Package 'fuzzySurvival'

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

x, f	A fuzzy_number object (f is a fuzzy number, x is generic in the underlying function).
level	Alpha level(s) at which to evaluate the cut, in ⁠[0, 1]⁠.
at	Points at which to evaluate the membership function.
by	Spacing of the level grid used by membership() and defuzz(); 1/by levels are used in total.

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

f	A fuzzy_number.
level	Alpha level of the representative.
method	"midpoint" for the middle of the alpha-cut, "lower" or "upper" for its end points. "midpoint" at level = 1 is the middle of the core, i.e. the most plausible value.

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

x	A fuzzy_number, a list column holding one, or a numeric vector (promoted to a crisp fuzzy number).

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

...	fuzzy_number objects (or numeric vectors, which are promoted to crisp fuzzy numbers) of equal length, or of length one.

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

x	A fuzzy_number, or an object with a fuzzy_number method (such as the fuzzy survival curve returned by fkm_fuzzy_surv()).
method	Defuzzification rule; see Details. A vector of rules returns one row per rule.
level	Alpha level, only used by method = "midrange".
by	Spacing of the level grid used for the integrals.
...	Further arguments, such as times when x is a fitted estimator.

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

x	A fcox fit.
method	Defuzzification rule; see Details. A vector of rules returns one row per rule.
level	Alpha level, only used by method = "midrange".
by	Spacing of the level grid used for the integrals.
conf.int	Confidence level of the (fuzzy) confidence interval.
exponentiate	Return defuzzified hazard ratios instead of coefficients.

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

x	An fkm fit.
method	Defuzzification rule; see Details. A vector of rules returns one row per rule.
level	Alpha level, only used by method = "midrange".
by	Spacing of the level grid used for the integrals.
times	Times at which the survival probability is evaluated; defaults to the distinct times of the fit.

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

x	A fuzzy_number.

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

formula	A model formula such as fuzzy_surv(time, status) ~ age + arm.
data	A data frame containing the variables of formula.
time, status	Accepted instead of formula.
covariates	A data frame or matrix of covariates, used instead of the right hand side of formula.
levels	Alpha levels at which the model is fitted.
ties, ...	Passed to survival::coxph() when the model is refitted at each end of the alpha-cuts.

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

beta	a fuzzy_number of regression coefficients,
se	standard errors of the level 1 fit,
var	variance of the level 1 fit,
fit	the crisp survival::coxph() fit at level 1,
p.value	a fuzzy_number of attainable p-values.

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

formula	A model formula, e.g. fuzzy_surv(time, status) ~ group, or a fuzzy_surv object.
data	A data frame containing the variables of formula.
time, status	Accepted as an alternative to formula: a vector of fuzzy or crisp times, and a vector of failure indicators.
group	Optional grouping variable; fits a separate curve per level.
levels	Alpha levels at which the bounds are evaluated. Level 0 (the support, i.e. the completely vague case) and level 1 (the core, i.e. the fully plausible case) are always included.
conf.type	Transformation used for the confidence bands of the two bounds, as in survival::survfit().
conf.int	Confidence level of the bands.
...	Further arguments passed to survival::survfit() or survival::coxph().

Value

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

time	the distinct times at which the curves are evaluated.
lower, upper	matrices of the lower and upper survival bounds, one row per alpha level.
levels	the alpha levels of those rows.
surv	the defuzzified curve (see defuzz()).
n.risk, n.event	numbers at risk and of events.

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

x	An fkm fit.
times	Times at which the survival probability is evaluated. Defaults to the distinct times of the fit.

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

formula	A model formula fuzzy_surv(time, status) ~ group.
data	A data frame containing the variables of formula.
time, status, group	Accepted instead of formula.
levels	Alpha levels at which the statistic is evaluated.
...	Further arguments, ignored.

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")

---

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

l, cl, cr, r	Left and right end points and core end points of the trapezoid; must satisfy ⁠l <= cl <= cr <= r⁠. All must be finite.
label	Optional character label (e.g. a variable name).
a, m, b	Left end point, mode and right end point of a triangle.
lo, hi	End points of a flat (crisp-valued on the interval) fuzzy number.
x	Values of crisp fuzzy numbers.
acut	Function of a vector of alpha levels returning a list with matrices lower and upper of dimension n by length(alpha).
support	Optional n by 2 matrix of support end points; by default the alpha-cut at level 0 is used.

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

f	A fuzzy_number.
level	Alpha level(s).

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

time	A fuzzy_number vector of event times, a plain numeric vector of crisp times, or a list of length n of fuzzy numbers.
status	Numeric vector of failure indicators. Values in ⁠[0, 1]⁠ give the degree of belief that the event occurred; values outside ⁠[0, 1]⁠ are an error.
...	Additional components stored in the result, e.g. a grouping or stratification variable.

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))

---

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

e1, e2

fuzzy_number objects or numeric vectors (promoted to crisp fuzzy numbers). Their lengths must be equal, or one of them must be one.

Value

A fuzzy_number of the common length.

Examples

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

---

Is a fuzzy number crisp?

Description

Is a fuzzy number crisp?

Usage

is_crisp(f, tol = 1e-08)

Arguments

f	A fuzzy_number.
tol	Tolerance for comparing the end points of the alpha-cuts.

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

x	Any object.

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

x	An fkm fit (a grouped fit for type = "diff").
type	"surv" for the survival curve, "diff" for the difference between the first two groups, "table" for the numbers at risk.
levels	Alpha levels whose bounds are drawn.
cols	Colours for the lower and upper bound.
fill	Alpha transparency of the shaded region between the bounds.
xlab, ylab, main	Axis labels and title.
lty	Line type of the defuzzified curve.
legend	Whether to draw a legend.
times	Times at which the difference is evaluated.
...	Further graphical parameters.

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

x	A fuzzy_number.
y	Unused, for compatibility with the generic.
n	Maximum number of fuzzy numbers to draw.
main, xlab, ylab	Plot labels.
...	Further graphical parameters.

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

n	Number of subjects.
beta	Named vector of log hazard ratios, one per covariate column.
mu, sigma, shape	Weibull parameters: $T = \exp(\mu + \sigma W)$ with $W$ Weibull of the given shape.
cov_mean, cov_sd	Mean and standard deviation of the covariates; a two-level covariate (a treatment arm) is created when any name of beta contains "arm".
censor_max, censor_rate	Censoring: a uniform cut-off on censor_max or an exponential rate censor_rate on the observed time scale.
exam_every	Spacing of the examinations, in time units. A failure between two examinations is reported as a vague time spanning them.
prop_vague	Proportion of the failures that are reported as vague (interval) times rather than exact ones.
prop_uncertain	Proportion of the censored records whose failure status is uncertain; their degree of belief is drawn uniformly on ⁠[0.4, 0.9]⁠.
seed	Optional seed for reproducibility.

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")

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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

object	A fcox fit.
conf.int	Confidence level of the reported intervals.
exponentiate	Report hazard ratios instead of coefficients.
digits	Number of significant digits.
method	Defuzzification rule, see defuzz().
...	Further arguments, ignored by summary().
x	A fcox fit.
type	Either "hr" for a forest plot of the fuzzy hazard ratios, or "membership" for their membership functions.
xlab, main, legend	Plot labels, and whether to draw a legend.

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

object	An fkm fit.
times	Times at which to report the curves.
digits	Number of significant digits.
method	Defuzzification rule, see defuzz().
conf.int	Confidence level of the reported bands.
alpha	Alpha level of the reported band, 0 being the widest.
...	Further arguments, ignored by summary().
x	The data frame returned by summary(), used by its print method.

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.

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