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Reliability and Survivial Analysis


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Table Of Contents

Copyright 20032005 Minitab Inc. All rights reserved. 3

Table Of ContentsTest Plans ............... ................ ................ ................. ................ ................ ................ ................... ................ ................ ........... 7

Test Plans Overview................. ................ ................ ................ ............... ................ .................... ................ ................ ..... 7Failure Censoring ............................................................................................................................................................. 7Time Censoring ................................................................................................................................................................ 8Type I and Type II Errors.................................................................................................................................................. 8Demonstration Test Plans ................................................................................................................................................ 8Estimation Test Plans..................................................................................................................................................... 12

Accelerated Life Test Plans............................................................................................................................................ 16Distribut ion Analysis.. ................ ................ ................ ................ ................ ................ .................. ................ ................ ......... 23

Distribution Analysis Overview....................................................................................................................................... 23Estimation methods........................................................................................................................................................ 23Distribution Analysis Data............................................................................................................................................... 24Goodness-of-fit statistics................................................................................................................................................ 24Stacked vs. Unstacked data........................................................................................................................................... 25

Arbitrarily Censored Data ............................................................................................................................................... 25Right Censored Data...................................................................................................................................................... 64

Growth Curves................................................................................................................................................................... 115Growth Curve Overview ............................................................................................................................................... 115Data - Growth Curves............... ................. ................ ................ ................. ................ .................... ................ .............. 115Growth curves - exact data........ ................ ................ ................ ................. ................ .................... ................ .............. 115Growth curves - interval data ....................................................................................................................................... 116Growth curves - grouped interval data ......................................................................................................................... 117Using Cost or Frequency Columns .............................................................................................................................. 118Using Time and Retirement Columns .......................................................................................................................... 118Parametric Growth Curve............................................................................................................................................. 118Nonparametric Growth Curve....................................................................................................................................... 130

Accelerated Life Testing .................................................................................................................................................... 141

Regression with Life Data Overview ............................................................................................................................ 141Accelerated Life Testing............................................................................................................................................... 141Worksheet Structure for Regression with Life Data ..................................................................................................... 142To perform accelerated life testing with uncensored/right censored data .................................................................... 142To perform accelerated life testing with uncensored/arbitrarily censored data ............................................................ 143Transforming the accelerating variable ........................................................................................................................ 143Percentiles and survival probabilities ........................................................................................................................... 144

Accelerated Life Testing - Censor ................................................................................................................................ 144Accelerated Life Testing - Estimate.............................................................................................................................. 144To estimate percentiles and survival probabilities........................................................................................................ 145

Accelerated Life Testing - Graphs................................................................................................................................ 145To modify the relation plot ............................................................................................................................................ 145Relation plot..... ................ ................ ................ ............... ................ ................ ..................... ................ ................ ......... 146Probability plot for each accelerating level based on fitted model................................................................................ 146Probability plots ............................................................................................................................................................ 146

Accelerated Life Testing - Options ............................................................................................................................... 147Accelerated Life Testing - Results................................................................................................................................ 147Accelerated Life Testing - Storage ............................................................................................................................... 147Example of Accelerated Life Testing............................................................................................................................ 148


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Copyright 20032005 Minitab Inc. All rights reserved.4

Output ............... ................ ................ ................ ................ ................ ................ ..................... ................ ................ ....... 150Regression with Life Data.................................................................................................................................................. 151

Regression with Life Data Overview ............................................................................................................................ 151Regression with Life Data ............................................................................................................................................ 151Data - Regression with Life Data.................................................................................................................................. 151Uncensored/arbitrarily censored data .......................................................................................................................... 152Uncensored/right censored data .................................................................................................................................. 153Failure times................................................................................................................................................................. 153To perform regression with uncensored/right censored data ....................................................................................... 154To perform regression with uncensored/arbitrarily censored data ............................................................................... 154Estimating the model parameters................................................................................................................................. 154Factor variables and reference levels .......................................................................................................................... 154Multiple degrees of freedom test .................................................................................................................................. 155Regression with Life Data - Censor.............................................................................................................................. 155Regression with Life Data - Estimate ........................................................................................................................... 155To estimate percentiles and survival probabilities........................................................................................................ 156Regression with Life Data - Graphs ............................................................................................................................. 156Probability plots for regression with life data ................................................................................................................ 156To draw a probability plot of the residuals.................................................................................................................... 156Regression with Life Data - Options............................................................................................................................. 157To control estimation of the parameters....................................................................................................................... 157To change the reference factor level ............................................................................................................................ 157Regression with Life Data - Results ............................................................................................................................. 157To perform multiple degrees of freedom tests.............................................................................................................. 158Regression with Life Data - Storage............................................................................................................................. 158Example of Regression with Life Data ......................................................................................................................... 158Default output ............................................................................................................................................................... 161

Probit Analysis ................................................................................................................................................................... 163Probit Analysis Overview.............................................................................................................................................. 163Probit Analysis.............................................................................................................................................................. 163Data - Probit Analysis................................................................................................................................................... 163To perform a probit analysis......................................................................................................................................... 164Probit model and distribution function .......................................................................................................................... 164Estimating the model parameters................................................................................................................................. 165Factor variables and reference levels .......................................................................................................................... 165Natural response rate................................................................................................................................................... 165Percentiles.. ................ ............... ................ ................ ................ ................ ................ .................... ................ ............... 166Survival and cumulative probabilities ........................................................................................................................... 166Probit Analysis - Estimate............................................................................................................................................. 166To request survival probabilities................................................................................................................................... 167Probit Analysis - Graphs............................................................................................................................................... 167To draw a survival plot......... ............... ................ ................ ................ ................ ................... ................. ................ ...... 167Probability plots ............................................................................................................................................................ 167Survival plots ................................................................................................................................................................ 168Probit Analysis - Options .............................................................................................................................................. 168To control estimation of the parameters....................................................................................................................... 168Probit Analysis - Results............................................................................................................................................... 168To modify the table of percentiles ................................................................................................................................ 169Probit Analysis - Storage.............................................................................................................................................. 169


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Table Of Contents


Example of a Probit Analysis........................................................................................................................................ 170Probit Analysis - Output................................................................................................................................................ 173

References - Reliability and Survival Analysis................................................................................................................... 175Index .................................................................................................................................................................................. 177


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Test Plans


Test Plans

Test Plans OverviewUse Minitab's test planning commands to determine the sample size and testing time needed to estimate modelparameters or to demonstrate that you have met specified reliability requirements.

A test plan includes: The number of units you need to test

A stopping rule the amount of time you must test each unit or the number of failures that must occur

Success criterion the number of failures allowed while the test still passes (for example, every unit runs for thespecified amount of time and there are no failures)

Three kinds of test plans are available: demonstration, estimation, and accelerated life.

Demonstration test plans

Use demonstration test plans to determine the sample size or testing time needed to demonstrate, with some level ofconfidence, that the reliability exceeds a given standard.

There are two types of demonstration tests:

Substantiation tests provide statistical evidence that a redesigned system has suppressed or significantly reduced aknown cause of failure. You are testing:

H0: The redesigned system is no different from the old system.H1: The redesigned system is better than the old system.

Reliability tests provide statistical basis that a reliability specification has been achieved. You are testing:

H0: The system reliability is less than or equal to a goal value.

H1: The system reliability is greater than a goal value.

You can rewrite these hypotheses in terms of the scale (Weibull or exponential distributions) or location (otherdistributions), a percentile, the reliability at a particular time, or the mean time to failure (MTTF). For example, you can testwhether or not the MTTF for a redesigned system is greater than the MTTF for the old system.

Minitab provides an m-failure test plan for substantiation and reliability testing. If more than mfailures occur in an m-failuretest, the test fails.

Estimation test plans

Use estimation test plans to determine the number of test units that you need to estimate percentiles or reliabilities with aspecified degree of precision. Estimation test plans are similar to classical sample-size problems, but computations aremore intensive because the data are usually censored. Use estimation test plans to answer questions such as:

How many units must I test to estimate the 10th percentile with a 95% lower confidence bound within 100 hours of theestimate?

How long must I run the test to estimate the reliability at 500 hours with a 95% lower confidence bound within 0.05 ofthe estimate?

Accelerated life test plans

Use accelerated life test plans to determine the number of units to test and how to allocate those units across stresslevels for an accelerated life test or to determine the standard error for the parameter you wish to estimate given a fixednumber of test units. Use accelerated life test plans to answer questions such as:

How many units must I test to estimate the 10th percentile with a 95% upper confidence bound within 100 hours of theestimate?

What is the best allocation of 20 units across 3 stress levels in order to estimate the reliability at 1000 hours?

Twenty units are available for testing. What standard error can you expect for the estimate of the 500-hour reliability?To obtain an accelerated test plan, you provide the stress values and, optionally, the proportionate allocation of test units.Minitab evaluates the resulting plans and displays the "best" plans with respect to minimizing the variance.

Failure CensoringFailure censoring is useful for:

Testing lower percentiles For any percentile, increasing the test duration improves the precision of your estimate.However, you will see little improvement in precision when you run a test far beyond the estimated percentile. Forexample, if you estimate the 10th percentile, you obtain important gains in precision by running the test until around


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15% of the units fail, but little improvement by running the test longer. In fact, running the test beyond 15% of the unitsfailing could bias your estimate of the 10th percentile.

Replacing test units If you have a limited number of test positions, you can use failure censoring to determine whento replace unfailed units. For example, if you want to estimate the 10th percentile, but can only test 5 units at a time,you may want to replace all 5 units after the first failure in each group. In this case, you are failure-censoring when20% of the units in each group have failed.

Time CensoringTesting all units to failure in a life test usually does not make sense, especially if you are only interested in the lowerpercentiles of the distribution. For any percentile of interest, the precision of your results depends on:

Test duration

Sample size

To minimize cost, you need to balance the test duration and sample size. For a given precision, Minitab displays a list ofsample sizes for each censoring time you provide. As time increases, the sample size decreases. Choose the time andsample size combination that minimizes costs.

For an accelerated life test plan, you only need to provide one set of censor times. Each time in the set corresponds to thecensor time at a stress level. The first time corresponds to the lowest stress level, the second time corresponds to thesecond stress level, and so on.

Type I and Type II ErrorsHypothesis tests have four possible outcomes:

Null Hypothesis (H0)

Decision True False

Fail to reject H0: Correct decision

p = 1

Type II error

p = Reject H0: Type I error

p = Correct decision

p = 1

The outcome of the test depends on whether the null hypothesis (H0) is true or false and whether you reject or fail to rejectit.

When H0 is true and you reject it, you make a Type I error. The probability (p) of making a Type I error is called alpha

(), or the level of significanceof the test. When H0 is false and you fail to reject it, you make a Type II error. The probability (p) of making a Type II error is called

beta().

The powerof a test is the probability of correctly rejecting H0 when it is false. In other words, power is the likelihood thatyou will identify a significant effect when one exists.

Demonstration Test Plans


Stat > Reliability/Survival > Demonstration Test Plans

Use to demonstrate that you have met a reliability specification or that a redesigned system has improved reliability. In ademonstration test, you verify that only a certain number of failures occur in a set amount of test time.

Dialog box items

Minimum Value to be Demonstrated

Scale (Weibull or expo) or location (other dists): Choose to demonstrate the minimum scale for the Weibull andexponential distributions or the minimum location for other distributions, then enter the scale or location value.

Percentile: Choose to demonstrate the minimum percentile. In Percentile, enter a percentile. The percentile should bein units of time. In Percent, enter a percent associated with the percentile. The percent must be a number between 0and 1 or a percentage between 0 and 100.

Reliability: Choose to demonstrate the minimum reliability. In Reliability, enter the reliability. The reliability must be anumber between 0 and 1. In Time, enter the time associated with the reliability.

MTTF: Choose to demonstrate the mean time to failure (MTTF), then enter the MTTF.


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Maximum number of failures allowed: Enter one or more maximum number of failures your test allows.

Sample sizes: Choose to enter the number of units available for testing. Enter one or more sample sizes.

Testing times for each unit: Choose to enter the amount of time available for testing. Enter one or more test durations.

Note Each combination of maximum number of failures allowed and sample size or testing time will result in one testplan. You may wish to request several test plans and compare the results.

Distribution Assumptions

Distribution: Choose one of seven common distributions: Weibull (default), exponential, smallest extreme value,normal, lognormal, logistic, and loglogistic.

Shape (Weibull) or scale (other distributions): Enter the shape (Weibull) or scale (other distributions). For anexponential distribution, Minitab assumes a shape value of one. See Specifying planning values.

To determine testing time or sample size for a demonstration test

1 Choose Stat > Reliability/Survival > Demonstration Test Plans.

2 UnderMinimum Value to be Demonstrated, choose one of the following:

Scale (Weibull or expo) or location (other dists) to provide the scale of Weibull or exponential distributions orthe location of other distributions, then enter the scale or location.

Percentile, then enter the percentile. In Percent, enter a number between 0 and 100 for the associated percent.

Reliability, then enter a reliability value between zero and one. In Time, enter the time.

MTTF to provide the mean time to failure, then enter the time.3 In Maximum number of failures allowed, enter a number greater than or equal to zero. See m-failure test plan.

4 UnderSpecify values for one of the following, choose either:

Sample sizes, then enter the number of units available for testing.

Testing times for each unit, then enter each unit's test duration.

Note Each combination of maximum number of failures allowed and sample size or testing time will result in one testplan. You may wish to request several test plans and compare the results.

5 UnderDistribution Assumptions, choose any distribution from Distribution. Then, enter an estimate of the shape orscale in Shape (Weibull) or scale (other dists). See estimating the shape or scale.

6 If you like, use any dialog box options, then click OK.

Choosing Between a 0-Failure and an M-Failure Test

Use the table below to choose between a 0-failure and an m-failure test.

A 0-failure test... An m-failure test (m > 0)...

Usually reduces total test time for highly reliable items. May reduce total test time if you can run the testssequentially. For example, if you are testing 3 units in a 1-failure test and the first 2 units pass, you do not have totest the third.

Is more practical when failures are unlikely in a reasonableamount of time.

May not be feasible for highly reliable units.

Does not let you check the assumptions of the test design.

You cannot estimate the shape (Weibull distribution) orscale (other distributions) to compare it to the assumedvalue.

You can estimate the scale (Weibull or exponential

distribution) or location (other distributions), but yourestimate may be conservative.

Allows you to check the assumptions of the test design.

You can estimate the shape (Weibull distribution) orscale (other distributions) and compare it to theassumed value.

You can obtain a more accurate estimate of the scale

(Weibull or exponential distribution) or location (otherdistributions).

Does not make sense when you are likely to have at leastone failure.

Has a better chance of passing than a 0-failure test whenyou have a marginally improved design.

M-Failure Test Plan

In an m-failure test plan, the test is successful if no more than mfailures occur. For example, ifm= 3, a test passes if 0, 1,2, or 3 failures occur among Nidentical systems that are tested independently and have the same failure distribution.

Assumptions of the m-failure test plan:


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For the Weibull distribution, you know the shape parameter and wish to demonstrate the scale parameter.

For the exponential distribution, you wish to demonstrate the scale parameter. The shape parameter is one.

For the extreme value, normal, lognormal, logistic, and loglogistic distributions, you know the scale parameter andwish to demonstrate the location parameter.

For more information, see Choosing between a 0-failure and m-failure test.

Estimating the shape or scale

When running a demonstration test, it is common to have a good estimate of the shape (Weibull distribution) or scale(other distributions) parameter because this parameter is often not impacted by a redesign. However, if your assumptionsregarding this value are wrong, your demonstration test plan will be flawed.

You should consider rerunning the analysis using a range of reasonable values for the assumed parameter to see howthe assumed value is impacting your conclusions.

Increasing Power

The power of a test is the probability of correctly rejecting H0 when it is false. In a demonstration test, power is theprobability of correctly concluding that you have demonstrated a goal value.

You can increase the power of your demonstration test in two ways:

1 Reduce your goal value. As the improvement ratio increases, the power of the test increases. If the improvement ratio

is small, then the goal value is too large. Reduce the minimum value you want to demonstrate. This way, systems thathave improved or systems with high reliability values have a better chance of passing the m-failure test. However,reducing the minimum value yields a weaker conclusion about the reliability of the systems.

2 Increase the maximum number of failures allowed in the m-failure test.

Type I and Type II Errors in a Demonstration Test

The hypotheses for a demonstration test are:

H0: The system reliability is less than or equal to a goal value.

H1: The system reliability is greater than a goal value.

You can make either of these errors:

The test concludes that you have exceeded the goal value, but you really have not. (Type I error)

You have exceeded the goal value or a redesigned system has improved, but the test did not detect it. (Type II error)

Minitab provides testing times or sample sizes to control the Type I error (). You can adjust the Type I error by changingthe confidence level in the Options subdialog box. You can reduce the probability of a Type II error () by reducing theminimum value for the unknown parameter or by increasing the maximum number of failures your test allows. SeeIncreasing Power.

Demonstration Test Plans GraphsStat > Reliability/Survival > Demonstration Test Plans > Graphs

Use to draw a POP (probability of passing) graph to assist you in choosing a minimum value for the parameter you wantto demonstrate.

Dialog box items

Probability of passing the demonstration test: Check to display a POP graph.

Show different sample sizes/testing times overlaid on the same page: Choose to display different sample sizes or

testing times overlaid on the same page.Show different test plans overlaid on the same page: Choose to display different test plans overlaid on the samepage.

Minimum X scale: Enter a value for the minimum x-axis scale.

Maximum X scale: Enter a value for the maximum x-axis scale.

POP Graph

Use a POP (Probability of Passing) graph to choose a minimum value for the parameter you wish to demonstrate, so thata system with high reliability has a high probability of passing the m-failure test.


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Test Plans


The curve that appears on this graph shows you the likelihood of actually passing the demonstration test that youspecified in the dialog box. The likelihood that the test will pass depends on:

How much the unit's life has truly improved. (The more the unknown true life has improved over the hypothesizedvalue, the more likely the test will pass.)

The number of failures allowed.

Testing time and sample size combinations.

Minitab uses the sample size and corresponding testing time to control the Type I error (). You can adjust the Type Ierror by changing the confidence level in the Options subdialog box. You can reduce the probability of a Type II error ()by choosing the minimum value of the unknown parameter. See Type I and Type II errors in a Demonstration Test.

The POP graph is a plot of the power of your test (probability of passing your test) against the improvement ratio or theimprovement amount. By increasing power, you are reducing the chance of making a Type II error. See Increasing Power.

Note Minitab displays the likelihood of passing as a percent. To re-scale this as a probability, you must edit thedisplayed graph. Select the y-axis, right-click, and choose Edit > Y Scale. Click the Type tab and chooseProbability.

Demonstration Test Plans OptionsStat > Reliability/Survival > Demonstration Test Plans > Options

You can enter a confidence level that Minitab will use for all confidence intervals.

Dialog box items

Confidence level: Enter a number between 0 and 100. The default is 95.0.

Example of creating a demonstration test plan

The reliability goal for a turbine engine combustor is a 1 percentile of at least 2000 cycles. The number of cycles to failuretends to follow a Weibull distribution with shape = 3. You can accumulate up to 8000 test cycles on each combustor. Youmust determine the number of combustors needed to demonstrate the reliability goal using a 1-failure test plan.

1 Choose Stat > Reliability/Survival > Demonstration Test Plans.

2 Choose Percentile, then enter2000. In Percent, enter1.

3 In Maximum number of failures allowed, enter1.

4 Choose Testing times for each unit, then enter8000.

5 From Distribution, choose Weibull. InShape (Weibull) or scale (other dists), enter3. Click OK.

Session window output


Reliability Test Plan

Distribution: Weibull, Shape = 3

Percentile Goal = 2000, Target Confidence Level = 95%

Actual

Failure Testing Sample Confidence

Test Time Size Level

1 8000 8 95.2122


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Graph window output

Interpreting the results

You must test 8 combustors for 8000 cycles to demonstrate with 95.2% confidence that the first percentile is at least 2000cycles.

The graph shows the likelihood of actually passing the test that you specified. Here,

The probability that your 1-failure test will pass increases steadily as the improvement ratio increases from zero to two.

If the improvement ratio is greater than about two, the test has an almost certain chance of passing. If the (unknown) true first percentile was 4000, then the improvement ratio = 4000/2000 = 2, and the probability of

passing the test would be about 0.88. If you reduced the value to be demonstrated to 1600, then the improvementratio would increase to 2.5 and the probability of passing the test would increase to around 0.96. By reducing thevalue to be demonstrated, you would increase the probability of passing the test. However, you would also be makinga less powerful statement about the reliability of the turbine engine combustor.

Estimation Test Plans


Stat > Reliability/Survival > Estimation Test Plans

Use to determine the number of test units that you need to estimate percentiles or reliabilities with a specified degree ofprecision.

The data you collect can be: Uncensored or complete

Right-censored

Interval-censored

A time-censored or failure-censored test plan often gives precise results while minimizing your testing costs.

Dialog box items

Parameter to be Estimated

Percentile for percent: Choose to estimate a percentile, then enter the percent.


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Test Plans


Reliability at time: Choose to estimate the reliability at a specified time, then enter the time.

Precisions as distances from bound of CI to estimate: Choose to estimate the precision between the estimate andlower bound or the estimate and upper bound, then enter the precision value. See Choosing the precision whenestimating a percentile or Choosing the precision when estimating a reliability.

Assumed distribution: Choose one of seven common distributions: Weibull (default), exponential, smallest extremevalue, normal, lognormal, logistic, and loglogistic.

Specify planning values for two of the following: Specify one value for the exponential distribution or two values for

the other distributions. See Specifying Planning Values.

Shape (Weibull) or scale (other distributions): Enter the shape (Weibull) or scale (other distributions). For theexponential distribution, Minitab does not expect an entry because there is no shape parameter.

Scale (Weibull or expo) or location (other dists): Enter the scale (Weibull or exponential) or location (otherdistributions).

Percentile: Enter a percentile. In Percent, enter a percent associated with the percentile.

Percentile: Enter a second percentile. In Percent, enter a percent associated with the percentile.

To use an estimation test plan for estimating a percentile

1 Choose Stat > Reliability/Survival > Estimation Test Plans.

2 UnderParameter to be Estimated, choose Percentile for percent, then enter a percent between 0 and 100.

3 From Precisions as distances from bound of CI to estimate, choose whether you wish to provide the desired

precision from the upper or lower bound to the estimate, then enter the precision. See Choosing the precision whenestimating a percentile.

4 From Distribution, choose one of the available distributions.

5 In Specify planning values for two of the following, complete two of the following:

In Shape (Weibull) or scale (other distributions), enter the shape or scale.

In Scale (Weibull or expo) or location (other dists), enter the scale or location.

In Percentile, enter the percentile. In Percent, enter the percent. If you enter planning values for two percentiles,they must be different.

6 Click Right Cens orInterval Cens to add any censoring information, then click OK.


To use an estimation test plan for estimating a reliability

1 Choose Stat > Reliability/Survival > Estimation Test Plans.2 UnderParameter to be Estimated, choose Reliability at time, then enter the time.

3 From Precisions as distances from bound of CI to estimate, choose whether you wish to provide the desiredprecision from the upper or lower bound to the estimate, then enter the precision. See Choosing the precision whenestimating a reliability.

4 From Distribution, choose one of the available distributions.


In Shape (Weibull) or scale (other distributions), enter the shape or scale.

In Scale (Weibull or expo) or location (other dists), enter the scale or location.

In Percentile, enter the percentile. In Percent, enter the percent. If you enter planning values for two percentiles,they must be different.

6 Click Right Cens orInterval Cens to add any censoring information, then click OK.


Determining sample size for estimating scale or location parameters

You may want to approximate the sample size needed to estimate the scale parameter (Weibull or exponentialdistribution) or the location parameter (other distributions). To do this, use an estimation test plan to obtain the samplesize needed to estimate the corresponding percentile of the distribution. For example, estimating the location parameterfor the normal distribution is equivalent to estimating the 50th percentile of that distribution.

Use the following table to determine the percent that corresponds to the scale or location parameter for the chosendistribution. Entering this value in Percentile for percent in the Estimation Test Plan dialog box will result in theapproximate sample size that you need for estimating the scale of location parameter.


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Distribution Parameter to estimate Percent

Normal 0.5

Lognormal exp() 0.5

Logistic 0.5

Loglogistic exp() 0.5

Extreme value 1 e-1

Weibull 1 e-1

Exponential 1 e-1

Choosing the precision when estimating a percentile

The precision is based on the width around the confidence interval for the parameter you are estimating. The wider yourconfidence interval, the fewer units you need to test.

For example, if you want to estimate the 10th percentile of your failure time distribution, and the lower bound is to be nomore than 25 hours less than your estimate, choose Lower bound and enter 25 as your desired precision in Samplesizes or precisions as distances from bound of CI to estimate.

You may want to enter a range of values for the precision, to see its impact on your sample size.

Choosing the precision when estimating a reliabilityThe precision is based on the width around the confidence interval for the parameter you are estimating. The wider yourconfidence interval, the fewer units you need to test.

For example, if you want to estimate the reliability of your units at 200 hours, and the lower bound is to be a reliability thatis no more than 0.025 below your estimate, choose Lower bound and enter 0.025 as your desired precision in Samplesizes or precisions as distances from bound of CI to estimate.

You may want to enter a range of values for the precision, to see its impact on your sample size.

Specifying Planning Values

To create a test plan, you need information about the data you expect to collect. You can obtain planning informationfrom:

Design specifications

Expert opinions Prior studies or small pilot studies

For an estimation test plan, you must do one of the following:

Provide planning values for both unknown parameters (scale and shape or location and scale). Alternatively, you canprovide planning values for one or two of the percentiles and Minitab will calculate the value of the unknownparameters.

Provide a planning value for the unknown scale (Weibull or exponential distribution) or location (other distributions)parameter when the shape (Weibull distribution) or scale (other distributions) is known.

For an accelerated life test plan, you must provide the shape (Weibull distribution) or scale, and planning values for one ofthe following:

Percentiles at two different stress levels

One percentile and the intercept

One percentile and the slope

The intercept and the slope

Note The slope represents the activation energy when the Arrhenius relationship is chosen and the assumeddistribution is Weibull, exponential, lognormal, or loglogistic.

Estimation Test Plans Right CensoringStat > Reliability/Survival > Estimation Test Plans > Right Censoring

Use the right-censoring options if your data are censored in either of these ways:


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Test Plans


Time-censored Test each unit for a preset amount of time.

Failure-censored Test the units until a preset proportion of failures occur.

Your data can be either singly censored or multiply censored:

Singly-censored All of the test units run for the same amount of time or until the same percent of units fail. Unitssurviving at the end of the study are considered censored data.

Multiply-censored Test units are censored at different times or in groups where a different percent of units are

allowed to fail.Dialog box items

Type of Censoring

Time censor at: Choose for time-censored data, then enter the censoring time. If your data will be singly censored,enter one or more censoring times. If your data will be multiply censored, enter one or more columns of censoringtimes. Each row in a column represents a group of test units. See Time Censoring.

Failure censor at percent of units failed: Choose for failure-censored data, then enter the percent of failures at whichto begin censoring. If your data will be singly censored, enter one or more percents. If your data will be multiplycensored, enter one or more columns of percents. Each row in a column represents a group of test units. See FailureCensoring.

Allocation for Multiple Groups

Equal percent per group: Choose to run the same percentage of units for each group.

Percent of units run in each group: Choose to change the percentage of units run for each group, then enter thepercentages.

Estimation Test Plans Interval CensoringStat > Reliability/Survival > Estimation Test Plans > Interval Censoring

Use interval censoring when you will be inspecting units for failures at pre-set intervals. You can space these intervalsequally in time or in log time; or set intervals so that the expected number of failures in each is the same.

Dialog box items

Number of Inspections: Enter the number of inspections.

Inspection times

Equally spaced: Choose for equally spaced inspection times. In Last inspection time,enter the last inspection time.

Equal probability: Choose for the expected proportion of failures to be the same in each interval. In Total percent offailures,enter the expected percent of failures for the entire test.

Equally spaced in log time: Choose for equally spaced log inspection times. In First inspection time and Lastinspection time, enter the first and last times.

Estimation Test Plans OptionsStat > Reliability/Survival > Estimation Test Plans > Options

You can assume a known shape or scale parameter.

You can also enter a confidence level that Minitab will use for all confidence intervals.

Dialog box items

Assume shape (Weibull) or scale (other distributions) is known: Check if you know the shape (Weibull) or scale(other distributions) parameter. This results in a smaller sample size because Minitab assumes that you do not need toestimate this parameter. For the exponential distribution, Minitab assumes a known shape parameter of one.

Confidence level: Enter the confidence level. The default is 95.0.

Example of creating an estimation test plan

You want to run a life test to estimate the 5th percentile for the life of a metal component used in a switch. You can run thetest for 100,000 cycles.

You expect about 5% of the units to fail by 40,000 cycles, 15% by 100,000 cycles, and the life to follow the Weibulldistribution. You want the lower bound of your confidence interval to be within 20,000 cycles of your estimate.

1 Choose Stat > Reliability/Survival > Estimation Test Plans.

2 UnderParameter to be Estimated, choose Percentile for percent, then enter5.

3 From Precisions as distances from bound of CI to estimate, choose Lower bound, then enter20000.


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4 From Assumed distribution, choose Weibull.

5 UnderSpecify planning values for two of the following, do the following:

In the first Percentile, enter40000. In Percent, enter5.

In the second Percentile, enter100000. In Percent, enter15.

6 Click Right Cens.

7 UnderType of Censoring, choose Time censor at, then enter100000. Click OK in each dialog box.



Type I right-censored data (Single Censoring)

Estimated parameter: 5th percentile

Calculated planning estimate = 40000

Target Confidence Level = 95%

Planning Values

Percentile values 40000, 100000 for percents 5, 15

Planning distribution: Weibull

Scale = 423612, Shape = 1.25859

Actual

Censoring Sample Confidence

Time Precision Size Level

100000 20000 74 95.0516


To estimate the 5th percentile with a lower confidence bound within 20,000 cycles of the estimate, you must test 74components for 100,000 cycles.

Accelerated Life Test Plans

Accelerated Life Test Plans

Stat > Reliability/Survival > Accelerated Life Test PlansUse accelerated life test plans to determine the number of test units and how to allocate these units across stress levelsfor an accelerated life test.

The data you collect can be:

Uncensored or complete

Right-censored

Interval-censored

A time-censored or failure-censored test plan often gives precise results while minimizing testing costs.

Dialog box items

Parameter to be Estimated

Percentile for percent: Choose to estimate a percentile, then enter the percent.

Reliability at time: Choose to estimate the reliability at a specified time, then enter the time.

Sample sizes or precisions as distances from bound of CI to estimate: Choose Sample size, Lower bound, or Upperbound, and enter either the sample size or the precision value. See Choosing the precision when estimating a percentileor Choosing the precision when estimating a reliability.

Distribution: Choose one of seven common distributions: Weibull (default), exponential, smallest extreme value, normal,lognormal, logistic, and loglogistic.

Relationship: Choose linear (no transformation, the default), Arrhenius, inverse temperature, or loge (power)transformation for the accelerating variable. See Transforming the Accelerating Variable.

Shape (Weibull) or scale (other distributions): Enter the shape (Weibull) or scale (other distributions). For theexponential distribution, Minitab does not expect an entry because there is no shape parameter.


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Test Plans


Specify planning values for two of the following: Specify planning values for two of the model parameters. If youchoose to specify planning values for two percentiles, they must be at different stress levels. See Specifying PlanningValues.

Percentile: Enter a percentile. In Percent, enter a percent associated with the percentile. In Stress, enter the stresslevel.

Percentile: Enter a second percentile. In Percent, enter a percent associated with the percentile. In Stress, enter thestress level.

Intercept: Enter the intercept for the relationship with the accelerating variable. See Choosing the Slope and Intercept.

Slope: Enter the slope for the relationship with the accelerating variable. See Choosing the Slope and Intercept.

To use an accelerated life test plan for estimating a percentile

1 Choose Stat > Reliability/Survival > Accelerated Life Test Plans.

2 UnderParameter to be Estimated, choose Percentile for percent, then enter a percent between 0 and 100.

3 From Sample sizes or precisions as distances from bound of CI to estimate, choose one of the following:

Sample size, then enter the number of units available to test.

Upper bound, then enter the desired precision from the estimate to the upper bound. See Choosing the precisionwhen estimating a percentile.

Lower bound, then enter the desired precision from the lower bound to the estimate. See Choosing the Precisionwhen estimating a percentile.

4 From Distribution, choose one of the available distributions. From Relationship, choose one of the availablerelationships. See Transforming the Accelerating Variable.

5 In Shape (Weibull) or scale (other distributions), enter the shape or scale.


In Percentile, enter the percentile. In Percent, enter the percent. In Stress, enter the stress level. If you enterplanning values for two percentiles, they must be at different stress levels.

In Intercept, enter the intercept. See Choosing the Slope and Intercept.

In Slope, enter the slope. See Choosing the Slope and Intercept.

7 Click Stresses. In Design stress, enter the design stress. In Test stresses, enter the levels of the test stresses. Youcan type the design or level of test stresses, enter a stored constant, or enter a column. Columns must be the samelength.

8 If your data are censored, click Right Cens orInterval Cens to add censoring information, then click OK.


To use an accelerated life test plan for estimating a reliability

1 Choose Stat > Reliability/Survival > Accelerated Life Test Plans.

2 UnderParameter to be Estimated, choose Reliability at time, then enter the time.

3 From Sample sizes or precisions as distances from bound of CI to estimate, choose one of the following:

Sample size, then enter the number of units available to test.

Upper bound, then enter the desired precision from the estimate to the upper bound. See Choosing the precisionwhen estimating a reliability.

Lower bound, then enter the desired precision from the lower bound to the estimate. See Choosing the precisionwhen estimating a reliability.

4 From Distribution, choose one of the available distributions. From Relationship, choose one of the availablerelationships. See Transforming the Accelerating Variable.

5 In Shape (Weibull) or scale (other distributions), enter the shape or scale.


In Percentile, enter the percentile. In Percent, enter the percent. In Stress, enter the stress level. If you enterplanning values for two percentiles, they must be at different stress levels.

In Intercept, enter the intercept. See Choosing the Slope and Intercept.

In Slope, enter the slope. See Choosing the Slope and Intercept.

7 Click Stresses. In Design stress, enter the design stress. In Test stresses, enter the levels of the test stresses. Youcan type the design or level of test stresses, enter a stored constant, or enter a column. Columns must be the samelength.

8 If your data are censored, click Right Cens orInterval Cens to add censoring information, then click OK.


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Accelerated Life Test Models

Relationship Model

Arrhenius Y = 0 + 1 [11604.83/ C + 273.16)] +

Inverse temperature Y = 0 + 1 [1/( C + 273.16)] +

Loge (power) Y = 0 + 1 log(accelerating variable) +

Linear Y = 0 + 1 accelerating variable +

where:

Y = failure time or log failure time.

0 = y-intercept (constant)

1 = regression coefficient

= reciprocal of the shape parameter (Weibull distribution) or the scale parameter (other distributions).

= random error term.

Note The slope, 1, is the activation energy in Arrhenius models when the assumed distribution is Weibull,exponential, lognormal, or loglogistic.

Choosing the Slope and Intercept

If you have previously used accelerated life tests for similar experiments, you can use historical estimates of the slope andintercept as planning values. See Accelerated Life Test Models.

Efficiency and Accuracy of Accelerated Life Test Plans

Minitab evaluates the efficiency of each plan and ranks them in order. Efficiency is measured in terms of the variance ofthe parameter you want to estimate. It is possible, however, for a highly efficient test plan (one with small variance) toproduce results that are not accurate. In particular, the results of an accelerated life test are based on obtaining enoughfailures at each stress level to accurately estimate the parameter of interest. To obtain accurate parameter estimates, acommon rule of thumb is that the expected number of failures at each of the test stresses should be at least four or five.

By default, Minitab displays three different test plans.

Accelerated Life Test Plans Stress LevelsStat > Reliability/Survival > Accelerated Life Test Plans > Stresses

You must enter the design and test stress levels. By default, Minitab will determine an "optimal" allocation of units acrossstress levels. Alternatively, you can provide the allocation. See Searching for the Optimum Proportions.

Dialog box items

Design stress: Enter the stress level for normal use conditions.

Test stresses: Enter one or more fixed test stress levels. You can type the stress levels, enter stored constants, or entercolumns. Type or enter stored constants if the test stress levels are for a single test plan. Use columns for a set of teststress levels for a series of test plans. Each column represents a separate set of test stresses.

User Defined Allocations for each Stress Level

Percent Allocations: Enter the percent of units to test at each test stress level. You can type the percent allocations,enter stored constants, or enter columns. Type or enter stored constants if the percent allocations are for a single test

plan and sum to 100%. Use columns for a set of allocations for a series of test plans. Columns must be the samelength as the columns of test stresses and must sum to 100%.

Search for the Best Allocation for each Stress Level: Check to have Minitab find the optimal allocation for each stresslevel.

Step length in search: Enter a value between 0.01 and 0.5 to use to go through the range of each test stress. Thedefault is 0.05. See Prefixed Ranges and Default Steps.

Number of "best" plans to output: Enter the number of test plans for Minitab to display. The default is 3. SeeEfficiency and Accuracy of Accelerated Life Test Plans.


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Test Plans


Searching for the Optimum Proportions

The most efficient plan is only the most efficient in the specified search space. Minitab can find the most efficient or"optimum" allocation of test units in two ways:

You specify the search space as a finite set or sets of proportions. Each column represents a different test plan.Minitab ranks those test plans according to their efficiency.

Minitab searches for the optimum proportions in ranges. That is, Minitab uses a step to go from one candidate group

of proportions to another. The default step length is 0.05, but you can increase or reduce the length.

Pre-fixed Ranges and Default Steps

Minitab chooses the pre-fixed ranges for the proportionate allocation of test units based on the following criteria:

More test units are assigned to the lowest test stress.

Either a large or small number of test units exists at the middle stresses.

The proportionate allocation of units at a test stress is not too small relative to the others.

The ranges change as the number of stresses change:

For a two-stress design, the ranges for the proportions at the lowest and highest test stress are RL = [0.05, 0.85] andRH = [0.075, 0.5], respectively.

For a three-stress design, the lowest test stress range is RL = [0.333, 0.683]. The other test stresses have a commonrange of R = [0.040, 0.333].

In general, if your design has K test stresses, the range for the proportions at the

lowest test stress is RL = [1/K, 1/K + 0.35]

middle stresses is R = [(1- 1/K - 0.350)/2K, 1/K]

highest test stress is R = [(1- 1/K - 0.350)/2K, 1/K] and chosen so that the complete set of proportionate allocationssums to one

Accelerated Life Test Plans Right CensoringStat > Reliability/Survival > Accelerated Life Test Plans > Right Censoring

Use the right-censoring options if your data will be censored in either of these ways:

Time-censored Test each unit for a preset amount of time, which can be different for each stress level.

Failure-censored Test the units until a preset proportion of failures occurs. The proportion can be different for eachstress level.

Dialog box items

Type of Censoring

Time censor for each stress level: Choose for time-censored data, then enter the censoring time for each stresslevel, in order, from the lowest to the highest. See Time Censoring.

Failure censor at percent of units failed for each stress level: Choose for failure-censored data, then enter thepercent of failures at which to begin censoring for each stress level, in order, from the lowest to the highest. See FailureCensoring.

Accelerated Life Test Plans Interval CensoringStat > Reliability/Survival > Accelerated Life Test Plans > Interval Censoring

Use interval censoring when you will be inspecting units for failures at pre-set intervals. You can space these intervalsequally in time or in log time; or set intervals so that the expected number of failures in each is the same.

Dialog box itemsNumber of inspections for each stress level: Enter the number of inspections for each stress level, in order, from thelowest stress level to the highest stress level. You must have the same number entries as you have test stress levels.

Inspection Times

Equally spaced: Choose for equally spaced inspection times. In Last inspection time for each stress level, enterthe last inspection time from the lowest stress level to the highest stress level. You must have the same number entriesas you have test stress levels.

Equal probability: Choose for the expected proportion of failures to be the same in each interval. In Total percent offailures at each stress level, enter the expected percent of failures in the entire test from the lowest stress level to thehighest stress level. You must have the same number entries as you have test stress levels.


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Equally spaced in log time: Choose for equally spaced log inspection times. In First inspection time for eachstress level and Last inspection time for each stress level,enter the first and last times from the lowest stress levelto the highest stress level. You must have the same number entries as you have test stress levels.

Accelerated Life Test Plans OptionsStat > Reliability/Survival > Accelerated Life Test Plans > Options

You can assume a known shape or scale parameter.

You can also enter a confidence level that Minitab will use for all confidence intervals.

Dialog box items

Assume shape (Weibull) or scale (other distributions) is known: Check if you know the shape (Weibull) or scale(other distributions) parameter. This results in a smaller sample size because Minitab assumes that you do not need toestimate this parameter. For an exponential distribution, Minitab assumes a knownshape parameter of one.

Confidence level: Enter the confidence level. The default is 95.0.

Example of creating an accelerated life test plan

You want to plan an accelerated life test to estimate the 1000-hour reliability of an incandescent light bulb at the designvoltage of 110 volts. You have 20 light bulbs available to test until failure. To accelerate failures, you will run the test at120 volts and 130 volts.

You believe that a power relationship will adequately model the relationship between failure time and voltage. Historicaldata indicate that a lognormal distribution with a scale of 50 appropriately models light bulb failure. The planning valuesare 1200 for the 50th percentile at 110 volts and 600 for the 50th percentile at 120 volts.1 Choose Stat > Reliability/Survival > Accelerated Life Test Plans.

2 UnderParameter to be Estimated, choose Reliability at time, then enter1000.

3 In Sample sizes or precisions as distances from bound of CI to estimate, choose Sample size, then enter20.

4 From Distribution, choose Lognormal. From Relationship, choose Loge (Power).

5 In Shape (Weibull) or scale (other distributions), enter50.

6 Under Specify planning values for two of the following, do the following:

In the first Percentile, enter1200. In Percent, enter50. In Stress, enter110.

In the second Percentile, enter600. In Percent, enter50. In Stress, enter120.

7 Click Stresses.

8 In Design stress, enter110. In Test stresses, enter120 130. Click OK in each dialog box.


Accelerated Life Testing Test Plans

Uncensored data

Power model

Estimated parameter: Reliability at time = 1000

Calculated planning estimate = 0.501455

Design stress value = 110

Target Confidence Level = 95%

Planning Values

Percentile values = 1200, 600 for percents = 50, 50 at stresses = 110, 120

Planning distribution: Lognormal base e

Intercept = 44.5349, Slope = -7.96617 and Scale = 50

Selected test plans: "Optimum" allocations test plans

Total available sample units = 20


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Test Plans


1st Best "Optimum" Allocations Test Plan

Test Percent Percent Sample Expected

Stress Failure Alloc Units Failures

120 100 65.7524 13 13

130 100 34.2476 7 7

Standard error of the parameter of interest = 0.283150

2nd Best "Optimum" Allocations Test Plan



120 100 65 13 13

130 100 35 7 7


3rd Best "Optimum" Allocations Test Plan



120 100 70 14 14

130 100 30 6 6



To estimate the 1000-hour reliability at the design voltage of 110 volts, test 13 units until failure at 120 volts and 7 unitsuntil failure at 130 volts.


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Distribution Analysis



Distribution Analysis OverviewUse Minitab's distribution analysis commands to understand the lifetime characteristics of a product, part, person, ororganism. For instance, you might want to estimate how long a part is likely to last under different conditions, or how longa patient will survive after a certain type of surgery.

Your goal is to estimate the failure-time distribution of a product. You do this by estimating percentiles, survivalprobabilities, cumulative failure probabilities, and distribution parameters and by drawing survival plots, cumulative failureplots, or hazard plots. You can use either parametric or nonparametric estimates. Parametric estimates are based on anassumed parametric distribution, while nonparametric estimates assume no parametric distribution.

Choosing a distribution analysis command

How do you know which distribution analysis command to use? You need to consider two things: 1) the type of censoringyou have, and 2) whether or not you can assume a parametric distribution for your data.

CensoringLife data are often censored or incomplete in some way. Suppose you are testing how long a certain partlasts before wearing out and plan to cut off the study at a certain time. Any parts that did not fail before the study endedare censored, meaning their exact failure time is unknown. In this case, the failure is known only to be "on the right," orafter the present time. This type of censoring is called right censoring. Similarly, all you may know is that a part failedbefore a certain time (left censoring), or within a certain interval of time (interval censoring).

Use the right-censoring commands when you have exact failures and right censored data.

Use the arbitrary-censoring commands when your data are arbitrarily censored to include both exact failures and avaried censoring scheme, including right-censoring, left-censoring, and interval-censoring.

For details on creating worksheets for censored data, see Distribution Analysis Data.

Distribution Life data can be described using a variety of distributions. Once you have collected your data, you can usethe commands in this chapter to select the best distribution to use for modeling your data, and then estimate the variety offunctions that describe that distribution. These methods are called parametric because you assume the data follow aparametric distribution. If you cannot find a distribution that fits your data, Minitab provides nonparametric estimates ofthe same functions.

Use the parametric distribution analysis commands when you can assume your data follow a parametric distribution.

Use the nonparametric distribution analysis commands when you cannot assume a parametric distribution.

Estimation methods

Minitab provides both parametric and nonparametric methods to estimate functions. If a parametric distribution fits yourdata, then use the parametric estimates. If no parametric distribution adequately fits your data, then use thenonparametric estimates.

For parametric estimates, you can choose either the least squares method or the maximum likelihood method. Fornonparametric estimates, available methods depend on the type of censoring.

Estimation methods

Estimate Method Results Available with

Parametric(assumesparametricdistribution)

Maximumlikelihood

Distributionparameters, survival,cumulative failure,hazard, andpercentile estimates

Right-censored parametricdistribution analysis

Arbitrary-censored parametricdistribution analysis

Least-squaresestimation

Distributionparameters, survival,cumulative failure,hazard, andpercentile estimates

Right-censored parametricdistribution analysis

Arbitrary-censored parametricdistribution analysis


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Nonparametric(no distributionassumed)

Kaplan-Meier

Survival, cumulativefailure, and hazardestimates

Right-censorednonparametric distributionanalysis

Right-censored distributionoverview plot

Actuarial Survival, cumulativefailure, hazard, anddensity estimates,median residuallifetimes

Right-censorednonparametric distributionanalysis

Arbitrary-censorednonparametric distributionanalysis

Right-censored distributionoverview plot

Arbitrary-censoreddistribution overview plot

Turnbull Survival andcumulative failureestimates

Arbitrary-censorednonparametric distribution

analysis Right-censored distribution

overview plot

Distribution Analysis DataThe data you gather for the distribution analysis commands are individual failure times. For example, you might collectfailure times for units running at a given temperature. You might also collect samples of failure times under differenttemperatures, or under different combinations of stress variables.

Life data are often censored or incomplete in some way. Suppose you are monitoring air conditioner fans to find out thepercentage of fans that fail within a three-year warranty period. This table describes the types of observations you canhave.

Type of observation Description Example

Exact failure time You know exactly when thefailure occurred.

The fan failed at exactly 500 days.

Right censored You only know that the failureoccurred after a particular time.

The fan had not yet failed at 500 days.

Left censored You only know that the failureoccurred before a particular time.

The fan failed sometime before 500days.

Interval censored You only know that the failureoccurred between two particulartimes.

The fan failed sometime between 475and 500 days.

How you set up your worksheet depends, in part, on the type of censoring you have:

When your data consist of exact failures and right-censored observations, see Distribution analysis (right censoreddata).

When your data have exact failures and a varied censoring scheme, including right-censoring, left-censoring, andinterval-censoring, see Distribution analysis (arbitrarily censored data).

Goodness-of-fit statisticsMinitab displays up to two goodnessoffit statistics to help you compare the fit of distributions.

AndersonDarling statistic for the maximum likelihood and least squares estimation methods.


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Pearson correlation coefficient for the least squares estimation method.

The AndersonDarling statistic is a measure of how far the plot points fall from the fitted line in a probability plot. Thestatistic is a weighted squared distance from the plot points to the fitted line with larger weights in the tails of the

distribution. Minitab uses an adjusted AndersonDarling statistic, because the statistic changes when a different plot pointmethod is used. A smaller AndersonDarling statistic indicates that the distribution fits the data better.

The Pearson correlation measures the strength of the linear relationship between the X and Y variables on a probabilityplot. The correlation will range between 0 and 1, with higher values indicating a better fitting distribution.

Stacked vs. Unstacked dataIn unstacked data, each sample is in a separate column. Alternatively, you can stack all the data in one column and add acolumn of grouping indicators that define each sample. Like censoring indicators, grouping indicators can be numbers ortext.

Here is the same data set structured both ways:

Unstacked Data Stacked Data

Drug A

20

30

43

51

57

82

85

89

Drug B

2

3

6

14

24

26

27

31

Drug

20

30

43

51

57

82

85

89

2

3

6

14

24

26

27

31

Group

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

B

Note You cannot analyze more than one column of stacked data at a time, so the grouping indicators must be in onecolumn.

Arbitrarily Censored Data

Distribution ID Plot

Parametric distribution analysis commands

You can use all parametric distribution analysis commands for both right-censored and arbitrarily-censored data. Thecommands include Parametric Distribution Analysis, which performs the full analysis, and creates a Distribution ID Plotand Distribution Overview Plot. These graphs are often used before the full analysis to help choose a distribution or viewsummary information.

Command Description

Distribution ID Plot

Right Censored

Arbitrarily Censored

Draws probability plots from your choice of eleven common distributions:smallest extreme value, Weibull, 3-parameter Weibull, exponential, 2-parameter exponential, normal, lognormal, 3-parameter lognormal, logistic,loglogistic, and 3-parameter loglogistic. These plots help you determinewhich, if any, of the parametric distributions best fits your data.

Distribution Overview Plot

Right Censored


Draws a probability plot, probability density function, survival plot, andhazard plot in separate regions on the same graph. These help you assessthe fit of the chosen distribution and view summary graphs of your data.


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Parametric Distribution Analysis

Right Censored


Fits one of eleven common parametric distributions to your data, then usesthat distribution to estimate percentiles, survival probabilities, andcumulative failure probabilities. Also draws survival, cumulative failure,hazard, and probability plots.

Distribution ID Plot (Arbitrary Censoring)

Stat > Reliability/Survival > Distribution Analysis (Arbitrary Censoring) > Distribution ID PlotUse Distribution ID Plot (Arbitrary Censoring) to determine which distribution best fits your data by comparing how closelythe plot points lie to the best-fit lines of a probability plot.

Minitab also provides two goodness-of-fit measures to help you assess how the distribution fits your data:

Anderson-Darling for the least squares and maximum likelihood estimation methods

Pearson correlation coefficient for the least squares estimation method

You can display up to 50 samples on each plot. All the samples display on a single plot, with different colors and symbols.

Dialog box items

Start variables: Enter the columns of start times. You can enter up to 50 columns (50 different samples).

End variables: Enter the columns of end times. You can enter up to 50 columns (50 different samples).

Frequency columns (optional): Enter the columns of frequency data.

By variable: If all of the samples are stacked in one column, check By variable, then enter a column of grouping

indicators.

Use all distributions: Choose to have Minitab fit all eleven distributions.

Specify: Choose to fit up to four distributions.

Distribution 1: Check and choose one of eleven distributions: smallest extreme value, Weibull (default), 3-parameterWeibull, exponential, 2-parameter exponential, normal, lognormal, 3-parameter lognormal, logistic, loglogistic, or 3-parameter loglogistic.

Distribution 2: Check and choose one of eleven distributions: smallest extreme value, Weibull, 3-parameter Weibull,exponential, 2-parameter exponential, normal, lognormal (default), 3-parameter lognormal, logistic, loglogistic, or 3-parameter loglogistic.

Distribution 3: Check and choose one of eleven distributions: smallest extreme value, Weibull, 3-parameter Weibull,exponential (default), 2-parameter exponential, normal, lognormal, 3-parameter lognormal, logistic, loglogistic, or 3-parameter loglogistic.

Distribution 4: Check and choose one of eleven distributions: smallest extreme value, Weibull, 3-parameter Weibull,exponential, 2-parameter exponential, normal (default), lognormal, 3-parameter lognormal, logistic, loglogistic, or 3-

parameter loglogistic.

Distribution Analysis (Arbitrarily Censored Data)

When your data consist of exact failures and a varied censoring scheme, including right-, left- and interval-censored data,your data is arbitrarily-censored. For general information on life data and censoring, see Distribution Analysis Data.

You can enter up to 50 samples per analysis. Minitab estimates the functions independently for each sample, unless youassume a common shape (Weibull) or scale (other distributions). All the samples display on a single plot, with differentcolors and symbols, which helps you compare the various functions between samples.

Minitab analyzes systems with one cause of failure or multiple causes of failure. For systems that have more than onecause of failure, see Multiple Failure Modes (Arbitrarily Censored Data).

Enter your data in table form, using a Start column and End column:

For this observation... Enter in the Start Column... Enter in the End Column...

Exact failure time Failure time Failure time

Right censored Time that the failure occurred after Missing value symbol ''

Left censored Missing value symbol '' Time before which the failure occurred

Interval censored Time at start of interval during which thefailure occurred

Time at end of interval during which thefailure occurred


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This data set illustrates tabled data. For observations with corresponding columns of frequency, see Using frequencycolumns.

Start End

* 10000 Left censored at 10000 hours.

10000 20000

20000 30000

30000 30000 Exact failures at 30000 hours.

30000 40000

40000 50000

50000 50000

50000 60000 Interval censored between 50000 and 60000hours.

60000 70000

70000 80000

80000 90000

90000 * Right censored at 90000 hours.

When you have more than one sample, you can use separate columns for each sample. Alternatively, you can stack allthe samples in one column, then set up a column of grouping indicators, which can be numbers or text. For an illustration,see Stacked vs. Unstacked data.

To make a distribution ID plot (arbitrarily censored data)

1 Choose Stat > Reliability/Survival > Distribution Analysis (Arbitrary Censoring) > Distribution ID Plot.2 In Start variables, enter up to 50 columns of start times.

3 In End variables, enter up to 50 column of end times. The first start column is paired with the first end column, thesecond start column is paired with the second end column, and so on.

4 If you have frequency columns, enter them in Frequency columns.

5 If all of the samples are stacked in one column, check By variable, and enter a column of grouping indicators in thebox.

6 Do one of the following:

Choose Use all distributions to create probability plots for all eleven distributions. Choose Specify to create up to four probability plots with the distributions of your choice.

7 If you like, use any of the dialog box options, then click OK.

Distribution ID Plot (Arbitrary Censoring) OptionsStat > Reliability/Survival > Distribution Analysis (Arbitrary Censoring) > Distribution ID Plot > OptionsYou can choose the method used to estimate the parameters. You can also estimate percentiles for specified percents,specify the x-axis minimum and maximum, and add your own title.

Dialog box items

Estimation Method

Least Squares (failure time(X) on rank(Y)): Choose to estimate the distribution parameters using the least squares(XY) method, which are estimated by fitting a regression line to the points in a probability plot.

Maximum Likelihood: Choose to estimate the distribution parameters using the maximum likelihood method, whichare estimated by maximizing the likelihood function.

Estimate percentiles for these percents: Enter the additional percents for which you want to estimate percentiles. Youcan enter individual percents (0 < P < 100) or a column of percents.

Show graphs of different variables or by levels: Choose to display the graphs overlaid on the same graph or onseparate graphs.

Use default values: Choose to use the default values for the minimum and maximum X scale.

Use: Choose to enter your own values for the X scale minimum and maximum.

Minimum X scale: Enter a value for the minimum X scale.


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Maximum X scale: Enter a value for the maximum X scale.

Title: To replace the default title with your own title, type the desired text in this box.

Example of a Distribution ID Plot for arbitrarily-censored data

Suppose you work for a company that manufactures tires. You are interested in finding out how many miles it takes forvarious proportions of the tires to "fail," or wear down to 2/32 of an inch of tread. You are especially interested in knowing

how many of the tires last past 45,000 miles. You plan to get this information by using Parametric Distribution Analysis(Arbitrary Censoring), which requires you to specify the distribution for your data. Distribution ID Plot Arbitrary Censoringcan help you choose that distribution.

You inspect each good tire at regular intervals (every 10,000 miles) to see if the tire has failed, then enter the data into theMinitab worksheet.

1 Open the worksheet TIREWEAR.MTW.

2 Choose Stat > Reliability/Survival > Distribution Analysis (Arbitrary Censoring) > Distribution ID Plot.

3 InStart variables, enterStart. In End variables, enterEnd.

4 In Frequency columns, enterFreq.

5 Choose Specify. Leave the first three distributions at the default. From Distribution 4, choose Smallest extremevalue. Click OK.


Distribution ID Plot: Start = Start and End = End

Using frequencies in Freq

Goodness-of-Fit

Anderson-Darling Correlation

Distribution (adj) Coefficient

Weibull 2.387 0.948

Lognormal 2.960 0.880

Exponential 6.411 *

Smallest Extreme Value 2.325 0.998

Table of Percentiles

Standard 95% Normal CI

Distribution Percent Percentile Error Lower Upper

Weibull 1 15065.1 4005.60 8946.43 25368.5

Lognormal 1 19317.3 1249.06 17018.0 21927.3

Exponential 1 497.954 15.2114 469.015 528.678

Smallest

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