Reliability and Survival Methods

Transcription

1 Version 11 Reliability and Survival Methods The real voyage of discovery consists not in seeking new landscapes, but in having new eyes. Marcel Proust JMP, A Business Unit of SAS SAS Campus Drive Cary, NC 27513

2 The correct bibliographic citation for this manual is as follows: SAS Institute Inc JMP 11. Cary, NC: SAS Institute Inc. JMP 11 Copyright 2013, SAS Institute Inc., Cary, NC, USA ISBN All rights reserved. Produced in the United States of America. For a hard-copy book: No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the publisher, SAS Institute Inc. For a Web download or e-book: Your use of this publication shall be governed by the terms established by the vendor at the time you acquire this publication. The scanning, uploading, and distribution of this book via the Internet or any other means without the permission of the publisher is illegal and punishable by law. Please purchase only authorized electronic editions and do not participate in or encourage electronic piracy of copyrighted materials. Your support of others rights is appreciated. U.S. Government Restricted Rights Notice: Use, duplication, or disclosure of this software and related documentation by the U.S. government is subject to the Agreement with SAS Institute and the restrictions set forth in FAR , Commercial Computer Software-Restricted Rights (June 1987). SAS Institute Inc., SAS Campus Drive, Cary, North Carolina st printing, September, 2013 SAS Publishing provides a complete selection of books and electronic products to help customers use SAS software to its fullest potential. For more information about our e-books, e-learning products, CDs, and hard-copy books, visit the SAS Publishing Web site at support.sas.com/publishing or call SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. indicates USA registration. Other brand and product names are registered trademarks or trademarks of their respective companies.

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5 Altered source versions must be plainly marked as such, and must not be misrepresented as being the original software. The name of the author may not be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE AUTHOR AS IS AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. R software is Copyright , R Foundation for Statistical Computing. MATLAB software is Copyright , The MathWorks, Inc. Protected by U.S. and international patents. See MATLAB and Simulink are registered trademarks of The MathWorks, Inc. See trademarks for a list of additional trademarks. Other product or brand names may be trademarks or registered trademarks of their respective holders.

6 Get the Most from JMP Whether you are a first-time or a long-time user, there is always something to learn about JMP. Visit JMP.com to find the following: live and recorded webcasts about how to get started with JMP video demos and webcasts of new features and advanced techniques details on registering for JMP training schedules for seminars being held in your area success stories showing how others use JMP a blog with tips, tricks, and stories from JMP staff a forum to discuss JMP with other users

7 Contents 1 Learn about JMP Documentation and Additional Resources Formatting Conventions JMP Documentation JMP Documentation Library JMP Help Additional Resources for Learning JMP Tutorials Sample Data Tables Learn about Statistical and JSL Terms Learn JMP Tips and Tricks Tooltips JMP User Community JMPer Cable JMP Books by Users The JMP Starter Window Introduction to Reliability and Survival Lifetime and Failure Analysis Life Distribution Fit Distributions to Lifetime Data Life Distribution Platform Overview Example of the Life Distribution Platform Launch the Life Distribution Platform The Life Distribution Report Event Plot Compare Distributions Report Statistics Report

8 8 Competing Cause Report Life Distribution Platform Options Additional Examples of the Life Distribution Platform Omit Competing Causes Change the Scale Statistical Details Nonparametric Fit Parametric Distributions Fit Life by X Fit Single-Factor Models to Time-to Event Data Fit Life by X Platform Overview Example of the Fit Life by X Platform Launch the Fit Life by X Platform The Fit Life by X Report Summary of Data Scatterplot Nonparametric Overlay Comparisons Results Custom Relationship Fit Life by X Platform Options Additional Examples of the Fit Life by X Platform Capacitor Example Custom Relationship Example Recurrence Analysis Model the Frequency of Cost of Recurrent Events over Time Recurrence Analysis Overview Example of the Recurrence Analysis Platform Launch the Recurrence Analysis Platform Recurrence Analysis Platform Options Fit Model Additional Examples of the Recurrence Analysis Platform Bladder Cancer Recurrences Example

9 9 Diesel Ship Engines Example Degradation Model Product Deterioration over Time Degradation Platform Overview Example of the Degradation Platform Launch the Degradation Platform The Degradation Report Model Specification Simple Linear Path Nonlinear Path Inverse Prediction Prediction Graph Degradation Platform Options Model Reports Model Lists Reports Destructive Degradation Stability Analysis Reliability Forecast Forecast Product Failure Using Production and Failure Data Reliability Forecast Platform Overview Example Using the Reliability Forecast Platform Launch the Reliability Forecast Platform The Reliability Forecast Report Observed Data Report Life Distribution Report Forecast Report Reliability Forecast Platform Options Reliability Growth Model System Reliability as Changes are Implemented Reliability Growth Platform Overview Example Using the Reliability Growth Platform

10 10 Launch the Reliability Growth Platform Time to Event Timestamp Event Count Phase By Data Table Structure The Reliability Growth Report Observed Data Report Reliability Growth Platform Options Fit Model Script Fit Model Options Crow AMSAA Fixed Parameter Crow AMSAA Piecewise Weibull NHPP Reinitialized Weibull NHPP Piecewise Weibull NHPP Change Point Detection Additional Examples of the Reliability Growth Platform Piecewise NHPP Weibull Model Fitting with Interval-Censored Data Piecewise Weibull NHP Change Point Detection with Time in Dates Format Statistical Details for the Reliability Growth Platform Statistical Details for the Crow-AMSAA Report Statistical Details for the Piecewise Weibull NHPP Change Point Detection Report Reliability Block Diagram Engineer System Reliabilities Reliability Block Diagram Platform Overview Example Using the Reliability Block Diagram Platform Add Components Align Shapes Connect Shapes Configure Components The Reliability Block Diagram Window Preview Window

11 11 Reliability Block Diagram Platform Options Red Triangle Options for Designs and Library Workspace Options Distribution Properties for Components Configuration Settings Enter Nonparametric Distribution Data Profilers Available for Reliability Block Diagrams Distribution Profiler Remaining Life Distribution Profiler Reliability Profiler Quantile Profiler Density Profiler Hazard Profiler Birnbaum s Component Importance Remaining Life BCI Mean Time to Failure Print Algebraic Reliability Formula Survival Analysis Analyze Survival Time Data Survival Analysis Overview Example of Survival Analysis Launch the Survival Platform The Survival Plot Survival Platform Options Exponential, Weibull, and Lognormal Plots and Fits Fitted Distribution Plots Competing Causes Additional Examples of Survival Analysis Example of Competing Causes Example of Interval Censoring Statistical Reports for Survival Analysis

12 12 11 Fit Parametric Survival Fit Survival Data Using Regression Models Fit Parametric Survival Overview Example of Parametric Regression Survival Fitting Launch the Fit Parametric Survival Platform The Parametric Survival Fit Report Fit Parametric Survival Options Nonlinear Parametric Survival Models Additional Examples of Fitting Parametric Survival Arrhenius Accelerated Failure LogNormal Model Interval-Censored Accelerated Failure Time Model Analyze Censored Data Using the Nonlinear Platform Right-Censored Data Weibull Loss Function Using the Nonlinear Platform Fitting Simple Survival Distributions Using the Nonlinear Platform Statistical Details for Fit Parametric Survival Loss Formulas for Survival Distributions Fit Proportional Hazards Fit Survival Data Using Semi-Parametric Regression Models Fit Proportional Hazards Overview Example of the Fit Proportional Hazards Platform Risk Ratios for One Nominal Effect with Two Levels Launch the Fit Proportional Hazards Platform The Fit Proportional Hazards Report Fit Proportional Hazards Options Example Using Multiple Effects and Multiple Levels Risk Ratios for Multiple Effects and Multiple Levels A References Index

13 Chapter 1 Learn about JMP Documentation and Additional Resources This chapter includes the following information: book conventions JMP documentation JMP Help additional resources, such as the following: other JMP documentation tutorials indexes Web resources Figure 1.1 The JMP Help Home Window on Windows

14 Contents Formatting Conventions JMP Documentation JMP Documentation Library JMP Help Additional Resources for Learning JMP Tutorials Sample Data Tables Learn about Statistical and JSL Terms Learn JMP Tips and Tricks Tooltips JMP User Community JMPer Cable JMP Books by Users The JMP Starter Window

15 Chapter 1 Learn about JMP 21 Formatting Conventions Formatting Conventions The following conventions help you relate written material to information that you see on your screen. Sample data table names, column names, pathnames, filenames, file extensions, and folders appear in Helvetica font. Code appears in Lucida Sans Typewriter font. Code output appears in Lucida Sans Typewriter italic font and is indented farther than the preceding code. Helvetica bold formatting indicates items that you select to complete a task: buttons check boxes commands list names that are selectable menus options tab names text boxes The following items appear in italics: words or phrases that are important or have definitions specific to JMP book titles variables Features that are for JMP Pro only are noted with the JMP Pro icon. For an overview of JMP Pro features, visit Note: Special information and limitations appear within a Note. Tip: Helpful information appears within a Tip. JMP Documentation JMP offers documentation in various formats, from print books and Portable Document Format (PDF) to electronic books (e-books). Open the PDF versions from the Help > Books menu or from the JMP online Help footers.

16 22 Learn about JMP Chapter 1 JMP Documentation All books are also combined into one PDF file, called JMP Documentation Library, for convenient searching. Open the JMP Documentation Library PDF file from the Help > Books menu. e-books are available at Amazon, Safari Books Online, and in the Apple ibookstore. You can also purchase printed documentation on the SAS website: JMP Documentation Library The following table describes the purpose and content of each book in the JMP library. Document Title Document Purpose Document Content Discovering JMP Using JMP Basic Analysis If you are not familiar with JMP, start here. Learn about JMP data tables and how to perform basic operations. Perform basic analysis using this document. Introduces you to JMP and gets you started creating and analyzing data. Covers general JMP concepts and features that span across all of JMP, including importing data, modifying columns properties, sorting data, and connecting to SAS. Describes these Analyze menu platforms: Distribution Fit Y by X Matched Pairs Tabulate How to approximate sampling distributions using bootstrapping is also included.

17 Chapter 1 Learn about JMP 23 JMP Documentation Document Title Document Purpose Document Content Essential Graphing Profilers Design of Experiments Guide Find the ideal graph for your data. Learn how to use interactive profiling tools, which enable you to view cross-sections of any response surface. Learn how to design experiments and determine appropriate sample sizes. Describes these Graph menu platforms: Graph Builder Overlay Plot Scatterplot 3D Contour Plot Bubble Plot Parallel Plot Cell Plot Treemap Scatterplot Matrix Ternary Plot Chart Also covers how to create background and custom maps. Covers all profilers listed in the Graph menu. Analyzing noise factors is included along with running simulations using random inputs. Covers all topics in the DOE menu.

18 24 Learn about JMP Chapter 1 JMP Documentation Document Title Document Purpose Document Content Fitting Linear Models Specialized Models Multivariate Methods Learn about Fit Model platform and many of its personalities. Learn about additional modeling techniques. Read about techniques for analyzing several variables simultaneously. Describes these personalities, all available within the Analyze menu Fit Model platform: Standard Least Squares Stepwise Generalized Regression Mixed Model MANOVA Loglinear Variance Nominal Logistic Ordinal Logistic Generalized Linear Model Describes these Analyze > Modeling menu platforms: Partition Neural Model Comparison Nonlinear Gaussian Process Time Series Response Screening The Screening platform in the Analyze > Modeling menu is described in Design of Experiments Guide. Describes these Analyze > Multivariate Methods menu platforms: Multivariate Cluster Principal Components Discriminant Partial Least Squares

19 Chapter 1 Learn about JMP 25 JMP Documentation Document Title Document Purpose Document Content Quality and Process Methods Reliability and Survival Methods Consumer Research Read about tools for evaluating and improving processes. Learn to evaluate and improve reliability in a product or system and analyze survival data for people and products. Learn about methods for studying consumer preferences and using that insight to create better products and services. Describes these Analyze > Quality and Process menu platforms: Control Chart Builder and individual control charts Measurement Systems Analysis Variability / Attribute Gauge Charts Capability Pareto Plot Diagram Describes these Analyze > Reliability and Survival menu platforms: Life Distribution Fit Life by X Recurrence Analysis Degradation Reliability Forecast Reliability Growth Reliability Block Diagram Survival Fit Parametric Survival Fit Proportional Hazards Describes these Analyze > Consumer Research menu platforms: Categorical Factor Analysis Choice Uplift Item Analysis

20 26 Learn about JMP Chapter 1 Additional Resources for Learning JMP Document Title Document Purpose Document Content Scripting Guide JSL Syntax Reference Learn about taking advantage of the powerful JMP Scripting Language (JSL). Read about many JSL functions on functions and their arguments, and messages that you send to objects and display boxes. Covers a variety of topics, such as writing and debugging scripts, manipulating data tables, constructing display boxes, and creating JMP applications. Includes syntax, examples, and notes for JSL commands. Note: The Books menu also contains two reference cards that can be printed: The Menu Card describes JMP menus, and the Quick Reference describes JMP keyboard shortcuts. JMP Help JMP Help is an abbreviated version of the documentation library that provides targeted information. You can open JMP Help in several ways: On Windows, press the F1 key to open the Help system window. Get help on a specific part of a data table or report window. Select the Help tool from the Tools menu and then click anywhere in a data table or report window to see the Help for that area. Within a JMP window, click the Help button. Search and view JMP Help on Windows using the Help > Help Contents, Search Help, and Help Index options. On Mac, select Help > JMP Help. Search the Help at (English only). Additional Resources for Learning JMP In addition to JMP documentation and JMP Help, you can also learn about JMP using the following resources: Tutorials (see Tutorials on page 27) Sample data (see Sample Data Tables on page 27) Indexes (see Learn about Statistical and JSL Terms on page 27)

21 Chapter 1 Learn about JMP 27 Additional Resources for Learning JMP Tutorials Tip of the Day (see Learn JMP Tips and Tricks on page 28) Web resources (see JMP User Community on page 28) JMPer Cable technical publication (see JMPer Cable on page 28) Books about JMP (see JMP Books by Users on page 29) JMP Starter (see The JMP Starter Window on page 29) You can access JMP tutorials by selecting Help > Tutorials. The first item on the Tutorials menu is Tutorials Directory. This opens a new window with all the tutorials grouped by category. If you are not familiar with JMP, then start with the Beginners Tutorial. It steps you through the JMP interface and explains the basics of using JMP. The rest of the tutorials help you with specific aspects of JMP, such as creating a pie chart, using Graph Builder, and so on. Sample Data Tables All of the examples in the JMP documentation suite use sample data. Select Help > Sample Data to do the following actions: Open the sample data directory. Open an alphabetized list of all sample data tables. Find a sample data table within a category. Sample data tables are installed in the following directory: On Windows: C:\Program Files\SAS\JMP\<version_number>\Samples\Data On Macintosh: \Library\Application Support\JMP\<version_number>\Samples\Data In JMP Pro, sample data is installed in the JMPPRO (rather than JMP) directory. Learn about Statistical and JSL Terms The Help menu contains the following indexes: Statistics Index Provides definitions of statistical terms. Scripting Index Lets you search for information about JSL functions, objects, and display boxes. You can also edit and run sample scripts from the Scripting Index.

22 28 Learn about JMP Chapter 1 Additional Resources for Learning JMP Learn JMP Tips and Tricks When you first start JMP, you see the Tip of the Day window. This window provides tips for using JMP. To turn off the Tip of the Day, clear the Show tips at startup check box. To view it again, select Help > Tip of the Day. Or, you can turn it off using the Preferences window. See the Using JMP book for details. Tooltips JMP provides descriptive tooltips when you place your cursor over items, such as the following: Menu or toolbar options Labels in graphs Text results in the report window (move your cursor in a circle to reveal) Files or windows in the Home Window Code in the Script Editor Tip: You can hide tooltips in the JMP Preferences. Select File > Preferences > General (or JMP > Preferences > General on Macintosh) and then deselect Show menu tips. JMP User Community The JMP User Community provides a range of options to help you learn more about JMP and connect with other JMP users. The learning library of one-page guides, tutorials, and demos is a good place to start. And you can continue your education by registering for a variety of JMP training courses. Other resources include a discussion forum, sample data and script file exchange, webcasts, and social networking groups. To access JMP resources on the website, select Help > JMP User Community. JMPer Cable The JMPer Cable is a yearly technical publication targeted to users of JMP. The JMPer Cable is available on the JMP website:

23 Chapter 1 Learn about JMP 29 Additional Resources for Learning JMP JMP Books by Users Additional books about using JMP that are written by JMP users are available on the JMP website: The JMP Starter Window The JMP Starter window is a good place to begin if you are not familiar with JMP or data analysis. Options are categorized and described, and you launch them by clicking a button. The JMP Starter window covers many of the options found in the Analyze, Graph, Tables, and File menus. To open the JMP Starter window, select View (Window on the Macintosh) > JMP Starter. To display the JMP Starter automatically when you open JMP on Windows, select File > Preferences > General, and then select JMP Starter from the Initial JMP Window list. On Macintosh, select JMP > Preferences > Initial JMP Starter Window.

24 30 Learn about JMP Chapter 1 Additional Resources for Learning JMP

25 Chapter 2 Introduction to Reliability and Survival Lifetime and Failure Analysis This book describes a number of methods and tools that are available in JMP to help you evaluate and improve reliability in a product or system and analyze survival data for people and products: The Life Distribution platform enables you to analyze the lifespan of a product, component, or system to improve quality and reliability. This analysis helps you determine the best material and manufacturing process for the product, thereby increasing the quality and reliability of the product. For more information, see Chapter 3, Life Distribution. The Fit Life by X platform helps you analyze lifetime events when only one factor is present. You can choose to model the relationship between the event and the factor using various transformations, or create a custom transformation of your data. For more information, see Chapter 4, Fit Life by X. The Recurrence Analysis platform analyzes event times like the other Reliability and Survival methods, but the events can recur several times for each unit. Typically, these events occur when a unit breaks down, is repaired, and then put back into service after the repair. For more information, see Chapter 5, Recurrence Analysis. The Degradation platform analyzes degradation data to predict pseudo failure times. These pseudo failure times can then be analyzed by other reliability platforms to estimate failure distributions. For more information, see Chapter 6, Degradation. The Reliability Forecast platform helps you predict the number of future failures. The analysis estimates the parameters for a life distribution using production dates, failure dates, and production volume. For more information, see Chapter 7, Reliability Forecast. The Reliability Growth platform models the change in reliability of a single repairable system over time as improvements are incorporated into its design. For more information, see Chapter 8, Reliability Growth. The Reliability Block Diagram platform displays the reliability relationship between a system's components and, if reliability distributions are given to the components, analytically obtains the reliability behavior. For more information, see Chapter 9, Reliability Block Diagram. The Survival platform computes survival estimates for one or more groups. It can be used as a complete analysis or is useful as an exploratory analysis to gain information for more complex model fitting. For more information, see Chapter 10, Survival Analysis.

26 26 Introduction to Reliability and Survival Chapter 2 The Fit Parametric Survival platform fits the time to event variable using linear regression models that can involve both location and scale effects. The fit is performed using several distributions. For more information, see Chapter 11, Fit Parametric Survival. The Fit Proportional Hazards platform fits the Cox proportional hazards model, which assumes a multiplying relationship between predictors and the hazard function. For more information, see Chapter 12, Fit Proportional Hazards.

27 Chapter 3 Life Distribution Fit Distributions to Lifetime Data The Life Distribution platform helps you discover distributional properties of time-to-event data. In one graph, you can compare common distributions (such as Weibull, Fréchet, and extreme values) and decide which distribution best fits your data. Analyzing multiple causes of failure and censored data are other important options in Life Distribution. Figure 3.1 Distributional Fits and Comparisons

28 Contents Life Distribution Platform Overview Example of the Life Distribution Platform Launch the Life Distribution Platform The Life Distribution Report Event Plot Compare Distributions Report Statistics Report Competing Cause Report Life Distribution Platform Options Additional Examples of the Life Distribution Platform Omit Competing Causes Change the Scale Statistical Details Nonparametric Fit Parametric Distributions

29 Chapter 3 Life Distribution 29 Life Distribution Platform Overview Life Distribution Platform Overview Life data analysis, or life distribution analysis, is the process of analyzing the lifespan of a product, component, or system to improve quality and reliability. For example, you can observe failure rates over time to predict when a computer component might fail. This analysis then helps you determine the best material and manufacturing process for the product, increasing the quality and reliability of the product. Decisions on warranty periods and advertising claims can also be more accurate. With the Life Distribution platform, you can analyze censored data in which some time observations are unknown. And if there are potentially multiple causes of failure, you can analyze the competing causes to estimate which cause is more influential. You can use the Reliability Test Plan and Reliability Demonstration calculators to choose the appropriate sample sizes for reliability studies. The calculators are found at DOE > Sample Size and Power. Example of the Life Distribution Platform Suppose you have failure times for 70 engine fans, with some of the failure times censored. You want to fit a distribution to the failure times and then estimate various measurements of reliability. 1. Open the Fan.jmp sample data table in the Reliability folder. 2. Select Analyze > Reliability and Survival > Life Distribution. 3. Select Time and click Y, Time to Event. 4. Select Censor and click Censor. 5. Click OK. The Life Distribution report window appears. 6. In the Compare Distribution report, select Lognormal distribution and the corresponding Scale radio button. A probability plot appears in the report window (Figure 3.2).

30 30 Life Distribution Chapter 3 Example of the Life Distribution Platform Figure 3.2 Probability Plot cumulative distribution estimate pointwise confidence interval of the cumulative distribution estimate simultaneous confidence intervals In the probability plot, the data points fit reasonably well along the red line. Below the Compare Distributions report, the Statistics report appears (Figure 3.3). This report provides statistics on model comparisons, nonparametric and parametric estimates, profilers, and more.

31 Chapter 3 Life Distribution 31 Example of the Life Distribution Platform Figure 3.3 Statistics Report The parameter estimates are provided for the distribution. The profilers are useful for visualizing the fitted distribution and estimating probabilities and quantiles. In the preceding report, the Quantile Profiler tells us that the median estimate is 25, hours. Launch the Life Distribution Platform Launch the Life Distribution platform by selecting Analyze > Reliability and Survival > Life Distribution. Figure 3.4 Life Distribution Launch Window

32 32 Life Distribution Chapter 3 Example of the Life Distribution Platform The Life Distribution launch window contains the following options: Y, Time to Event The time to event (such as the time to failure) or time to censoring. With interval censoring, specify two Y variables, where one Y variable gives the lower limit and the other Y variable gives the upper limit for each unit. For details about censoring, see Event Plot on page 33. Censor Identifies censored observations. In your data table, 1 indicates a censored observation; 0 indicates an uncensored observation. Failure Cause The column that contains multiple failure causes. This data helps you identify the most influential causes. If a Failure Cause column is selected, then a section is added to the window: Check boxes appear that allow the failure mode to use ZI Distributions, TH Distributions, DS Distributions, Fixed Parameter Models or Bayesian models for the analysis. Distribution specifies the distribution to fit for each failure cause. Select one distribution to fit for all causes; select Individual Best to let the platform automatically choose the best fit for each cause; or select Manual Pick to manually choose the distribution to fit for each failure cause after JMP creates the Life Distribution report. You can also change the distribution fits on the Life Distribution report itself. Comparison Criterion is an option only when you choose the Individual Best distribution fit. Select the method by which JMP chooses the best distribution: Akaike Information Criterion Corrected (AICc), Bayesian Information Criterion (BIC), or -2Loglikelihood. You can change the method later in the Model Comparisons report. (See Model Comparisons on page 36 for details.) Censor Indicator in Failure Cause Column identifies the indicator used in the Failure Cause column for observations that did not fail. Select this option and then enter the indicator in the box that appears. See Meeker and Escobar (1998, chap. 15) for a discussion of multiple failure causes. Omit Competing Causes on page 42 also illustrates how to analyze multiple causes. Freq Frequencies or observation counts when there are multiple units. If the value is 0 or a positive integer, then the value represents the frequencies or counts of observations for each row when there are multiple units recorded. Label Is an identifier other than the row number. These labels appear on the y axis in the event plot. Censor Code Identifies censored observations. By default, 1 indicates censored observations; all other values or missing values are uncensored observations. JMP attempts to detect the censor code and display it in the list. You can select another code from the list or select Other to specify a code.

33 Chapter 3 Life Distribution 33 The Life Distribution Report Select Confidence Interval Method Defines the method for computing confidence intervals for the parameters. The default is Wald, but you can select Likelihood instead. For the Custom Estimation report, for Wald type interval methods, the computations for the confidence intervals for the cumulative distribution function (cdf) start with Wald confidence intervals on the standardized variable. Next, the intervals are transformed to the cdf scale. The confidence intervals given in the other graphs and profilers are transformed Wald intervals. Joint confidence intervals for the parameters of a two-parameter distribution are shown in the log-likelihood contour plots. They are based on approximate likelihood ratios for the parameters. For more information, refer to Statistical Details on page 46. The Life Distribution Report Event Plot The report window contains three main sections: Event Plot on page 33 Compare Distributions Report on page 35 Statistics Report on page 36 When you select a Failure Cause column on the launch window, the report window also contains the following reports: Cause Combination Report on page 39 Individual Causes Report on page 40 Click the Event Plot disclosure icon to see a plot of the failure or censoring times. The following Event Plot for the Censor Labels.jmp sample data table shows a mix of censored data (Figure 3.5).

34 34 Life Distribution Chapter 3 The Life Distribution Report Figure 3.5 Event Plot for Mixed-Censored Data failure (top line) interval censored (next two lines) right censored (middle two lines) left censored (bottom two lines) In the Event Plot, the dots represent time measurements. The failed and was removed from observation. Other line styles indicate the type of censored data. line indicates that the unit Right-Censored Values indicates that the unit did not fail during the study. The unit will fail in the future, but you do not know when. In the data table, right-censored values have one time value and a censor value or two time values. Left-Censored Values indicates that the unit failed during the study, but you do not know when (for example, a unit stopped working before inspection). In a data table, left-censored values have two time values; the left value is missing, the right value is the end time. Interval-Censored Values indicates that observations were recorded at regular intervals until the unit failed. The failure could have occurred anytime after the last observation was recorded. Interval censoring narrows down the failure time, but left censoring only tells you that the unit failed. Figure 3.6 shows the data table for mixed-censored data.

35 Chapter 3 Life Distribution 35 The Life Distribution Report Figure 3.6 Censored Data Types right censored (rows three and four) left censored (rows one and two) interval censored (rows five and six) Compare Distributions Report The Compare Distributions report lets you fit different distributions to the data and compare the fits. The default plot contains the nonparametric estimates (Kaplan-Meier-Turnbull) with their confidence intervals. When you select a distribution method, the following events occur: The distributions are fit and a Parametric Estimate report appears. See Parametric Estimate on page 37. The cumulative distribution estimates appear on the probability plot (in Figure 3.7, the magenta and yellow lines). The nonparametric estimates (or horizontal blue lines) also appear initially (unless all data is right censored). For right-censored data, the plot shows no nonparametric estimates. The shaded regions indicate confidence intervals for the cumulative distributions. A profiler for each distribution shows the cumulative probability of failure for a given period of time. Figure 3.7 shows an example of the Compare Distributions report.