Statistical Modeling of Multivariate Destructive Degradation Tests with Blocking

In degradation tests, the test units are usually divided into several groups, with each group tested simultaneously in a test rig. Each rig constitutes a rig-layer block from the perspective of design of experiments. Within each rig, the test units measured at the same time further form a gauge-layer block. Due to the uncontrollable factors among test rigs and the common errors incurred for each measurement, the degradation measurements of the test units may differ among various blocks. On the other hand, the degradation should be more homogeneous within a block. Motivated by an application of emerging contaminants (ECs), this study proposes a multivariate statistical model to account for the two-layer block effects in destructive degradation tests. A multivariate Wiener process is first used to model the correlation among different dimensions of degradation. The rig-layer block effect is modeled by a one-dimensional frailty motivated by the degradation physics, while the gauge-layer block effect at each measurement epoch is captured by a common additive measurement error. We develop an Expectation-Maximization (EM) algorithm to obtain the point estimates of the model parameters and construct confidence intervals for the parameters. A procedure is proposed to test significance of the block effects in the degradation data. Through a case study on an EC degradation dataset, we show the existence of the two-layer block effects from the test. By making use of the proposed model, decision makers can readily make risk assessment of each contaminant and determine the minimal water treatment time for removal of the contaminants.