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 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. Supplementary materials for this article are available online.

在退化试验中，试验单元通常被划分为若干组，每组在同一试验台（test rig）中同步开展测试。从试验设计的视角来看，每个试验台构成一个台架层区组（rig-layer block）。在每个试验台内，同一时刻被测量的试验单元进一步构成一个测量层区组（gauge-layer block）。由于试验台间存在不可控因素，且每次测量均会引入共性误差，不同区组内试验单元的退化测量结果可能存在差异；反之，同一区组内的退化结果应具备更高的同质性。本研究受新兴污染物（ECs）的应用场景启发，针对破坏性退化试验中的两层区组效应，提出了一种多元统计模型。首先采用多元维纳过程（Wiener Process）对退化不同维度间的相关性进行建模；台架层区组效应基于退化物理机制，采用一维脆弱因子（frailty）进行建模，而各测量时刻下的测量层区组效应，则通过共性加性测量误差进行刻画。本文推导了期望-最大化（Expectation-Maximization, EM）算法以获取模型参数的点估计，并构建了参数的置信区间；同时提出了一种用于检验退化数据中区组效应显著性的流程。通过针对EC退化数据集的案例研究，本文验证了试验中两层区组效应的存在。借助所提出的模型，决策者可便捷地开展各类污染物的风险评估，并确定去除污染物所需的最小水处理时长。本文的补充材料可在线获取。