Statistical Analysis of Reliability Data

Martin J. Crowder, Alan Kimber, T. Sweeting, R. Smith

• Publisher: Routledge
• Year: 2017
• Print ISBN: 9780412594809
• eBook ISBN: 9781351414616
• Topic: Business
• Subtopic: Operations
• Index: Business

Written for those who have taken a first course in statistical methods, this book takes a modern, computer-oriented approach to describe the statistical techniques used for the assessment of reliability.

1

Statistical concepts in reliability

1.1 INTRODUCTION

Reliability is a word with many different connotations. When applied to a human being, it usually refers to that person’s ability to perform certain tasks according to a specified standard. By extension, the word is applied to a piece of equipment, or a component of a larger system, to mean the ability of that equipment or component to fulfil what is required of it. The original use of the term was purely qualitative. For instance, aerospace engineers recognized the desirability of having more than one engine on an aeroplane without any precise measurements of failure rate. As used today, however, reliability is almost always a quantitative concept, and this implies the need for methods of measuring reliability.

There are a number of reasons why reliability needs to be quantitative. Perhaps the most important is economic since to improve reliability costs money, and this can be justified only if the costs of unreliable equipment are measured. For a critical component whose successful operation is essential to a system, reliability may be measured as the probability that the component is operating successfully, and the expected cost of an unreliable component measured as the product of the probability of failure and the cost of failure. In a more routine application where components are allowed to fail but must then be repaired, the mean time between failures (MTBF) is a critical parameter. In either case, the need for a probabilistic definition of reliability is apparent.

Another reason for insisting on quantitative definitions of reliability is that different standards of reliability are required in different applications. In an unmanned satellite, for instance, a comparatively high failure rate may be considered acceptable, but in cases where human life is at stake, the probability of failure must be very close to zero. The applications where the very highest reliability are required are those in the nuclear industry, where it is common to insist on a bound of around 10⁻⁹ for the expected number of failures per reactor per year. Whether it is possible to guarantee such reliability in practice is another matter.

Whatever the specific application, however, reliability needs to be measured, and this implies the need for statistical methods. One well-known author on reliability has gone to considerable lengths to emphasize the distinction between physical quantities such as elastic modulus or electrical resistance, and reliability which by its nature is not a physical quantity. This is certainly an important distinction, but it does not mean that reliability cannot be measured, merely that the uncertainty of measurements is in most cases much higher than would be tolerated in a physical experiment. Consequently, it is important in estimating reliability to make a realistic appraisal of the uncertainty in any estimate quoted.

1.2 RELIABILITY DATA

This book is concerned primarily with methods for statistical analysis of data on reliability. The form of these data necessarily depends on the application being considered. The simplest case consists of a series of experimental units tested to a prescribed standard, and then classified as failures or survivors. The number of failures typically follows a Binomial or Hypergeometric distribution, from which it is possible to make inferences about the failure rate in the whole population.

More sophisticated applications usually involve a continuous measure of failure, such as failure load or failure time. This leads us to consider the distribution of the failure load or failure time, and hence to employ statistical techniques for estimating that distribution. There are a number of distinctions to be made here.

1. Descriptive versus inferential statistics.

In some applications, it is sufficient to use simple measures such as the mean and variance, survivor function or hazard function, and to summarize these with a few descriptive statistics or graphs. In other cases, questions requiring more sophisticated methods arise, such as determining a confidence interval for the mean or a specified quantile of failure time, or testing a hypothesis about the failure-time distribution.

2. Uncensored versus censored data.

It is customary to stop an experiment before all the units have failed, in which case only a lower bound is known for the failure load or failure time of the unfailed units. Such data are called right-censored. In other contexts, only an upper bound for the failure time may be known (left-censored) or it may be known only that failure occurred between two specified times (interval-censored data).

3. Parametric versus nonparametric methods.

Many statistical methods take the form of fitting a parametric family such as the Normal, Lognormal or Weibull distribution. In such cases it is important to have an efficient method for estimating the parameters, but also to have ways of assessing the fit of a distribution. Other statistical procedures do not require any parametric form. For example, the Kaplan-Meier estimator (section 2.11) is a method of estimating the distribution function of failure time, with data subject to right-censoring, without any assumption about a parametric family. At a more advanced level, proportional hazards analysis (Chapter 5) is an example of a semiparametric procedure, in which some variables (those defining the proportionality of two hazard functions) are assumed to follow a parametric family, but others (the baseline hazard function) are not.

4. Single samples versus data with covariates.

Most texts on reliability concentrate on single samples – for example, suppose we have a sample of lifetimes of units tested under identical conditions, then one can compute the survival curve or hazard function for those units. In many contexts, however, there are additional explanatory variables or covariates. For example, there may be several different samples conducted with slightly different materials or under different stresses or different ambient conditions. This leads us into the study of reliability models which incorporate the covariates. Such techniques for the analysis of survival data are already very well established in the context of medical data, but for some reason are not nearly so well known among reliability practitioners. One of the aims of this book is to emphasize the possibilities of using such models for reliability data (especially in Chapters 4 and 5).

5. Univariate and multivariate data.

Another kind of distinction concerns the actual variable being measured – is it a single lifetime and therefore a scalar, or are there a vector of observations, such as lifetimes of different components or failure stresses in different directions, relevant to the reliability of a single unit? In the latter case, we need multivariate models for failure data, a topic developed in Chapter 7.

6. Classical versus ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Statistical concepts in reliability
• 2 Probability distributions in reliability
• 3 Statistical methods for single samples
• 4 Regression models for reliability data
• 5 Proportional hazards modelling
• 6 The Bayesian approach
• 7 Multivariate models
• 8 Repairable systems
• 9 Models for system reliability
• Appendix: The Delta method
• References
• Author index
• Subject index