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| The test is for the current (amperage) of a laser device. Failure is defined as a 10% increases in amperage. | | The test is for the current (i.e., amperage) of a laser device. Failure is defined as a 10% increases in amperage. |
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| * For the power regression model | | * The following picture shows the parameters of the linear regression model for each unit. |
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| [[Image:DA_linear_model.png|center|]] | | [[Image:DA_linear_model.png|center|]] |
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| * The predicted pseudo failure times. | | * The predicted pseudo failure times are shown next. |
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| [[Image:DA_linear_extrapolated.png|center|]] | | [[Image:DA_linear_extrapolated.png|center|]] |
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| * The fitted Weibull distribution | | * The next picture shows the parameters of the fitted Weibull distribution. |
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| [[Image:DA_linear_plot.png|center|550px|]] | | [[Image:DA_linear_plot.png|center|550px|]] |
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Degradation Data Analysis with a Linear Regression Model
|
This example compares the results for a degradation analysis with a linear regression model.
Reference Case
The data set is from Example 13.10 on page 338 in the book Statistical Methods for Reliability Data by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998.
Data
The test is for the current (i.e., amperage) of a laser device. Failure is defined as a 10% increases in amperage.
| Time
|
Unit Number
|
| 1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
15
|
| 250 |
0.47 |
0.71 |
0.71 |
0.36 |
0.27 |
0.36 |
0.36 |
0.46 |
0.51 |
0.41 |
0.44 |
0.4 |
0.3 |
0.44 |
0.51
|
| 500 |
0.93 |
1.22 |
1.17 |
0.62 |
0.61 |
1.39 |
0.92 |
1.07 |
0.93 |
1.79 |
1 |
0.8 |
0.74 |
0.7 |
0.83
|
| 750 |
2.11 |
1.9 |
1.73 |
1.36 |
1.11 |
1.95 |
1.21 |
1.42 |
1.57 |
2.38 |
1.57 |
1.4 |
1.52 |
1.05 |
1.29
|
| 1000 |
2.72 |
2.3 |
1.99 |
1.95 |
1.77 |
2.86 |
1.46 |
1.77 |
1.96 |
3 |
1.96 |
1.7 |
1.85 |
1.35 |
1.52
|
| 1250 |
3.51 |
2.87 |
2.53 |
2.3 |
2.06 |
3.46 |
1.93 |
2.11 |
2.59 |
3.84 |
2.51 |
3 |
2.39 |
1.8 |
1.91
|
| 1500 |
4.34 |
3.75 |
2.97 |
2.95 |
2.58 |
3.81 |
2.39 |
2.4 |
3.29 |
4.5 |
2.84 |
3.6 |
2.95 |
2.55 |
2.27
|
| 1750 |
4.91 |
4.42 |
3.3 |
3.39 |
2.99 |
4.53 |
2.68 |
2.78 |
3.61 |
5.25 |
3.47 |
4 |
3.51 |
2.83 |
2.78
|
| 2000 |
5.48 |
4.99 |
3.94 |
3.79 |
3.38 |
5.35 |
2.94 |
3.02 |
4.11 |
6.26 |
4.01 |
4.4 |
3.92 |
3.39 |
3.42
|
| 2250 |
5.99 |
5.51 |
4.16 |
4.11 |
4.05 |
5.92 |
3.42 |
3.29 |
4.6 |
7.05 |
4.51 |
4.8 |
5.03 |
3.72 |
3.78
|
| 2500 |
6.72 |
6.07 |
4.45 |
4.5 |
4.63 |
6.17 |
4.09 |
3.75 |
4.91 |
7.8 |
4.8 |
5.2 |
5.47 |
4.09 |
4.11
|
| 2750 |
7.13 |
6.64 |
4.89 |
4.72 |
5.24 |
7.7 |
4.58 |
4.16 |
5.34 |
8.32 |
5.2 |
5.5 |
5.84 |
4.83 |
4.38
|
| 3000 |
8 |
7.16 |
5.27 |
4.98 |
5.62 |
8.61 |
4.84 |
4.76 |
5.84 |
8.93 |
5.66 |
6 |
6.5 |
5.41 |
4.63
|
| 3250 |
8.92 |
7.78 |
5.69 |
5.28 |
6.04 |
9.15 |
5.11 |
5.16 |
6.4 |
9.55 |
6.2 |
6.2 |
6.94 |
5.76 |
5.38
|
| 3500 |
9.49 |
8.42 |
6.02 |
5.61 |
6.32 |
9.95 |
5.57 |
5.46 |
6.84 |
10.5 |
6.54 |
7 |
7.39 |
6.14 |
5.84
|
| 3750 |
9.87 |
8.91 |
6.45 |
5.95 |
7.1 |
10.49 |
6.11 |
5.81 |
7.2 |
11.3 |
6.96 |
7.4 |
7.85 |
6.51 |
6.16
|
| 4000 |
10.94 |
9.28 |
6.88 |
6.14 |
7.59 |
11.01 |
7.17 |
6.24 |
7.88 |
12.2 |
7.42 |
7.9 |
8.09 |
6.88 |
6.62
|
Result
The book provides a linear regression model, [math]\displaystyle{ y=b+at\,\! }[/math], but the parameters of the model are not provided. Only the predicted failure times are given. These are: 3702, 4194, 5847, 6172, 5301, 3592, 6051, 6538, 5110, 3306, 5326, 4995, 4721, 5689, and 6102. A Weibull distribution is used to fit the data; however, the model parameters are not provided. Only a probability plot is given.
Results in Weibull++
- The following picture shows the parameters of the linear regression model for each unit.
- The predicted pseudo failure times are shown next.
- The predicted failure times in Weibull++ are slightly different from the values given in the book. Since the book does not provide the model parameters, we couldn’t compare the Weibulll++ results with the results in the book.
- The next picture shows the parameters of the fitted Weibull distribution.
