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. A numeric vector indicating the probabilities $$p$$ of the $$B_p$$-lives (quantiles) to be considered. A character string specifying the bound(s) to be computed. Confidence level of the interval. 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"
)