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Register a new userExperimental Designs for Accelerated Degradation Tests Based on Linear Mixed Effects Models
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Added by
Helmi Shat
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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Accelerated degradation tests are used to provide accurate estimation of lifetime properties of highly reliable products within a relatively short testing time. There data from particular tests at high levels of stress (e.,g. temperature, voltage, or vibration) are extrapolated, through a physically meaningful model, to obtain estimates of lifetime quantiles under normal use conditions. In this work, we consider repeated measures accelerated degradation tests with multiple stress variables, where the degradation paths are assumed to follow a linear mixed effects model which is quite common in settings when repeated measures are made. We derive optimal experimental designs for minimizing the asymptotic variance for estimating the median failure time under normal use conditions when the time points for measurements are either fixed in advance or are also to be optimized.
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Accelerated degradation tests are used to provide accurate estimation of lifetime characteristics of highly reliable products within a relatively short testing time. Data from particular tests at high levels of stress (e.g., temperature, voltage, or vibration) are extrapolated, through a physically meaningful statistical model, to attain estimates of lifetime quantiles at normal use conditions. The gamma process is a natural model for estimating the degradation increments over certain degradation paths, which exhibit a monotone and strictly increasing degradation pattern. In this work, we derive first an algorithm-based optimal design for a repeated measures degradation test with single failure mode that corresponds to a single response component. The univariate degradation process is expressed using a gamma model where a generalized linear model is introduced to facilitate the derivation of an optimal design. Consequently, we extend the univariate model and characterize optimal designs for accelerated degradation tests with bivariate degradation processes. The first bivariate model includes two gamma processes as marginal degradation models. The second bivariate models is expressed by a gamma process along with a mixed effects linear model. We derive optimal designs for minimizing the asymptotic variance for estimating some quantile of the failure time distribution at the normal use conditions. Sensitivity analysis is conducted to study the behavior of the resulting optimal designs under misspecifications of adopted nominal values.
Accelerated degradation testing (ADT) is one of the major approaches in reliability engineering which allows accurate estimation of reliability characteristics of highly reliable systems within a relatively short time. The testing data are extrapolated through a physically reasonable statistical model to obtain estimates of lifetime quantiles at normal use conditions. The Gamma process is a natural model for degradation, which exhibits a monotone and strictly increasing degradation path. In this work, optimal experimental designs are derived for ADT with two response components. We consider the situations of independent as well as dependent marginal responses where the observational times are assumed to be fixed and known. The marginal degradation paths are assumed to follow a Gamma process where a copula function is utilized to express the dependence between both components. For the case of independent response components the optimal design minimizes the asymptotic variance of an estimated quantile of the failure time distribution at the normal use conditions. For the case of dependent response components the $D$-criterion is adopted to derive $D$-optimal designs. Further, $D$- and $c$-optimal designs are developed when the copula-based models are reduced to bivariate binary outcomes.
The ability to generate samples of the random effects from their conditional distributions is fundamental for inference in mixed effects models. Random walk Metropolis is widely used to conduct such sampling, but such a method can converge slowly for medium dimension problems, or when the joint structure of the distributions to sample is complex. We propose a Metropolis Hastings (MH) algorithm based on a multidimensional Gaussian proposal that takes into account the joint conditional distribution of the random effects and does not require any tuning, in contrast with more sophisticated samplers such as the Metropolis Adjusted Langevin Algorithm or the No-U-Turn Sampler that involve costly tuning runs or intensive computation. Indeed, this distribution is automatically obtained thanks to a Laplace approximation of the original model. We show that such approximation is equivalent to linearizing the model in the case of continuous data. Numerical experiments based on real data highlight the very good performances of the proposed method for continuous data model.
This paper arises from collaborative research the aim of which was to model clinical assessments of upper limb function after stroke using 3D kinematic data. We present a new nonlinear mixed-effects scalar-on-function regression model with a Gaussian process prior focusing on variable selection from large number of candidates including both scalar and function variables. A novel variable selection algorithm has been developed, namely functional least angle regression (fLARS). As they are essential for this algorithm, we studied the representation of functional variables with different methods and the correlation between a scalar and a group of mixed scalar and functional variables. We also propose two new stopping rules for practical usage. This algorithm is able to do variable selection when the number of variables is larger than the sample size. It is efficient and accurate for both variable selection and parameter estimation. Our comprehensive simulation study showed that the method is superior to other existing variable selection methods. When the algorithm was applied to the analysis of the 3D kinetic movement data the use of the non linear random-effects model and the function variables significantly improved the prediction accuracy for the clinical assessment.
The functional linear model is a popular tool to investigate the relationship between a scalar/functional response variable and a scalar/functional covariate. We generalize this model to a functional linear mixed-effects model when repeated measurements are available on multiple subjects. Each subject has an individual intercept and slope function, while shares common population intercept and slope function. This model is flexible in the sense of allowing the slope random effects to change with the time. We propose a penalized spline smoothing method to estimate the population and random slope functions. A REML-based EM algorithm is developed to estimate the variance parameters for the random effects and the data noise. Simulation studies show that our estimation method provides an accurate estimate for the functional linear mixed-effects model with the finite samples. The functional linear mixed-effects model is demonstrated by investigating the effect of the 24-hour nitrogen dioxide on the daily maximum ozone concentrations and also studying the effect of the daily temperature on the annual precipitation.
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