Unit 10: Planning Life Tests Ramón V. León

Notes largely based on “Statistical Methods for Reliability Data” by W.Q. Meeker and L. A. Escobar, Wiley, 1998 and on their class notes. 11/2/2004

Unit 10 - Stat 567 - Ramón León

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Unit 10 Objectives • Explain the basic ideas behind planning a life test • Use simulation to anticipate the results, analysis, and precision for a proposed test plan • Explain large-sample approximate methods to assess precision of future results from a reliability study • Compute sample size needed to achieve a degree of precision • Assess tradeoffs between sample size and length of a study. • Illustrate the use of simulation to calibrate the easier-touse large-sample approximate methods 11/2/2004

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Basic Ideas in Test Planning • The enormous cost of reliability studies makes it essential to do careful planning. Frequently asked questions include: – How many units do I need to test in order to estimate the .1 quantile of life? – How long do I need to run the life test?

• Clearly, more test units and more time will buy more information and thus more precision in estimation • To anticipate the results from a test plan and to respond to the questions above, it is necessary to have some planning information about the life distribution to be estimated 11/2/2004

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Engineering Planning Values and Assumed Distribution for Planning an Insulation Life Test

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Simulation as a Tool for Test Planning • Use assumed model and planning values of model parameters to simulate data from the proposed study • Analyze the data perhaps under different assumed models • Assess precision provided • Simulate many times to assess actual sample-to-sample differences • Repeat with different sample sizes to gauge needs • Repeat with different input planning values to assess sensitivity to these inputs. Any surprises? 11/2/2004

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Simulations of Insulation Life Tests

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Trade-offs Between Test Length and Sample Size

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Assessing the Variability of the Estimates

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Simulations of Insulation Life Test-Continued

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Motivation for Use of Large-Sample Approximations of Test Plan Properties Asymptotic methods provide: • Simple expressions giving precision of a specified estimator as a function of sample size • Simple expressions giving needed sample size as a function of specified precision of a specified estimator • Simple tables and graphs that will allow easy assessments of tradeoffs in test planning decisions like sample size and test length • Can be fine tuned with simulation evaluation 11/2/2004

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Asymptotic Variances

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Delta Method for Two Parameters ∂g (θ ) , ∂θ1

∂g (θ ) v1 ∂θ 2 v12

∂g (θ ) v12 ∂θ1 = v2 ∂g (θ ) ∂θ 2

∂g (θ ) ∂g (θ ) ∂g (θ ) ∂g (θ ) ∂g (θ ) ∂θ1 v1 + v12 , v12 + v2 = ∂θ 2 ∂θ1 ∂θ 2 ∂θ1 ∂g (θ ) ∂θ 2 ∂g (θ ) ∂g (θ ) ∂g (θ ) ∂g (θ ) ∂g (θ ) ∂g (θ ) v2 v1 + v12 v12 + = + ∂θ 2 ∂θ 2 ∂θ 2 ∂θ1 ∂θ1 ∂θ1 2

2

∂g (θ ) ∂g (θ ) ∂g (θ ) ∂g (θ ) v1 + 2 v12 + v2 ∂θ1 ∂θ1 ∂θ 2 ∂θ 2 11/2/2004

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Example

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Sample Size Determination for Positive Functions of the Parameters

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Sample Size Determination for Positive Functions of the Parameters-Continued

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Sample Size Needed to Estimate the Mean of an Exponential Distribution Used to Describe Insulation Life

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Sample Size Needed to Estimate the Mean of an Exponential Distribution Used to Describe Insulation LifeContinued

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Derivation.1 − θti t TTT − e n − θi −r θ θ L = ∏ e e = ∏ θ i =1 i = r +1 ⇒ r

l = − r log θ − TTTθ −1 ⇒ ∂l = − rθ −1 + TTTθ −2 ∂θ ⇒ ∂ 2l = rθ −2 − 2TTTθ −3 ∂θ 2 11/2/2004

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Derivation.2 ∂ 2l E − 2 = E 2TTTθ −3 − Rθ −2 ∂θ

= 2θ −3 ( E [TTT ]) − θ −2 E [ R ] = 2θ −3 (θ E [ R ]) − θ −2 E [ R ] = θ −2 ( E [ R ]) = θ −2 ( n × P [T ≤ tc ]) = θ −2 ( npc ) t − c = nθ 1 − e θ −2

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Location-Scale Distributions and Single Right Censoring Asymptotic Variance-Covariance

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Location-Scale Distributions and Single Right Censoring Fisher Information Elements

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Table of Information Matrix Elements and Variance Factors

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Large-Sample Asymptotic Variance for Estimators of Functions of Location-Scale Parameters

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Proportion failing by censoring time tc

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Figures for Sample Sizes to Estimate Weibull, Lognormal, and Loglogistic Quantiles

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Generalization: Location-Scale Parameters and Multiple Censoring

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Test Plans to Demonstrate Conformity with a Reliability Standard

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Minimum Sample Size Reliability Demonstration Test Plans

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Justification for the Weibull Zero-Failures Test Plan

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Justification for the Weibull Zero-Failures Test Plan (Continued)

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Additional Comments on Zero Failure Test Plans

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Other Topics in Chapter 10 • Uncertainty in planning values and sensitivity analysis • Sample size to estimate unrestricted functions of the parameters, the mean of an exponential, the hazard function of a location-scale distribution • Test planning for non-location-scale distributions 11/2/2004

Unit 10 - Stat 567 - Ramón León

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Notes largely based on “Statistical Methods for Reliability Data” by W.Q. Meeker and L. A. Escobar, Wiley, 1998 and on their class notes. 11/2/2004

Unit 10 - Stat 567 - Ramón León

1

Unit 10 Objectives • Explain the basic ideas behind planning a life test • Use simulation to anticipate the results, analysis, and precision for a proposed test plan • Explain large-sample approximate methods to assess precision of future results from a reliability study • Compute sample size needed to achieve a degree of precision • Assess tradeoffs between sample size and length of a study. • Illustrate the use of simulation to calibrate the easier-touse large-sample approximate methods 11/2/2004

Unit 10 - Stat 567 - Ramón León

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1

Basic Ideas in Test Planning • The enormous cost of reliability studies makes it essential to do careful planning. Frequently asked questions include: – How many units do I need to test in order to estimate the .1 quantile of life? – How long do I need to run the life test?

• Clearly, more test units and more time will buy more information and thus more precision in estimation • To anticipate the results from a test plan and to respond to the questions above, it is necessary to have some planning information about the life distribution to be estimated 11/2/2004

Unit 10 - Stat 567 - Ramón León

3

Engineering Planning Values and Assumed Distribution for Planning an Insulation Life Test

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4

2

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5

Simulation as a Tool for Test Planning • Use assumed model and planning values of model parameters to simulate data from the proposed study • Analyze the data perhaps under different assumed models • Assess precision provided • Simulate many times to assess actual sample-to-sample differences • Repeat with different sample sizes to gauge needs • Repeat with different input planning values to assess sensitivity to these inputs. Any surprises? 11/2/2004

Unit 10 - Stat 567 - Ramón León

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3

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4

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5

Simulations of Insulation Life Tests

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Trade-offs Between Test Length and Sample Size

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6

Assessing the Variability of the Estimates

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Simulations of Insulation Life Test-Continued

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7

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Motivation for Use of Large-Sample Approximations of Test Plan Properties Asymptotic methods provide: • Simple expressions giving precision of a specified estimator as a function of sample size • Simple expressions giving needed sample size as a function of specified precision of a specified estimator • Simple tables and graphs that will allow easy assessments of tradeoffs in test planning decisions like sample size and test length • Can be fine tuned with simulation evaluation 11/2/2004

Unit 10 - Stat 567 - Ramón León

16

8

Asymptotic Variances

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Delta Method for Two Parameters ∂g (θ ) , ∂θ1

∂g (θ ) v1 ∂θ 2 v12

∂g (θ ) v12 ∂θ1 = v2 ∂g (θ ) ∂θ 2

∂g (θ ) ∂g (θ ) ∂g (θ ) ∂g (θ ) ∂g (θ ) ∂θ1 v1 + v12 , v12 + v2 = ∂θ 2 ∂θ1 ∂θ 2 ∂θ1 ∂g (θ ) ∂θ 2 ∂g (θ ) ∂g (θ ) ∂g (θ ) ∂g (θ ) ∂g (θ ) ∂g (θ ) v2 v1 + v12 v12 + = + ∂θ 2 ∂θ 2 ∂θ 2 ∂θ1 ∂θ1 ∂θ1 2

2

∂g (θ ) ∂g (θ ) ∂g (θ ) ∂g (θ ) v1 + 2 v12 + v2 ∂θ1 ∂θ1 ∂θ 2 ∂θ 2 11/2/2004

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Example

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Sample Size Determination for Positive Functions of the Parameters

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Sample Size Determination for Positive Functions of the Parameters-Continued

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Sample Size Needed to Estimate the Mean of an Exponential Distribution Used to Describe Insulation Life

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Sample Size Needed to Estimate the Mean of an Exponential Distribution Used to Describe Insulation LifeContinued

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12

Derivation.1 − θti t TTT − e n − θi −r θ θ L = ∏ e e = ∏ θ i =1 i = r +1 ⇒ r

l = − r log θ − TTTθ −1 ⇒ ∂l = − rθ −1 + TTTθ −2 ∂θ ⇒ ∂ 2l = rθ −2 − 2TTTθ −3 ∂θ 2 11/2/2004

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Derivation.2 ∂ 2l E − 2 = E 2TTTθ −3 − Rθ −2 ∂θ

= 2θ −3 ( E [TTT ]) − θ −2 E [ R ] = 2θ −3 (θ E [ R ]) − θ −2 E [ R ] = θ −2 ( E [ R ]) = θ −2 ( n × P [T ≤ tc ]) = θ −2 ( npc ) t − c = nθ 1 − e θ −2

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13

Location-Scale Distributions and Single Right Censoring Asymptotic Variance-Covariance

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Location-Scale Distributions and Single Right Censoring Fisher Information Elements

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14

Table of Information Matrix Elements and Variance Factors

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15

Large-Sample Asymptotic Variance for Estimators of Functions of Location-Scale Parameters

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Proportion failing by censoring time tc

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16

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Figures for Sample Sizes to Estimate Weibull, Lognormal, and Loglogistic Quantiles

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Generalization: Location-Scale Parameters and Multiple Censoring

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Test Plans to Demonstrate Conformity with a Reliability Standard

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Minimum Sample Size Reliability Demonstration Test Plans

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Justification for the Weibull Zero-Failures Test Plan

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Justification for the Weibull Zero-Failures Test Plan (Continued)

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Additional Comments on Zero Failure Test Plans

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Other Topics in Chapter 10 • Uncertainty in planning values and sensitivity analysis • Sample size to estimate unrestricted functions of the parameters, the mean of an exponential, the hazard function of a location-scale distribution • Test planning for non-location-scale distributions 11/2/2004

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