The major challenge for assessing system reliability or decision-making regarding maintenance or replacement policies is that many systems are composed of, or can be viewed as, complex systems of components (or of subsystems) whose behavior within the system can be quite complicated due to omponent interactions and dependencies. This challenge requires model selection that incorporates "physics-of-failure" considerations into the model, integrates component reliability information into system reliability models that realistically models the censoring for these types of systems, and accounts for uncertainties in the model. Crucial to the decision making process is the proper model choice (model selection) as well as the integration of component reliability information into the system reliability assessment where in many cases the component information obtained from system data involves complicated censoring mechanisms. The major goal of this proposed project is to study dynamic load-sharing reliability models and decision-making with model selection for such models. Specifically, the aims of this project are: 1. to propose dynamic load-sharing models for reliability systems that incorporate "physics-of-failure" considerations and the dynamic interactions and dependencies among components or subsystems; 2. to obtain probabilistic properties of these load-sharing systems, in particular, to derive the system life distribution; 3. to examine data-accrual schemes for such systems and to develop statistical inference procedures for these load-sharing systems that also account for censoring; and 4. to develop decision-making strategies in the context of reliability systems when there are several competing models, leading in particular to decision-making with model selection or, possibly, model-averaging.
The assessment of system reliability requires accurate prediction of system failure. This is also essential to decision making regarding maintenance or replacement policies and is especially important for key systems or equipment in attempting to prevent catastrophic failures during critical operations. The merit of this proposed project emanates from the fact that the dynamic load-sharing reliability models will take into account model component interactions and dependencies. These generic reliability models will also be useful for other complex systems, such as in physical, biological, and medical sciences. For example, for a mechanical system under increasing load, (such as a composite under tensile loading where fiber segments are components or a routing system under increasing traffic, where the nodes are components) the load-sharing rule describes how the load or traffic is transferred/redistributed from failed components to working components and takes into consideration the "physics-of-failure." The proposed models have the potential to synthesize "physics-of-failure" and "statistical reliability" concepts to describe how damage to the system contributes to system failure. The statistical inference aspects of this project, including the model selection and the decision-making portion, will address important problems pertaining to the estimation of model parameters based on complex data for these dynamic models. This inference problem has not been dealt with extensively in the reliability literature, hence this project is expected to provide significant advances on this direction. The results of the investigations are expected to impact engineering and other sciences by providing novel and more realistic models for system failure and inference procedures for prediction of failure and for making maintenance and replacement decisions.
The major research themes of the research project were the (i) statistical modeling and analysis of complex reliability systems; (ii) improving decision making for such systems or in multiple testing situations or when there are multiple model choices; and (iii) dynamic survival and reliability models. Load-sharing systems were developed to model complex materials such as fibrous composites. The major project outcomes obtained for such complex load-sharing systems and dynamic survival and reliability models were: (a) The joint distribution of the working and failed components when the system is subjected to a static load. (b) The statistical behavior of component failures and the nonparametric estimation of the distribution of the component breaking strengths as the system is subjected to an increasing load (dynamic load). (c) The aforementioned estimation procedure is used to analyze fibrous composite data and illustrates how a priori knowledge concerning the strength can be incorporated into the analysis. (d) Statistical inference procedures, such as estimation, confidence interval, and hypothesis tests, for the finite-dimensional (parametric) and infinite-dimensional (nonparametric) parameters and their functionals of the dynamic reliability models, in particular, with regards to recurrent event data such as those arising in the medical, public health, biological, engineering, economic and sociological fields. Multiple testing issues, such controlling false positive and false negative rates, are highly important in the analysis of large data sets, especially for informatics purposes. For example, in a gene expression data, one often is interested in determining the few important significantly differentially expressed genes from the many irrelevant genes (background effects). Related issues involve the sample size allocation to detect significant effects, for example, the sizes of the effects of the differentially expressed genes. Major outcomes that were obtained in the project pertaining to this topic are: (a) A simple model two point mixture model where one component in the mixture corresponds to the background effects and the second to the significant effects. An optimal statistical test is obtained to determine if there are any significant effects present. Surprisingly, the test statistic can have an asymptotic distribution which is heavy tailed, a stable law which is not a normal distribution. (b) A mathematical result is obtained indicating how the multiple tests should be dynamically allocated their Type I (a false discovery) error levels to achieve overall optimality in detecting significant genes, subject to control of false discovery rate or family-wise error rate. The optimal Type I error level allocations exploits the different detection abilities of the multiple tests due to varied effect sizes or heterogeneity in the variability of the data utilized for each test. Interestingly, the optimal allocation dictates that larger Type I error levels be targeted to those tests where the differential gains in detection abilities, attenuated by the allocated Type I error level, is highest. These are usually the tests with moderate detection abilities. (c) A general notion of compound p-value or significance value was introduced, through the notion of a decision process, which is general in nature and allowable even with randomized test procedures. This notion of p-value is particularly useful in dealing with multiple testing settings. Results of the project are expected to provide more useful and better models for decision making and statistical inference for the failure phenomena in complex systems that arise in materials science and other networks where components or subsystems experience load-sharing. At the same time, the applicability of the results are not confined to the engineering sciences as they are also highly applicable in genomics and proteomics, where the rapid advance in miniaturization and computer technology have spawned the generation of massive high-dimensional data sets whose proper treatment requires dynamic multiple testing and multiple decision making. Important real-life decisions (e.g., development of new drugs; performing surgeries based on results of multiple mammograms; etc.) are to be based on information from these massive data sets, and so the importance of being able to access the appropriate information from them is paramount. The project was also instrumental in developing and training new researchers in the statistical sciences. It enabled the principal investigators to provide support to some doctoral students while they were performing their research. The development and training of new researchers is of very high importance to the infrastructure of science since it helps in guaranteeing the continuance of research, specifically statistical research, in the years to come.
