Statistical Intervals
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Preface to First Edition vii
Acknowledgments x
1 Introduction, Basic Concepts, and Assumptions 1
1.1 Statistical Inference 2
1.2 Different Types of Statistical Intervals: An Overview 2
1.3 The Assumption of Sample Data 3
1.4 The Central Role of Practical Assumptions Concerning Representative Data 4
1.5 Enumerative Versus Analytic Studies 5
1.6 Basic Assumptions for Enumerative Studies 7
1.7 Considerations in the Conduct of Analytic Studies 10
1.8 Convenience and Judgment Samples 11
1.9 Sampling People 12
1.10 Infinite Population Assumptions 13
1.11 Practical Assumptions: Overview 14
1.12 Practical Assumptions: Further Example 14
1.13 Planning the Study 17
1.14 The Role of Statistical Distributions 17
1.15 The Interpretation of Statistical Intervals 18
1.16 Statistical Intervals and Big Data 19
1.17 Comment Concerning Subsequent Discussion 19
2 Overview of Different Types of Statistical Intervals 21
2.1 Choice of a Statistical Interval 21
2.2 Confidence Intervals 23
2.3 Prediction Intervals 24
2.4 Statistical Tolerance Intervals 26
2.5 Which Statistical Interval Do I Use? 27
2.6 Choosing a Confidence Level 28
2.7 Two-Sided Statistical Intervals Versus One-Sided Statistical Bounds 29
2.8 The Advantage of Using Confidence Intervals Instead of Significance Tests 30
2.9 Simultaneous Statistical Intervals 31
3 Constructing Statistical Intervals Assuming a Normal Distribution Using Simple Tabulations 33
3.1 Introduction 34
3.2 Circuit Pack Voltage Output Example 35
3.3 Two-Sided Statistical Intervals 36
3.4 One-Sided Statistical Bounds 38
4 Methods for Calculating Statistical Intervals for a Normal Distribution 43
4.1 Notation 44
4.2 Confidence Interval for the Mean of a Normal Distribution 45
4.3 Confidence Interval for the Standard Deviation of a Normal Distribution 45
4.4 Confidence Interval for a Normal Distribution Quantile 46
4.5 Confidence Interval for the Distribution Proportion Less (Greater) Than a Specified Value 47
4.6 Statistical Tolerance Intervals 48
4.7 Prediction Interval to Contain a Single Future Observation or the Mean of m Future Observations 50
4.8 Prediction Interval to Contain at least k of m Future Observations 51
4.9 Prediction Interval to Contain the Standard Deviation of m Future Observations 52
4.10 The Assumption of a Normal Distribution 53
4.11 Assessing Distribution Normality and Dealing with Nonnormality 54
4.12 Data Transformations and Inferences from Transformed Data 57
4.13 Statistical Intervals for Linear Regression Analysis 60
4.14 Statistical Intervals for Comparing Populations and Processes 62
5 Distribution-Free Statistical Intervals 65
5.1 Introduction 66
5.2 Distribution-Free Confidence Intervals and One-Sided Confidence Bounds for a Quantile 68
5.3 Distribution-Free Tolerance Intervals and Bounds to Contain a Specified Proportion of a Distribution 78
5.4 Prediction Intervals to Contain a Specified Ordered Observation in a Future Sample 81
5.5 Distribution-Free Prediction Intervals and Bounds to Contain at Least k of m Future Observations 84
6 Statistical Intervals for a Binomial Distribution 89
6.1 Introduction to Binomial Distribution Statistical Intervals 90
6.2 Confidence Intervals for the Actual Proportion Nonconforming in the Sampled Distribution 92
6.3 Confidence Interval for the Proportion of Nonconforming Units in a Finite Population 102
6.4 Confidence Intervals for the Probability that the Number of Nonconforming Units in a Sample is Less than or Equal to (or Greater than) a Specified Number 104
6.5 Confidence Intervals for the Quantile of the Distribution of the Number of Nonconforming Units 105
6.6 Tolerance Intervals and One-Sided Tolerance Bounds for the Distribution of the Number of Nonconforming Units 107
6.7 Prediction Intervals for the Number Nonconforming in a Future Sample 108
7 Statistical Intervals for a Poisson Distribution 115
7.1 Introduction 116
7.2 Confidence Intervals for the Event-Occurrence Rate of a Poisson Distribution 117
7.3 Confidence Intervals for the Probability that the Number of Events in a Specified Amount of Exposure is Less than or Equal to (or Greater than) a Specified Number 124
7.4 Confidence Intervals for the Quantile of the Distribution of the Number of Events in a Specified Amount of Exposure 125
7.5 Tolerance Intervals and One-Sided Tolerance Bounds for the Distribution of the Number of Events in a Specified Amount of Exposure 127
7.6 Prediction Intervals for the Number of Events in a Future Amount of Exposure 128
8 Sample Size Requirements for Confidence Intervals on Distribution Parameters 135
8.1 Basic Requirements for Sample Size Determination 136
8.2 Sample Size for a Confidence Interval for a Normal Distribution Mean 137
8.3 Sample Size to Estimate a Normal Distribution Standard Deviation 141
8.4 Sample Size to Estimate a Normal Distribution Quantile 143
8.5 Sample Size to Estimate a Binomial Proportion 143
8.6 Sample Size to Estimate a Poisson Occurrence Rate 144
9 Sample Size Requirements for Tolerance Intervals, Tolerance Bounds, and Related Demonstration Tests 148
9.1 Sample Size for Normal Distribution Tolerance Intervals and One-Sided Tolerance Bounds148
9.2 Sample Size to Pass a One-Sided Demonstration Test Based on Normally Distributed Measurements 150
9.3 Minimum Sample Size For Distribution-Free Two-Sided Tolerance Intervals and One-Sided Tolerance Bounds 152
9.4 Sample Size for Controlling the Precision of Two-Sided Distribution-Free Tolerance In-tervals and One-Sided Distribution-Free Tolerance Bounds 153
9.5 Sample Size to Demonstrate that a Binomial Proportion Exceeds (is Exceeded by) a Specified Value 154
10 Sample Size Requirements for Prediction Intervals 164
10.1 Prediction Interval Width: The Basic Idea 164
10.2 Sample Size for a Normal Distribution Prediction Interval 165
10.3 Sample Size for Distribution-Free Prediction Intervals for k of m Future Observations 170
11 Basic Case Studies 172
11.1 Demonstration that the Operating Temperature of Most Manufactured Devices will not Exceed a Specified Value 173
11.2 Forecasting Future Demand for Spare Parts 177
11.3 Estimating the Probability of Passing an Environmental Emissions Test 180
11.4 Planning a Demonstration Test to Verify that a Radar System has a Satisfactory Prob-ability of Detection 182
11.5 Estimating the Probability of Exceeding a Regulatory Limit 184
11.6 Estimating the Reliability of a Circuit Board 189
11.7 Using Sample Results to Estimate the Probability that a Demonstration Test will be Successful 191
11.8 Estimating the Proportion within Specifications for a Two-Variable Problem 194
11.9 Determining the Minimum Sample Size for a Demonstration Test 195
12 Likelihood-Based Statistical Intervals 197
12.1 Introduction to Likelihood-Based Inference 198
12.2 Likelihood Function and Maximum Likelihood Estimation 200
12.3 Likelihood-Based Confidence Intervals for Single-Parameter Distributions 203
12.4 Likelihood-Based Estimation Methods for Location-Scale and Log-Location-Scale Distri-butions 206
12.5 Likelihood-Based Confidence Intervals for Parameters and Scalar Functions of Parameters212
12.6 Wald-Approximation Confidence Intervals 216
12.7 Some Other Likelihood-Based Statistical Intervals 224
13 Nonparametric Bootstrap Statistical Intervals 226
13.1 Introduction 227
13.2 Nonparametric Methods for Generating Bootstrap Samples and Obtaining Bootstrap Estimates 227
13.3 Bootstrap Operational Considerations 231
13.4 Nonparametric Bootstrap Confidence Interval Methods 233
14 Parametric Bootstrap and Other Simulation-Based Statistical Intervals 245
14.1 Introduction 246
14.2 Parametric Bootstrap Samples and Bootstrap Estimates 247
14.3 Bootstrap Confidence Intervals Based on Pivotal Quantities 250
14.4 Generalized Pivotal Quantities 253
14.5 Simulation-Based Tolerance Intervals for Location-Scale or Log-Location-Scale Distribu-tions 258
14.6 Simulation-Based Prediction Intervals and One-Sided Prediction Bounds for k of m Fu-ture Observations from Location-Scale or Log-Location-Scale Distributions 260
14.7 Other Simulation and Bootstrap Methods and Application to Other Distributions and Models 263
15 Introduction to Bayesian Statistical Intervals 270
15.1 Bayesian Inference: Overview 271
15.2 Bayesian Inference: an Illustrative Example 274
15.3 More About Specification of a Prior Distribution 283
15.4 Implementing Bayesian Analyses Using Markov Chain Monte Carlo Simulation 286
15.5 Bayesian Tolerance and Prediction Intervals 291
16 Bayesian Statistical Intervals for the Binomial, Poisson and Normal Distributions 297
16.1 Bayesian Intervals for the Binomial Distribution 298
16.2 Bayesian Intervals for the Poisson Distribution 306
16.3 Bayesian Intervals for the Normal Distribution 311
17 Statistical Intervals for Bayesian Hierarchical Models 321
17.1 Bayesian Hierarchical Models and Random Effects 322
17.2 Normal Distribution Hierarchical Models 323
17.3 Binomial Distribution Hierarchical Models 325
17.4 Poisson Distribution Hierarchical Models 328
17.5 Longitudinal Repeated Measures Models 329
18 Advanced Case Studies 335
18.1 Confidence Interval for the Proportion of Defective Integrated Circuits 336
18.2 Confidence Intervals for Components of Variance in a Measurement Process 339
18.3 Tolerance Interval to Characterize the Distribution of Process Output in the Presence of Measurement Error 344
18.4 Confidence Interval for the Proportion of Product Conforming to a Two-Sided Specification345
18.5 Confidence Interval for the Treatment Effect in a Marketing Campaign 348
18.6 Confidence Interval for the Probability of Detection with Limited Hit-Miss Data 349
18.7 Using Prior Information to Estimate the Service-Life Distribution of a Rocket Motor 353
Epilogue 357
A Notation and Acronyms 360
B Generic Definition of Statistical Intervals and Formulas for Computing Coverage Probabilities 367
B.1 Introduction 367
B.2 Two-sided Confidence Intervals and One-sided Confidence Bounds for Distribution Pa-rameters or a Function of Parameters 368
B.3 Two Sided Control-the-Center Tolerance Intervals to Contain at Least a Specified Pro-portion of a Distribution 371
B.4 Two Sided Tolerance Intervals to Control Both Tails of a Distribution 374
B.5 One-Sided Tolerance Bounds 377
B.6 Two-sided Prediction Intervals and One-Sided Prediction Bounds for Future Observations378
B.7 Two-Sided Simultaneous Prediction Intervals and One-Sided Simultaneous Prediction Bounds 381
B.8 Calibration of Statistical Intervals 383
C Useful Probability Distributions 384
C.1 Probability Distribution and R Computations 384
C.2 Important Characteristics of Random Variables 385
C.3 Continuous Distributions 388
C.4 Discrete Distributions 398
D General Results from Statistical Theory and Some Methods Used to Construct Sta-tistical Intervals 404
D.1 cdfs and pdfs of Functions of Random Variables 405
D.2 Statistical Error Propagation-The Delta Method 409
D.3 Likelihood and Fisher Information Matrices 410
D.4 Convergence in Distribution 413
D.5 Outline of General ML Theory 415
D.6 The CDF pivotal method for constructing confidence intervals 419
D.7 Bonferroni approximate statistical intervals 424
E Pivotal Methods for Constructing Parametric Statistical Intervals 427
E.1 General definition and examples of pivotal quantities 428
E.2 Pivotal Quantities for the Normal Distribution 428
E.3 Confidence intervals for a Normal Distribution Based on Pivotal Quantities 429
E.4 Confidence Intervals for Two Normal Distributions Based on Pivotal Quantities 432
E.5 Tolerance Intervals for a Normal Distribution Based on Pivotal Quantities 432
E.6 Normal Distribution Prediction Intervals Based on Pivotal Quantities 434
E.7 Pivotal Quantities for Log-Location-Scale Distributions 436
F Generalized Pivotal Quantities 440
F.1 Definition of Generalized Pivotal Quantities 440
F.2 A Substitution Method to Obtain GPQs 441
F.3 Examples of GPQs for Functions of Location-Scale Distribution Parameters 441
F.4 Conditions for Exact Intervals Derived from GPQs 443
G Distribution-Free Intervals Based on Order Statistics 446
G.1 Basic Statistical Results Used in this Appendix 446
G.2 Distribution-Free Confidence Intervals and Bounds for a Distribution Quantile 447
G.3 Distribution-Free Tolerance Intervals to Contain a Given Proportion of a Distribution 448
G.4 Distribution-Free Prediction Interval to Contain a Specified Ordered Observation From a Future Sample 449
G.5 Distribution-Free Prediction Intervals and Bounds to Contain at Least k of m Future Observations From a Future Sample 451
H Basic Results from Bayesian Inference Models 455
H.1 Basic Statistical Results Used in this Appendix 455
H.2 Bayes' Theorem 456
H.3 Conjugate Prior Distributions 456
H.4 Jeffreys Prior Distributions 459
H.5 Posterior Predictive Distributions 463
H.6 Posterior Predictive Distributions Based on Jeffreys Prior Distributions 465
I Probability of Successful Demonstration 468
I.1 Demonstration Tests Based on a Normal Distribution Assumption 468
I.2 Distribution-Free Demonstration Tests 469
J Tables 471
References 508
Subject Index 525
Preface to Second Edition
Overview
The first edition of Statistical Intervals was published twenty-five years ago. We believe the book successfully met its goal of providing a comprehensive overview of statistical intervals for practitioners and statisticians and we have received much positive feedback. Despite, and perhaps because of this, there were compelling reasons for a second edition. In developing this second edition, Bill Meeker and Gerry Hahn have been most fortunate to have a highly qualified colleague, Luis Escobar, join them.
The new edition aims to:
- Improve or expand on various previously presented statistical intervals, using methods developed since the first edition was published.
- Provide general methods for constructing statistical intervals-some of which have recently been developed or refined-for important situations beyond those previously considered.
- Provide a webpage that gives up-to-date information about available software for calculating statistical intervals, as well as other important up-to-date information.
- Provide, via technical appendices, some of the theory underlying the intervals presented in this book.
In addition to updating the original chapters, this new edition includes new chapters on
- Likelihood-based statistical intervals (Chapter 12).
- Nonparametric bootstrap statistical intervals (Chapter 13).
- Parametric bootstrap and other simulation-based statistical intervals (Chapter 14).
- An introduction to Bayesian statistical intervals (Chapter 15).
- Bayesian statistical intervals for the binomial, Poisson, and normal distributions (Chapter 16).
- Statistical intervals for Bayesian hierarchical models (Chapter 17).
The new edition also includes an additional chapter on advanced case studies (Chapter 18). This chapter further illustrates the use of the newly introduced more advanced general methods for constructing statistical intervals. In totality, well over half of this second edition is new material-an indication of how much has changed over the past twenty-five years.
The first edition tended to focus on simple methods for constructing statistical intervals in commonly encountered situations and relied heavily on tabulations, charts, and simple formulas. The new edition adds methodology that can be readily implemented using easy-to-access software and allows more complicated problems to be addressed.
The purpose and audience for the book, however, remain essentially the same and what we said in the preface to the first edition (see below) still holds. We expect the book to continue to appeal to practitioners and statisticians who need to apply statistical intervals and hope that this appeal will be enhanced by the addition of the new and updated material. In addition, we expect the new edition to have added attraction to those interested in the theory underlying the construction of statistical intervals. With this in mind, we have extended the book title to read Statistical Intervals: A Guide for Practitioners and Researchers.
We have added many new applications to illustrate the use of the methods that we present. As in the first edition, all of these applications are based on real data. In some of these, however, we have changed the names of the variables or the scale of the data to protect sensitive information.
Elaboration on New Methods
Chapters 3 and 4 continue to describe (and update) familiar classical statistical methods for confidence intervals, tolerance intervals, and prediction intervals for situations in which one has a simple random sample from an underlying population or process that can be adequately described by a normal distribution. The interval procedures in these chapters have the desirable property of being "exact"-their coverage probabilities (i.e., the probability that the interval constructed using the procedure will include the quantity it was designed to include) are equal to their nominal confidence levels.
For distributions other than the normal, however, we must often resort to the use of approximate procedures for setting statistical intervals. Such procedures have coverage probabilities that usually differ from their (desired or specified) nominal confidence levels. Seven new chapters (Chapters 12-18) describe and illustrate the use of procedures for constructing intervals that are usually approximate. These procedures also have applicability for constructing statistical intervals in more complicated situations involving, for example, nonlinear regression models, random-effects models, and censored, truncated, or correlated data, building on the significant recent research in these areas. At the time of the first edition, such advanced methods were not widely used because they were not well known, and tended to be computationally intensive for the then available computing capabilities. Also, their statistical properties had not been studied carefully. Therefore, we provided only a brief overview of such methods in Chapter 12 of the first edition. Today, such methods are considered state of the art and readily achievable computationally. The new methods generally provide coverage probabilities that are closer to the nominal confidence level than the computationally simple Wald-approximation (also known as normal-approximate) methods that are still commonly used today to calculate statistical intervals in some popular statistical computing packages.
Other major changes in the new edition include updates to Chapters 5-7:
- Chapter 5 (on distribution-free statistical intervals) includes recently developed methods for interpolation between order statistics to provide interval coverage probabilities that are closer to the nominal confidence level.
- Chapters 6 and 7 (on statistical intervals for the binomial and Poisson distributions, respectively) now include approximate procedures with improved coverage probability properties for constructing statistical intervals for discrete distributions.
In addition, we have updated the discussion in the original chapters in numerous places. For example, Chapter 1 now includes a section on statistical intervals and big data.
New Technical Appendices
Some readers of the first edition indicated that they would like to see the theory, or at least more technical justification, for the statistical interval procedures. In response, we added a series of technical appendices that provide details of the theory upon which most of the intervals are based and how their statistical properties can be computed. These appendices also provide readers additional knowledge useful in generalizing the methods and adapting them to situations not covered in this book. We maintain, however, our practitioner-oriented focus by placing such technical material into appendices.
The new appendices provide:
- Generic definitions of statistical intervals and development of formulas for computing coverage probabilities (Appendix B).
- Properties of probability distributions that are important in data analysis applications or useful in constructing statistical intervals (Appendix C).
- Some generally applicable results from statistical theory and their use in constructing statistical intervals, including an outline of the general maximum likelihood theory concepts used in Chapter 12 and elsewhere (Appendix D).
- An outline of the theory for constructing statistical intervals for parametric distributions based on pivotal quantities used in Chapters 3, 4, and 14 (Appendix E).
- An outline of the theory for constructing statistical intervals for parametric distributions based on generalized pivotal quantities used in Chapter 14 (Appendix F).
- An outline of the theory for constructing distribution-free intervals based on order statistics, as presented in Chapter 5 (Appendix G).
- Some basic results underlying the construction of the Bayesian intervals used in Chapters 15, 16, and 17 (Appendix H).
- Derivation of formulas to compute the probability of successfully passing a (product) demonstration test based on statistical intervals described in Chapter 9 (Appendix I).
Similar to the first edition, Appendices A and J of the new edition provide, respectively, a summary of notation and acronyms and important tabulations for constructing statistical intervals.
Computer Software
Many commercial statistical software products (e.g., JMP, MINITAB, SAS, and SPSS) compute statistical intervals. New versions of these packages with improved capabilities for constructing statistical intervals, such as those discussed in this book, are released periodically. Therefore,instead of directly discussing current features of popular software packages-which might become rapidly outdated-we provide this information in an Excel spreadsheet accessible from the book's webpage and plan to update this webpage to keep it current.
In many parts of this book we show how to use the open-source R system (http://www.r-project.org/) as a sophisticated calculator to compute statistical intervals. To supplement the capabilities in R, we have developed an R package StatInt that contains some additional...
