Man Ho Ling

Accelerated Degradation Analysis for the Quality of a System Based on the Gamma Process

As most systems these days are highly reliable with long lifetimes, failures of systems become rare; consequently, traditional failure time analysis may not be able to provide a precise assessment of the system reliability. In this regard, a degradation measure, as a percentage of the initial value, is an alternate way of describing the system health. This paper presents accelerated degradation analysis that characterizes the health and quality of systems with monotonic and bounded degradation. The maximum likelihood estimates (MLEs) of the model parameters are derived, based on a gamma process, time-scale transformation, and a power link function for associating the covariates. Then, methods of estimating the reliability, the mean and median lifetime, the conditional reliability, and the remaining useful life of systems under normal use conditions are all described. Moreover, approximate confidence intervals for the parameters of interest are developed based on the observed Fisher information matrix. A model validation metric with exact power is introduced. A Monte Carlo simulation study is carried out for evaluating the performance of the proposed methods. For an illustration of the proposed model, and the methods of inference developed here, a numerical example involving light intensity of light emitting diodes (LED) is analyzed.

Gamma lifetimes and one-shot device testing analysis

Abstract Gamma distribution is widely used to model lifetime data in reliability and survival analysis. In the context of one-shot device testing, encountered commonly in testing devices such as munitions, rockets, and automobile air-bags, either left- or right-censored data are collected instead of actual lifetimes of the devices under test. The destructive nature of one-shot devices makes it difficult to collect sufficient lifetime information on the devices. For this reason, accelerated life-tests are commonly used in which the test devices are subjected to conditions in excess of its normal use-condition in order to induce more failures, so as to obtain more lifetime information within a relatively short period of time. In this paper, we discuss the analysis of one-shot device testing data under accelerated life-tests based on gamma distribution. Both scale and shape parameters of the gamma distribution are related to stress factors through log–linear link functions. Since lifetimes of devices under this test are censored, the EM algorithm is developed here for the estimation of the model parameters. The inference on the reliability at a specific mission time as well as on the mean lifetime of the devices is also developed. Moreover, by using missing information principle, the asymptotic variance–covariance matrix of the maximum likelihood estimates under the EM framework is determined, and is then used to construct asymptotic confidence intervals for the parameters of interest. For the reliability at a specific mission time and also for the mean lifetime of the devices, transformation approaches are proposed for the construction of confidence intervals. These confidence intervals are then compared through a simulation study in terms of coverage probabilities and average widths. Recommendations are then made for an appropriate approach for the construction of confidence intervals for different sample sizes and different levels of reliability. A distance-based statistic is suggested for testing the validity of the model to an observed data. Finally, since current status data with covariates in survival analysis and one-shot device testing data with stress factors in reliability analysis share the same data structure, a real data from a toxicological study is used to illustrate the developed methods.

Confidence intervals for a difference between proportions based on paired data

We construct several explicit asymptotic two-sided confidence intervals (CIs) for the difference between two correlated proportions using the method of variance of estimates recovery (MOVER). The basic idea is to recover variance estimates required for the proportion difference from the confidence limits for single proportions. The CI estimators for a single proportion, which are incorporated with the MOVER, include the Agresti-Coull, the Wilson, and the Jeffreys CIs. Our simulation results show that the MOVER-type CIs based on the continuity corrected Phi coefficient and the Tango score CI perform satisfactory in small sample designs and spare data structures. We illustrate the proposed CIs with several real examples.

Exact and approximate unconditional confidence intervals for proportion difference in the presence of incomplete data

Confidence interval (CI) construction with respect to proportion/rate difference for paired binary data has become a standard procedure in many clinical trials and medical studies. When the sample size is small and incomplete data are present, asymptotic CIs may be dubious and exact CIs are not yet available. In this article, we propose exact and approximate unconditional test-based methods for constructing CI for proportion/rate difference in the presence of incomplete paired binary data. Approaches based on one- and two-sided Walds tests will be considered. Unlike asymptotic CI estimators, exact unconditional CI estimators always guarantee their coverage probabilities at or above the pre-specified confidence level. Our empirical studies further show that (i) approximate unconditional CI estimators usually yield shorter expected confidence width (ECW) with their coverage probabilities being well controlled around the pre-specified confidence level; and (ii) the ECWs of the unconditional two-sided-test-based CI estimators are generally narrower than those of the unconditional one-sided-test-based CI estimators. Moreover, ECWs of asymptotic CIs may not necessarily be narrower than those of two-sided-based exact unconditional CIs. Two real examples will be used to illustrate our methodologies.

Multiple-Stress Model for One-Shot Device Testing Data Under Exponential Distribution

Left- and right-censored life time data arise naturally in one-shot device testing. An experimenter is often interested in identifying the effects of several stress variables on the lifetime of a device, and furthermore multiple-stress experiments controlling simultaneously several variables, result in reducing the experimental time as well as the cost of the experiment. Here, we present an expectation-maximization (EM) algorithm for developing inference on the reliability at a specific time, as well as the mean lifetime of the device based on one-shot device testing data under the exponential distribution when there are multiple stress factors. We use the log-linear link function for this purpose. Unlike in the typical EM algorithm, it is not necessary to obtain maximum likelihood estimates (MLEs) of the parameters at each step of the iteration. By using the one-step Newton-Raphson method, we observe that the convergence occurs quickly. We also use the jackknife technique to reduce the bias of the estimate obtained from the EM algorithm. In addition, we discuss the construction of confidence intervals for some reliability characteristics by using the asymptotic properties of the MLEs based on the observed Fisher information matrix, as well as by the jackknife technique, the parametric bootstrap methods, and a transformation technique. Finally, we present an example to illustrate all the inferential methods developed here.

EM algorithm for one-shot device testing under the exponential distribution

The EM algorithm is a powerful technique for determining the maximum likelihood estimates (MLEs) in the presence of binary data since the maximum likelihood estimators of the parameters cannot be expressed in a closed-form. In this paper, we consider one-shot devices that can be used only once and are destroyed after use, and so the actual observation is on the conditions rather than on the real lifetimes of the devices under test. Here, we develop the EM algorithm for such data under the exponential distribution for the lifetimes. Due to the advances in manufacturing design and technology, products have become highly reliable with long lifetimes. For this reason, accelerated life tests are performed to collect useful information on the parameters of the lifetime distribution. For such a test, the Bayesian approach with normal prior was proposed recently by Fan et al. (2009). Here, through a simulation study, we show that the EM algorithm and the mentioned Bayesian approach are both useful techniques for analyzing such binary data arising from one-shot device testing and then make a comparative study of their performance and show that, while the Bayesian approach is good for highly reliable products, the EM algorithm method is good for moderate and low reliability situations.

EM Algorithm for One-Shot Device Testing With Competing Risks Under Weibull Distribution

This paper is an extension of the work of Balakrishnan and Ling [1] introducing a competing risk model into a one-shot device testing analysis under an accelerated life-test setting. An Expectation Maximization (EM) algorithm is then developed for the estimation of the model parameters. An extensive Monte Carlo simulation study is carried out to assess the performance of the EM algorithm and then compare the obtained results with the initial estimates obtained by the Inequality Constrained Least Squares (ICLS) method of estimation. Finally, we apply the EM algorithm to a clinical data, ED01, to illustrate the method of inference developed here.

Best Constant-Stress Accelerated Life-Test Plans With Multiple Stress Factors for One-Shot Device Testing Under a Weibull Distribution

We discuss here the design of constant-stress accelerated life-tests for one-shot device testing by assuming a Weibull distribution as a lifetime model. Because there are no explicit expressions for the maximum likelihood estimators of the model parameters and their variances, we adopt the asymptotic approach here to develop an algorithm for the determination of optimal allocation of devices, inspection frequency, and the number of inspections at each stress level, by assuming a Weibull distribution with non-constant scale and shape parameters as the lifetime distribution. The asymptotic variance of the estimate of reliability of the device at a specified mission time is minimized subject to a pre-fixed experimental budget, and a termination time. Examples are provided to illustrate the proposed algorithm for the determination of the best test plan. A sensitivity analysis of the best test plan is also carried out to examine the effect of misspecification of the model parameters.

A Bayesian Approach for One-Shot Device Testing With Exponential Lifetimes Under Competing Risks

This paper considers a competing risk model for a one-shot device testing analysis under an accelerated life test setting. Due to the consideration of competing risks, the joint posterior distribution becomes quite complicated. The Metropolis-Hastings sampling method is used for the estimation of the posterior means of the variables of interest. A simulation study is carried out to assess the Bayesian approach with different priors, and also to compare it with the EM algorithm for maximum likelihood estimation. Finally, an example from a tumorigenicity experiment is presented.

Learning analytics for monitoring students participation online: Visualizing navigational patterns on learning management system

With the increasing use of blended learning approaches in classroom, various kinds of technologies are incorporated to provide digital teaching and learning resources to support students. These resources are often centralized in learning management systems (LMSs), which also store valuable learning data of students. The data could assist teachers in their pedagogical decision making but they are often not well utilized. This paper proposes the use of data mining and visualization techniques as learning analytics to provide a more comprehensive overview of students’ learning online based on log data from LMSs . The focus of this study is the discovery of frequent navigational patterns by sequential pattern mining techniques and the demonstration of how presentation of patterns through hierarchical clustering and sunburst visualization could facilitate the interpretation of patterns. The data in this paper were collected from a blended statistics course for undergraduate students.