Analysis of single classification experiments based on censored samples from the two-parameter Weibull distribution

Abstract

Methodology and supporting tables are given for the detailed analysis of a one way layout or single classification experiment in which the data comprise censored samples from two parameter Weibull populations. The complete analysis includes a test for the equality of shape parameters and, when the shape parameters can be assumed homogeneous, 1. (i) an “omnibus” test for the equality of scale parameters 2. (ii) a fixed range multiple comparison test for the equality of scale parameters 3. (iii) interval estimation of the common shape parameter and 4. (iv) interval estimation of an arbitrary percentile of one of the populations. Two omnibus tests are considered, viz., the likelihood ratio test and a test analogous to the F ratio test of the analysis of variance. This latter test utilizes the ratio of the two maximum likelihood estimates of the shape parameter obtained with and without the condition that the scale parameters are distinct. Numerical values of key percentage points of the relevant distributions required for carrying out the analysis, obtained by Monte Carlo sampling, cover the range from k = 2 samples of size n = 5 censored at order statistic number r = 3 to the case k = r = n = 10. The power of the shape parameter ratio test is shown to depend on a single scale invariant symmetric function of the scale parameters. The power of the likelihood ratio test is governed by both this same function and by one additional scale invariant symmetric function. The acceptance probability is tabled for choices of four values of each of these two functions for eight n, r, k combinations in the range from k = 2, n = 5, r = 3 to k = r = n = 10. The shape parameter ratio test is found to be more powerful than the likelihood ratio test against the alternative hypothesis that k − 1 of k scale parameters are homogeneous. The likelihood ratio test is superior against more diffuse alternatives. The distribution of the likelihood ratio test statistic is compared to its asymptotic chi square distribution. A factor less than unity is found for each n, r, k whereby the likelihood ratio test statistic may be corrected into good accord with its asymptotic distribution. Using critical values from the asymptotic distribution with the uncorrected likelihood ratio test statistic is shown to result in a substantially higher than nominal error rate when the sample size is small and/or the amount of censoring is high. A numerical example is given of the analysis of a one way layout and of three alternative methods for the selection of sample size in single classification experiments.