Publications

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This document outlines a data-driven probabilistic approach to setting product acceptance testing limits. Product Specification (PS) limits are testing requirements for assuring that the product meets the product requirements. After identifying key manufacturing and performance parameters for acceptance testing, PS limits should be specified for these parameters, with the limits selected to assure that the unit will have a very high likelihood of meeting product requirements (barring any quality defects that would not be detected in acceptance testing). Because the settings for which the product requirements must be met is typically broader than the production acceptance testing space, PS limits should account for the difference between the acceptance testing setting relative to the worst-case setting. We propose an approach to setting PS limits that is based on demonstrating margin to the product requirement in the worst-case setting in which the requirement must be met. PS limits are then determined by considering the overall margin and uncertainty associated with a component requirement and then balancing this margin and uncertainty between the designer and producer. Specifically, after identifying parameters critical to component performance, we propose setting PS limits using a three step procedure: 1. Specify the acceptance testing and worst-case use-settings, the performance characteristic distributions in these two settings, and the mapping between these distributions. 2. Determine the PS limit in the worst-case use-setting by considering margin to the requirement and additional (epistemic) uncertainties. This step controls designer risk, namely the risk of producing product that violates requirements. 3. Define the PS limit for product acceptance testing by transforming the PS limit from the worst-case setting to the acceptance testing setting using the mapping between these distributions. Following this step, the producer risk is quantified by estimating the product scrap rate based on the projected acceptance testing distribution. The approach proposed here provides a framework for documenting the procedure and assumptions used to determine PS limits. This transparency in procedure will help inform what actions should occur when a unit violates a PS limit and how limits should change over time.

There are many statistical challenges in the design and analysis of margin testing for product qualification. To further complicate issues, there are multiple types of margins that can be considered and there are often competing experimental designs to evaluate the various types of margin. There are two major variants of margin that must be addressed for engineered components: performance margin and design margin. They can be differentiated by the specific regions of the requirements space that they address. Performance margin are evaluated within the region where all inputs and environments are within requirements, and it expresses the difference between actual performance and the required performance of the system or component. Design margin expresses the difference between the maximum (or minimum) inputs and environments where the component continues to operate as intended (i.e. all performance requirements are still met), and the required inputs and conditions. The model Performance = f(Inputs, Environments? + ϵ (1) can be used to help frame the overall set of margin questions. The interdependence of inputs, environments, and outputs should be considered during the course of development in order to identify a complete test program that addresses both performance margin and design margin questions. Statistical methods can be utilized to produce a holistic and efficient program, both for qualitative activities that are designed to reveal margin limiters and for activities where margin quantification is desired. This paper discusses a holistic framework and taxonomy for margin testing and identifies key statistical challenges that may arise in developing such a program.

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This paper proposes a tolerance bound approach for determining sample sizes. With this new methodology we begin to think of sample size in the context of uncertainty exceeding margin. As the sample size decreases the uncertainty in the estimate of margin increases. This can be problematic when the margin is small and only a few units are available for testing. In this case there may be a true underlying positive margin to requirements but the uncertainty may be too large to conclude we have sufficient margin to those requirements with a high level of statistical confidence. Therefore, we provide a methodology for choosing a sample size large enough such that an estimated QMU uncertainty based on the tolerance bound approach will be smaller than the estimated margin (assuming there is positive margin). This ensures that the estimated tolerance bound will be within performance requirements and the tolerance ratio will be greater than one, supporting a conclusion that we have sufficient margin to the performance requirements. In addition, this paper explores the relationship between margin, uncertainty, and sample size and provides an approach and recommendations for quantifying risk when sample sizes are limited.

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The use of the Transfer Function approach to Model-Assisted Probability of Detection (MAPOD) is explored by inspecting airplane lap joint specimen sets with multiple site fatigue damage. Four specimen sets representing a matrix with structure type and defect types as the two axes were inspected using ultrasonic linear array. Probability of Detection (POD) curves were generated in each quadrant for 4 inspectors. POD curves in the fourth quadrant, which represents actual in-service crack detection in real airplane structures were predicted from other quadrant curves. Predicted POD curves were compared to observed curves. © 2012 American Institute of Physics.

This paper presents some statistical concepts and techniques for refining the expression of uncertainty arising from: a) random variability (aleatory uncertainty) of a random quantity; and b) contributed epistemic uncertainty due to limited sampling of the random quantity. The treatment is tailored to handling experimental uncertainty in a context of model validation and calibration. Two particular problems are considered. One involves deconvolving random measurement error from measured random response. The other involves exploiting a relationship between two random variates of a system and an independently characterized probability density of one of the variates.

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