Rank Regression for Parametric Lifetime Distributions

This function fits a regression model to a linearized parametric lifetime distribution for complete and (multiple) right-censored data. The parameters are determined in the frequently used (log-)location-scale parameterization.

For the Weibull, estimates are additionally transformed such that they are in line with the parameterization provided by the stats package (see Weibull).

# S3 method for default
rank_regression(
  x,
  y,
  status,
  distribution = c("weibull", "lognormal", "loglogistic", "sev", "normal", "logistic",
    "weibull3", "lognormal3", "loglogistic3", "exponential", "exponential2"),
  conf_level = 0.95,
  direction = c("x_on_y", "y_on_x"),
  control = list(),
  options = list(),
  ...
)

Arguments

x

A numeric vector which consists of lifetime data. Lifetime data could be every characteristic influencing the reliability of a product, e.g. operating time (days/months in service), mileage (km, miles), load cycles.

y

A numeric vector which consists of estimated failure probabilities regarding the lifetime data in x.

status

A vector of binary data (0 or 1) indicating whether a unit is a right censored observation (= 0) or a failure (= 1).

distribution

Supposed distribution of the random variable.

conf_level

Confidence level of the interval.

direction

Direction of the dependence in the regression model.

control

A list of control parameters (see optim).

control is in use only if a three-parametric distribution was specified. If this is the case, optim (always with method = "L-BFGS-B" and control$fnscale = -1) is called to determine the threshold parameter (see r_squared_profiling).

options

A list of named options. See 'Options'.

...

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

Value

A list with classes wt_model, wt_rank_regression and wt_model_estimationwhich contains:

• coefficients : A named vector of estimated coefficients (parameters of the assumed distribution). Note: The parameters are given in the (log-)location-scale-parameterization.

• confint : Confidence intervals for the (log-)location-scale parameters. For threshold distributions no confidence interval for the threshold parameter can be computed. If direction = "y_on_x", back-transformed confidence intervals are provided.

• shape_scale_coefficients : Only included if distribution is "weibull" or "weibull3" (parameterization used in Weibull).

• shape_scale_confint : Only included if distribution is "weibull" or "weibull3". Approximated confidence intervals for scale \(\eta\) and shape \(\beta\) (and threshold \(\gamma\) if distribution = "weibull3").

• varcov : Only provided if options = list(conf_method = "HC") (default). Estimated heteroscedasticity-consistent (HC) variance-covariance matrix for the (log-)location-scale parameters.

• r_squared : Coefficient of determination.

• data : A tibble with columns x, status and prob.

• distribution : Specified distribution.

• direction : Specified direction.

Details

The confidence intervals of the parameters are computed on the basis of a heteroscedasticity-consistent (HC) covariance matrix. Here it should be said that there is no statistical foundation to determine the standard errors of the parameters using Least Squares in context of Rank Regression. For an accepted statistical method use maximum likelihood.

If options = list(conf_method = "Mock"), the argument distribution must be one of "weibull" and "weibull3". The approximated confidence intervals for the Weibull parameters can then only be estimated on the following confidence levels (see 'References' (Mock, 1995)):

• conf_level = 0.90

• conf_level = 0.95

• conf_level = 0.99

Options

Argument options is a named list of options:

Name	Value
conf_method	"HC" (default) or "Mock"

References

• Mock, R., Methoden zur Datenhandhabung in Zuverlässigkeitsanalysen, vdf Hochschulverlag AG an der ETH Zürich, 1995

• Meeker, William Q; Escobar, Luis A., Statistical methods for reliability data, New York: Wiley series in probability and statistics, 1998

See also

rank_regression

Examples

# Vectors:
obs <- seq(10000, 100000, 10000)
status_1 <- c(0, 1, 1, 0, 0, 0, 1, 0, 1, 0)

cycles <- alloy$cycles
status_2 <- alloy$status

# Example 1 - Fitting a two-parametric weibull distribution:
tbl_john <- estimate_cdf(
  x = obs,
  status = status_1,
  method = "johnson"
)

rr <- rank_regression(
  x = tbl_john$x,
  y = tbl_john$prob,
  status = tbl_john$status,
  distribution = "weibull",
  conf_level = 0.90
)

# Example 2 - Fitting a three-parametric lognormal distribution:
tbl_kaplan <- estimate_cdf(
  x = cycles,
  status = status_2,
  method = "kaplan"
)

rr_2 <- rank_regression(
  x = tbl_kaplan$x,
  y = tbl_kaplan$prob,
  status = tbl_kaplan$status,
  distribution = "lognormal3"
)