Example of an Accelerated Life Test Design
In this example, suppose that you need to design an accelerated life test for a mechanical component. The single acceleration factor is torque, and the normal use stress is 35 Nm (newton meters). You want to estimate the B10 life, which is the life at which 10% of the units fail at the normal use stress.
Your test plan has the following characteristics:
• A total of 100 units are available for testing.
• The life distribution is assumed to be Weibull.
• The life-stress relationship is given by the logarithmic transformation.
• You will test at torque three stress levels: 50, 75, and 100 Nm.
• You have some prior knowledge to help you guess failure times at the test levels. More details appear in “Obtain Prior Estimates”.
• The test will continue for 5000 cycles.
• You will monitor the process for failures on a continuous basis.
Obtain Prior Estimates
To create an accelerated life test design, you need to provide prior estimates of the parameters. Here is a convenient approach to obtaining prior estimates:
1. Use your process knowledge to create a table of hypothetical but likely, failure times at a small number of stress levels.
2. Use the Fit Life by X platform to fit a model and obtain estimates of the model parameters.
3. Use these estimates as your prior values for creating a design using the ALT Design platform.
Following the approach outlined above, you create a data table containing estimates of the number of failure cycles for a balanced design. Your table consists of five units at each of the three stress levels that you will use in your design. Your data table is Torque Prior.jmp, found in the Design Experiment sub folder.
1. Select Help > Sample Data Library and open Design Experiment/Torque Prior.jmp.
2. Select Analyze > Reliability and Survival > Fit Life by X.
3. Select Cycles and click Y, Time to Event.
4. Select Torque and click X.
5. From the Relationship list, select Log.
6. From the Distribution list, select Weibull.
Figure 22.2 Fit Life by X Launch Window
7. Click OK.
8. Scroll down to the Weibull Results report and open the Covariance Matrix report.
Figure 22.3 Fit Life by X Model for Prior Data
The model for the mean is given in the Estimates report. The Estimate column contains the parameter estimates for the intercept (β0), the linear coefficient (β1), and the scale (σ). The estimated variances and covariances for the parameter estimates are given in the Covariance Matrix report. You will use these parameter estimates and their covariances as your prior values in constructing your ALT design.
Enter Basic Specifications
1. Select DOE > Special Purpose > Accelerated Life Test Design.
Notice that Design for one accelerating factor and Continuous Monitoring are selected by default.
2. Click Continue.
3. Under Factor Name, click X1 and type Torque.
Notice that the Number of Levels is set to 3 by default.
4. Select Log under Factor Transformation.
5. Enter 35 first for the High Usage Condition and then for the Low Usage Condition.
Setting both the high and low usage conditions at the common value of 35 indicates that 35 represents the normal usage condition.
Figure 22.4 Completed ALT Specification Window
6. Click Continue.
7. Enter 50, 75, and 100 for the Torque Level Values.
8. Ensure that Weibull is selected as the Distribution Choice.
Enter Prior Information and Remaining Specifications
1. Under Prior Mean, enter your prior acceleration model parameter estimates, rounding the values shown in the Estimates report in Figure 22.3 to two decimal places:
‒ Next to Intercept, type 15.88.
‒ Next to Torque, type -1.87.
‒ Next to scale, type 0.05.
2. Open the Prior Variance Matrix outline.
3. De-select the option Ignore prior variance. Generate the local design for the prior mean.
4. Enter the estimated covariances from the prior acceleration model, rounding the values shown in the Covariance Matrix report in Figure 22.3 to five decimal places:
‒ In the row and column for Intercept, click on 0.10000 and type 0.04570.
‒ In the row for Intercept and column for Torque, click on 0.00000 and type -0.01062.
‒ In the row for Intercept and column for scale, click on 0.00000 and type -0.00015.
‒ In the row and column for Torque, click on 0.10000 and type 0.00248.
‒ In the row for Torque and column for scale, click on 0.00000 and type 0.00002.
‒ In the row and column for scale, click on 0.10000 and type 0.00013.
Figure 22.5 Completed Prior Specification Outline
The test will be conducted over 5000 cycles. However, you are interested in predictions for as many as 10,000 cycles.
5. Under Diagnostic Choices, enter 10,000 for both boxes for Time range of interest. Leave the Probability of interest value set to 0.1.
This indicates that you are interested in estimating the time by which 10% of the units fail (B10 life).
6. Under Design Choices, enter 5000 for Length of test.
7. Enter 100 for Number of units under test.
Figure 22.11 shows the completed Accelerated Life Test Plan outline.
Figure 22.6 Completed Design Details Window
Create the Design
1. Click Continue.
Figure 22.7 Balanced Design Diagnostics
The number of runs in a balanced design appears in the Candidate Runs outline. The Parameter Variance for Balanced Design outline shows the covariance matrix for the parameters for this design. The Distribution Profiler also appears. When you obtain the optimal design, you can compare the Parameter Variance and Distribution Profiler results for the optimal design to those for the balanced design to see the reduction in uncertainty.
2. From the ALT Plan red triangle menu, select ALT Optimality Criterion > Make Probability I-Optimal Design.
This tells JMP to use the Probability optimality criterion when creating the design in the next step. For details about this criterion, see “Make Probability I-Optimal Design”.
3. Click Make Design.
The optimal experimental design is shown, along with other results.
Figure 22.8 Optimal Design
The optimal design is computed based on the levels of the test runs, the total number of units to be tested, and the prior information that you specified. The optimal design consists of testing the following number of units at each torque level:
‒ 72 units at 50 Nm
‒ 8 units at 75 Nm
‒ 20 units at 100 Nm
Example of Augmenting an Accelerated Life Test Design
This example shows how to use the Accelerated Life Test Design platform to augment an existing design.
In this example, 50 capacitor units are tested at three temperatures (85o, 105o, and 125o Celsius) for 1500 hours. The results are recorded in the Capacitor ALT.jmp sample data table. The resulting model is used to predict the fraction of the population that is failing at 100,000 hours at normal use temperature of 25o Celsius.
Review Current Predictions
1. Select Help > Sample Data Library and open Design Experiment/Capacitor ALT.jmp.
2. Click the green triangle to run the Fit Life by X table script.
3. In the Distribution Profiler, found in the Comparisons report on the Distribution tab, do the following:
‒ Click 105 above Temp and change it to 25.
‒ Click 750.5 above Hours and change it to 100,000.
Figure 22.9 Distribution Profiler for Capacitor Model
Based on your current study, the predicted fraction of the population that fails at 25o at 100,000 hours is 0.00358, with a confidence interval of 0.00056 to 0.02268. You want to estimate the failure fraction more precisely. To decrease the width of the confidence interval, augment your study with additional tests.
Augment the Design
You want to augment your design to obtain more precise estimates of the predicted failure fractions. Your original design used temperature settings of 85, 105, and 125. In your augmented design, you want to test at temperature values of 90, 110, and 125. Note that two of these settings are new. To augment the design with optimally selected runs, follow these steps:
1. Select DOE > Special Purpose > Accelerated Life Test Design.
2. Select Design for one accelerating factor and click Continue.
3. Enter Temp for Factor Name.
4. Enter 5 for Number of Levels.
Even though your augmented runs span only three levels (90, 110, and 125), you must specify the levels used in the original experiment as well, making a total of five levels. The Factor Transformation is set to Arrhenius Celsius by default.
5. Enter 25 for both Low Usage Condition and High Usage Condition.
6. Click Continue.
7. Enter 85, 90, 105, 110, and 125 for the Temp Level Values.
There are three levels from the original experiment (85, 105, and 125). The augmented design will have two new levels (90 and 110) and one of the levels from the first experiment (125). All levels must be listed.
8. Ensure that Weibull is selected as the Distribution Choice.
9. Under Prior Mean, enter the acceleration model parameter estimates from the Fit Life by X Estimates report, found in the Weibull Results report on the Statistics tab.
Figure 22.10 Parameter Estimates and Fitted Model from Weibull Results Report
‒ Enter -35.200 for Intercept.
‒ Enter 1.389 for Temp.
This is an estimate of the activation energy and is the coefficient of the inverse temperature, measured in degrees Kelvin, multiplied by Boltsmann’s constant.
‒ Enter 1.305 for scale.
For the Weibull distribution, JMP uses a parameterization that depends on a location parameter μ and scale parameter σ. In terms of the usual α and β parameterization, the scale parameter is σ = 1/β. See “JMP’s Weibull Parameterization”.
In the Accelerated Life Test Plan window, you could specify uncertainty about your prior means in the Prior Variance Matrix outline. In this example, do not make any changes to the Prior Variance Matrix outline. Your design is created assuming that the Prior Means are the true parameter values.
10. Under Diagnostic Choices, enter 100,000 for both boxes for Time range of interest. Leave the Probability of interest value set to 0.1.
11. Under Design Choices, enter 1500 for Length of test.
The test will be conducted over 1500 hours, which was the length of the original design.
12. Enter 250 for Number of units under test.
The previous experiment tested 150 units, and the augmented experiment will test an additional 100 units, for a total of 250.
Figure 22.11 shows the completed Accelerated Life Test Plan outline.
Figure 22.11 Completed Design Details Window
13. Click Continue.
14. To account for the units at each setting of Temp in the previous experiment, enter the following under Candidate Runs.
‒ For the Temperatures of 85, 105, and 125, enter Minimum Units of 50 for each.
15. From the ALT Plan red triangle menu, select ALT Optimality Criterion > Make Probability I-Optimal Design.
This tells JMP to use the Probability optimality criterion when creating the design in step 16. For details about this criterion, see “Make Probability I-Optimal Design”.
16. Click Make Design.
The optimal experimental design is shown, along with other results.
Figure 22.12 Optimal Design
The optimal design is computed based on the levels of the test runs, the minimum number of units under test, the total number of units to be tested (this is the information in the Candidate Runs outline), and other information that you specified earlier. The optimal design consists of testing the following number of units at each temperature level:
• 50 units at 85o. Since the previous experiment already tested 50 units at 85o, no additional units are needed.
• 89 units at 90o. The next experiment will test 89 units at 90o.
• 50 units at 105o. Since the previous experiment test 50 units at 105o, no additional units are needed.
• 0 units at 110o. The next experiment will not test any units at this level.
• 61 units at 125o. Since the previous experiment test 50 units at 125o, 11 additional units are needed.
Compare the Augmented Design to the Original Study
1. In the Distribution Profiler, enter 25 for Temperature and 100,000 for Time. The estimate of the fraction of the population that is failing is 0.00357, with a 95% confidence interval of 0.00093 to 0.01361. This interval is narrower than the one from the initial experiment, which ranges from 0.003575 to 0.02268 (Figure 22.9).
Figure 22.13 Distribution Profiler for Temp = 25 and Time = 100000
Accelerated Life Test Plan Window
The Accelerated Life Test Plan window is updated as you work through the design steps. The outlines that appear, separated by buttons that update the window, follow the flow in Figure 22.14.
Figure 22.14 Accelerated Life Test Plan Flow
This section describes the outlines in the Accelerated Life Test Plan flow.
Specify the Design Structure
Select the model structure and the type of inspection that describe your design.
Figure 22.15 Initial ALT Design Window
Select one of the first three options to indicate the number of acceleration factors in your design. If you have two factors, indicate whether you want to fit a main effects model or a model that contains an interaction term for the two factors.
Monitoring at Intervals
Assumes that units will be inspected for failures at intervals. Failure times are interval censored. Enter the number of inspections, the time of the first inspection, and a time between inspections. For inspection intervals that are irregular, you can change the inspection times later in the Design Choices outline.
Continuous Monitoring
Assumes that exact failure times are recorded. Failure times beyond the length of the test are right censored.
Specify Acceleration Factors
Specify details about the acceleration factor or factors.
Figure 22.16 ALT Specification Window
Factor Name
Enter a name for each acceleration factor.
Number of Levels
For each acceleration factor, enter the number of proposed levels that you want to include in the experiment.
Factor Transformation
Select a transformation for each acceleration factor. This transformation describes the life-stress relationship, which is the manner in which the life distribution changes across stress levels. The transformations are Arrhenius Celsius, Reciprocal, Log, Square Root, and Linear.
Low Usage Condition
For each acceleration factor, enter a lower bound for its value in typical usage conditions.
Note: The Low Usage Condition and High Usage Condition values can be identical.
High Usage Condition
For each acceleration factor, enter an upper bound for its value in typical usage conditions.
Specify Design Details
Specify the factor levels, details of the prior distribution, the time range and probability of interest, and the length of the test and number of units to be tested.
Figure 22.17 Distribution Details
Factors
Enter the settings for the acceleration factors.
Distribution Choice
Select a life distribution (Weibull or LogNormal) for each acceleration factor. For more information, see “Statistical Details”.
Prior Mean
Enter prior estimates of the acceleration model parameters. The prior estimates are hyperparameters in a Bayesian prior distribution. The Prior Mean values can be a best guess based on subject matter knowledge or they can be based on a previous study.
Prior Variance Matrix
Enter values for the variances and covariances of the prior distribution for the acceleration model parameters. These variances and covariances reflect uncertainty relative to the prior estimates of the acceleration model parameters. Large variances indicate greater uncertainty.
Ignore prior variance
Select this option to ignore the prior variances and covariances for the prior distribution. When these variances and covariances are ignored, the design is created by treating the values entered under Prior Mean as the true parameter values. This design is close to optimal if the prior mean parameters are close to the true values. However, this design is not robust to misspecification of the parameter estimates. If you are unsure about your prior estimates, use the Prior Variance Matrix to reflect your uncertainty.
Diagnostic Choices
Specify values used to construct Time I-Optimal and Probability I-Optimal designs. See “ALT Optimality Criterion”. Enter values for the following:
Time range of interest
Specify the time interval over which you want to estimate the fraction of the population that is failing. Enter the lower value in the left box and the upper value in the right box. If you are interested in a specific time point, enter that value in both boxes.
Probability of interest
Specify the failure fraction for which you want an estimate of time. For example, if you want to estimate the time at which 10% of the units fail, then enter 0.10.
Design Choices
Specify values relating to the length of the test, inspection intervals, and the number of units being tested. Enter values for the following:
Length of test
(Available only for Continuous Monitoring.) Length of time during which units will be on test. When you make the design table, record each unit’s failure time or whether it was right censored.
Inspection Times
(Available only for Monitoring at Intervals.) Times at which inspections are conducted. When you make the design table, these times are used to construct Start Time and End Time columns. The number of units failing in each interval is recorded.
Number of units under test
The number of units in the experiment.
‒ If you are designing an initial experiment, enter the number of units that you plan to test.
‒ If you are augmenting a previous experiment, enter the number of units tested in the previous experiment plus the number of units for the next experiment.
Review Balanced Design Diagnostics and Update Specifications
Three new outlines are added to the window. Two of these give results for a balanced design, enabling you to compare results for a balanced design to those for an optimal design:
Figure 22.18 Additional Outline Nodes
Candidate Runs
Enter the minimum and maximum number of units allowed at each setting of the acceleration factors. If you are augmenting a previous experiment, for each setting, include the number of units already run at that setting in the Minimum Units.
Parameter Variance for Balanced Design
Gives a matrix proportional to the covariance matrix for the estimates of the acceleration model parameters for the balanced design.
Denote the matrix of first partial derivatives of the model with respect to the parameters, θ, by X. Denote the error variance by σ2. Under general conditions, the least squares estimator of θ is asymptotically unbiased with asymptotic covariance matrix given as follows:
The Parameter Variance for Balanced Design outline gives (X‘X)-1, where derivatives are calculated numerically. The calculation assumes that the values specified as Prior Mean are the true parameter values.
For more information, see “Nonlinear Models” in the “Nonlinear Designs” chapter.
Distribution Profiler
Provides a way to explore failure probabilities, based on the balanced design, as you vary the acceleration factors and time. The probabilities are based on the assumption that the values specified as Prior Mean are the true parameter values.
Click Update Profiler to update the profiler if changes are made to Distribution Choice, Prior Mean, or Design Choices.
Create and Assess the Optimal Design
Click Make Design to create the optimal design and see results that address the quality of the design. By default, the optimal design is D-optimal. You can change the optimality criterion by selecting the red triangle option ALT Optimality Criterion.
Three new outlines are added to the window: Design, Parameter Variance for Optimal Design, and Optimality Criteria.
Figure 22.19 Design Outlines
Design
Gives the number of units to be tested at each design setting. For a single factor, the first column gives the levels of the factor. For two factors, the first two columns give the design settings.
N Units
Number of units to be tested at each design setting.
Expected Failures
Expected number of failures for the design setting. The expected number is computed using the prior model specification.
All Censored Probability
Probability that none of the units tested at the design setting will fail. The probability is computed using the prior model specification.
Note: The expected failures and censoring probabilities enable you to judge whether your prior specifications are reasonable.
Parameter Variance for Optimal Design
Gives a matrix proportional to the covariance matrix for the estimates of the acceleration model parameters for the optimal design. The calculation assumes that the values specified as Prior Mean are the true parameter values. For more information, see “Parameter Variance for Balanced Design”.
Note: Compare the values in the Parameter Variance for Optimal Design matrix to those in the Parameter Variance for Balanced Design matrix to determine the extent to which the optimal design reduces the variance of estimates.
Optimality Criteria
Values for three optimality criteria appear:
D Criterion
D-optimality of the design. See “Make D-Optimal Design”.
Quantile Criterion
Time I-optimality of the design. See “Make Time I-Optimal Design”.
Probability Criterion
Probability I-optimality of the design. See “Make Probability I-Optimal Design”.
Update the Design and Create Design Tables
You can view the design, create a data table that summarizes the design, or create a data table where you can record your experimental results.
Make Design
Updates the optimal design if any changes are made to the Distribution Choice, Prior Mean, Prior Variance Matrix, or Design Choices.
Make Test Plan
Creates a data table where each row corresponds to a distinct design setting. The table shows the acceleration factor design settings and the number of units to include at those design settings.
Make Table
Creates a table that you can use for recording your failure-time data.
‒ For Continuous Monitoring, the table contains a row for each unit to be tested and the design settings for that unit. Record each unit’s failure time in the Failure Time column or whether the observation was right censored in the Censored column.
‒ For Monitoring at Intervals, the table contains a row for each design setting and time interval combination. The time intervals are defined by the Start Time and End Time columns, which are based on the Inspection Times entered in the Design Choices outline. For each setting and time interval, record the number failing in the Number Failing column.
Platform Options
The Accelerated Life Test Plan red triangle menu contains the following options:
Simulate Responses
Adds simulated responses to the table when you click Make Table. The simulated responses are created by taking random draws from the chosen distribution at the parameter values specified under Prior Mean. If a simulated response exceeds the specified test length, the observation is censored at the test length value.
ALT Optimality Criterion
Gives three choices for design optimality:
Make D-Optimal Design
Creates a Bayesian D-optimal design if the number of Monte Carlo spheres is greater than 0. The optimality criterion is the expectation of the logarithm of the determinant of the information matrix with respect to the prior distribution. If the number of Monte Carlo spheres is 0, then the design is a locally D-optimal design. It follows that D-optimality focuses on precise estimates of the coefficients.
Make Time I-Optimal Design
Creates a design that minimizes the average prediction variance with respect to the prior distribution when predicting the time to failure over the Time range of interest at the failure probability specified in the Diagnostic Choices outline. See “Diagnostic Choices”.
Make Probability I-Optimal Design
Creates a design that minimizes the average prediction variance with respect to the prior distribution when predicting the failure probability over the Time range of interest specified in the Diagnostic Choices outline. See “Diagnostic Choices”.
Advanced Options
Gives the N Monte Carlo Spheres option, which enables you to set the number of nonzero radius values used in the integration. To find a nonlinear design that optimizes a given optimality criterion, JMP must minimize the integral of the log of the determinant of the Fisher information matrix with respect to the prior distribution of the parameters. Such an integral must be calculated numerically. For details about how the integration is performed, see “Nonlinear Design Options” in the “Nonlinear Designs” chapter.
Tip: By default N Monte Carlo Spheres is set to four. Higher values result in better numerical accuracy but with more computation time.
Save Script to Script Window
Creates the script for the design that you specified in the Accelerated Life Test Plan window and places it in an open script window.
Statistical Details
In the ALT design platform, you can select either a lognormal or Weibull failure distribution. The parameterizations for the probability density function (pdf) and cumulative distribution function (cdf) for each distribution are given in this section. For additional detail on the Weibull distribution, see the Life Distribution chapter in the Reliability and Survival Methods book.
Lognormal
Lognormal distributions are used commonly for failure times when the range of the data is several powers of 10. This distribution is often conceptualized as the multiplicative product of many small positive independently and identically distributed random variables. This distribution is appropriate when the logarithms of the data values appear normally distributed. The pdf is usually characterized by strong right-skewness.
The lognormal family is parameterized by a location parameter, μ, and a shape parameter, σ. The lognormal pdf and cdf are given as follows, where the logarithm is to the base e:
,The functions
and
are the pdf and cdf, respectively, for the standard normal distribution (N(0,1)).
Weibull
The Weibull distribution can be used to model failure time data with either an increasing or a decreasing hazard rate. It is used frequently in reliability analysis because of its tremendous flexibility in modeling many different types of data, based on the values of the shape parameter.
The Weibull pdf and cdf are commonly represented as follows:
where α is a scale parameter, and β is a shape parameter. The Weibull distribution reduces to an exponential distribution when β = 1.
JMP’s Weibull Parameterization
An alternative parameterization is commonly used in the literature and in JMP. In the JMP parameterization, σ is the scale parameter and μ is the location parameter. These are related to the α and β parameterization as follows:
and
With these parameters, the pdf and the cdf of the Weibull distribution are expressed as a log-transformed smallest extreme value distribution (SEV) using a location-scale parameterization with μ = log(α) and σ = 1/β:
where
and
are the pdf and cdf, respectively, for the standardized smallest extreme value (μ = 0, σ = 1) distribution.
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