Estimation of Failure Probabilities

This function applies a non-parametric method to estimate the failure probabilities of complete data taking (multiple) right-censored observations into account.

estimate_cdf(x, ...)

# S3 method for wt_reliability_data
estimate_cdf(
  x,
  methods = c("mr", "johnson", "kaplan", "nelson"),
  options = list(),
  ...
)

Arguments

x

A tibble with class wt_reliability_data returned by reliability_data.

...

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

methods

One or multiple methods of "mr", "johnson", "kaplan" or "nelson" used for the estimation of failure probabilities. See 'Details'.

options

A list of named options. See 'Options'.

Value

A tibble with class wt_cdf_estimation containing the following columns:

• id : Identification for every unit.

• x : Lifetime characteristic.

• status : Binary data (0 or 1) indicating whether a unit is a right censored observation (= 0) or a failure (= 1).

• rank : The (computed) ranks. Determined for methods "mr" and "johnson", filled with NA for other methods or if status = 0.

• prob : Estimated failure probabilities, NA if status = 0.

• cdf_estimation_method : Specified method for the estimation of failure probabilities.

Details

One or multiple techniques can be used for the methods argument:

• "mr" : Method Median Ranks is used to estimate the failure probabilities of failed units without considering censored items. Tied observations can be handled in three ways (See 'Options'):

  • "max" : Highest observed rank is assigned to tied observations.

  • "min" : Lowest observed rank is assigned to tied observations.

  • "average" : Mean rank is assigned to tied observations.

  Two formulas can be used to determine cumulative failure probabilities F(t) (See 'Options'):

  • "benard" : Benard's approximation for Median Ranks.

  • "invbeta" : Exact Median Ranks using the inverse beta distribution.

• "johnson" : The Johnson method is used to estimate the failure probabilities of failed units, taking censored units into account. Compared to complete data, correction of probabilities is done by the computation of adjusted ranks. Two formulas can be used to determine cumulative failure probabilities F(t) (See 'Options'):

  • "benard" : Benard's approximation for Median Ranks.

  • "invbeta" : Exact Median Ranks using the inverse beta distribution.

• "kaplan" : The method of Kaplan and Meier is used to estimate the survival function S(t) with respect to (multiple) right censored data. The complement of S(t), i.e. F(t), is returned. In contrast to the original Kaplan-Meier estimator, one modification is made (see 'References').

• "nelson" : The Nelson-Aalen estimator models the cumulative hazard rate function in case of (multiple) right censored data. Equating the formal definition of the hazard rate with that according to Nelson-Aalen results in a formula for the calculation of failure probabilities.

Options

Argument options is a named list of options:

Method	Name	Value
mr	mr_method	"benard" (default) or "invbeta"
mr	mr_ties.method	"max" (default), "min" or "average"
johnson	johnson_method	"benard" (default) or "invbeta"

References

NIST/SEMATECH e-Handbook of Statistical Methods, 8.2.1.5. Empirical model fitting - distribution free (Kaplan-Meier) approach, NIST SEMATECH, December 3, 2020

Examples

# Reliability data:
data <- reliability_data(
  alloy,
  x = cycles,
  status = status
)

# Example 1 - Johnson method:
prob_tbl <- estimate_cdf(
  x = data,
  methods = "johnson"
)

# Example 2 - Multiple methods:
prob_tbl_2 <- estimate_cdf(
  x = data,
  methods = c("johnson", "kaplan", "nelson")
)

# Example 3 - Method 'mr' with options:
prob_tbl_3 <- estimate_cdf(
  x = data,
  methods = "mr",
  options = list(
    mr_method = "invbeta",
    mr_ties.method = "average"
  )
)
#> The 'mr' method only considers failed units (status == 1) and does not retain intact units (status == 0).

# Example 4 - Multiple methods and options:
prob_tbl_4 <- estimate_cdf(
  x = data,
  methods = c("mr", "johnson"),
  options = list(
    mr_ties.method = "max",
    johnson_method = "invbeta"
  )
)
#> The 'mr' method only considers failed units (status == 1) and does not retain intact units (status == 0).