Package {fuzzySurvival}

Title:	Fuzzy Survival Analysis with Vague Event Times and Fuzzy Censoring
Version:	0.1.0
Description:	Fuzzy survival analysis for data in which event times and/or failure indicators are vague, for instance when a failure is only known to have occurred during the second quarter of a year, or when it is uncertain whether a subject lost to follow-up had in fact failed. Vague information is represented by fuzzy numbers. The package provides a fuzzy product-limit estimator obtained from nested alpha-cuts of the fuzzy survival function, a fuzzy Cox proportional hazards model with fuzzy coefficient and hazard ratio estimates, and a fuzzy log-rank test. Type-reduction utilities for summarizing fuzzy estimates by single numbers, a simulation engine for validation studies, plotting, print and summary methods, and alpha-cut based confidence bands are provided. The bounds are computed at the alpha-cuts of the fuzzy event times, so the alpha-cut intervals returned for the survival function, the regression coefficients and the test statistic are nested by construction.
License:	MIT + file LICENSE
Encoding:	UTF-8
Depends:	R (≥ 4.1.0)
Imports:	stats, graphics, grDevices, survival (≥ 3.2-13)
Suggests:	testthat (≥ 3.0.0), knitr, rmarkdown, covr
VignetteBuilder:	knitr
Config/testthat/edition:	3
LazyData:	true
Config/roxygen2/version:	8.1.0
NeedsCompilation:	no
Packaged:	2026-10-06 04:55:37 UTC; IUD
Author:	Mukul Bijalwan [aut, cre], Gunjan Aggrwal [aut], Mukul Jain [aut]
Maintainer:	Mukul Bijalwan <mukulbijalwan555@gmail.com>
Repository:	CRAN
Date/Publication:	2026-10-15 17:50:09 UTC

Alpha-cuts, support, core and membership of fuzzy numbers

Description

alpha_cuts() evaluates the nested alpha-cuts, support() and core() return the 0-cut and the 1-cut, and membership() recovers the membership function from the alpha-cut function.

Usage

alpha_cuts(x, level = c(0, 0.25, 0.5, 0.75, 1))

support(f)

core(f)

membership(x, at, by = 0.001)

Arguments

Details

membership() works by inverting the alpha-cut function: for a value s the membership is the highest level at which s still lies inside the alpha-cut, i.e. \mu(s) = \sup\{\alpha : f^-_\alpha \le s \le f^+_\alpha\}. Because the cut family is nested this is the usual definition, computed on a grid of 1/by levels.

Value

alpha_cuts() and support() / core() return data frames with columns id, level, lower, upper; membership() returns an n by length(at) matrix.

Examples

fn <- fuzzy_number_trapezoid(3, 5, 7, 9)
alpha_cuts(fn, c(0, 0.25, 1))
support(fn)
membership(fn, c(2, 4, 6, 8, 10))

Crisp representatives of fuzzy numbers

Description

Turns a fuzzy number into a numeric vector, e.g. to hand over to the functions of survival, or to compare the fuzzy estimators with their crisp counterparts.

Usage

as_crisp(f, level = 1, method = c("midpoint", "lower", "upper"))

Arguments

Value

A numeric vector with the same length as f.

Examples

as_crisp(fuzzy_number_trapezoid(3, 5, 7, 9))
as_crisp(fuzzy_number_trapezoid(3, 5, 7, 9), level = 1, method = "lower")

Coerce to a fuzzy_number, unwrapping data frame columns

Description

A fuzzy_number placed in a data frame becomes a list column holding one object, so that the columns of every subject stay together. This helper accepts either a fuzzy_number or such a list column and always returns a fuzzy_number, which is what the user-facing functions call internally.

Usage

as_fuzzy_number(x)

Arguments

Value

A fuzzy_number.

Examples

d <- data.frame(t = I(list(fuzzy_number_interval(4, 6))))
d$t
alpha_cuts(d$t, 1)
is_crisp(d$t)

Combine fuzzy numbers

Description

Combine fuzzy numbers

Usage

## S3 method for class 'fuzzy_number'
c(...)

Arguments

Value

A fuzzy_number of length max(length(...)).

Examples

c(fuzzy_number_triangle(1, 2, 3), fuzzy_number_crisp(5))

Defuzzification of fuzzy numbers

Description

A fuzzy number carries a whole family of alpha-cuts; to obtain a single crisp value a defuzzification (type reduction) rule is needed. All the rules implemented here are the classical type-reduction rules of fuzzy set theory, written on the level-valued-set (alpha-cut) representation used by the package. Writing [l_j, r_j] for the alpha-cut at level \alpha_j, s_j = (l_j + r_j)/2 for its middle and \Delta_j = \alpha_j - \alpha_{j-1} for the width of the level strip:

Usage

defuzz(
  x,
  method = c("midrange", "centroid", "gmirl", "sgmrl", "ms", "msm", "yager", "min",
    "max"),
  level = 0.5,
  by = 0.001,
  ...
)

Arguments

Details

midrange

s(alpha), the middle of the alpha-cut at level alpha (mean of range).

centroid

\sum_j \alpha_j s_j \Delta_j / \sum_j \alpha_j \Delta_j, the centre of gravity of the membership function.

gmirl

graded mean integration representation level, \sum_j \alpha_j^2 s_j \Delta_j / \sum_j \alpha_j^2 \Delta_j.

sgmrl

simplified graded mean, \sum_j \alpha_j s_j / \sum_j \alpha_j.

ms

smallest element of the supremum cut, l_1.

msm

smallest mean of the supremum, (l_1 + l_0)/2.

yager

Yager's ranking; for a continuous membership function it reduces to the middle of the core, (l_1 + r_1)/2.

min, max

the smallest and largest element of the support.

For a symmetric fuzzy number the centroid, gmirl, sgmrl, yager and midrange rules all return its centre of symmetry, and min/max its symmetric end points, which makes the rules easy to check.

Passing several rules at once returns one row per rule, which is a quick way of comparing the type reductions on the same fuzzy number.

Value

A numeric vector with one value per fuzzy number, or a matrix with one row per requested rule when method has length greater than one.

See Also

fuzzy_spread(), fkm_fuzzy_surv()

Examples

fn <- fuzzy_number_trapezoid(3, 5, 7, 9)
defuzz(fn, "centroid")
defuzz(fn, "midrange", level = 0.5)

# a symmetric fuzzy number: every centred rule gives 5
defuzz(fuzzy_number_triangle(2, 5, 8), c("centroid", "gmirl", "sgmrl"))

Defuzzification of fuzzy Cox coefficients

Description

Defuzzification of fuzzy Cox coefficients

Usage

defuzz.fcox(
  x,
  method = c("midrange", "centroid", "gmirl", "sgmrl", "ms", "msm", "yager", "min",
    "max"),
  level = 0.5,
  by = 0.001,
  conf.int = 0.95,
  exponentiate = FALSE
)

Arguments

Value

A data frame with the point estimate and the fuzzy interval.

Examples

data(fuzzy_dfs)
fit <- fcox(fuzzy_surv(ft, fs) ~ age + arm, data = fuzzy_dfs)
defuzz(fit, "centroid", exponentiate = TRUE)

Defuzzification of an fkm fit

Description

Defuzzification of an fkm fit

Usage

defuzz.fkm(
  x,
  method = c("midrange", "centroid", "gmirl", "sgmrl", "ms", "msm", "yager", "min",
    "max"),
  level = 0.5,
  by = 0.001,
  times = NULL
)

Arguments

Value

A numeric vector with one value per requested time.

Examples

data(fuzzy_dfs)
fit <- fkm(fuzzy_surv(ft, fs) ~ 1, data = fuzzy_dfs)
defuzz(fit, "centroid", times = c(6, 12, 24))

Exponential of a fuzzy number

Description

exp() is increasing, so the alpha-cuts are mapped end point by end point.

Usage

## S3 method for class 'fuzzy_number'
exp(x)

Arguments

Value

A fuzzy_number.

Examples

exp(fuzzy_number_trapezoid(-1, 0, 1, 2))

Fuzzy Cox proportional hazards model

Description

A Cox model for vague event times. At each alpha level the model is fitted with survival::coxph() to the lower and to the upper end of the time cuts, and the two fits bound the regression coefficients. The alpha-cut of the coefficient of a covariate is

\beta_\alpha = [\,\min_{a \le \alpha} \hat\beta^-_a, \max_{a \le \alpha} \hat\beta^+_a\,],

where the cumulative min/max over the levels up to \alpha is what makes the intervals nested, so the result is a fuzzy number rather than a family of unrelated intervals. Hazard ratios are obtained by exponentiating the fuzzy coefficients (an increasing, hence membership-preserving map).

Usage

fcox(
  formula,
  data,
  time,
  status,
  covariates = NULL,
  levels = NULL,
  ties = "efron",
  ...
)

## S3 method for class 'formula'
fcox(
  formula,
  data,
  time,
  status,
  covariates = NULL,
  levels = NULL,
  ties = "efron",
  ...
)

## Default S3 method:
fcox(
  formula,
  data,
  time,
  status,
  covariates = NULL,
  levels = NULL,
  ties = "efron",
  ...
)

Arguments

Details

Confidence intervals combine the fuzzy coefficient with the standard error of the crisp (level 1) fit, and the p-values are the smallest and largest values attainable given the fuzzy coefficient and that standard error.

Value

An object of class fcox, a list with components

See Also

defuzz(), fkm(), flogrank()

Examples

data(fuzzy_dfs)
fit <- fcox(fuzzy_surv(ft, fs) ~ age + arm, data = fuzzy_dfs)
summary(fit)
defuzz(fit, "centroid", exponentiate = TRUE)

Fuzzy Kaplan-Meier estimator

Description

The fuzzy Kaplan-Meier estimator of the survival function for vague event times and/or vague failure indicators.

Usage

fkm(
  formula,
  data,
  time,
  status,
  group = NULL,
  levels = NULL,
  conf.type = c("log", "log-log", "plain"),
  conf.int = 0.95,
  ...
)

## S3 method for class 'formula'
fkm(
  formula,
  data,
  time,
  status,
  group = NULL,
  levels = NULL,
  conf.type = c("log", "log-log", "plain"),
  conf.int = 0.95,
  ...
)

## Default S3 method:
fkm(
  formula,
  data,
  time,
  status,
  group = NULL,
  levels = NULL,
  conf.type = c("log", "log-log", "plain"),
  conf.int = 0.95,
  ...
)

Arguments

Value

An object of class fkm, a list with, among others, the components

Method

Let \tilde t_i be the fuzzy event time of subject i. For an alpha level \alpha the set of data compatible with "the true times have membership at least \alpha" is D_\alpha = \prod_i [\tilde t^-_i(\alpha), \tilde t^+_i(\alpha)]. Replacing every vague time by the lower end of its alpha-cut makes the failures happen as early as possible, so the resulting Kaplan-Meier curve \hat S^-_\alpha is the lower bound of the survival function, and the upper end gives the upper bound \hat S^+_\alpha. The alpha-cut of the fuzzy survival function is then

\hat S_\alpha(t) = [\, \hat S^-_\alpha(t), \hat S^+_\alpha(t) \,].

Because the time cuts are nested, the risk sets of \hat S^-_\alpha shrink as \alpha grows, hence \hat S^-_\alpha is non-decreasing and \hat S^+_\alpha non-increasing in \alpha: the reported intervals are nested alpha-cuts of a proper fuzzy number, and they collapse onto the classical Kaplan-Meier curve when the data are crisp.

Vague failure indicators are handled by the fractional estimator: a subject with degree of belief \pi that the event occurred contributes a fractional number of events d_j += \pi, which reduces exactly to the classical estimator when all \pi are 0 or 1.

See Also

fkm_fuzzy_surv(), fcox(), flogrank()

Examples

data(fuzzy_dfs)

# overall fuzzy Kaplan-Meier curve
fit <- fkm(fuzzy_surv(ft, fs) ~ 1, data = fuzzy_dfs)
summary(fit, times = c(6, 12, 24))

# a defuzzified curve, and the alpha-cut bounds
plot(fit)
defuzz(fit, "centroid", times = 12)

# by treatment arm
fit2 <- fkm(fuzzy_surv(ft, fs) ~ arm, data = fuzzy_dfs)
plot(fit2, type = "diff")

Fuzzy survival curve as fuzzy numbers

Description

Turns the fuzzy survival curve of an fkm() fit into a vector of fuzzy numbers, one per requested time, so that defuzz(), alpha_cuts() and membership() can be applied to the survival probability itself.

Usage

fkm_fuzzy_surv(x, times = NULL)

Arguments

Value

A fuzzy_number whose alpha-cut at level alpha is ⁠[S^-_alpha(t), S^+_alpha(t)]⁠.

See Also

defuzz(), fkm()

Examples

data(fuzzy_dfs)
fit <- fkm(fuzzy_surv(ft, fs) ~ 1, data = fuzzy_dfs)
s <- fkm_fuzzy_surv(fit, times = c(6, 12, 24))
alpha_cuts(s, c(0, 0.5, 1))
membership(s, 0.6)

Fuzzy log-rank test for a difference between two groups

Description

The log-rank statistic is computed on the lower and on the upper end of the time cuts at each alpha level, giving a fuzzy chi-square statistic and hence a fuzzy p-value. When the data are crisp the bounds coincide with the classical log-rank test.

Usage

flogrank(formula, data, time, status, group = NULL, levels = NULL, ...)

Arguments

Value

An object of class flogrank, a list with the fuzzy statistic statistic, the fuzzy p-value p.value, the crisp level 1 fit fit and the observed and expected number of events per group.

See Also

fkm(), fcox()

Examples

data(fuzzy_dfs)
fit <- flogrank(fuzzy_surv(ft, fs) ~ arm, data = fuzzy_dfs)
fit
defuzz(fit$p.value, "centroid")

Arithmetic on fuzzy numbers

Description

Point-wise extension of the arithmetic operations through the alpha-cut representation (Zadeh's extension principle), evaluated on a grid of 1/by levels. Division is only defined for cuts that do not contain zero.

Usage

## S3 method for class 'fuzzy_number'
Ops(e1, e2)

Arguments

Value

A fuzzy_number of the common length.

Examples

x <- fuzzy_number_trapezoid(1, 2, 3, 4)
x + 10
x * x

Simulated disease-free survival with vague event times

Description

A synthetic example dataset of 240 subjects with a treatment arm, age and sex, generated by sim_fuzzy_surv() with quarterly examinations. Use it to see the whole workflow: fkm(), fcox() and flogrank().

Usage

fuzzy_dfs

Format

A data frame with 240 rows and the following columns:

id

subject identifier.

age

age in years.

arm

treatment arm, "Treated" or "Control".

sex

"Male" or "Female".

ft

fuzzy event time: an interval fuzzy number when the failure was only known to occur between two examinations, a crisp fuzzy number otherwise.

fs

failure indicator, strictly between 0 and 1 when the status of a censored record is uncertain.

true_time, true_status, observed

the unobserved quantities used to generate the fuzzy ones, useful for checking coverage.

See Also

sim_fuzzy_surv(), fkm(), fcox()

Examples

data(fuzzy_dfs)
str(fuzzy_dfs)
fit <- fkm(fuzzy_surv(ft, fs) ~ arm, data = fuzzy_dfs)
plot(fit, type = "diff")

Vague (fuzzy) quantities on the real line

Description

A fuzzy number is a fuzzy set on the real line described by its nested alpha-cuts

F_\alpha = [\, f^-_\alpha, f^+_\alpha\,],\quad 0 < \alpha \le 1,

where f^-_\alpha is non-decreasing and f^+_\alpha is non-increasing in \alpha, F_0 is the support and F_1 is the core. Every object of class fuzzy_number stores such an alpha-cut function; the membership function is recovered numerically from it, so that both representations are always consistent.

Usage

fuzzy_number_trapezoid(l, cl, cr, r, label = NULL)

fuzzy_number_triangle(a, m, b, label = NULL)

fuzzy_number_interval(lo, hi, label = NULL)

fuzzy_number_crisp(x, label = NULL)

fuzzy_number_from_acut(acut, support = NULL, label = NULL)

Arguments

Details

The package uses fuzzy numbers for two kinds of vague information in survival data: vague (interval-like) event times such as "the failure occurred in the second quarter of the year", and vague failure indicators such as "lost to follow-up, possibly with an unobserved failure".

Value

An object of class fuzzy_number: a list with components acut (alpha-cut function), support (n by 2), label and n.

Standard shapes

fuzzy_number_trapezoid()

trapezoidal, ⁠(l, cl, cr, r)⁠.

fuzzy_number_triangle()

triangular, a special case.

fuzzy_number_interval()

flat core: the value is plausible to degree one on the whole interval (a "vague period").

fuzzy_number_crisp()

a point mass; a degenerate fuzzy number.

A crisp vector is accepted wherever a fuzzy number is expected and is promoted with fuzzy_number_crisp(), so a fuzzy estimator reproduces the classical one exactly on crisp data.

See Also

alpha_cuts(), membership(), defuzz(), fkm(), fcox()

Examples

# "the failure occurred in the second quarter" (months 4 to 6)
q2 <- fuzzy_number_interval(4, 6)
alpha_cuts(q2, c(0, 0.5, 1))

# a rounded vague time: certainly between 3 and 9, surely at 5-7
fuzzy_number_trapezoid(3, 5, 7, 9)

# membership functions
fuzzy_number_triangle(0, 1, 2)

Width (diffusion) of fuzzy numbers

Description

Width (diffusion) of fuzzy numbers

Usage

fuzzy_spread(f, level = 0)

Arguments

Value

Numeric vector or matrix of widths ⁠f^+_alpha - f^-_alpha⁠.

Examples

fuzzy_spread(fuzzy_number_triangle(1, 2, 3), c(0, 0.5, 1))

Survival data with vague times or vague failure indicators

Description

fuzzy_surv() is the fuzzySurvival analogue of survival::Surv(): it bundles a vector of (possibly fuzzy) times with a vector of failure indicators, so that the estimators can be used inside model formulas.

Usage

fuzzy_surv(time, status, ...)

Arguments

Details

The failure indicator may itself be vague. A numeric status in ⁠[0, 1]⁠ is read as a degree of belief that the subject failed, and subjects whose indicator is strictly between 0 and 1 are handled by the fractional Kaplan-Meier estimator (they enter the risk set with a fractional weight). This models records such as "lost to follow-up, last seen alive, but an unobserved event cannot be excluded".

Value

An object of class fuzzy_surv, a list with components time (a fuzzy_number), status (numeric) and n.

See Also

fkm(), fcox()

Examples

# crisp times
fuzzy_surv(c(2, 5, 7, 9), c(1, 0, 1, 1))

# vague event times: "failed during the second quarter" for subject 2
tm <- fuzzy_number_crisp(c(2, 6, 9))
tm[2] <- fuzzy_number_interval(4, 6)
fuzzy_surv(tm, c(1, 1, 1))

# vague censoring: subject 2 failed with degree of belief 0.5
fuzzy_surv(c(2, 5, 7, 9), c(1, 0.5, 1, 1))

Is a fuzzy number crisp?

Description

Is a fuzzy number crisp?

Usage

is_crisp(f, tol = 1e-08)

Arguments

Value

A logical vector of length length(f).

Examples

is_crisp(c(fuzzy_number_crisp(3), fuzzy_number_interval(1, 2)))

Test whether an object is a fuzzy number

Description

Test whether an object is a fuzzy number

Usage

is_fuzzy_number(x)

is.fuzzy_number(x)

Arguments

Value

TRUE if x inherits from fuzzy_number.

Examples

is.fuzzy_number(fuzzy_number_triangle(1, 2, 3))
is.fuzzy_number(1:3)

Plot fuzzy Kaplan-Meier curves

Description

Draws the alpha-cut bounds of the fuzzy survival function. With type = "diff" the fuzzy difference of two groups' curves is drawn instead, the fuzzy counterpart of a survival difference plot.

Usage

## S3 method for class 'fkm'
plot(
  x,
  type = c("surv", "diff", "table"),
  levels = c(0, 0.25, 0.5, 0.75, 1),
  cols = c("#2166AC", "#B2182B"),
  fill = 80,
  xlab = "Time",
  ylab = "Survival probability",
  main = "Fuzzy Kaplan-Meier estimate",
  lty = 1,
  legend = TRUE,
  times = NULL,
  ...
)

Arguments

Value

NULL invisibly; called for the side effect.

Examples

data(fuzzy_dfs)
fit <- fkm(fuzzy_surv(ft, fs) ~ 1, data = fuzzy_dfs)
plot(fit)

fit2 <- fkm(fuzzy_surv(ft, fs) ~ arm, data = fuzzy_dfs)
plot(fit2, type = "diff")

Plot the membership function of fuzzy numbers

Description

Plot the membership function of fuzzy numbers

Usage

## S3 method for class 'fuzzy_number'
plot(
  x,
  y,
  n = 20,
  main = "Membership function",
  xlab = "Value",
  ylab = "Membership",
  ...
)

Arguments

Value

NULL invisibly; called for the side effect.

Examples

plot(c(fuzzy_number_trapezoid(0, 1, 2, 4), fuzzy_number_interval(5, 6)))

Simulate survival data with vague event times and vague indicators

Description

Generates true event times from a Weibull model, applies censoring, and then coarsens the information the way a real record does: the failure is only known to have occurred between two examinations, and a fraction of the censored records carries an uncertain failure status.

Usage

sim_fuzzy_surv(
  n = 200,
  beta = c(age = 0.3, arm = -0.6),
  mu = log(24),
  sigma = 0.6,
  shape = 1.4,
  cov_mean = 0,
  cov_sd = 1,
  censor_max = NULL,
  censor_rate = 1/40,
  exam_every = 3,
  prop_vague = 0.6,
  prop_uncertain = 0.2,
  seed = NULL
)

Arguments

Details

The true times and statuses are returned alongside the fuzzy ones, so that the coverage of the fuzzy estimators can be checked directly.

Value

A data frame with the covariates, the fuzzy time column ft, the fuzzy status column fs, and the true_time/true_status used to generate them.

See Also

fkm(), fcox()

Examples

set.seed(1)
d <- sim_fuzzy_surv(n = 60, beta = c(age = 0.4, arm = -0.5))
head(d[, c("arm", "fs")])
fit <- fkm(fuzzy_surv(ft, fs) ~ arm, data = d)
plot(fit, type = "diff")

Summaries and plots of a fuzzy Cox fit

Description

Summaries and plots of a fuzzy Cox fit

Usage

## S3 method for class 'fcox'
summary(
  object,
  conf.int = 0.95,
  exponentiate = FALSE,
  digits = 4,
  method = "centroid",
  ...
)

## S3 method for class 'fcox'
print(x, conf.int = 0.95, ...)

## S3 method for class 'fcox'
coef(object, exponentiate = FALSE, ...)

## S3 method for class 'fcox'
plot(
  x,
  type = c("hr", "membership"),
  conf.int = 0.95,
  xlab = "Hazard ratio",
  main = "Fuzzy Cox model",
  legend = TRUE,
  ...
)

Arguments

Value

summary() returns a list with the coefficient table and the number of subjects and events; the plot method returns NULL invisibly.

Examples

data(fuzzy_dfs)
fit <- fcox(fuzzy_surv(ft, fs) ~ age + arm, data = fuzzy_dfs)
summary(fit)
plot(fit)

Summaries of a fuzzy Kaplan-Meier fit

Description

Summaries of a fuzzy Kaplan-Meier fit

Usage

## S3 method for class 'fkm'
summary(
  object,
  times = NULL,
  digits = 4,
  method = "centroid",
  conf.int = 0.95,
  alpha = 0,
  ...
)

## S3 method for class 'fkm_summary'
print(x, ...)

Arguments

Value

A data frame with one row per time and columns for the alpha-cut bounds lower/upper, the defuzzified value surv, the numbers at risk, of events and censored, and the band on the defuzzified curve.