Ramón V. León

Bayesian Modeling of Accelerated Life Tests with Random Effects

We show how to use Bayesian methods to make inferences from an accelerated life test where the test units come from different groups (such as batches) and the group effect is random and significant both statistically and practically. Our approach can handle multiple random effects and several accelerating factors. We illustrate the method with an application concerning pressure vessels wrapped in Kevlar 49 fibers, where the fiber of each vessel comes from a single spool and the spool effect is random. We show how Bayesian analysis using Markov chain Monte Carlo (MCMC) methods is used to answer questions of interest in accelerated life tests with random effects that are not as easily answered with more traditional frequentist methods. For example, we can predict the lifetime of a pressure vessel wound with a Kevlar 49 fiber either from a spool used in the accelerated life test or from another random spool from the population of spools. We comment on the implications that this analysis has on the estimates of reliability (and safety) for the Space Shuttle, which has a system of 22 such pressure vessels. Our approach is implemented in the freely available WinBUGS software so that readers can apply the method to their own data.

Effect of Not Having Homogeneous Test Units in Accelerated Life Tests

In accelerated life tests (ALTs), test units are run at higher stress levels in order to experience more failures. A Weibull regression model can, in this case, be used to infer failure behavior at the normal stress level. In practice, statistical analyses usually do not take batch differences into account even when they are present. To correctly include batch differences in the analysis, one needs to use a regression model with random effects. In this paper, we use such a model to show that ignoring batch differences in modeling can result in overly precise estimates of quantiles and probabilities of failure at the normal stress level, as well as overly precise predictions of the failure time for a new unit at the normal stress level.

Estimating Upper Percentiles of Strand Thickness for Oriented Strand Board

Statistical reliability methods are applied to estimate the upper percentiles of strand thickness for the face layers of oriented strand board (OSB) panels manufactured from six mills in the Eastern United States. The influence of thick strands on OSB properties (thickness swell, TS; internal bond, IB; and modulus of rupture, MOR) has been well documented. However, there is an absence in the literature of characterizing wood strand thickness for OSB mills in the context of statistical reliability methods. With induced percentile left censoring for improved model fitting, bootstrapping methods are employed for better estimating the upper percentiles and confidence intervals for strand thickness. The upper percentiles of flakes may be costly, damage equipment, or cause dimensional instability in OSB. The distributions of wood strands were nonnormal, and best-fit distributions varied from the log-logistic, largest extreme value, lognormal, and Weibull. The mean and median strand thicknesses for all mills wer...

Comparison of Two Wood Plastic Composite Extruders Using Bootstrap Confidence Intervals on Measurements of Sample Failure Data

ABSTRACT Wood plastic composite (WPC) boards are an emerging engineered wood composite that is a substitute for solid wood and other wood composite materials used for exterior applications, primarily decking. We are interested in understanding the strength of these boards and estimating the lower percentiles of failure under perpendicular pressure. The strength of WPC is determined by the perpendicular pressure required to permanently deform a board (modulus of elasticity, MOE) and the perpendicular pressure required to rupture the board (modulus of rupture, MOR). Two WPC production-size extrusion lines at the same facility are compared in this article by comparing the distributions of pressure to failure for samples of WPC extruded from each line. Parametric bootstrapping is used to calculate confidence intervals of the 1st, 5th, and 10th percentiles of the MOE and the MOR from each line. Furthermore, both parametric and nonparametric bootstrapping are performed to estimate confidence intervals on the differences between the two lines for the 1st, 5th, and 10th percentiles of the MOE and the MOR. A statistical difference between the strength of the WPC extruded from the two lines is found in the MOR.

Improving estimates of critical lower percentiles by induced censoring

In this article, we present an approach based on induced censoring for improving the estimation of critical lower percentiles. We validate this technique via simulation results and practical industrial insights. Data from product components that have at least two aging periods (e.g., bathtub failure rate) is investigated. When such data are improperly fit by certain reliability distributions, estimates of lower percentiles are impacted by longer-lasting failures, resulting in larger root mean square errors (RMSE) and bias. In lieu of utilizing a more complex bathtub model, we propose induced right censoring of data at various points to substantially reduce RMSE and bias of lower percentile estimates. A technique for finding optimal or near optimal censoring points is discussed and two real world examples illustrate how this works in practice.

Estimation of Error From Treating Travel Time as Additional Repair Time

We consider the unavailability of a two-unit parallel system with one traveling repair person, and common statistically independent exponential unit life lengths. The unavailability can be approximated by treating travel time as an additional repair time for each unit failure before returning to the duplex state, where the two units are functioning. This is an approximation because travel time extends the repair time only for the first unit failure before returning to the duplex state. We derive error bounds for this overestimation for arbitrary travel and repair time distributions. The error bounds show that the percent overestimation has a theoretical maximum of 27% (and can be as low as zero when the travel time distribution is degenerate at zero). Thus, this paper has important contributions for practice because it provides insight to the errors that occurred due to using an approximation.

Robustly Estimating Lower Percentiles When Observations Are Costly

ABSTRACT This article illustrates the effective use of Bayesian methods for the estimation of lower percentiles for the breaking strengths of materials. The method is presented in conjunction with the technique of median censoring, which censors reliability data at a point just slightly larger than the median. As a result, median censoring allows users to safeguard against different failure mechanisms with greater weight placed on the smaller observations when model fitting. Additionally, Bayesian techniques may enable one to use less data from the production line for destructive testing.

Detecting Changes in Field Reliability Using Data from a Complex Factory Screen

New products are sometimes screened in the factory before shipment to the field. We show how to use failure time data from one of these screens to predict early field reliability. These reliability predictions are more economical and timely than reliability estimates that wait for data from field tracking studies. Further, these predictions are continuously responsive to the changes in reliability that frequently occur during the manufacture of a new product, and therefore, can form the basis of a rapid-feedback system for manufacturing process control.