Design of Experiment (DOE) has been widely applied on improving product performance. It is an important part of Design for Six Sigma (DFSS). However, due to its limitation on data requirement and model assumptions, it is not popularly used in life test. In this presentation, a method combining regular DOE technique with proper life data analysis method is presented. This method can be used to identify factors that affect product life and also can be used to optimize design variables to improve product reliability.

1. Reliability DOE: The Proper
Analysis Approach for Life Data
可靠性DOE:寿命数据的正确分
析方法
Huairui Guo, Ph.D.
郭怀瑞,博士
©2011 ASQ & Presentation Guo
Presented live on May 11th, 2011
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3. Reliability DOE:
The Proper Analysis Approach
for Life Data
可靠性DOE:
寿命数据的正确分析方法
Huairui Guo, Ph.D.
郭怀瑞, 博士
Document Revision: 1.0.1
©1992-2011 ReliaSoft Corporation - ALL RIGHTS RESERVED

4. 2
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ReliaSoft 简介
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5. 3
常用词中英文对照表
ANOVA: 方差分析 Censored Data: 删失数据
DOE: 实验设计 MLE: 极大似然估计
Factor: 因子 Likelihood function: 似然函数
Level: 水平 Life Characteristic: 寿命特征量
2-Level Factorial Design: 两 Life-Factor Relationship: 寿命-
水平因子实验 因子关系
2-Level Fractional Factorial Life-Stress Relationship:寿命-
Design: 两水平部分因子实验 应力关系
Response: 反应 Likelihood Ratio Test: 似然比
Main Effect: 主效应 检验
Interaction Effect: 交互效应 Probability density function
(pdf): 概率密度函数
Coefficient: 系数
Mean Squares (MS): 均方差
Critical Value: 关键值
Mean Squares of Error: (MSE):
Outlier: 离群值
残方差

6. 4
Introduction Example
引例
Consider an experiment
to improve the reliability
of fluorescent lights.
Five factors A-E are
investigated in the
experiment. A 25-2 design
with factor generators
D=AC and E=BC was
conducted*.
Objective: To identify
significant factors and
adjust them to improve
life.
*Taguchi, 1987, p. 930.

7. 5
Introduction Example (cont’d)
引例(继续)
A B C D E Failure Time
-1 -1 -1 1 1 14~16 20+
-1 -1 1 -1 -1 18~20 20+
-1 1 -1 1 -1 8~10 10~12
-1 1 1 -1 1 18~20 20+
1 -1 -1 -1 1 20+ 20+
1 -1 1 1 -1 12~14 20+
1 1 -1 -1 -1 16~18 20+
1 1 1 1 1 12~14 14~16
Two replicates at each treatment.
Inspections were conducted every two days.
Results have interval data and suspensions.

8. 6
Traditional DOE Approach
传统的DOE方法
Assumes that the response (life) is normally
distributed.
Treats suspensions as failures.
Uses the middle point of the interval data as
the failure time.
Problem: The above assumptions and
adjustments are incorrect and do not apply to
life data.

9. 7
EDUCATION
Life Data Analysis
寿命数据分析简介

10. 8
Life Data Types
寿命数据类型
Complete Data
Censored Data
Right Censored (Suspended)
Interval Censored

11. 9
Complete and Censored Data
完全数据和删失数据
Complete Data
Censored Data
Right Censored
?
Interval Censored
?

12. 10
Complete Data: Example
完全数据例子
For example, if we tested five units and they all failed, we would
then have complete information as to the time of each failure in
the sample.

13. 11
Right Censored (Suspended) Data: Example
右删失数据 (终止): 例子
Imagine we tested five units and three failed. In this scenario,
our data set is composed of the times-to-failure of the three units
that failed and the running time of the other two units without
failure.
This is the most common censoring scheme and is used
extensively in the analysis of field data.

14. 12
Interval Censored Data: Example
区间删失数据: 例子
Imagine we are running a test on five units and inspecting them
every 100 hr. If a unit failed between inspections, we do not
know exactly when it failed, but rather that it failed between
inspections. This is also called “inspection data”.

15. 13
Censored Data Analysis Example
删失数据计算例子
100 pumps operated for three months.
One failed during the first month.
One failed during the second month.
Two failed during the third month.
What is the average time-to-failure?
1(1)  1(2)  2(3)
 2.25?
4
You can’t answer this question without
assuming a model for the data.

16. 14
Common Distributions Used in Reliability
可靠性中常用的分布
Weibull distribution pdf:

 1 t
t  
 
f (t )    e  
  
 
Lognormal distribution pdf:
2
1  ln( t )   
1  
2 

f (t )  e 
 t 2
Exponential distribution pdf:
 t 
1  
f (t )  e m
m

17. 15
Parameter Estimation:
Maximum Likelihood Estimation (MLE)
极大似然参数估计
Statistical (non-graphical) approach to parameter
estimation.
Given a data set, estimates the parameters that
maximize the probability that the data belong to that
distribution and that set of parameters.
Constructs likelihood function as product of densities,
assuming independence.
Uses calculus to find the values that maximize the likelihood
function.
Has elegant statistical properties when the sample size is
large.

18. 16
MLE Concept
极大似然参数估计概念
Which model is more likely if two values are observed:
-3 and 3?

19. 17
Likelihood Function: Complete Data
似然函数: 完全数据
If T is a continuous random variable with pdf:
f (T ;1 ,2 , , k )
where 1, 2, … , k are k unknown parameters that need to be estimated,
and we conduct an experiment and obtain N independent observations,
T1, T2, … , TN, then the likelihood function is given by:
N
L(1 , 2 , ,k T1 , T2 , , TN )   f (Ti ;1 , 2 , , k )
i 1
For a one-parameter distribution with a single parameter  and data of
10, 20, 30, the likelihood of the function would be:
L( 10,20,30)  f (10) f (20) f (30)

20. 18
Likelihood Function: Complete Data (cont‘d)
似然函数: 完全数据 (继续)
The logarithmic likelihood function is:
  ln L(1 , 2 , , k T1 , T2 , , TN )
N
  ln( f (Ti ;1 , 2 , , k ))
i 1
The maximum likelihood estimators (MLE) of 1, 2, … , k are obtained
by maximizing either L or .
By maximizing , which is much easier to work with than L, the
maximum likelihood estimators (MLE) of 1, 2, … , k are the
simultaneous solutions of k equations such that:

 0, i  1, 2,...k
i

21. 19
Likelihood Function: Right Censored Data
似然函数: 右删失数据
The likelihood function for M suspension times,
S1,S2,…,SM, is given by:
L(1 ,  2 ,...,  k | S1 , S 2 ,..., S M )
M
  1  F  S j ;1 ,  2 ,...,  k  
 
j 1
M
   R  S j ;1 ,  2 ,...,  k  
 
j 1

22. 20
Likelihood Function: Interval Data
似然函数: 区间数据
The likelihood function for P intervals, IL1 , IU1; IL2 , IU2;…;
ILP , IUP, is given by:
L(1 ,  2 ,..., k | I L1 , IU 1 , I L 2 , IU 2 ,..., I LP , IUP )
P
   F  IUl ;1 ,  2 ,...,  k   F  I Ll ;1 ,  2 ,...,  k  
 
l 1

23. 21
The Complete Likelihood Function
完整的似然函数
After completing the likelihood function for the different types of
data, the likelihood function (without the constant) can now be
expressed in its complete form:
N M
L   f Ti ;1 , 2 ,...,  k    R  S j ;1 , 2 ,...,  k  
 
i 1 j 1
P
  F  IUl ;1 , 2 ,...,  k   F  I Ll ;1 , 2 ,...,  k  
 
l 1

24. 22
MLE Parameter Estimation
极大似然解
The logarithmic likelihood function is:
  ln L(1 ,2 , , k T1 , , TN , S1 ,..., S N , IU 1 , I L1 ,..., IUP , I LP )
The maximum likelihood estimators (MLE) of 1, 2, … , k
are the simultaneous solutions of k equations such that:

 0, i  1, 2,...k
i

25. 23
EDUCATION
Combining Reliability and DOE
可靠性和DOE的结合
可靠性和DOE的结合

26. 24
Combining Reliability and DOE: Life-Factor Relationship
可靠性和DOE的结合: 寿命因子关系
可靠性和DOE的结合:
The graphic shows an example where life decreases when a factor is
changed from the low level to the high level.
It is seen that the pdf changes in scale only. The scale of the pdf is
compressed at the high level.
The failure mode remains the same. Only the time of occurrence
decreases at the high level.

27. 25
Life-Factor Relationship Simplify: Life Characteristic
简化寿命-因子关系:寿命特征量
Instead of considering the entire scale of the pdf, the life characteristic
can be chosen to investigate the effect of potential factors on life.
The life characteristic for the 3 commonly used distributions are:
Weibull:  Lognormal:  Exponential: m

28. 26
Life-Factor Relationship
寿命-因子关系
Using the life characteristic, the model to investigate
the effect of factors on life can be expressed as:
 '  0  1 x1  2 x2  ...  12 x1 x2  ...
where:
 '  ln( ) or '  or  '  ln( m)
xj : jth factor value
Note that a logarithmic transformation is applied to
the life characteristics of the Weibull and exponential
distributions.
This is because  and m can take only positive values.

29. 27
MLE Based on Life-Factor Relationship
基于寿命-因子关系的极大似然解
Life-Factor Relationship i'   0  1 xi1   2 xi 2  ...  12 xi1 xi 2  ...
N
Failure Time Data L f   f (Ti ; i,  )
i 1
M
Suspension Data LS   R( S j ; i,  )
j 1
P
Interval Data LI    F ( IUl ; i,  )  F ( I Ll ; i,  )
l 1
MLE
0 , 1 , 2 ,... and  for lognormal

30. 28
Testing Effect Significance: Likelihood Ratio Test
检验效应的显著性: 似然比检验
Life-factor relationship is
i'   0  1 xi1   2 xi 2  ...  12 xi1 xi 2  ...
Likelihood ratio test
L(effect k removed )
LR(effect k )  2 ln
L( full Model )
If LR(effect k )  1,
2
then effect k is significant or active.

31. 29
Fluorescent Lights R-DOE: Data and Design
荧光灯可靠性DOE: 数据和实验设计
The design is identical to traditional DOE.
Data entered includes suspensions and interval data.

32. 30
Fluorescent Lights R-DOE: Results
荧光灯可靠性DOE: 结果
Life is assumed to follow the Weibull distribution.

33. 31
Fluorescent Lights R-DOE: Analyzing Model Fit
荧光灯可靠性DOE: 模型拟合分析
Residual Probability Plot
When using the Weibull distribution for life, the residuals from the
life-factor relationship should follow the extreme value distribution
with a mean of zero.

34. 32
Fluorescent Lights R-DOE: Analyzing Model Fit
(cont’d)
荧光灯的可靠性DOE: 模型拟合分析(继续)
Plot of residuals against run order
There should be no outliers or pattern.

35. 33
Fluorescent Lights R-DOE: Interpreting the Results
荧光灯的可靠性DOE: 理解结果
From the results, factors A,B, D and E are significant at the risk Level of
0.10. Therefore, attention should be paid to these factors.
In order to improve the life, factor A and E should be set to the high
level; while factors B and D should be set to the low level.
MLE Information
Term Coefficient
A:A 0.1052
B:B -0.2256
C:C -0.0294
D:D -0.2477
E:E 0.1166

36. 34
EDUCATION
Traditional DOE Approach
传统的DOE方法
传统的DOE方法

37. 35
Traditional DOE Approach: Model
传统的DOE方法: 模型
Traditional DOE uses ANOVA models.
y  0  1 x1  2 x2  12 x1 x2  ...
ˆ
…coefficients are estimated using least squares.
A B C D E Failure Time
-1 -1 -1 1 1 14~16 20+
-1 -1 1 -1 -1 18~20 20+
-1 1 -1 1 -1 8~10 10~12
-1 1 1 -1 1 18~20 20+
1 -1 -1 -1 1 20+ 20+
1 -1 1 1 -1 12~14 20+
1 1 -1 -1 -1 16~18 20+
1 1 1 1 1 12~14 14~16
For the first observation:
y1  0  1  (1)   2  (1)  3  (1)   4  (1)  5  (1)
ˆ
…assuming that the interactions are absent.

38. 36
Traditional DOE Approach: Effect Significance
传统的DOE方法: 效应显著性检验
The ANOVA model is
yi   0  1 xi1   2 xi 2  ...   k xik  12 xi1 xi 2  ...
ˆ
F test
MS k
F0 (  k ) 
MS E
If F0 (  k )  f critical
then effect k is significant or active.

39. 37
Fluorescent Lights Example: Traditional DOE
Approach
荧光灯例子: 传统DOE分析方法
Suspensions are treated as failures.
Mid-points are used as failure times for interval data.
Life is assumed to follow the normal distribution.

40. 38
Fluorescent Lights Example: Traditional DOE
Approach Results
荧光灯例子:传统DOE分析结果
B and D come out to be significant using traditional DOE approach.
A, B, D and E were found to be significant using R-DOE.
Tradition DOE fails to identify A and E as an important factor at a
significance level of 0.1.

41. 39
Where to Get More Information
哪里可以找到更多的信息
1. http://www.itl.nist.gov/div898/handbook/
2. www.Weibull.com

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