Beta Binomial Confidence Bounds for Quantiles and Probabilities

Source: R/confidence_intervals.R

This function computes the non-parametric beta binomial confidence bounds (BB) for quantiles and failure probabilities.

confint_betabinom(x, ...)

# S3 method for wt_model
confint_betabinom(
  x,
  b_lives = c(0.01, 0.1, 0.5),
  bounds = c("two_sided", "lower", "upper"),
  conf_level = 0.95,
  direction = c("y", "x"),
  ...
)

Arguments

x

A list with class wt_model (and further classes) returned by rank_regression.

...

Further arguments passed to or from other methods. Currently not used.

b_lives

A numeric vector indicating the probabilities \(p\) of the \(B_p\)-lives (quantiles) to be considered.

bounds

A character string specifying the bound(s) to be computed.

conf_level

Confidence level of the interval.

direction

A character string specifying the direction of the confidence interval. "y" for failure probabilities or "x" for quantiles.

Value

A tibble with class wt_confint containing the following columns:

  • x : An ordered sequence of the lifetime characteristic regarding the failed units, starting at min(x) and ending up at max(x). With b_lives = c(0.01, 0.1, 0.5) the 1%, 10% and 50% quantiles are additionally included in x, but only if the specified probabilities are in the range of the estimated probabilities.

  • rank : Interpolated ranks as a function of probabilities, computed with the converted approximation formula of Benard.

  • prob : An ordered sequence of probabilities with specified b_lives included.

  • lower_bound : Provided, if bounds is one of "two_sided" or "lower". Lower confidence limits with respect to direction, i.e. limits for quantiles or probabilities.

  • upper_bound : Provided, if bounds is one of "two_sided" or "upper". Upper confidence limits with respect to direction, i.e. limits for quantiles or probabilities.

  • cdf_estimation_method : Method for the estimation of failure probabilities which was specified in estimate_cdf.

Further information is stored in the attributes of this tibble:

  • distribution : Distribution which was specified in rank_regression.

  • bounds : Specified bound(s).

  • direction : Specified direction.

  • model_estimation : Input list with class wt_model.

Details

The procedure is similar to the Median Ranks method but with the difference that instead of finding the probability for the j-th rank at the 50% level the probability (probabilities) has (have) to be found at the given confidence level.

Examples

# Reliability data preparation:
## Data for two-parametric model:
data_2p <- reliability_data(
  shock,
  x = distance,
  status = status
)

## Data for three-parametric model:
data_3p <- reliability_data(
  alloy,
  x = cycles,
  status = status
)

# Probability estimation:
prob_tbl_2p <- estimate_cdf(
  data_2p,
  methods = "johnson"
)

prob_tbl_3p <- estimate_cdf(
  data_3p,
  methods = "johnson"
)

prob_tbl_mult <- estimate_cdf(
  data_3p,
  methods = c("johnson", "mr")
)
#> The 'mr' method only considers failed units (status == 1) and does not retain intact units (status == 0).

# Model estimation with rank_regression():
rr_2p <- rank_regression(
  prob_tbl_2p,
  distribution = "weibull"
)

rr_3p <- rank_regression(
  prob_tbl_3p,
  distribution = "lognormal3",
  conf_level = 0.90
)

rr_lists <- rank_regression(
  prob_tbl_mult,
  distribution = "loglogistic3",
  conf_level = 0.90
)

# Example 1 - Two-sided 95% confidence interval for probabilities ('y'):
conf_betabin_1 <- confint_betabinom(
  x = rr_2p,
  bounds = "two_sided",
  conf_level = 0.95,
  direction = "y"
)

# Example 2 - One-sided lower/upper 90% confidence interval for quantiles ('x'):
conf_betabin_2_1 <- confint_betabinom(
  x = rr_2p,
  bounds = "lower",
  conf_level = 0.90,
  direction = "x"
)

conf_betabin_2_2 <- confint_betabinom(
  x = rr_2p,
  bounds = "upper",
  conf_level = 0.90,
  direction = "x"
)

# Example 3 - Two-sided 90% confidence intervals for both directions using
# a three-parametric model:
conf_betabin_3_1 <- confint_betabinom(
  x = rr_3p,
  bounds = "two_sided",
  conf_level = 0.90,
  direction = "y"
)

conf_betabin_3_2 <- confint_betabinom(
  x = rr_3p,
  bounds = "two_sided",
  conf_level = 0.90,
  direction = "x"
)

# Example 4 - Confidence intervals if multiple methods in estimate_cdf, i.e.
# "johnson" and "mr", were specified:

conf_betabin_4 <- confint_betabinom(
  x = rr_lists,
  bounds = "two_sided",
  conf_level = 0.99,
  direction = "y"
)