# Degradation Data Analysis with a Power Regression Model

 Degradation Data Analysis with a Power Regression Model

This example validates the results for a degradation analysis with a power regression model in Weibull++ degradation folios.

Reference Case

The data set is from Example 8.1 on page 336 in the book Life Cycle Reliability Engineering by Dr. Guangbin Yang, John Wiley & Sons, 2007.

Data

The following table shows the percent transconductance degradation data taken at different times for five units of a MOS field-effect transistor. The failure criterion is defined as a degradation greater than or equal to 15%.

Time 1 2 3 4 5
100 1.05 0.58 0.86 0.6 0.62
200 1.4 0.9 1.25 0.6 0.64
300 1.75 1.2 1.45 0.6 1.25
400 2.1 1.75 1.75 0.9 1.3
500 2.1 2.01 1.75 0.9 0.95
600 2.8 2 2 1.2 1.25
700 2.8 2 2 1.5 1.55
800 2.8 2 2 1.5 1.9
900 3.2 2 2.3 1.5 1.25
1000 3.4 2.3 2.3 1.7 1.55
1200 3.8 2.6 2.6 2.1 1.5
1400 4.2 2.9 2.8 2.1 1.55
1600 4.2 3.2 3.15 1.8 1.9
1800 4.5 3.6 3.2 2.1 1.85
2000 4.9 3.8 3.2 2.1 2.2
2500 5.6 4.2 3.8 2.4 2.2
3000 5.9 4.4 3.8 2.7 2.5
3500 6.3 4.8 4 2.7 2.2
4000 6.6 5 4.2 3 2.8
4500 7 5.6 4.4 3 2.8
5000 7.8 5.9 4.6 3 2.8
6000 8.6 6.2 4.9 3.6 3.1
7000 9.1 6.8 5.2 3.6 3.1
8000 9.5 7.4 5.8 4.2 3.1
9000 10.5 7.7 6.1 4.6 3.7
10000 11.1 8.4 6.3 4.2 4.4
12000 12.2 8.9 7 4.8 3.7
14000 13 9.5 7.2 5.1 4.4
16000 14 10 7.6 4.8 4.4
18000 15 10.4 7.7 5.3 4.1
20000 16 10.9 8.1 5.8 4.1
25000 18.5 12.6 8.9 5.7 4.7
30000 20.3 13.2 9.5 6.2 4.7
35000 22.1 15.4 11.2 8 6.4
40000 24.2 18.1 14 10.9 9.4

Result

In the book, the following equation is used: $ln(y) = \beta_{1} + \beta_{2} ln(t)\,\!$. It in fact is a power equation $y = bt^{a}\,\!$ with $ln(b) = \beta_{1}\,\!$ and $a = \beta_{2}\,\!$. This degradation equation is used for each test unit to predict the pseudo failure time, and then a lognormal distribution is used to model the pseudo failure times. The results are:

• The parameters of the power regression model for each unit are:
• For unit 1: $\beta_{1}\,\!$ = -2.413 , $\beta_{2}\,\!$ = 0.524
• For unit 2: $\beta_{1}\,\!$ = -2.735 , $\beta_{2}\,\!$ = 0.525
• For unit 3: $\beta_{1}\,\!$ = -2.056 , $\beta_{2}\,\!$ = 0.424
• For unit 4: $\beta_{1}\,\!$ = -2.796 , $\beta_{2}\,\!$ = 0.465
• For unit 5: $\beta_{1}\,\!$ = -2.217 , $\beta_{2}\,\!$ = 0.383
• The predicted pseudo failure times are: 17,553; 31,816; 75,809; 138,229.
• The parameters of the fitted lognormal distribution are: Ln-Mean = 11.214, Ln-Std = 1.085.

Results in Weibull++

• The following picture shows the parameters of the power regression model for each unit.

• The predicted pseudo failure times are shown next.

• The next picture shows the parameters of the fitted lognormal distribution.