Lukas Bütikofer1, Bogna Stawarczyk2, Malgorzata Roos3. 1. Division of Biostatistics, Institute of Social and Preventive Medicine, University of Zurich, Switzerland. 2. Department of Prosthodontics, Dental School, Ludwig-Maximilians University, Munich, Germany. 3. Division of Biostatistics, Institute of Social and Preventive Medicine, University of Zurich, Switzerland. Electronic address: mroos@ifspm.uzh.ch.
Abstract
OBJECTIVES: 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. CONCLUSION: 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.
OBJECTIVES: 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. CONCLUSION: 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.