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Package {CompRiskRel}

Type: Package
Title: Reliability and Competing Risks Analysis under Hybrid Censoring
Version: 0.1.0
Description: Generalized computational algorithms for competing risks analysis, stress-strength reliability modeling, optimal designs, and reliability acceptance sampling plans under various hybrid censoring schemes. Includes data generation routines, Maximum Likelihood Estimation (MLE) with seven optimization algorithms ('Newton-Raphson', 'BFGS', 'BFGSR', 'BHHH', 'SANN', 'CG', and 'Nelder-Mead'), Bayesian inference via Gibbs sampling and Metropolis-Hastings MCMC, Importance Sampling, and 'Lindley' asymptotic approximation. Visualization functions generate histograms, dot plots, and autocorrelation plots for model validation. Methodology and design principles are based on 'Balakrishnan', 'Cramer', and 'Kundu' (2023, "Hybrid Censoring Know-How: Designs and Implementations", Academic Press, ISBN:978-0-12-398387-9).
License: GPL-3
Encoding: UTF-8
RoxygenNote: 7.3.3
Imports: stats, graphics
Suggests: testthat (≥ 3.0.0)
NeedsCompilation: no
Packaged: 2026-07-25 17:29:46 UTC; shikhar tyagi
Author: Shikhar Tyagi ORCID iD [aut, cre], Arvind Pandey [aut], Bhupendra Singh [aut], Vrijesh Tripathi [aut]
Maintainer: Shikhar Tyagi <shikhar1093tyagi@gmail.com>
Repository: CRAN
Date/Publication: 2026-08-05 06:30:11 UTC

CompRiskRel: Generalized Reliability and Competing Risks Analysis under Hybrid Censoring

Description

Provides comprehensive tools for data generation, Maximum Likelihood Estimation (MLE), Bayesian estimation (Gibbs sampling, Metropolis-Hastings), Importance Sampling, Lindley approximation, stress-strength reliability estimation, optimal designs, and reliability acceptance sampling plans under Type-I, Type-II, and generalized (progressive) hybrid censoring schemes.

Author(s)

Maintainer: Shikhar Tyagi shikhar1093tyagi@gmail.com (ORCID)

Authors:

Bayesian Inference via Gibbs Sampling for Censored Data Models

Description

Bayesian Inference via Gibbs Sampling for Censored Data Models

Usage

bayes_gibbs_censored(log_posterior, init_par, n_sim = 5000, burn_in = 1000)

Arguments

log_posterior

Function taking parameter vector and returning log posterior density.

init_par

Initial parameter vector.

n_sim

Total number of MCMC iterations.

burn_in

Number of initial iterations to discard as burn-in.

Value

S3 object of class bayes_fit with parameter chains, posterior means, and credible intervals.

Examples

log_post <- function(th) {
  if (th[1] <= 0) return(-Inf)
  dexp(th[1], rate = 1, log = TRUE) +
    sum(dexp(c(0.5, 1.2, 0.8), rate = th[1], log = TRUE))
}
bayes_gibbs_censored(
  log_post, init_par = c(1.0),
  n_sim = 1000, burn_in = 200
)

Bayesian Inference via Metropolis-Hastings MCMC Algorithm

Description

Bayesian Inference via Metropolis-Hastings MCMC Algorithm

Usage

bayes_mh_censored(
  log_posterior,
  init_par,
  n_sim = 5000,
  burn_in = 1000,
  proposal_sd = 0.1
)

Arguments

log_posterior

Function evaluating log-posterior density.

init_par

Initial parameter values.

n_sim

Total MCMC sample size.

burn_in

Discarded burn-in samples.

proposal_sd

Proposal standard deviation vector.

Value

S3 object of class bayes_fit.

Examples

log_post <- function(th) {
  if (th[1] <= 0) return(-Inf)
  dexp(th[1], rate = 1, log = TRUE) +
    sum(dexp(c(0.5, 1.2, 0.8), rate = th[1], log = TRUE))
}
bayes_mh_censored(
  log_post, init_par = c(1.0),
  n_sim = 1000, burn_in = 200
)

Bayesian Reliability Acceptance Sampling Plan for Exponential lifetimes (10.5.3)

Description

Bayesian Reliability Acceptance Sampling Plan for Exponential lifetimes (10.5.3)

Usage

bayesian_sampling_plan_exp(
  n,
  T_star,
  r,
  prior_shape = 2,
  prior_rate = 1,
  c_accept = 1.2
)

Arguments

n

Sample size.

T_star

Termination time limit.

r

Failure count limit.

prior_shape

Shape parameter of Inverse-Gamma prior.

prior_rate

Rate parameter of Inverse-Gamma prior.

c_accept

Acceptance threshold constant.

Value

S3 object of class sampling_plan with posterior risk and decision criteria.

Examples

bayesian_sampling_plan_exp(
  n = 20, T_star = 1.0, r = 10,
  prior_shape = 2, prior_rate = 1, c_accept = 1.2
)

Generate Data for Competing Risks Analysis

Description

Generate Data for Competing Risks Analysis

Usage

gen_competing_risks(
  pdf1,
  cdf1,
  pdf2,
  cdf2,
  lower = 0,
  upper = Inf,
  n,
  censoring_type = c("type1_hybrid", "type2_hybrid", "prog_hybrid"),
  r = NULL,
  T_star = NULL,
  R_plan = NULL,
  seed = NULL
)

Arguments

pdf1

Density function for Cause 1.

cdf1

Cumulative distribution function for Cause 1.

pdf2

Density function for Cause 2.

cdf2

Cumulative distribution function for Cause 2.

lower

Lower bound of support.

upper

Upper bound of support.

n

Total sample size.

censoring_type

Type of censoring ("type1_hybrid", "type2_hybrid", "prog_hybrid").

r

Number of failures required.

T_star

Time limit.

R_plan

Progressive plan (if applicable).

seed

Optional random seed.

Value

A list containing observed failure times, causes of failure (1 or 2), and censoring flags.

Examples

gen_competing_risks(
  pdf1 = function(x) dexp(x, rate = 1),
  cdf1 = function(x) pexp(x, rate = 1),
  pdf2 = function(x) dexp(x, rate = 1.5),
  cdf2 = function(x) pexp(x, rate = 1.5),
  lower = 0, upper = 10, n = 25,
  censoring_type = "type1_hybrid",
  r = 15, T_star = 1.0, seed = 123
)

Generate Data under Generalized Progressive Hybrid Censoring Scheme

Description

Generate Data under Generalized Progressive Hybrid Censoring Scheme

Usage

gen_gen_prog_hybrid(
  pdf,
  cdf,
  lower = 0,
  upper = Inf,
  n,
  m,
  k,
  T_star,
  R_plan,
  seed = NULL
)

Arguments

pdf

Probability density function.

cdf

Cumulative distribution function.

lower

Lower bound of support.

upper

Upper bound of support.

n

Total sample size.

m

Target number of failures.

k

Threshold failure count.

T_star

Termination time limit.

R_plan

Progressive censoring plan vector of length m.

seed

Optional integer random seed for reproducibility.

Value

A list containing observed failure times, progressive removal plan, and censoring details.

Examples

gen_gen_prog_hybrid(
  pdf = function(x) dexp(x, rate = 1),
  cdf = function(x) pexp(x, rate = 1),
  lower = 0, upper = 10, n = 20, m = 10,
  k = 5, T_star = 1.2, R_plan = rep(1, 10),
  seed = 123
)

Generate Data for Stress-Strength Reliability Models

Description

Generate Data for Stress-Strength Reliability Models

Usage

gen_stress_strength(
  pdf_X,
  cdf_X,
  pdf_Y,
  cdf_Y,
  lower_X = 0,
  upper_X = Inf,
  lower_Y = 0,
  upper_Y = Inf,
  n_X,
  n_Y,
  censoring_type = c("type1_hybrid", "type2_hybrid", "prog_hybrid"),
  r_X = NULL,
  T_X = NULL,
  r_Y = NULL,
  T_Y = NULL,
  seed = NULL
)

Arguments

pdf_X

Density function for stress X.

cdf_X

Cumulative distribution function for stress X.

pdf_Y

Density function for strength Y.

cdf_Y

Cumulative distribution function for strength Y.

lower_X

Lower bound for X.

upper_X

Upper bound for X.

lower_Y

Lower bound for Y.

upper_Y

Upper bound for Y.

n_X

Sample size for stress X.

n_Y

Sample size for strength Y.

censoring_type

Censoring scheme.

r_X

Minimum required failures for X.

T_X

Time cutoff for X.

r_Y

Minimum required failures for Y.

T_Y

Time cutoff for Y.

seed

Optional random seed.

Value

A list containing generated stress dataset X and strength dataset Y.

Examples

gen_stress_strength(
  pdf_X = function(x) dexp(x, rate = 1.2),
  cdf_X = function(x) pexp(x, rate = 1.2),
  pdf_Y = function(y) dexp(y, rate = 0.8),
  cdf_Y = function(y) pexp(y, rate = 0.8),
  n_X = 20, n_Y = 20,
  censoring_type = "type1_hybrid",
  r_X = 12, T_X = 1.2, r_Y = 12, T_Y = 1.5, seed = 123
)

Generate Data under Type-I Hybrid Censoring Scheme

Description

Generate Data under Type-I Hybrid Censoring Scheme

Usage

gen_type1_hybrid(pdf, cdf, lower = 0, upper = Inf, n, r, T_star, seed = NULL)

Arguments

pdf

Probability density function.

cdf

Cumulative distribution function.

lower

Lower bound of the support of the distribution.

upper

Upper bound of the support of the distribution.

n

Total sample size.

r

Minimum required number of failures.

T_star

Fixed termination time.

seed

Optional integer random seed for reproducibility.

Value

A list containing:

observed_times

Vector of observed failure/censoring times.

censor_status

Binary indicator (1 for failure, 0 for censored at T_star).

n_failures

Number of observed failures.

termination_time

Effective termination time of the test.

scheme

Character string indicating censoring scheme.

Examples

gen_type1_hybrid(
  pdf = function(x) dexp(x, rate = 1),
  cdf = function(x) pexp(x, rate = 1),
  lower = 0, upper = 10, n = 20, r = 10,
  T_star = 1.5, seed = 123
)

Generate Data under Type-II Hybrid Censoring Scheme

Description

Generate Data under Type-II Hybrid Censoring Scheme

Usage

gen_type2_hybrid(pdf, cdf, lower = 0, upper = Inf, n, r, T_star, seed = NULL)

Arguments

pdf

Probability density function.

cdf

Cumulative distribution function.

lower

Lower bound of the support of the distribution.

upper

Upper bound of the support of the distribution.

n

Total sample size.

r

Minimum required number of failures.

T_star

Target observation time limit.

seed

Optional integer random seed for reproducibility.

Value

A list containing observed failure times, censor statuses, and termination details.

Examples

gen_type2_hybrid(
  pdf = function(x) dexp(x, rate = 1),
  cdf = function(x) pexp(x, rate = 1),
  lower = 0, upper = 10, n = 20, r = 10,
  T_star = 1.0, seed = 123
)

Importance Sampling for Posterior Estimation

Description

Importance Sampling for Posterior Estimation

Usage

importance_sampling_censored(
  log_target,
  proposal_pdf,
  rproposal,
  n_sim = 10000
)

Arguments

log_target

Function evaluating log target density (unnormalized log posterior).

proposal_pdf

Density function of proposal distribution.

rproposal

Random generation function for proposal distribution.

n_sim

Sample size of proposal draws.

Value

List with posterior mean estimates, normalized importance weights, and effective sample size.

Examples

importance_sampling_censored(
  log_target = function(th) dexp(th, rate = 2, log = TRUE),
  proposal_pdf = function(th) dexp(th, rate = 1),
  rproposal = function(n) rexp(n, rate = 1),
  n_sim = 1000
)

Lindley Asymptotic Approximation for Posterior Expectations

Description

Lindley Asymptotic Approximation for Posterior Expectations

Usage

lindley_approx_censored(log_lik, log_prior, init_par)

Arguments

log_lik

Log-likelihood function of parameter vector.

log_prior

Log-prior density function of parameter vector.

init_par

Maximum likelihood estimate or posterior mode.

Value

List with Lindley posterior estimates and standard errors.

Examples

lindley_approx_censored(
  log_lik = function(th) -sum((c(1.2, 0.8, 1.5) - th[1])^2),
  log_prior = function(th) dexp(th[1], rate = 1, log = TRUE),
  init_par = c(1.1)
)

Maximum Likelihood Estimation for Competing Risks Analysis

Description

Maximum Likelihood Estimation for Competing Risks Analysis

Usage

mle_competing_risks(data, pdf1, cdf1, pdf2, cdf2, init_par, method = "BFGS")

Arguments

data

Object of class comp_risk_rel_data generated by gen_competing_risks.

pdf1

Density function for Cause 1.

cdf1

CDF function for Cause 1.

pdf2

Density function for Cause 2.

cdf2

CDF function for Cause 2.

init_par

Vector of initial parameter values for both causes c(theta1, theta2).

method

Optimization method.

Value

S3 object of class mle_fit.

Examples

dat <- gen_competing_risks(
  pdf1 = function(x) dexp(x, rate = 1),
  cdf1 = function(x) pexp(x, rate = 1),
  pdf2 = function(x) dexp(x, rate = 1.5),
  cdf2 = function(x) pexp(x, rate = 1.5),
  lower = 0, upper = 10, n = 25,
  censoring_type = "type1_hybrid",
  r = 15, T_star = 1.0, seed = 123
)
mle_competing_risks(
  data = dat,
  pdf1 = function(x, th) dexp(x, rate = th[1]),
  cdf1 = function(x, th) pexp(x, rate = th[1]),
  pdf2 = function(x, th) dexp(x, rate = th[2]),
  cdf2 = function(x, th) pexp(x, rate = th[2]),
  init_par = c(0.8, 1.2), method = "BFGS"
)

Maximum Likelihood Estimation under Generalized Progressive Hybrid Censoring

Description

Maximum Likelihood Estimation under Generalized Progressive Hybrid Censoring

Usage

mle_gen_prog_hybrid(
  data,
  pdf,
  cdf,
  init_par,
  R_plan = NULL,
  method = "BFGS",
  lower = -Inf,
  upper = Inf
)

Arguments

data

Object of class comp_risk_rel_data.

pdf

Density function.

cdf

CDF function.

init_par

Initial parameter values.

R_plan

Progressive removal vector.

method

Maximization algorithm.

lower

Lower parameter bound.

upper

Upper parameter bound.

Value

S3 object of class mle_fit.

Examples

dat <- gen_gen_prog_hybrid(
  pdf = function(x) dexp(x, rate = 1),
  cdf = function(x) pexp(x, rate = 1),
  lower = 0, upper = 10, n = 20, m = 10, k = 5,
  T_star = 1.2, R_plan = rep(1, 10), seed = 123
)
mle_gen_prog_hybrid(
  data = dat,
  pdf = function(x, theta) dexp(x, rate = theta[1]),
  cdf = function(x, theta) pexp(x, rate = theta[1]),
  init_par = c(0.8), R_plan = rep(1, 10), method = "BFGS"
)

Maximum Likelihood Estimation under Type-I Hybrid Censoring

Description

Maximum Likelihood Estimation under Type-I Hybrid Censoring

Usage

mle_type1_hybrid(
  data,
  pdf,
  cdf,
  init_par,
  method = "BFGS",
  lower = -Inf,
  upper = Inf
)

Arguments

data

Object of class comp_risk_rel_data or list with observed_times, censor_status, termination_time, and total sample size n.

pdf

Parametric probability density function of time x and vector parameter theta.

cdf

Parametric cumulative distribution function of time x and vector parameter theta.

init_par

Initial parameter vector for numerical optimization.

method

Maximization routine: "NR", "BFGS", "BFGSR", "BHHH", "SANN", "CG", or "NM".

lower

Lower bound for parameter vector.

upper

Upper bound for parameter vector.

Value

S3 object of class mle_fit with parameter estimates, log-likelihood, SE, and information criteria.

Examples

dat <- gen_type1_hybrid(
  pdf = function(x) dexp(x, rate = 1),
  cdf = function(x) pexp(x, rate = 1),
  lower = 0, upper = 10, n = 20, r = 10,
  T_star = 1.5, seed = 123
)
mle_type1_hybrid(
  data = dat,
  pdf = function(x, theta) dexp(x, rate = theta[1]),
  cdf = function(x, theta) pexp(x, rate = theta[1]),
  init_par = c(0.8), method = "BFGS"
)

Maximum Likelihood Estimation under Type-II Hybrid Censoring

Description

Maximum Likelihood Estimation under Type-II Hybrid Censoring

Usage

mle_type2_hybrid(
  data,
  pdf,
  cdf,
  init_par,
  method = "BFGS",
  lower = -Inf,
  upper = Inf
)

Arguments

data

Object of class comp_risk_rel_data or list.

pdf

Parametric density function.

cdf

Parametric CDF function.

init_par

Initial parameter values.

method

Maximization method ("NR", "BFGS", "BFGSR", "BHHH", "SANN", "CG", "NM").

lower

Lower parameter bounds.

upper

Upper parameter bounds.

Value

S3 object of class mle_fit.

Examples

dat <- gen_type2_hybrid(
  pdf = function(x) dexp(x, rate = 1),
  cdf = function(x) pexp(x, rate = 1),
  lower = 0, upper = 10, n = 20, r = 10,
  T_star = 1.0, seed = 123
)
mle_type2_hybrid(
  data = dat,
  pdf = function(x, theta) dexp(x, rate = theta[1]),
  cdf = function(x, theta) pexp(x, rate = theta[1]),
  init_par = c(0.8), method = "BFGS"
)

Optimal Design Selection for Hybrid Censoring Schemes

Description

Optimal Design Selection for Hybrid Censoring Schemes

Usage

optimal_design_hybrid(
  n,
  r_candidates,
  T_candidates,
  pdf,
  cdf,
  par,
  criterion = c("D-optimal", "A-optimal", "cost")
)

Arguments

n

Sample size.

r_candidates

Candidate failure count choices.

T_candidates

Candidate time limit choices.

pdf

Probability density function of lifetime distribution.

cdf

Cumulative distribution function of lifetime distribution.

par

Model parameters.

criterion

Optimization criterion: "D-optimal" (maximize determinant of Fisher Information), "A-optimal" (minimize trace of inverse Fisher Information), or "cost".

Value

List containing optimal choice of (r, T) and associated information measure.

Examples

optimal_design_hybrid(
  n = 20, r_candidates = c(5, 10, 15), T_candidates = c(0.5, 1.0, 1.5),
  pdf = function(x, th) dexp(x, rate = th[1]),
  cdf = function(x, th) pexp(x, rate = th[1]),
  par = c(1.0), criterion = "D-optimal"
)

Plot Method for Objects of Class comp_risk_rel_data

Description

Plot Method for Objects of Class comp_risk_rel_data

Usage

## S3 method for class 'comp_risk_rel_data'
plot(x, ...)

Arguments

x

Object of class comp_risk_rel_data.

...

Additional graphical parameters.

Value

No return value, called for side effects.

Diagnostic Plots for Censored Data Schemes

Description

Produces 3 diagnostic plots (Histogram with PDF overlay, Dot plot, and ACF plot) to verify sample distributional properties and independence.

Usage

plot_diagnostics(data, pdf = NULL, main = "Censoring Diagnostic Plots")

Arguments

data

Object of class comp_risk_rel_data or a numeric vector of failure times.

pdf

Optional probability density function to overlay on histogram.

main

Overall plot title.

Value

No return value, called for side effects (renders diagnostic plots).

Examples

dat <- gen_type1_hybrid(
  pdf = function(x) dexp(x, rate = 1), cdf = function(x) pexp(x, rate = 1),
  lower = 0, upper = 10, n = 20, r = 10, T_star = 1.5, seed = 123
)
plot_diagnostics(dat, pdf = function(x) dexp(x, rate = 1))

Print Method for Bayesian Fit Objects

Description

Print Method for Bayesian Fit Objects

Usage

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

Arguments

x

Object of class bayes_fit.

...

Unused parameters.

Value

No return value, called for side effects.

Print Method for MLE Fit Objects

Description

Print Method for MLE Fit Objects

Usage

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

Arguments

x

Object of class mle_fit.

...

Unused additional parameters.

Value

No return value, called for side effects.

Print Method for Sampling Plan Objects

Description

Print Method for Sampling Plan Objects

Usage

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

Arguments

x

Object of class sampling_plan.

...

Unused parameters.

Value

No return value, called for side effects.

Reliability Acceptance Sampling Plan for Exponential Distribution (10.5.1)

Description

Reliability Acceptance Sampling Plan for Exponential Distribution (10.5.1)

Usage

sampling_plan_exponential(
  n,
  T_star,
  r,
  alpha = 0.05,
  beta = 0.1,
  theta0,
  theta1
)

Arguments

n

Sample size placed on test.

T_star

Fixed test termination time.

r

Required failure limit.

alpha

Producer's risk.

beta

Consumer's risk.

theta0

Acceptable quality level (AQL) mean lifetime.

theta1

Rejectable quality level (RQL) mean lifetime.

Value

S3 object of class sampling_plan with acceptance constant c, operating characteristic (OC) values, and decision rule.

Examples

sampling_plan_exponential(
  n = 20, T_star = 1.0, r = 10,
  alpha = 0.05, beta = 0.10,
  theta0 = 2.0, theta1 = 0.8
)

Reliability Acceptance Sampling Plan for Weibull Distribution (10.5.2)

Description

Reliability Acceptance Sampling Plan for Weibull Distribution (10.5.2)

Usage

sampling_plan_weibull(
  n,
  T_star,
  r,
  beta_shape,
  alpha = 0.05,
  beta = 0.1,
  theta0,
  theta1
)

Arguments

n

Sample size.

T_star

Termination time.

r

Minimum failure limit.

beta_shape

Known shape parameter of Weibull distribution.

alpha

Producer's risk.

beta

Consumer's risk.

theta0

AQL scale parameter.

theta1

RQL scale parameter.

Value

S3 object of class sampling_plan.

Examples

sampling_plan_weibull(
  n = 25, T_star = 1.5, r = 12, beta_shape = 1.5,
  alpha = 0.05, beta = 0.10, theta0 = 3.0, theta1 = 1.0
)

Stress-Strength Reliability Estimation R = P(X < Y)

Description

Stress-Strength Reliability Estimation R = P(X < Y)

Usage

stress_strength_rel(
  data_X,
  data_Y,
  pdf_X,
  cdf_X,
  pdf_Y,
  cdf_Y,
  init_par_X,
  init_par_Y,
  method = "BFGS"
)

Arguments

data_X

Object of class comp_risk_rel_data for stress X.

data_Y

Object of class comp_risk_rel_data for strength Y.

pdf_X

Density function for X.

cdf_X

CDF function for X.

pdf_Y

Density function for Y.

cdf_Y

CDF function for Y.

init_par_X

Initial parameter value for X.

init_par_Y

Initial parameter value for Y.

method

Optimization method.

Value

S3 object containing parameter estimates, estimated reliability R = P(X < Y), standard error, and asymptotic confidence interval.

Examples

dat <- gen_stress_strength(
  pdf_X = function(x) dexp(x, rate = 1.2), cdf_X = function(x) pexp(x, rate = 1.2),
  pdf_Y = function(y) dexp(y, rate = 0.8), cdf_Y = function(y) pexp(y, rate = 0.8),
  lower_X = 0, upper_X = 10, lower_Y = 0, upper_Y = 10,
  n_X = 20, n_Y = 20, censoring_type = "type1_hybrid",
  r_X = 12, T_X = 1.2, r_Y = 12, T_Y = 1.5, seed = 123
)
stress_strength_rel(
  data_X = dat$data_X, data_Y = dat$data_Y,
  pdf_X = function(x, th) dexp(x, rate = th[1]), cdf_X = function(x, th) pexp(x, rate = th[1]),
  pdf_Y = function(y, th) dexp(y, rate = th[1]), cdf_Y = function(y, th) pexp(y, rate = th[1]),
  init_par_X = c(1.0), init_par_Y = c(0.7), method = "BFGS"
)

Summary Method for Bayesian Fit Objects

Description

Summary Method for Bayesian Fit Objects

Usage

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

Arguments

object

Object of class bayes_fit.

...

Unused parameters.

Value

Summary table with posterior estimates and credible intervals.

Summary Method for MLE Fit Objects

Description

Summary Method for MLE Fit Objects

Usage

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

Arguments

object

Object of class mle_fit.

...

Unused additional parameters.

Value

A matrix summary of parameter estimates, standard errors, z-values, and p-values.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.