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Steven E. Rigdon, PhD, is Professor in the Department of Biostatistics at Saint Louis University. He is also Distinguished Research Professor Emeritus at Southern Illinois University Edwardsville. His research interests include spatial disease surveillance and reliability assessment.
Rong Pan, PhD, is Associate Professor of Industrial Engineering at the School of Computing, Informatics, and Decision Systems Engineering at Arizona State University. His research interests include failure time data analysis, design of experiments, multivariate statistical quality control, time series analysis, and control.
Douglas C. Montgomery, PhD, is Regents Professor of Industrial Engineering and ASU Foundation Professor of Engineering at Arizona State University. His research interests include industrial statistics and design of experiments.
Laura J. Freeman, PhD, is Research Associate Professor of Statistics and Director of the Intelligent Systems Division of the National Security Institute at Virginia Tech. Her research interests include design of experiments, leveraging experimental methods in emerging technology research with a focus in cyber-physical systems, artificial intelligence (AI), and machine learning.
Preface xiii
About the Companion Website xv
Part I Reliability 1
1 Reliability Concepts 3
1.1 Definitions of Reliability 3
1.2 Concepts for Lifetimes 4
1.3 Censoring 10
Problems 14
2 Lifetime Distributions 17
2.1 The Exponential Distribution 17
2.2 The Weibull Distribution 22
2.3 The Gamma Distribution 25
2.4 The Lognormal Distribution 28
2.5 Log Location and Scale Distributions 30
2.5.1 The Smallest Extreme Value Distribution 31
2.5.2 The Logistic and Log-Logistic Distributions 33
Problems 35
3 Inference for Parameters of Life Distributions 39
3.1 Nonparametric Estimation of the Survival Function 39
3.1.1 Confidence Bounds for the Survival Function 42
3.1.2 Estimating the Hazard Function 44
3.2 Maximum Likelihood Estimation 46
3.2.1 Censoring Contributions to Likelihoods 46
3.3 Inference for the Exponential Distribution 50
3.3.1 Type II Censoring 50
3.3.2 Type I Censoring 54
3.3.3 Arbitrary Censoring 55
3.3.4 Large Sample Approximations 56
3.4 Inference for the Weibull 58
3.5 The SEV Distribution 59
3.6 Inference for Other Models 60
3.6.1 Inference for the GAM(¿, a) Distribution 61
3.6.2 Inference for the Log Normal Distribution 61
3.6.3 Inference for the GGAM(¿, ¿, a) Distribution 62
3.7 Bayesian Inference 67
3.a Kaplan-Meier Estimate of the Survival Function 80
3.a.1 The Metropolis-Hastings Algorithm 82
Problems 83
Part II Design of Experiments 89
4 Fundamentals of Experimental Design 91
4.1 Introduction to Experimental Design 91
4.2 A Brief History of Experimental Design 93
4.3 Guidelines for Designing Experiments 95
4.4 Introduction to Factorial Experiments 101
4.4.1 An Example 103
4.4.2 The Analysis of Variance for a Two-Factor Factorial 105
4.5 The 2k Factorial Design 114
4.5.1 The 22 Factorial Design 115
4.5.2 The 23 Factorial Design 119
4.5.3 A Singe Replicate of the 2k Design 124
4.5.4 2k Designs are Optimal Designs 129
4.5.5 Adding Center Runs to a 2k Design 133
4.6 Fractional Factorial Designs 135
4.6.1 A General Method for Finding the Alias Relationships in Fractional Factorial Designs 142
4.6.2 De-aliasing Effects 145
Problems 147
5 Further Principles of Experimental Design 157
5.1 Introduction 157
5.2 Response Surface Methods and Designs 157
5.3 Optimization Techniques in Response Surface Methodology 160
5.4 Designs for Fitting Response Surfaces 165
5.4.1 Classical Response Surface Designs 165
5.4.2 Definitive Screening Designs 171
5.4.3 Optimal Designs in RSM 175
Problems 176
Part III Regression Models for Reliability Studies 185
6 Parametric Regression Models 187
6.1 Introduction to Failure-Time Regression 187
6.2 Regression Models with Transformations 188
6.2.1 Estimation and Confidence Intervals for Transformed Data 189
6.3 Generalized Linear Models 198
6.4 Incorporating Censoring in Regression Models 205
6.4.1 Parameter Estimation for Location Scale and Log-Location Scale Models 205
6.4.2 Maximum Likelihood Method for Log-Location Scale Distributions 206
6.4.3 Inference for Location Scale and Log-Location Scale Models 207
6.4.4 Location Scale and Log-Location Scale Regression Models 208
6.5 Weibull Regression 208
6.6 Nonconstant Shape Parameter 228
6.7 Exponential Regression 233
6.8 The Scale-Accelerated Failure-Time Model 234
6.9 Checking Model Assumptions 236
6.9.1 Residual Analysis 237
6.9.2 Distribution Selection 243
Problems 245
7 Semi-parametric Regression Models 249
7.1 The Proportional Hazards Model 249
7.2 The Cox Proportional Hazards Model 251
7.3 Inference for the Cox Proportional Hazards Model 255
7.4 Checking Assumptions for the Cox PH Model 264
Problems 265
Part IV Experimental Design for Reliability Studies 269
8 Design of Single-Testing-Condition Reliability Experiments 271
8.1 Life Testing 272
8.1.1 Life Test Planning with Exponential Distribution 273
8.1.1.1 Type II Censoring 273
8.1.1.2 Type I Censoring 274
8.1.1.3 Large Sample Approximation 275
8.1.1.4 Planning Tests to Demonstrate a Lifetime Percentile 276
8.1.1.5 Zero Failures 279
8.1.2 Life Test Planning for Other Lifetime Distributions 281
8.1.3 Operating Characteristic Curves 282
8.2 Accelerated Life Testing 286
8.2.1 Acceleration Factor 287
8.2.2 Physical Acceleration Models 288
8.2.2.1 Arrhenius Model 288
8.2.2.2 Eyring Model 289
8.2.2.3 Peck Model 290
8.2.2.4 Inverse Power Model 290
8.2.2.5 Coffin-Manson Model 290
8.2.3 Relationship Between Physical Acceleration Models and Statistical Models 291
8.2.4 Planning Single-Stress-Level ALTs 292
Problems 294
9 Design of Multi-Factor and Multi-Level Reliability Experiments 297
9.1 Implications of Design for Reliability 298
9.2 Statistical Acceleration Models 299
9.2.1 Lifetime Regression Model 299
9.2.2 Proportional Hazards Model 303
9.2.3 Generalized Linear Model 306
9.2.4 Converting PH Model with Right Censoring to GLM 309
9.3 Planning ALTs with Multiple Stress Factors at Multiple Stress Levels 311
9.3.1 Optimal Test Plans 313
9.3.2 Locality of Optimal ALT Plans 318
9.3.3 Comparing Optimal ALT Plans 319
9.4 Bayesian Design for GLM 322
9.5 Reliability Experiments with Design and Manufacturing Process Variables 326
Problems 336
A The Survival Package in R 339
B Design of Experiments using JMP 351
C The Expected Fisher Information Matrix 357
C.1 Lognormal Distribution 359
C.2 Weibull Distribution 359
C.3 Lognormal Distribution 361
C.4 Weibull Distribution 362
D Data Sets 363
E Distributions Used in Life Testing 375
Bibliography 381
Index 387
It is difficult to define reliability precisely because this term evokes many different meanings in different contexts. In the field of reliability engineering, we primarily deal with engineered devices and systems. Single-word descriptions may depict one or two aspects of reliability in an engineering application context, but they are inadequate for a technical definition of engineering reliability. So, how do engineers and technical experts define reliability?
There are some variations in the aforementioned definitions, but they all either explicitly or implicitly state the following characteristics of reliability:
We define a function, , to be the survival, or reliability, function, which is the probability of the product, service, or system being successfully operated under its normal operating condition at time ; in other words, the unit survived past time .
When an item fails, the "fix" sometimes involves making a repair to bring it back to a working condition. Another possibility is to discard the item and replace it with a working item. In general, the more complex a system is, the more likely we are to repair it, and the simpler it is the more likely we are to scrap it and replace it with a new item. For example, if the starter on our automobile fails, we would probably take out the old starter and replace it with a new one. In a case like this, the automobile is a repairable system, but the starter is nonrepairable since our fix has been to replace it entirely.
Since complex systems, which are usually repairable, are made up of component parts that are nonrepairable, we will focus in this book on nonrepairable items. If these nonrepairable items are designed and built to have high reliability, then the system should be reliable as well. For nonrepairable systems we are interested in studying the distribution of the time to the first (and only) failure, or more generally, the effect of predictor variables on this lifetime. This lifetime need not be measured in calendar time; it could be measured in operating time (for an item that is switched on and off periodically), miles driven (for a motor vehicle like a car or truck), copies made (for a copier or printer), or cycles (for an industrial machine). For nonrepairable systems, we study the occurrence of events in time, such as failures (and subsequent repairs) or recurrence of a disease or its symptoms. See Rigdon and Basu (2000) for a treatment of repairable systems.
The lifetime of a unit is a random variable that necessarily takes on nonnegative values. Usually, but not always, we think of as a continuous random variable taking on values in the interval . There are various forms that the distribution may take, many of which, including the exponential, Weibull and gamma, are presented in detail Chapter 2. Here we present the fundamental ideas and terms for continuous random variables.
Thus, probabilities for a continuous random variable are found as areas under the PDF. (See Figure 1.1a.) Note that and can be or . Since the probability that , that is, the probability that equals a particular value is equal to zero. This also implies that
See Figure 1.1b.
Figure 1.1 Properties of the PDF for a lifetime distribution.
Since the probability is 1 that is between and we have the property that
See Figure 1.1c. Also, since all probabilities must be nonnegative, the PDF must satisfy
The results in (1.1) and (1.2) are the fundamental properties for a PDF.
The development earlier makes no assumption about the possible values that the random variable can take on. For lifetimes, which must be nonnegative, we have for Thus, for lifetimes, the PDF must satisfy
The set of values for which the PDF of the random variable is positive is called the support of The support for a lifetime distribution is although for some distributions we exclude the possibility of .
Note that the PDF does not give probabilities directly; for example, does not give the probability that Rather, as an approximation we can write
(See Figure 1.2.) Thus, the PDF can be interpreted as
or equivalently
To be precise, the PDF is equal to the limit of the right side earlier as :
Figure 1.2 Approximation .
Note that we have changed the variable of integration from to in order to avoid confusion with the upper limit on the integral. For a lifetime distribution with support we have the result
As on the right side earlier, the integral goes to , which equals 1. Also, since the probability of having a negative lifetime is 0, the CDF must be zero for all Finally, since the CDF "accumulates" probability up to and including increasing can only increase (or hold constant) the CDF. Thus, for a lifetime distribution, the CDF must satisfy
Equation (1.4) shows how to get the CDF given the PDF. A formula for the reverse (getting the PDF from the CDF) can be obtained by differentiating both sides of (1.4) with respect to and applying the fundamental theorem of calculus:
In other words, the PDF is the derivative of the CDF.
In other words, is the probability that an item survives past time while is the probability that it fails at or before time (that is, that it doesn't survive past time ). Thus, and are related by
One of the most important concepts in lifetime analysis is the hazard function.
As an approximation, we can write
analogous to (1.3).
The probability in the definition of the hazard is a conditional probability; it is conditioned on survival to the beginning of the interval. This is a natural quantity to consider because it makes intuitive sense to talk about the failure probability of an item that is still working. It is conceptually more difficult to talk about the probability of an item failing if the item might or might not be working. If we replace the conditional probability in the definition of the hazard function with an unconditional probability, we get
which is equal to the PDF . Thus, the PDF is the (limit of) the probability of failing in a small interval when viewed before testing begins. The hazard is the (limit of) the probability of failing in a small interval for a unit that is known to be working.
The hazard function can be written as
Indeed, many books define the hazard function in this way. We choose to define the hazard as the limit of a conditional probability because this intuitive concept is helpful for understanding the failure mechanism.
To illustrate the difference between hazard and density, consider a discrete case, say, where items are placed on test and are observed every 1000?hours. Let denote the probability that an item fails in the th interval Suppose first that so that there is a probability of that a working unit will fail in any time interval. Thus, at the end of a time interval when we inspect those units still operating, we would expect that about one-tenth of them would fail. We could naturally ask the question "What is the probability that a unit fails in the th interval?" This is different from the question "What is the probability that a unit that is currently operating fails in the th...
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