Package 'stagelife'

Extract coefficients from a fitted two-stage life testing model

Description

Extract coefficients from a fitted two-stage life testing model

Usage

## S3 method for class 'slt_fit'
coef(object, ...)

Arguments

object	An "slt_fit" object from fit_slt().
...	Currently unused.

Value

The named numeric vector object$par (natural, positive scale; names and length depend on object$case, see slt_cases()).

---

Asymptotic (Wald) confidence intervals for a fitted two-stage life testing model

Description

Two Wald-type intervals are available: "normal" and "lognormal".

Usage

## S3 method for class 'slt_fit'
confint(object, parm, level = 0.95, type = c("lognormal", "normal"), ...)

Arguments

object	An "slt_fit" object from fit_slt().
parm	Optional character vector of parameter names to include (default: all).
level	Confidence level.
type	"lognormal" (default) or "normal".
...	Currently unused.

Details

"lognormal" (the default) builds the interval on the log(theta) scale (where the MLE is asymptotically normal by construction) and exponentiates, which guarantees a positive interval and is generally preferred for these positive shape/scale parameters. "normal" applies the delta-method standard error directly on the natural scale and can (rarely) produce a negative lower limit for small samples.

Value

A numeric matrix with one row per parameter and two columns (lower, upper).

Examples

set.seed(1)
d <- rslt(200, "weibull", c(0.05, 1.5), c(0.2, 1.5), tau1 = 5, prop_transfer = 1)
fit <- fit_slt(d, "weibull", "shape1", tau1 = 5)
confint(fit)

---

Fit a two-stage life testing model by maximum likelihood

Description

Fit the stage life testing model for given data using the cumulative-exposure method for one of 12 baseline distributions and one of three parameter cases.

Usage

fit_slt(
  data,
  family = "weibull",
  case = c("common", "shape1", "shape11"),
  tau1,
  init = NULL,
  n_starts = 4L,
  bootstrap = FALSE,
  control = list()
)

Arguments

data	A list with numeric vectors Y (stage-0 observed failure times) and Z (stage-1 observed failure times), as produced by rslt() (or assembled directly from real data).
family	One of slt_families().
case	One of "common", "shape1", "shape11" (see slt_cases()).
tau1	The stage-change time, used to shift Z back to the stage-1 origin.
init	Optional numeric vector of starting values on the natural scale, in the order given by case_info(case)$par_names (see slt_cases() for the parameter order for each case). If NULL a data-driven default is used.
n_starts	Number of (perturbed) starting values to try; the converged fit with the highest log-likelihood is returned.
bootstrap	If TRUE, use looser/faster optimiser control settings appropriate for repeated bootstrap refits.
control	Optional list overriding stats::optim() control parameters for both the Nelder-Mead and BFGS stages.

Details

Estimation uses a Nelder-Mead warm start followed by BFGS (with a numerical Hessian for inference), repeated from several perturbed starting values to guard against local optima; the converged fit with the highest log-likelihood is returned.

Value

An object of class "slt_fit", a list with elements including converged, family, case, tau1, par (natural-scale estimates), se, theta0, theta1, vcov_log, loglik, aic, bic. If no starting value converges, converged is FALSE and most other elements are absent.

Examples

set.seed(1)
d <- rslt(150, "weibull", c(0.05, 1.5), c(0.2, 1.5), tau1 = 5, prop_transfer = 1)
fit <- fit_slt(d, family = "weibull", case = "shape1", tau1 = 5)
summary(fit)

---

Two-stage cumulative-exposure log-likelihood

Description

Two-stage cumulative-exposure log-likelihood

Usage

loglik_slt(data, family, theta0, theta1, tau1)

Arguments

data	A list with numeric vectors Y (stage-0 observed failure times) and Z (stage-1 observed failure times), as returned by rslt().
family	A family key (see slt_families()) or family object.
theta0	Numeric c(lambda0, alpha0), baseline-stage parameters.
theta1	Numeric c(lambda1, alpha1), elevated-stage parameters.
tau1	The stage-change time used to shift Z back to the stage-1 time origin.

Value

The log-likelihood (a scalar), or -Inf for parameter values outside the support of the model.

Examples

set.seed(1)
d <- rslt(80, "weibull", c(0.05, 1.5), c(0.2, 1.5), tau1 = 5, prop_transfer = 1)
loglik_slt(d, "weibull", c(0.05, 1.5), c(0.2, 1.5), tau1 = 5)

---

Extract the log-likelihood of a fitted two-stage life testing model

Description

Extract the log-likelihood of a fitted two-stage life testing model

Usage

## S3 method for class 'slt_fit'
logLik(object, ...)

Arguments

object	An "slt_fit" object from fit_slt().
...	Currently unused.

Value

An object of class "logLik" (the fitted log-likelihood, NA_real_ if object did not converge, with "df"/"nobs" attributes set for use by stats::AIC()/stats::BIC()).

---

Fitted vs empirical CDF plot

Description

Plots the fitted population CDF (pslt()) against the empirical CDF of the observed data.

Usage

plot_cdf(fit)

Arguments

fit	A converged "slt_fit" object, fit to data from a simple-step-stress (rslt() with prop_transfer = 1) design.

Details

Assumes every survivor was transferred at tau1 (the classical simple-step-stress design, prop_transfer = 1 in rslt()) – see the note in slt_gof_ks() for why: the pooled Y/Z sample only has pslt() as its correct marginal CDF in that case. This applies to all four diagnostic plots in this file.

Value

A ggplot object.

Examples

set.seed(1)
d <- rslt(150, "weibull", c(0.05, 1.5), c(0.2, 1.5), tau1 = 5, prop_transfer = 1)
fit <- fit_slt(d, "weibull", "shape1", tau1 = 5)
plot_cdf(fit)

---

P-P (probability-probability) plot

Description

P-P (probability-probability) plot

Usage

plot_pp(fit)

Arguments

fit	A converged "slt_fit" object, fit to data from a simple-step-stress (rslt() with prop_transfer = 1) design.

Value

A ggplot object.

Examples

set.seed(1)
d <- rslt(150, "weibull", c(0.05, 1.5), c(0.2, 1.5), tau1 = 5, prop_transfer = 1)
fit <- fit_slt(d, "weibull", "shape1", tau1 = 5)
plot_pp(fit)

---

Q-Q (quantile-quantile) plot

Description

Q-Q (quantile-quantile) plot

Usage

plot_qq(fit)

Arguments

fit	A converged "slt_fit" object, fit to data from a simple-step-stress (rslt() with prop_transfer = 1) design.

Value

A ggplot object.

Examples

set.seed(1)
d <- rslt(150, "weibull", c(0.05, 1.5), c(0.2, 1.5), tau1 = 5, prop_transfer = 1)
fit <- fit_slt(d, "weibull", "shape1", tau1 = 5)
plot_qq(fit)

---

Fitted vs empirical survival curve

Description

Fitted vs empirical survival curve

Usage

plot_survival(fit)

Arguments

fit	A converged "slt_fit" object, fit to data from a simple-step-stress (rslt() with prop_transfer = 1) design.

Value

A ggplot object.

Examples

set.seed(1)
d <- rslt(150, "weibull", c(0.05, 1.5), c(0.2, 1.5), tau1 = 5, prop_transfer = 1)
fit <- fit_slt(d, "weibull", "shape1", tau1 = 5)
plot_survival(fit)

---

Diagnostic plots for a fitted two-stage life-testing model

Description

See plot_cdf() for an important caveat: these assume the classical simple-step-stress design (prop_transfer = 1 in rslt()).

Usage

## S3 method for class 'slt_fit'
plot(x, which = c("cdf", "survival", "pp", "qq", "all"), ...)

Arguments

x	A converged "slt_fit" object.
which	One of "cdf", "survival", "pp", "qq", or "all" (returns a named list of all four ggplot objects).
...	Currently unused.

Value

A ggplot object, or (if which = "all") a named list of four ggplot objects.

Examples

set.seed(1)
d <- rslt(150, "weibull", c(0.05, 1.5), c(0.2, 1.5), tau1 = 5, prop_transfer = 1)
fit <- fit_slt(d, "weibull", "shape1", tau1 = 5)
plot(fit, which = "cdf")

---

Print a fitted two-stage life testing model

Description

Print a fitted two-stage life testing model

Usage

## S3 method for class 'slt_fit'
print(x, digits = 4, ...)

Arguments

x	An "slt_fit" object from fit_slt().
digits	Number of digits to round to.
...	Currently unused.

Value

x, invisibly.

See Also

summary.slt_fit() for a more detailed summary with confidence limits.

---

Print a summary of a fitted two-stage life testing model

Description

Print a summary of a fitted two-stage life testing model

Usage

## S3 method for class 'summary.slt_fit'
print(x, digits = 4, ...)

Arguments

x	A "summary.slt_fit" object from summary.slt_fit().
digits	Number of digits to round to.
...	Currently unused.

Value

x, invisibly.

---

Population CDF/PDF/quantile function of a two-stage step-stress model

Description

pslt(), dslt() and qslt() implement the full stage life testing design, in which every unit switches from the baseline to the elevated stress at a common change-point tau1. Combined with the cumulative-exposure equivalent age slt_equiv_age(), the two branches (pre-/post-tau1) are pieced together into a single, continuous population CDF.

Usage

pslt(t, family, theta0, theta1, tau1)

dslt(x, family, theta0, theta1, tau1)

qslt(p, family, theta0, theta1, tau1)

Arguments

t, x	Numeric vector of times at which to evaluate the CDF/PDF.
family	A family key (see slt_families()) or family object.
theta0	Numeric c(lambda0, alpha0), baseline-stage parameters.
theta1	Numeric c(lambda1, alpha1), elevated-stage parameters.
tau1	Scalar change-point time (same argument as tau in slt_equiv_age()).
p	Numeric vector of probabilities at which to evaluate the quantile function.

Details

For a single unit switching from stage 0 to stage 1 at the fixed time tau1 (every unit switches – the classical simple-step-stress design, matching prop_transfer = 1 in rslt()), the CEM construction (see slt_equiv_age()) gives a population cdf

$F(t;\theta_0,\theta_1,\tau_1) = \begin{cases} F_0(t;\theta_0), & 0 < t \le \tau_1, \\ F_1(t-\tau_1+\nu_1;\ \theta_1), & t > \tau_1, \end{cases}$

where $\nu_1 = \nu(\tau_1;\theta_0,\theta_1)$ is the stage-1 equivalent age (slt_equiv_age()) at the moment of the switch. Differentiating – a constant shift of the argument by $\tau_1-\nu_1$, not a general change of variables, so no extra Jacobian factor arises – gives the matching population pdf

$f(t;\theta_0,\theta_1,\tau_1) = \begin{cases} f_0(t;\theta_0), & 0 < t \le \tau_1, \\ f_1(t-\tau_1+\nu_1;\ \theta_1), & t > \tau_1. \end{cases}$

$F$ is continuous at $\tau_1$ by construction of $\nu_1$ ($F_1(\tau_1-\tau_1+\nu_1;\theta_1) = F_1(\nu_1;\theta_1) = F_0(\tau_1;\theta_0)$, verified as an automatic consequence of the equal-reliability definition of $\nu_1$, not a separate assumption) – but $F$ is generally not differentiable at $\tau_1$ (the hazard rate jumps there), an intrinsic, well-known feature of the cumulative exposure model, not an artefact of this implementation. qslt() inverts $F$ directly (analytically, not numerically): for $p \le F_0(\tau_1;\theta_0)$ it returns $F_0^{-1}(p;\theta_0)$; otherwise it returns $\tau_1-\nu_1+ F_1^{-1}(p;\theta_1)$, using each stage's own quantile function (itself closed-form for every family in slt_families()).

Value

A numeric vector the same length as t/x/p.

See Also

slt_equiv_age() for nu1; loglik_slt() for the estimation log-likelihood; rslt() for simulating data under the actual random-withdrawal design; fit_slt() for maximum likelihood estimation.

Examples

pslt(c(1, 6, 10), "weibull", c(0.1, 1.5), c(0.3, 1.5), tau1 = 5)

---

Simulate data from the two-stage life-testing model

Description

Simulates data from the two-stage life-testing model of Laumen & Cramer (2019, Procedure 2.3).

Usage

rslt(
  n,
  family,
  theta0,
  theta1,
  tau1,
  design = c("typeP", "typeM"),
  prop_transfer = 0.5,
  R0_1 = NULL
)

Arguments

n	Number of units placed on test.
family	A family key (see slt_families()) or family object.
theta0	Numeric c(lambda0, alpha0), baseline-stage parameters.
theta1	Numeric c(lambda1, alpha1), elevated-stage parameters.
tau1	Time at which the transfer decision is made.
design	"typeP" (default, fixed target proportion) or "typeM" (fixed target count); see Details.
prop_transfer	Proportion (in ⁠[0, 1]⁠) of survivors at tau1 transferred to the elevated stage, for design = "typeP". Set to 1 for the classical simple-step-stress design.
R0_1	Target number of survivors transferred (capped by the number actually available), for design = "typeM".

Details

At the stage-change time tau1, ⁠R*_1⁠ of the units that have not yet failed are randomly selected and moved to the elevated stage (cumulative-exposure shifted); the remaining survivors continue failing under the original stage-0 conditions and are never observed under stage 1.

Two rules for generating ⁠R*_1⁠ are supported, following the paper:

"typeP"

A fixed target proportion pi1 of the survivors is transferred: ⁠R*_1 = floor(pi1 * (n - D1))⁠, where D1 is the (random) number of stage-0 failures observed before tau1. pi1 = 1 transfers every survivor, which is exactly the classical "simple step-stress" design (Remark 2.2(4) of the paper) – no separate function is needed for that special case.

"typeM"

A fixed target count R0_1 is transferred, capped by the number of available survivors: ⁠R*_1 = min(n - D1, R0_1)⁠.

Value

A list with Y (all stage-0 failure times: early failures before tau1 plus the non-transferred survivors, who keep failing under stage-0), Z (stage-1 observed failure times for the transferred units), n_early (D1, number failing before tau1) and n_transferred (⁠R*_1⁠).

Examples

set.seed(1)
d <- rslt(100, "weibull", theta0 = c(0.05, 1.5), theta1 = c(0.2, 1.5),
          tau1 = 5, design = "typeP", prop_transfer = 0.5)
str(d)

d_m <- rslt(100, "weibull", theta0 = c(0.05, 1.5), theta1 = c(0.2, 1.5),
            tau1 = 5, design = "typeM", R0_1 = 30)
str(d_m)

# The classical simple-step-stress design (every survivor transferred):
d_full <- rslt(100, "weibull", theta0 = c(0.05, 1.5), theta1 = c(0.2, 1.5),
               tau1 = 5, design = "typeP", prop_transfer = 1)
str(d_full)

---

Bobotas & Kateri (2015) simple step-stress data

Description

A simple step-stress life-testing data set on the charge detrapping and dielectric breakdown of nanocrystalline embedded high-k devices, with 15 units observed under the baseline stress and 23 under the elevated stress after a common change-point at tau1 = 600 (seconds).

Usage

slt_bobotas_kateri_2015

Format

A list with elements y0, y1, tau1, description, source (see slt_han_kundu_2014 for the structure).

Source

Bobotas, P. and Kateri, M. (2015). The step-stress tampered failure rate model under interval monitoring. Statistical Methodology, 27, 100-122.

See Also

slt_han_kundu_2014, slt_wang_fei_2003, slt_zhu_2010.

Examples

data(slt_bobotas_kateri_2015)
str(slt_bobotas_kateri_2015)

---

Bootstrap confidence intervals for a fitted two-stage life testing model

Description

Computes percentile, bootstrap-t, and log-scale bootstrap-t confidence intervals via either non-parametric or a parametric bootstrap.

Usage

slt_bootstrap_ci(
  fit,
  B = 1000,
  level = 0.95,
  method = c("nonparametric", "parametric"),
  design = c("typeP", "typeM"),
  prop_transfer = 1,
  R0_1 = NULL,
  n_cores = 1L
)

Arguments

fit	A converged "slt_fit" object.
B	Number of bootstrap replicates.
level	Confidence level.
method	"nonparametric" (default) or "parametric".
design	For method = "parametric", the design used to simulate new data via rslt(): "typeP" (fixed target proportion, the default) or "typeM" (fixed target count).
prop_transfer	Proportion transferred, for design = "typeP". Defaults to 1 (every survivor transferred, the classical simple-step-stress design, matching a typical published step-stress data set); set to the actual value used in your experiment if it was a genuine partial-withdrawal design.
R0_1	Target number transferred, for design = "typeM".
n_cores	Number of cores for parallel::mclapply() (ignored on Windows, where replicates run sequentially).

Details

Non-parametric bootstrap resamples (case resampling) the observed Y/Z failure times; parametric bootstrap simulates fresh data from the fitted model at the MLE. The parametric bootstrap is generally more efficient when the fitted model is (approximately) correctly specified – see slt_gof_ks() – but is more sensitive to misspecification; the non-parametric bootstrap is robust to misspecification but, for small samples, can only reproduce combinations of already-observed values.

The plain (natural-scale) bootstrap-t interval, like the asymptotic Wald interval from confint(..., type = "normal"), is an ordinary linear pivot (⁠estimate -/+ quantile * se⁠) and can occasionally produce a negative lower bound for a positive parameter, especially at small n or for a skewed bootstrap distribution. The log-scale bootstrap-t interval applies the same pivot construction on log(theta) and exponentiates the result, which guarantees a positive interval by construction – directly analogous to why confint(..., type = "lognormal") (the default) is generally preferred over type = "normal" for the asymptotic interval. The percentile interval is always positive already (it is built from quantiles of the always-positive bootstrap estimates themselves), so only bootstrap-t needed a log-scale counterpart.

Value

An object of class "slt_boot_ci": a list with percentile, boot_t, and log_boot_t data.frames (columns lower, upper, one row per parameter), B, B_converged, method, design, level, and the raw boot_estimates matrix.

Examples

set.seed(1)
d <- rslt(150, "weibull", c(0.05, 1.5), c(0.2, 1.5), tau1 = 5, prop_transfer = 1)
fit <- fit_slt(d, "weibull", "shape1", tau1 = 5)
slt_bootstrap_ci(fit, B = 100)

---

Identifiable parameter-sharing cases for two-stage estimation

Description

Describes the three parameter-sharing structures between the baseline-stage parameters ⁠theta0 = (lambda0, alpha0)⁠ and the elevated-stage parameters ⁠theta1 = (lambda1, alpha1)⁠ supported for estimation. A fourth, fully unrestricted structure (alpha0 and alpha1 both free) is deliberately not supported; see Details.

Usage

slt_cases()

Details

Four parameter-sharing structures are logically possible for the pair $\theta_0=(\lambda_0,\alpha_0)$, $\theta_1=(\lambda_1,\alpha_1)$:

$\begin{array}{lll} \texttt{full} : & \theta_0=(\lambda_0,\alpha_0), & \theta_1=(\lambda_1,\alpha_1) \quad\text{(4 free parameters)} \\ \texttt{common} : & \theta_0=(\lambda_0,\alpha), & \theta_1=(\lambda_1,\alpha) \quad\text{(3 free parameters, shared shape)} \\ \texttt{shape1} : & \theta_0=(\lambda_0,\alpha_0), & \theta_1=(\lambda_1,1) \quad\text{(3 free parameters)} \\ \texttt{shape11} : & \theta_0=(\lambda_0,1), & \theta_1=(\lambda_1,1) \quad\text{(2 free parameters)} \end{array}$

This package supports the last three; full is deliberately not offered (see "Why full is excluded" below).

shape1/shape11 fix the elevated-stage shape to the numeric value 1 – this is a structural constraint, not a claim that the elevated stage becomes Exponential. The two only coincide for the four families whose alpha = 1 boundary happens to be Exponential (weibull, expexponential, moexponential, gamma; see ?slt_dist); for the other eight families, alpha1 = 1 is simply that family's own (otherwise unnamed) member at shape 1 – e.g. Chen with its shape fixed at 1, not an Exponential elevated stage. Fixing the shape is still meaningful regardless: it reduces the free parameter count the same way for every family (see "Why full is excluded" below), and "is the elevated-stage shape exactly 1?" is a concrete, testable nested hypothesis via slt_lrt() whether or not that constraint has a textbook name.

Each case defines a bijection between its free-parameter vector par (on the natural, positive scale, in the order given by case_info(case)$par_names) and ⁠(theta0, theta1)⁠, implemented by slt_expand_params() and inverted by slt_reduce_params():

$\texttt{common}:\ (\lambda_0,\lambda_1,\alpha) \mapsto \big((\lambda_0,\alpha),\ (\lambda_1,\alpha)\big),$

$\texttt{shape1}:\ (\lambda_0,\alpha_0,\lambda_1) \mapsto \big((\lambda_0,\alpha_0),\ (\lambda_1,1)\big),$

$\texttt{shape11}:\ (\lambda_0,\lambda_1) \mapsto \big((\lambda_0,1),\ (\lambda_1,1)\big).$

fit_slt() optimises the log-likelihood (loglik_slt(), via neg_loglik_slt()) over log(par) in this order, then reports the fitted par under exactly these names – so, for instance, case = "shape1"'s second free parameter is always labelled alpha0 and is always the baseline-stage shape, never accidentally the elevated-stage rate or any other quantity. This parameter-index bookkeeping ( case_info()$par_names, slt_expand_params(), slt_reduce_params(), and fit_slt()'s internal default-initial-value heuristic all sharing one consistent order) is exactly the kind of place a silent mislabelling bug could hide, so it has been checked directly: the round trip slt_expand_params(slt_reduce_params(theta0, theta1, case), case) reproduces ⁠(theta0, theta1)⁠ exactly for all three cases, and a full simulate-then-refit check (large n, known theta0/theta1) recovers the true values under the correct parameter names for all three cases (see the package's test suite, test-fit.R).

full (both shapes free) is excluded because, with a single change-point and no replicated stress levels, its extra free parameter is empirically non-identifiable (a flat likelihood ridge, frequent non-convergence); shape1/shape11 remove this by fixing one or both shapes to 1, common by tying them together instead. shape11 is nested in both common and shape1 (the natural null model for slt_lrt()); shape1 and common are not nested in each other, so compare those two with slt_model_comparison() instead.

Value

A data.frame with columns case, n_par, parameters, description.

See Also

slt_expand_params() and slt_reduce_params() for the parameter mapping implementing each case; fit_slt() for estimation; slt_lrt() for the nested likelihood-ratio test; slt_model_comparison() for comparing non-nested cases.

Examples

slt_cases()

---

Baseline density, distribution and quantile functions for a distribution

Description

Returns the single-stage d/p/q functions, each of ⁠(x_or_p, lambda, alpha)⁠, for a baseline lifetime distribution. Useful for exploring a distribution in isolation from the two-stage step-stress structure. In every distribution, lambda > 0 is a rate/scale-type parameter and alpha > 0 is a shape/power parameter; see Details for the pdf/cdf of each and its literature source.

Usage

slt_dist(family)

Arguments

family

A family key from slt_families().

Details

Below, $F(x;\lambda,\alpha)$ and $f(x;\lambda,\alpha) = F'(x;\lambda,\alpha)$ denote the cdf and pdf for $x > 0$ (both are 0 for $x \le 0$). $\Phi$/$\phi$ are the standard normal cdf/pdf, and $\Gamma(\alpha, x) = \int_x^\infty t^{\alpha-1} e^{-t}\,dt$ is the upper incomplete gamma function.

Weibull (Weibull, 1951):

$F(x;\lambda,\alpha) = 1 - \exp(-\lambda x^\alpha), \qquad f(x;\lambda,\alpha) = \lambda\alpha x^{\alpha-1}\exp(-\lambda x^\alpha).$

Reduces to Exponential(rate = $\lambda$) at $\alpha = 1$.

Power Lindley (Ghitany, Al-Mutairi, Balakrishnan & Al-Enezi, 2013):

$F(x;\lambda,\alpha) = 1 - \frac{1+\lambda+\lambda x^\alpha}{1+\lambda} \exp(-\lambda x^\alpha), \qquad f(x;\lambda,\alpha) = \frac{\alpha\lambda^2}{1+\lambda} (1+x^\alpha)\,x^{\alpha-1}\exp(-\lambda x^\alpha).$

Reduces to (ordinary) Lindley($\lambda$) at $\alpha = 1$ (not Exponential). The quantile function inverts a Lindley-type equation via the $W_{-1}$ branch of the Lambert $W$ function (lamW::lambertWm1()): the survival function's numerator-times- exponential (product) structure reduces to the $w e^w = z$ form $W$ solves.

Exponentiated Lindley (Nadarajah, Bakouch & Tahmasbi, 2011): with baseline Lindley cdf/pdf $G_0(x;\lambda) = 1 - \frac{1+\lambda+\lambda x}{1+\lambda}\exp(-\lambda x)$ and $g_0(x;\lambda) = \frac{\lambda^2}{1+\lambda}(1+x)\exp(-\lambda x)$ (Power Lindley's $\alpha = 1$ case above),

$F(x;\lambda,\alpha) = G_0(x;\lambda)^\alpha, \qquad f(x;\lambda,\alpha) = \alpha\,g_0(x;\lambda)\,G_0(x;\lambda)^{\alpha-1}.$

Reduces to (ordinary) Lindley($\lambda$) at $\alpha = 1$ (not Exponential) – the same limit as Power Lindley above, of which this is the "exponentiate the cdf" sibling (exactly as Exponentiated Teissier and Power Muth are the two siblings of the Teissier baseline). Its quantile function reuses the same Lambert-$W$ baseline-Lindley inverter as Power Lindley, just inverting a different target probability.

Exponentiated Exponential (Gupta & Kundu, 1999):

$F(x;\lambda,\alpha) = \left(1-\exp(-\lambda x)\right)^\alpha, \qquad f(x;\lambda,\alpha) = \alpha\lambda\exp(-\lambda x) \left(1-\exp(-\lambda x)\right)^{\alpha-1}.$

Reduces to Exponential($\lambda$) at $\alpha = 1$.

Marshall-Olkin Exponential (Marshall & Olkin, 1997):

$F(x;\lambda,\alpha) = 1 - \frac{\alpha\,e^{-\lambda x}} {1-(1-\alpha)e^{-\lambda x}}, \qquad f(x;\lambda,\alpha) = \frac{\lambda\alpha\,e^{-\lambda x}} {\left(1-(1-\alpha)e^{-\lambda x}\right)^2}.$

Reduces to Exponential($\lambda$) at $\alpha = 1$. Constructed by a different mechanism than the exponentiated/power-transform families here – Marshall & Olkin's "tilt parameter" construction, applicable to any baseline survival function – with $\alpha$ as the tilt parameter (an odds-ratio-type parameter, interpretable via the family's proportional-odds property).

Lomax (Lomax, 1954):

$F(x;\lambda,\alpha) = 1 - (1+\lambda x)^{-\alpha}, \qquad f(x;\lambda,\alpha) = \alpha\lambda(1+\lambda x)^{-\alpha-1}.$

At $\alpha = 1$ this is simply the boundary member of the Lomax family itself (not a different named distribution).

Exponentiated Teissier (Sharma, Singh & Shekhawat, 2022): with baseline Teissier (1934) cdf/pdf $G_0(x;\lambda) = 1-\exp\{-(e^{\lambda x}-1-\lambda x)\}$ and $g_0(x;\lambda) = \lambda(e^{\lambda x}-1)\exp\{-(e^{\lambda x}-1-\lambda x)\}$,

$F(x;\lambda,\alpha) = G_0(x;\lambda)^\alpha, \qquad f(x;\lambda,\alpha) = \alpha\,g_0(x;\lambda)\,G_0(x;\lambda)^{\alpha-1},$

with quantile function (Sharma, Singh & Shekhawat, 2022, Proposition 4.2)

$Q(p;\lambda,\alpha) = \frac{1}{\lambda} \log\!\left(-W_{-1}\!\left(\frac{p^{1/\alpha}-1}{e}\right)\right),$

where $W_{-1}$ is the lower ($W_{-1}$) branch of the Lambert $W$ function (lamW::lambertWm1()). Reduces to baseline Teissier($\lambda$) at $\alpha = 1$ (the same limit as Power Muth below, of which this is the "exponentiate the cdf" sibling). The Teissier distribution is also known as the Muth distribution; see Power Muth below, both verified against the cited paper's own closed-form quantile.

Log-Logistic / Fisk (Fisk, 1961):

$F(x;\lambda,\alpha) = \frac{(\lambda x)^\alpha}{1+(\lambda x)^\alpha}, \qquad f(x;\lambda,\alpha) = \frac{\alpha\lambda(\lambda x)^{\alpha-1}} {\left(1+(\lambda x)^\alpha\right)^2}.$

The only family here with a non-monotonic (rise-then-fall) hazard rate for $\alpha > 1$; at $\alpha = 1$ it is simply Log-Logistic with shape 1 (not a different named distribution).

Chen (Chen, 2000):

$F(x;\lambda,\alpha) = 1 - \exp\left(\lambda\left(1-e^{x^\alpha}\right)\right), \qquad f(x;\lambda,\alpha) = \lambda\alpha x^{\alpha-1} e^{x^\alpha} \exp\left(\lambda\left(1-e^{x^\alpha}\right)\right).$

Hazard rate is increasing for $\alpha \ge 1$ and bathtub-shaped for $\alpha < 1$.

Log-normal: parameterised here as

$F(x;\lambda,\alpha) = \Phi\!\left(\frac{\log(\lambda x)}{\alpha}\right), \qquad f(x;\lambda,\alpha) = \frac{1}{\alpha x}\phi\!\left(\frac{\log(\lambda x)}{\alpha}\right),$

i.e. $\log X \sim N(-\log\lambda,\ \alpha^2)$: $1/\lambda$ is the median (not the mean) and $\alpha$ is the log-scale standard deviation, keeping "$\lambda$ a rate/scale parameter, $\alpha$ a shape parameter" consistent with every other family here. Built directly on stats::plnorm()-equivalent expressions (stats::pnorm()/stats::dnorm()/stats::qnorm()) rather than a hand-derived formula.

Gamma: standard shape-rate parameterisation,

$F(x;\lambda,\alpha) = 1 - \frac{\Gamma(\alpha,\lambda x)}{\Gamma(\alpha)}, \qquad f(x;\lambda,\alpha) = \frac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\lambda x}.$

Reduces to Exponential($\lambda$) at $\alpha = 1$. Built directly on stats::pgamma()/stats::dgamma()/stats::qgamma() rather than a hand-derived formula.

Power Muth (also known as Power Teissier; Jodra, Gomez, Jimenez-Gamero & Alba-Fernandez, 2017, generalising the baseline Muth/Teissier distribution of Muth, 1977, and Teissier, 1934, and itself studied via the Lambert $W$ function by Jodra, Jimenez-Gamero & Alba-Fernandez, 2015):

$F(x;\lambda,\alpha) = 1-\exp\left(-\left(e^{\lambda x^\alpha}-1-\lambda x^\alpha\right)\right), \qquad f(x;\lambda,\alpha) = \lambda\alpha x^{\alpha-1}\left(e^{\lambda x^\alpha}-1\right) \exp\left(-\left(e^{\lambda x^\alpha}-1-\lambda x^\alpha\right)\right),$

with quantile function

$Q(p;\lambda,\alpha) = \left(\frac{1}{\lambda} \log\!\left(-W_{-1}\!\left(\frac{p-1}{e}\right)\right)\right)^{1/\alpha},$

using the same Lambert-$W$ identity as Exponentiated Teissier above ($p$ in place of $p^{1/\alpha}$ there, since the power transform is applied to $x$ here rather than to the cdf). Reduces to baseline Teissier/Muth($\lambda$) at $\alpha = 1$ – the same limit as Exponentiated Teissier above, of which this is the "power the argument" sibling: both generalise the same baseline distribution, via $x \mapsto x^\alpha$ here versus $F_0 \mapsto F_0^\alpha$ there (exactly as Power Lindley and Exponentiated Exponential are, respectively, the "power the argument" and "exponentiate the cdf" generalisations of their own baselines).

Value

A list with functions d, p, q.

References

Chen, Z. (2000). A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statistics & Probability Letters, 49(2), 155-161.

Fisk, P. R. (1961). The graduation of income distributions. Econometrica, 29(2), 171-185.

Ghitany, M. E., Al-Mutairi, D. K., Balakrishnan, N., and Al-Enezi, L. J. (2013). Power Lindley distribution and associated inference. Computational Statistics & Data Analysis, 64, 20-33.

Gupta, R. D. and Kundu, D. (1999). Generalized exponential distributions. Australian & New Zealand Journal of Statistics, 41(2), 173-188.

Jodra, P., Gomez, H. W., Jimenez-Gamero, M. D., and Alba-Fernandez, M. V. (2017). The power Muth distribution. Mathematical Modelling and Analysis, 22(2), 186-201.

Jodra, P., Jimenez-Gamero, M. D., and Alba-Fernandez, M. V. (2015). On the Muth distribution. Mathematical Modelling and Analysis, 20(3), 291-310.

Lomax, K. S. (1954). Business failures: Another example of the analysis of failure data. Journal of the American Statistical Association, 49(268), 847-852.

Marshall, A. W. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84(3), 641-652.

Nadarajah, S., Bakouch, H. S., and Tahmasbi, R. (2011). A generalized Lindley distribution. Sankhya B, 73(2), 331-359.

Muth, E. J. (1977). Reliability models with positive memory derived from the mean residual life function. In The Theory and Applications of Reliability, Vol. 2 (C. P. Tsokos and I. N. Shimi, eds.), 401-435. Academic Press, New York.

Sharma, V. K., Singh, S. V., and Shekhawat, K. (2022). Exponentiated Teissier distribution with increasing, decreasing and bathtub hazard functions. Journal of Applied Statistics, 49(2), 371-393. doi:10.1080/02664763.2020.1813694

Teissier, G. (1934). Recherches sur le vieillissement et sur les lois de la mortalite. Annales de Physiologie et de Physicochimie Biologique, 10, 237-284.

Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18(3), 293-297.

See Also

slt_families() for the list of family keys; dslt(), pslt(), qslt(), rslt() for the two-stage (not baseline) population functions built from these; slt_equiv_age() for how a baseline distribution's cdf/quantile combine across the stage change.

Examples

fam <- slt_dist("weibull")
fam$p(1, lambda = 1, alpha = 2)
fam$q(0.5, lambda = 1, alpha = 2)

---

Equivalent age under the elevated stress

Description

Under the cumulative exposure model (CEM), a unit that has accumulated damage equivalent to surviving to time tau under stage-0 conditions has, under stage-1 conditions, already "used up" an equivalent age nu. This function computes nu.

Usage

slt_equiv_age(family, tau, theta0, theta1)

Arguments

family	A family key (see slt_families()) or family object.
tau	Non-negative scalar change-point time.
theta0	Numeric c(lambda0, alpha0), baseline-stage parameters.
theta1	Numeric c(lambda1, alpha1), elevated-stage parameters.

Details

The CEM's defining assumption is that only the total accumulated damage determines a unit's remaining life, not the stress history that produced it. Formalising "equal accumulated damage" as "equal reliability", a unit surviving to $\tau$ under the stage-0 distribution $F_0$ is equivalent to a unit that has survived to some age $\nu$ under the stage-1 distribution $F_1$, where $\nu$ solves

$S_1(\nu;\theta_1) = S_0(\tau;\theta_0) \quad\Longleftrightarrow\quad F_1(\nu;\theta_1) = F_0(\tau;\theta_0),$

i.e.

$\nu = \nu(\tau;\theta_0,\theta_1) = Q_1\!\left(F_0(\tau;\theta_0);\ \theta_1\right),$

where $Q_1 = F_1^{-1}$ is the stage-1 quantile function. This is exactly fam$quantile(fam$cdf(tau, theta0), theta1) below: plug the stage-0 cdf at tau into the stage-1 quantile function. $\nu$ is the building block for the population functions pslt()/dslt()/ qslt() and for the log-likelihood loglik_slt() – see their documentation for how it is used.

Value

The equivalent age (a non-negative scalar).

See Also

pslt() for how nu combines with tau1 into the population cdf; loglik_slt() for the resulting log-likelihood; slt_tau() for choosing tau as a stage-0 quantile.

Examples

slt_equiv_age("weibull", tau = 5, theta0 = c(0.1, 1.5), theta1 = c(0.3, 1.5))

---

Available baseline lifetime distributions

Description

Lists the two-parameter baseline lifetime distributions, theta = (lambda, alpha), that can be used as the family argument throughout the package. See slt_dist() for the pdf/cdf and literature source of each.

Usage

slt_families()

Value

A character vector of family keys.

See Also

slt_dist() for the mathematical definition and reference of each distribution; slt_cases() for the parameter-sharing cases these distributions are combined under.

Examples

slt_families()

---

Kolmogorov-Smirnov goodness-of-fit test with parametric-bootstrap correction for a fitted two-stage life testing model

Description

Performs the Kolmogorov-Smirnov goodness-of-fit test and reports both the naive p-value (for reference) and the standard parametric-bootstrap-corrected p-value.

Usage

slt_gof_ks(fit, B = 1000, n_cores = 1L)

Arguments

fit	A converged "slt_fit" object, fit to data from a simple-step-stress (rslt() with prop_transfer = 1) design.
B	Number of parametric bootstrap replicates.
n_cores	Number of cores for parallel::mclapply().

Details

The classical KS null distribution assumes a fully specified (not estimated) reference distribution; using it when the parameters were estimated from the same data makes the naive p-value anti-conservative (too small), a problem first noted by Lilliefors. The parametric bootstrap correction simulates B fresh data sets of the same size from the fitted model via rslt() (at the MLE), refits each with the same family/case, and takes the corrected p-value to be the proportion of bootstrap KS statistics at least as extreme as the one observed on the real data.

This test assumes every survivor was transferred at tau1 (the classical simple-step-stress design, prop_transfer = 1 in rslt()), because the single pooled reference distribution it uses (pslt()) is only the correct marginal CDF of the combined Y/Z sample in that case. Under genuine partial withdrawal, units that survive tau1 but are not transferred keep failing on stage 0 unmodified, which pslt() does not represent – pooling them into one KS statistic against pslt() would be a biased test. If fit$data came from a partial-withdrawal rslt() call (prop_transfer < 1 or design = "typeM"), this function is not appropriate; a Y-only and a Z-only KS test against their own (design-invariant) marginals would be the correct generalisation, but is not currently implemented.

Value

An object of class "slt_gof" with elements D (KS statistic), p_naive, p_bootstrap, B, B_converged.

Examples

set.seed(1)
d <- rslt(150, "weibull", c(0.05, 1.5), c(0.2, 1.5), tau1 = 5, prop_transfer = 1)
fit <- fit_slt(d, "weibull", "shape1", tau1 = 5)
slt_gof_ks(fit, B = 100)

---

Han & Kundu (2014) simple step-stress data

Description

A simple step-stress life-testing data set on the reliability of a solar lighting device, with 16 units observed under the baseline stress (normal operating temperature, 293K) and 15 under the elevated stress (353K) after a common change-point at tau1 = 500 (hours).

Usage

slt_han_kundu_2014

Format

A list with elements:

y0

Numeric vector of 16 stage-0 (baseline stress) failure times.

y1

Numeric vector of 15 stage-1 (elevated stress) failure times.

tau1

The change-point time.

description, source

Character strings.

Details

The original experiment placed n = 35 units on test, but only these 31 failure times are reported in the source paper – the remaining 4 units did not fail before the experiment ended. This package's model has no right-censoring mechanism (see ?stagelife-package), so those 4 units are not, and cannot be, represented here.

Source

Han, D. and Kundu, D. (2014). Inference for a step-stress model with competing risks for failure from the generalized exponential distribution under Type-I censoring. IEEE Transactions on Reliability, 64(1), 31-43.

See Also

slt_bobotas_kateri_2015, slt_wang_fei_2003, slt_zhu_2010 for the other three bundled data sets; fit_slt() for fitting a model to data in this format.

Examples

data(slt_han_kundu_2014)
fit <- fit_slt(list(Y = slt_han_kundu_2014$y0, Z = slt_han_kundu_2014$y1),
                  "weibull", "shape1", tau1 = slt_han_kundu_2014$tau1)
summary(fit)

---

Likelihood-ratio test between nested parameter-sharing cases

Description

Tests whether a reduced (nested) parameter-sharing case is adequate against a less-restricted alternative, via the likelihood-ratio statistic.

Usage

slt_lrt(fit_full, fit_reduced)

Arguments

fit_full, fit_reduced

Converged "slt_fit" objects for the same family, tau1 and data, with fit_reduced nested in fit_full.

Details

The only nested pairs among the three supported cases are ⁠("shape1", "shape11")⁠ and ⁠("common", "shape11")⁠ (both fix the remaining free shape parameter(s) to 1); see slt_cases(). This function tests H0: the reduced ("shape11") model is adequate, against the corresponding less-restricted alternative.

Value

An object of class "slt_lrt" with statistic, df, p.value.

Examples

set.seed(1)
d <- rslt(200, "weibull", c(0.05, 1), c(0.2, 1), tau1 = 5, prop_transfer = 1)
f1 <- fit_slt(d, "weibull", "shape1", tau1 = 5)
f0 <- fit_slt(d, "weibull", "shape11", tau1 = 5)
slt_lrt(f1, f0)

---

Compare several fitted models by information criteria

Description

Builds a ranked comparison table across any number of fitted models, by information criteria.

Usage

slt_model_comparison(fits, sort_by = c("aic", "bic", "loglik"))

Arguments

fits	A (preferably named) list of "slt_fit" objects.
sort_by	One of "aic" (default), "bic", "loglik".

Details

By default sorted by AIC, across any number of "slt_fit" objects – typically different family/case combinations fit to the same data set – for exploratory model selection on real data. This is a lightweight diagnostic table for a single data set, not a Monte Carlo simulation study.

Value

A data.frame with columns model, family, case, n_par, logLik, AIC, BIC, converged.

Examples

set.seed(1)
d <- rslt(200, "weibull", c(0.05, 1.5), c(0.2, 1.5), tau1 = 5, prop_transfer = 1)
fits <- list(
  "Weibull / shape1"  = fit_slt(d, "weibull", "shape1", tau1 = 5),
  "Weibull / shape11" = fit_slt(d, "weibull", "shape11", tau1 = 5),
  "Weibull / common"  = fit_slt(d, "weibull", "common", tau1 = 5)
)
slt_model_comparison(fits)

---

Probability of observing at least one stage-1 failure (Laumen & Cramer, Lemma 4.1)

Description

Computes the probability of observing at least one stage-1 failure, ⁠P(R*_1 > 0)⁠, implementing Lemma 4.1 of Laumen & Cramer (2019).

Usage

slt_prob_stage1(
  n,
  tau1,
  theta0,
  family = "weibull",
  design = c("typeP", "typeM"),
  pi1 = NULL,
  R0_1 = NULL
)

Arguments

n	Number of units placed on test.
tau1	Stage-change time.
theta0	Numeric c(lambda0, alpha0), baseline-stage parameters.
family	A family key (see slt_families()) or family object.
design	"typeP" (fixed target proportion pi1) or "typeM" (fixed target count R0_1), matching rslt().
pi1	Target proportion transferred, for design = "typeP".
R0_1	Target number transferred, for design = "typeM".

Details

Because the maximum likelihood estimate of the elevated-stage parameter only exists when at least one unit is actually transferred to stage 1 and fails there (⁠R*_1 > 0⁠), this is a useful quantity for planning a stage life test: choosing n, tau1, and pi1/R0_1 so that slt_prob_stage1() is comfortably close to 1.

⁠R*_1 = rho(D1) > 0⁠ iff ⁠D1 <= n*⁠, where ⁠n* = floor(n - 1/pi1)⁠ for the "typeP" design and ⁠n* = n - 1⁠ for the "typeM" design (with R0_1 >= 1), and D1 ~ Binomial(n, F0(tau1)).

Value

A list with n_star (the threshold on D1) and prob (⁠P(R*_1 > 0)⁠).

Examples

# Reproduces Table 5 of Laumen & Cramer (2019) exactly (n = 12,
# theta0 = 1 i.e. lambda0 = 1, tau1 = 0.5):
slt_prob_stage1(12, 0.5, c(1, 1), design = "typeP", pi1 = 0.25)  # n*=8,  prob=0.9864756680
slt_prob_stage1(12, 0.5, c(1, 1), design = "typeM", R0_1 = 6)    # n*=11, prob=0.9999862301

---

Reduce full ⁠(theta0, theta1)⁠ parameters to a case's free parameters

Description

The inverse of slt_expand_params(); mostly useful for extracting the "true" parameter vector matching a given case in simulation studies. For "common", theta0[2] (rather than an average) is used as the shared shape and a warning is issued if theta0[2] != theta1[2].

Usage

slt_reduce_params(theta0, theta1, case = c("common", "shape1", "shape11"))

Arguments

theta0, theta1	Numeric c(lambda, alpha) vectors.
case	One of "common", "shape1", "shape11".

Value

A named numeric vector matching case_info(case)$par_names.

Examples

slt_reduce_params(c(0.2, 1.5), c(0.4, 1.5), case = "common")

---

p-th quantile of the baseline-stage distribution

Description

Convenience wrapper typically used to set the change-point at a chosen quantile of the stage-0 lifetime distribution, e.g. tau <- slt_tau(family, theta0, p = 0.5) places the change-point at the stage-0 median.

Usage

slt_tau(family, theta0, p = 0.5)

Arguments

family	A family key (see slt_families()) or family object.
theta0	Numeric c(lambda0, alpha0), baseline-stage parameters.
p	Probability in ⁠(0, 1)⁠.

Value

The p-th quantile of the stage-0 distribution.

Examples

slt_tau("weibull", theta0 = c(0.1, 1.5), p = 0.5)

---

Wang & Fei (2003) simple step-stress data

Description

A simple step-stress life-testing data set on the reliability of electronic components under temperature stress, with 30 units observed under the baseline stress and 20 under the elevated stress after a common change-point at tau1 = 910 (hours).

Usage

slt_wang_fei_2003

Format

A list with elements y0, y1, tau1, description, source (see slt_han_kundu_2014 for the structure).

Source

Wang, R. and Fei, H. (2003). Uniqueness of the maximum likelihood estimate of the Weibull distribution tampered failure rate model. Communications in Statistics - Theory and Methods, 32(12), 2321-2338.

See Also

slt_han_kundu_2014, slt_bobotas_kateri_2015, slt_zhu_2010.

Examples

data(slt_wang_fei_2003)
str(slt_wang_fei_2003)

---

Zhu (2010) simple step-stress data

Description

A simple step-stress life-testing data set on light-bulb-filament fatigue (two sets of 32 bulbs pooled), with 33 units observed under the baseline stress and 20 under the elevated stress after a common change-point at tau1 = 94.

Usage

slt_zhu_2010

Format

A list with elements y0, y1, tau1, description, source (see slt_han_kundu_2014 for the structure).

Details

The original experiment used two sets of 32 bulbs (n = 64 total), but only these 53 failure times are bundled here – the remaining 11 units did not fail before their set's test ended. This package's model has no right-censoring mechanism (see ?stagelife-package), so those 11 units are not, and cannot be, represented here.

The source describes the nominal change-point as 96 hours, but this is inconsistent with the recorded data itself: the smallest stage-1 (post-change) observation is 94.38 (Set-II), which would be impossible if the change-point were 96 (a unit cannot fail on stage 1 before the stage change happens). tau1 = 94 – the only value consistent with both stages of the recorded data (every stage-0 observation is ⁠<= 91.56⁠; every stage-1 observation is ⁠>= 94.38⁠) – is used here instead.

Source

Zhu, Y. (2010). Optimal Design and Equivalency of Accelerated Life Testing Plans. PhD dissertation, Rutgers, The State University of New Jersey, School of Graduate Studies.

See Also

slt_han_kundu_2014, slt_bobotas_kateri_2015, slt_wang_fei_2003.

Examples

data(slt_zhu_2010)
str(slt_zhu_2010)

---

Summarise a fitted two-stage life testing model

Description

Summarise a fitted two-stage life testing model

Usage

## S3 method for class 'slt_fit'
summary(object, level = 0.95, ...)

Arguments

object	An "slt_fit" object from fit_slt().
level	Confidence level for the Wald (log-normal) confidence limits reported in the coefficient table.
...	Currently unused.

Value

An object of class "summary.slt_fit".

---

Variance-covariance matrix of a fitted two-stage life testing model's coefficients

Description

Computes the natural-scale variance-covariance matrix by the delta method, from the log-scale one fitted internally.

Usage

## S3 method for class 'slt_fit'
vcov(object, ...)

Arguments

object	An "slt_fit" object from fit_slt().
...	Currently unused.

Details

object$vcov_log (the inverse of the numerical Hessian of the negative log-likelihood in log(par)) is transformed to the natural scale by the delta method: with $\eta = \log(\theta)$, $d\theta/d\eta = \mathrm{diag}(\theta)$, so $\mathrm{Var}(\theta) \approx \mathrm{diag}(\theta)\, \mathrm{Var}(\eta)\,\mathrm{diag}(\theta)$.

Value

The natural-scale variance-covariance matrix, or NULL if the Hessian was not invertible at the optimum.

See Also

confint.slt_fit(), which uses this (or the log scale directly) to build a confidence interval.

---