title: Two regression methods for estimation of a two-parameter Weibull distribution for reliability of dental materials.
abstract: OBJECTIVE
  Comparison of estimation of the two-parameter Weibull distribution by two least squares (LS) methods with interchanged axes. Investigation of the influence of plotting positions and sample size. Derivation of 95% confidence intervals (95%CI) for Weibull parameters applicable in the context of LS estimation. Preparation of a free available Excel template for computation of point estimates and 95%CI for Weibull modulus (m) and characteristic strength (s).
  
  
  METHODS
  Monte Carlo simulation covering a wide range of Weibull parameters and sample sizes. Mathematical derivation of formulae for computation of 95%CI according to a Menon-type approach for both m and s. Empirical proof that the practically observed coverage agrees with the nominal one of 95%.
  
  
  RESULTS
  Relative and absolute performance of LS estimators depended on sample size, plotting positions and parameter to be estimated. For most situations they outperformed the corresponding Maximum Likelihood (ML) estimator in terms of bias, while precision was almost the same. Naïve Wald-type 95%CI based on standard errors of LS regression coefficients did not reach targeted coverage. An easy-to-apply alternative based on asymptotic standard errors (Menon 95%CI) resulted in excellent coverage.
  
  
  CONCLUSIONS
  Accuracy of the LS methods for Weibull modulus and characteristic strength essentially depend on plotting position and sample size. Large sample sizes (n≥30) support a credible Weibull parameters estimation. An important complement of the point estimates of Weibull parameters is provided by the Menon 95%CI. A free available Excel template considerably facilitating computation of point and interval estimates of Weibull parameters is provided.
authors.affiliation: Division of Biostatistics, Institute of Social and Preventive Medicine, University of Zurich, Switzerland.
authors.forename: Lukas
authors.initials: L
authors.lastname: Bütikofer
authors.affiliation: Department of Prosthodontics, Dental School, Ludwig-Maximilians University, Munich, Germany.
authors.forename: Bogna
authors.initials: B
authors.lastname: Stawarczyk
authors.affiliation: Division of Biostatistics, Institute of Social and Preventive Medicine, University of Zurich, Switzerland. Electronic address: mroos@ifspm.uzh.ch.
authors.forename: Malgorzata
authors.initials: M
authors.lastname: Roos
mesh_terms: D003764:Dental Materials / Q000737:chemistry*; D019232:Dental Restoration Failure / Q000706:statistics & numerical data*; D016013:Likelihood Functions; D009010:Monte Carlo Method; D012044:Regression Analysis; D015203:Reproducibility of Results; D016008:Statistical Distributions
chemical_list: D003764:Dental Materials
keywords: Confidence interval; Coverage; Failure probability; Hazen ranks; Least squares; Mean ranks; Median ranks; Plotting positions; Weibull characteristic strength; Weibull modulus