| Construction materials, such as lumber and steel, and manufactured materials, such as open link chain, may be tested for multiple physical strength properties. Any interrelationship between strength properties can make a system or structure less reliable. Strength variables are often measured through trials that have the potential for destroying the object of the test. Units destroyed by testing on an initial property are not available for further tests, so only one type of strength is measurable for each unit tested. Consequently, relationships among the various strength properties must be approximated. Experimenters have dealt with this problem using a technique called proof loading--stressing units up to a prescribed load, thereby destroying only the weaker units in the sample. Some units survive and can be tested on other variables.;Steiner and Wesolowsky (1995) approach the problem by recording the number of units that fail each of two proof loads. Hence, the experiment does not require sophisticated measuring equipment to record actual load at failure. The authors develop a two-stage process to estimate the correlation between the strength variables. They calculate the maximum likelihood estimator of the joint probability of failure from counts of units that fail the proof loads. Using an expression for the bivariate normal distribution due to Sheppard (1900), the authors solve for the correlation between the two variables. In a proof loading experiment, it is possible that the first proof loading yields survivors that are damaged, but not obviously so. Previous authors, including Steiner and Wesolowsky, assume that such damage does not occur.;We propose a distribution-free Bayesian approach for modeling the joint probability of failure on two proof loads. Prior probabilities of failure at a particular proof load can be realistically modeled using beta distributions, and a posterior distribution may be derived. One may then determine posterior moments for the joint probability of failure. In addition, the assumptions of normality and the absence of damage to survivors of proof loads are completely unnecessary. |
