMeRL_Surface_complete. The order-restricted functions BMeRL_Complete_ordered and BMeRL_Surface_complete_ordered additionally use the group variable disease.

Usage

complete

Format

A data frame with 120 rows and 4 variables:

id

Subject identifier.

time1

Lifetime of the first component, in months.

time2

Lifetime of the second component, in months.

disease

Group indicator used by the order-restricted functions: "Yes" for 50 subjects and "No" for 70 subjects.

Details

The paired lifetimes were simulated from Gumbel's bivariate exponential distribution with dependence parameter 0.5 and scaled to have mean lifetimes of 12 months in group "Yes" and 18 months in group "No". Thus, group "Yes" has the smaller median residual life. The observed times are rounded to 0.1 month, with 0.1 as the smallest recorded value.

Source

Simulated data generated for this package.

See Also

censored, which contains the same subjects under right censoring.

Examples

data(complete)
head(complete)
table(complete$disease)