Prediction intervals for load‐sharing systems in accelerated life testing

Based on accelerated lifetime experiments, we consider the problem of constructing prediction intervals for the time point at which a given number of components of a load‐sharing system fails. Our research is motivated by lab experiments with prestressed concrete beams where the tension wires fail successively. Due to an audible noise when breaking, the time points of failure could be determined exactly by acoustic measurements. Under the assumption of equal load sharing between the tension wires, we present a model for the failure times based on a birth process. We provide a model check based on a Q‐Q plot including a simulated simultaneous confidence band and four simulation‐free prediction methods. Three of the prediction methods are given by confidence sets where two of them are based on classical tests and the third is based on a new outlier‐robust test using sign depth. The fourth method uses the implicit function theorem and the δ‐method to get prediction intervals without confidence sets for the unknown parameter. We compare these methods by a leave‐one‐out analysis of the data on prestressed concrete beams. Moreover, a simulation study is performed to discuss advantages and drawbacks of the individual methods

Prediction intervals for the failure time of prestressed concrete beams

The aim is the prediction of the failure time of prestressed concrete beams under low cyclic load. Since the experiments last long for low load, accelerated failure tests with higher load are conducted. However, the accelerated tests are expensive so that only few tests are available. To obtain a more precise failure time prediction, the additional information of time points of breakage of tension wires is used. These breakage time points are modeled by a nonlinear birth process. This allows not only point prediction of a critical number of broken tension wires but also prediction intervals which express the uncertainty of the prediction

Variable selection for disease progression models: methods for oncogenetic trees and application to cancer and HIV

Abstract Background Disease progression models are important for understanding the critical steps during the development of diseases. The models are imbedded in a statistical framework to deal with random variations due to biology and the sampling process when observing only a finite population. Conditional probabilities are used to describe dependencies between events that characterise the critical steps in the disease process. Many different model classes have been proposed in the literature, from simple path models to complex Bayesian networks. A popular and easy to understand but yet flexible model class are oncogenetic trees. These have been applied to describe the accumulation of genetic aberrations in cancer and HIV data. However, the number of potentially relevant aberrations is often by far larger than the maximal number of events that can be used for reliably estimating the progression models. Still, there are only a few approaches to variable selection, which have not yet been investigated in detail. Results We fill this gap and propose specifically for oncogenetic trees ten variable selection methods, some of these being completely new. We compare them in an extensive simulation study and on real data from cancer and HIV. It turns out that the preselection of events by clique identification algorithms performs best. Here, events are selected if they belong to the largest or the maximum weight subgraph in which all pairs of vertices are connected. Conclusions The variable selection method of identifying cliques finds both the important frequent events and those related to disease pathways

Additional file 1 of Variable selection for disease progression models: methods for oncogenetic trees and application to cancer and HIV

Figure A.1. Illustrating example concerning the difference between largest and maximal cliques. Figure A.2. Cluster dendrogram of the L1-distances using the complete linkage approach to potentially restrict the number of parameter combinations. Figure A.3. Results of the univariate frequency method for the L 2-distances respectively cosine-distances. Figure A.4. Results of the missing six variable selection methods. Based on these graphics one can identify the best threshold. Figure A.5. Results of the simulation study for the two criteria sens and spec where Îą l =0.2. Figures A.6 and A.7. Results of sens vs. spec for 16 different data situations. Figure A.8. Remaining trees resulting from the variable selection process concerning the glioblastoma data set. Figure A.9. Scatterplots of true data vs. contaminated data. Figure A.10. Scatterplots of true data vs. data with 10% noise. Table B.1. List of the 32 parameter settings representing the different data situations that are investigated by our variable selection methods. Table B.2. List of events from the extended meningioma data set (39 additional variables with a random frequency of 0.5%) that were chosen by our variable selection methods using the thresholds from the simulation study. (PDF 409 kb

Trimmed likelihood estimators for lifetime experiments and their influence functions

We study the behaviour of trimmed likelihood estimators (TLEs) for lifetime models with exponential or lognormal distributions possessing a linear or nonlinear link function. In particular, we investigate the difference between two possible definitions for the TLE, one called original trimmed likelihood estimator (OTLE) and one called modified trimmed likelihood estimator (MTLE) which is the finite sample version of a form for location and linear regression used by Bednarski and Clarke {[}Trimmed likelihood estimation of location and scale of the normal distribution. Aust J Statist. 1993;35:141-153, Asymptotics for an adaptive trimmed likelihood location estimator. Statistics. 2002;36:1-8] and Bednarski et al. {[}Adaptive trimmed likelihood estimation in regression. Discuss Math Probab Stat. 2010;30:203-219]. The OTLE is always an MTLE but the MTLE may not be unique even in cases where the OLTE is unique. We compare especially the functional forms of both types of estimators, characterize the difference with the implicit function theorem and indicate situations where they coincide and where they do not coincide. Since the functional form of the MTLE has a simpler form, we use it then for deriving the influence function, again with the help of the implicit function theorem. The derivation of the influence function for the functional form of the OTLE is similar but more complicated