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Reliability DOE: 
                    Reliability DOE:
                  The Proper Analysis 
                  The Proper Analysis
                  pp
                 Approach for Life Data
                    Harry (Huairui) Guo, Ph.D.
                            ©2011 ASQ & Presentation Harry
                            Presented live on Apr 12th, 2012




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Reliability DOE:
The Proper Analysis Approach
       p       y      pp
                 for Life Data

         Harry (Huairui) Guo, Ph.D.


                                            Document Revision: 1.0.1

             ©1992-2011 ReliaSoft Corporation - ALL RIGHTS RESERVED
2



                                                                                         Who is ReliaSoft
    ReliaSoft is a world leading software company. We provide training, consulting
    and software tools for reliability and quality engineers around the world.




                          MSMT R li bilit Foundations
                               Reliability F  d ti          Advanced System
                                                            Ad        dS t
                                                            Reliability/Maintainability Analysis


                          Effective FMEA Series             Application of Fault Trees in
                                                            Reliability, Maintainability and Risk
                                                            Analysis
                          FRACAS Principles and
                          Applications
                                                            Simulation Modeling for Reliability and
                                                            Risk Analysis
                          RCM Principles and Applications

                                                            Reliability d M i
                                                            R li bili and Maintainability Analysis
                                                                                 i bili A l i
                          Standards Based Reliability       for Repairable Systems
                          Prediction

                                                            Fundamentals of Design for Reliability
                          Application of Reliability Growth (DFR)
                          Models in Developmental Testing
                          and Fielded Systems
                                                            Introduction to Reliability Concepts,
                                                            Principles and Applications
                          Advanced Accelerated Life
                          Testing Analysis
                                                            DOE: Experiment Design and Analysis



Have trained more than 15,000 engineers from about 3,000 companies and government agencies.
3



                             I t d ti Example
                             Introduction E l
     Consider an experiment
     to improve the reliability
     of fluorescent lights.
                     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.
4



               Introduction E
               I t d ti Example (cont’d)
                              l (   t’d)

    A     B      C      D      E        Failure Time
    -1
     1    -1
           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
                         1      1    18~20
                                     18 20        20+
     1    -1     -1     -1      1      20+        20+
     1    -1      1     1      -1    12~14        20+
     1     1     -1
                  1     -1
                         1     -1
                                1    16~18
                                     16 18        20+
     1     1      1      1      1    12~14       14~16


    Two replicates at each treatment.
    Inspections were conducted every two days
                                          days.
    Results have interval data and suspensions.
5



                T diti    l     A      h
                Traditional DOE Approach

Assumes that the response (life) is normally
d st buted
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.
6

                 EDUCATION




    Life Data Analysis
                  y
7



                                  Lif Data Types
                                  Life D t T

    Complete Data
    Censored Data
     Right Censored (Suspended)
     Interval Censored
8



               C   l t    dC       dD t
               Complete and Censored Data

Complete Data


Censored Data
    Right Censored

                                      ?

    Interval Censored

                            ?
9



                               Complete D t E
                               C   l t Data: Example
                                                  l

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.
    sample
10



     Right C
     Ri ht Censored (Suspended) Data: Example
                  d (S     d d) D t E      l

 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
                                          g
     extensively in the analysis of field data.
11



                   Interval Censored Data: Example
                   I t    lC       dD t E       l

 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
                         failed,
 inspections. This is also called “inspection data”.
12



             C
             Censored D t A l i E
                    d Data Analysis Example
                                         l
 100 pumps operated f th
                t d for three months.
                                 th
     One failed during the first month.
     One failed during the second month
                                   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.
       i       d l f th d t
13



C
Common Di t ib ti    U d i R li bilit
       Distributions Used in Reliability

 Weibull distribution pdf:
                                                    β
                                   β −1      ⎛t⎞
                         β⎛t⎞               −⎜ ⎟
                                             ⎜η ⎟
                  f (t) = ⎜ ⎟ e
                          ⎜η ⎟
                                             ⎝ ⎠
                         η⎝ ⎠

 Lognormal distribution pdf:
                                         1 ⎛ ln( t ) − μ ⎞
                                                             2

                              1         − ⎜              ⎟
                                         2⎝ σ
                  f (t ) =        e                      ⎠

                           σ t 2π
 Exponential distribution pdf:
                                ⎛ t ⎞
                          1    −⎜ ⎟
                  f (t ) = e    ⎝m⎠
                          m
14

                          Parameter Estimation:
            Maximum Likelihood Estimation (MLE)

 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
               independence.
     Uses calculus to find the values that maximize the likelihood
     function.
     function
     Has elegant statistical properties when the sample size is
        g
     large.
15



                                         MLE Concept
                                             C     t
 Which model is more likely if two values are observed:
     -3 and 3?
16



                   Likelihood F
                   Lik lih d Function: Complete Data
                                 ti    C   l t D t

 If T is a continuous random variable with pdf:

                               f (T ;θ1 , θ 2 ,K ,θ k )
where θ1, θ2, … , θk 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(θ1 , θ 2 ,K , θ k T1 , T2 ,K , TN ) = ∏ f (Ti ;θ1 , θ 2 ,K , θ k )
                                             i =1

 For a one-parameter distribution with a single parameter θ and data of
 10, 20, 30, the likelihood of the function would be:


              L (θ 10,20,30) = f (10) f (20) f (30)
17



Lik lih d F
Likelihood Function: C
               ti    Complete D t (
                         l t Data (cont‘d)
                                      t‘d)
 The logarithmic likelihood function is:

               Λ = ln L(θ1 , θ 2 ,K , θ k T1 , T2 ,K , TN )
                      N
                   = ∑ ln( f (Ti ;θ1 , θ 2 ,K , θ k ))
                      i =1
 The maximum likelihood estimators (MLE) of θ1, θ2, … , θk are obtained
 by maximizing either L or Λ.
 By maximizing Λ, which is much easier to work with than L, the
 maximum likelihood estimators (MLE) of θ1, θ2, … , θk are the
 simultaneous solutions of k equations such that:


                      ∂Λ
                           = 0, i = 1, 2,...k
                      ∂θ i
18



      Likelihood Function: Right Censored Data
      Lik lih d F    ti    Ri ht C      dD t

 The likelihood function for M suspension times,
 S1,S2,…,SM, is given by:


             L (θ1 , θ 2 ,..., θ k | S1 , S 2 ,..., S M )

             = ∏ ⎡1 − F ( S j ; θ1 , θ 2 ,..., θ k ) ⎤
                 M

                 ⎣
                 j =1
                                                     ⎦

             = ∏ ⎡ R ( S j ; θ1 , θ 2 ,..., θ k ) ⎤
                 M

                 ⎣
                 j =1
                                                  ⎦
19



                         Likelihood F
                         Lik lih d Function: I t
                                       ti    Interval Data
                                                    lD t

 The likelihood function for P intervals, IL1 , IU1; IL2 , IU2;…;
 ILP , IUP, is given by:


     L (θ1 , θ 2 ,..., θ k | I L1 , I U 1 , I L 2 , I U 2 ,..., I LP , I UP )
          P
     = ∏ ⎣ F ( I Ul ; θ1 , θ 2 ,..., θ k ) − F ( I Ll ; θ1 , θ 2 ,..., θ k ) ⎦
         ⎡                                                                   ⎤
         l =1
20



            Th C
            The Complete Lik lih d Function
                    l t Likelihood F   ti
 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:


     L = ∏ f (Ti ; θ1 , θ 2 ,..., θ k ) ⋅∏ ⎡ R ( S j ; θ1 , θ 2 ,..., θ k ) ⎤
            N                                M


           i =1
                                           ⎣ j =1
                                                                            ⎦
           P
        ⋅∏ ⎡ F ( IUl ; θ1 , θ 2 ,..., θ k ) − F ( I Ll ; θ1 , θ 2 ,..., θ k ) ⎤
           ⎣                                                                  ⎦
          l =1
21



                                      MLE Parameter Estimation
                                          P     t E ti ti

   The logarithmic likelihood function is:


Λ = ln L (θ1 , θ 2 ,K , θ k T1 ,K , TN , S1 ,..., S N , IU 1 , I L1 ,..., IUP , I LP )

   The maximum likelihood estimators (MLE) of θ1, θ2, … , θk
   are the simultaneous solutions of k equations such that:
                                        q

                         ∂Λ
                              = 0 i = 1, 2,...k
                                0,    12
                         ∂θ i
22

                             EDUCATION




     Combining Reliability and DOE
             g           y
23



Combining Reliability and DOE: Life-Factor Relationship




      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.
24



     Life-Factor Relationship Simplify: Life Characteristic




     Instead of considering the entire scale of the pdf, the life characteristic
                            g                       p ,
     can be chosen to investigate the effect of potential factors on life.
     The life characteristic for the 3 commonly used distributions are:
         Weibull: η      Lognormal: μ          Exponential: m
25



                                  Lif F t Relationship
                                  Life-Factor R l ti hi
     Using the life characteristic, the model to investigate
     the effect of factors on life can be expressed as:
            μ ' = β 0 + β1 x1 + β 2 x2 + ... + β12 x1 x2 + ...

                  where:
                   μ ' = ln(η )   or     μ' = μ     or μ ' = ln(m)
                   xj :   jth factor value

     Note that a logarithmic transformation is applied to
                   g                              pp
     the life characteristics of the Weibull and exponential
     distributions.
       This is because η and m can take only positive values.
26



     MLE B
         Based on Lif F t Relationship
             d    Life-Factor R l ti hi

 Life-Factor Relationship                      μi' = β0 + β1xi1 + β2 xi 2 + ...+ β12xi1xi 2 + ...
                                          N
 Failure Time Data L f = ∏ f (Ti ; μi′, σ )
                                        i =1

                                     M
 Suspension Data LS = ∏ R ( S j ; μi′, σ )
                                     j =1

                                    P
 Interval Data
 I t    lD t            LI = ∏ [ F ( IUl ; μi′, σ ) − F ( I Ll ; μi′, σ ) ]
                                   l =1


                                                   MLE


             β 0 , β1 , β 2 ,...    and σ for lognormal
27



T ti
Testing Eff t Si ifi
        Effect Significance: Lik lih d R ti T t
                             Likelihood Ratio Test

   Life-factor relationship is

             μ = β0 + β1xi1 + β2 xi 2 + ...+ β12xi1xi 2 + ...
              '
              i



   Likelihood ratio test
                                         L(effect k removed )
                  LR (effect k ) = −2 ln
                                            L( full Model )


  If LR (effect k ) > χ1,α
                        2



  then effect k is significant or active.
                                  active
28



     Fluorescent Lights R-DOE: D t and D i
     Fl        t Li ht R DOE Data d Design

 The design is identical to traditional DOE.
 Data entered includes suspensions and interval data.




                            FluorescentLights.rdoe
29



          Fl
          Fluorescent Lights R-DOE: R
                    t Li ht R DOE Results
                                      lt
 Life is assumed to follow the Weibull distribution.
30



Fluorescent Lights R-DOE: Analyzing Model Fit
Fl        t Li ht R DOE A l i       M d l

 Residual Probability Plot
     When using the Weibull distribution for life, the residuals from the
     life-factor
     life factor relationship should follow the extreme value distribution
     with a mean of zero.
31

     Fluorescent Lights R-DOE: Analyzing Model Fit
                   g               y g
                                          (cont’d)

 Plot of residuals against run order
     There should be no outliers or pattern.
32



Fl
Fluorescent Li ht R DOE I t
          t Lights R-DOE: Interpreting th R
                                  ti   the Results
                                               lt

 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
                          life,
 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
                                            0 0294
                            D:D            -0.2477
                            E:E             0.1166
33

                        EDUCATION




     Traditional DOE Approach
                      pp
34



                         Traditional DOE Approach: M d l
                         T diti    l     A      h Model

 Traditional DOE uses ANOVA models.

                y = β 0 + β1 x1 + β 2 x2 + β12 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
                                      1      -1
                                              1     16~18       20+
           1        1        1        1      1      12~14      14~16

 For the first observation:
       y1 = β0 + β1 × (−1) + β2 × (−1) + β3 × (−1) + β4 × (+1) + β5 × (+1)
       ˆ
 …assuming that the interactions are absent
  assuming                           absent.
35



 Traditional DOE Approach: Effect Significance
 T diti    l     A      h Eff t Si ifi

 The ANOVA model is
             yi = β 0 + β1 xi1 + β 2 xi 2 + ... + β k xik + β12 xi1 xi 2 + ...
             ˆ


 F test
                                     MS βk
                      F0 ( β k ) =
                                     MS E


If    F0 ( β k ) > f critical

then effect k is significant or active.
                                active
36

     Fluorescent Lights Example: Traditional DOE
                   g
                                       Approach
 Suspensions are treated as failures.
 Mid-points are used as failure times for interval data.
 Life is assumed to follow the normal distribution.
                                       distribution
37

          Fluorescent Lights Example: Traditional DOE
                        g        p
                                   Approach Results




     B and D come out to b significant using traditional DOE approach.
          d            t t be i ifi     t i t diti       l               h
     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.
38



                             Where to Get More Information
                             Wh    t G tM      I f    ti
1.
1    http://www.itl.nist.gov/div898/handbook/
     http://www itl nist gov/div898/handbook/
2.   www.Weibull.com




 3. http://www.reliawiki.org/index.php/ReliaSoft_Books
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                                                    info.

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                      R li S ft  b

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39

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The proper analysis approach for life data

  • 1. Reliability DOE:  Reliability DOE: The Proper Analysis  The Proper Analysis pp Approach for Life Data Harry (Huairui) Guo, Ph.D. ©2011 ASQ & Presentation Harry Presented live on Apr 12th, 2012 http://reliabilitycalendar.org/The_Re liability_Calendar/Webinars_ liability Calendar/Webinars ‐ _English/Webinars_‐_English.html
  • 2. ASQ Reliability Division  ASQ Reliability Division English Webinar Series English Webinar Series One of the monthly webinars  One of the monthly webinars on topics of interest to  reliability engineers. To view recorded webinar (available to ASQ Reliability  Division members only) visit asq.org/reliability ) / To sign up for the free and available to anyone live  webinars visit reliabilitycalendar.org and select English  Webinars to find links to register for upcoming events http://reliabilitycalendar.org/The_Re liability_Calendar/Webinars_ liability Calendar/Webinars ‐ _English/Webinars_‐_English.html
  • 3. Reliability DOE: The Proper Analysis Approach p y pp for Life Data Harry (Huairui) Guo, Ph.D. Document Revision: 1.0.1 ©1992-2011 ReliaSoft Corporation - ALL RIGHTS RESERVED
  • 4. 2 Who is ReliaSoft ReliaSoft is a world leading software company. We provide training, consulting and software tools for reliability and quality engineers around the world. MSMT R li bilit Foundations Reliability F d ti Advanced System Ad dS t Reliability/Maintainability Analysis Effective FMEA Series Application of Fault Trees in Reliability, Maintainability and Risk Analysis FRACAS Principles and Applications Simulation Modeling for Reliability and Risk Analysis RCM Principles and Applications Reliability d M i R li bili and Maintainability Analysis i bili A l i Standards Based Reliability for Repairable Systems Prediction Fundamentals of Design for Reliability Application of Reliability Growth (DFR) Models in Developmental Testing and Fielded Systems Introduction to Reliability Concepts, Principles and Applications Advanced Accelerated Life Testing Analysis DOE: Experiment Design and Analysis Have trained more than 15,000 engineers from about 3,000 companies and government agencies.
  • 5. 3 I t d ti Example Introduction E l Consider an experiment to improve the reliability of fluorescent lights. 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.
  • 6. 4 Introduction E I t d ti Example (cont’d) l ( t’d) A B C D E Failure Time -1 1 -1 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 1 1 18~20 18 20 20+ 1 -1 -1 -1 1 20+ 20+ 1 -1 1 1 -1 12~14 20+ 1 1 -1 1 -1 1 -1 1 16~18 16 18 20+ 1 1 1 1 1 12~14 14~16 Two replicates at each treatment. Inspections were conducted every two days days. Results have interval data and suspensions.
  • 7. 5 T diti l A h Traditional DOE Approach Assumes that the response (life) is normally d st buted 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.
  • 8. 6 EDUCATION Life Data Analysis y
  • 9. 7 Lif Data Types Life D t T Complete Data Censored Data Right Censored (Suspended) Interval Censored
  • 10. 8 C l t dC dD t Complete and Censored Data Complete Data Censored Data Right Censored ? Interval Censored ?
  • 11. 9 Complete D t E C l t Data: Example l 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. sample
  • 12. 10 Right C Ri ht Censored (Suspended) Data: Example d (S d d) D t E l 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 g extensively in the analysis of field data.
  • 13. 11 Interval Censored Data: Example I t lC dD t E l 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 failed, inspections. This is also called “inspection data”.
  • 14. 12 C Censored D t A l i E d Data Analysis Example l 100 pumps operated f th t d for three months. th One failed during the first month. One failed during the second month 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. i d l f th d t
  • 15. 13 C Common Di t ib ti U d i R li bilit Distributions Used in Reliability Weibull distribution pdf: β β −1 ⎛t⎞ β⎛t⎞ −⎜ ⎟ ⎜η ⎟ f (t) = ⎜ ⎟ e ⎜η ⎟ ⎝ ⎠ η⎝ ⎠ Lognormal distribution pdf: 1 ⎛ ln( t ) − μ ⎞ 2 1 − ⎜ ⎟ 2⎝ σ f (t ) = e ⎠ σ t 2π Exponential distribution pdf: ⎛ t ⎞ 1 −⎜ ⎟ f (t ) = e ⎝m⎠ m
  • 16. 14 Parameter Estimation: Maximum Likelihood Estimation (MLE) 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 independence. Uses calculus to find the values that maximize the likelihood function. function Has elegant statistical properties when the sample size is g large.
  • 17. 15 MLE Concept C t Which model is more likely if two values are observed: -3 and 3?
  • 18. 16 Likelihood F Lik lih d Function: Complete Data ti C l t D t If T is a continuous random variable with pdf: f (T ;θ1 , θ 2 ,K ,θ k ) where θ1, θ2, … , θk 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(θ1 , θ 2 ,K , θ k T1 , T2 ,K , TN ) = ∏ f (Ti ;θ1 , θ 2 ,K , θ k ) i =1 For a one-parameter distribution with a single parameter θ and data of 10, 20, 30, the likelihood of the function would be: L (θ 10,20,30) = f (10) f (20) f (30)
  • 19. 17 Lik lih d F Likelihood Function: C ti Complete D t ( l t Data (cont‘d) t‘d) The logarithmic likelihood function is: Λ = ln L(θ1 , θ 2 ,K , θ k T1 , T2 ,K , TN ) N = ∑ ln( f (Ti ;θ1 , θ 2 ,K , θ k )) i =1 The maximum likelihood estimators (MLE) of θ1, θ2, … , θk are obtained by maximizing either L or Λ. By maximizing Λ, which is much easier to work with than L, the maximum likelihood estimators (MLE) of θ1, θ2, … , θk are the simultaneous solutions of k equations such that: ∂Λ = 0, i = 1, 2,...k ∂θ i
  • 20. 18 Likelihood Function: Right Censored Data Lik lih d F ti Ri ht C dD t The likelihood function for M suspension times, S1,S2,…,SM, is given by: L (θ1 , θ 2 ,..., θ k | S1 , S 2 ,..., S M ) = ∏ ⎡1 − F ( S j ; θ1 , θ 2 ,..., θ k ) ⎤ M ⎣ j =1 ⎦ = ∏ ⎡ R ( S j ; θ1 , θ 2 ,..., θ k ) ⎤ M ⎣ j =1 ⎦
  • 21. 19 Likelihood F Lik lih d Function: I t ti Interval Data lD t The likelihood function for P intervals, IL1 , IU1; IL2 , IU2;…; ILP , IUP, is given by: L (θ1 , θ 2 ,..., θ k | I L1 , I U 1 , I L 2 , I U 2 ,..., I LP , I UP ) P = ∏ ⎣ F ( I Ul ; θ1 , θ 2 ,..., θ k ) − F ( I Ll ; θ1 , θ 2 ,..., θ k ) ⎦ ⎡ ⎤ l =1
  • 22. 20 Th C The Complete Lik lih d Function l t Likelihood F ti 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: L = ∏ f (Ti ; θ1 , θ 2 ,..., θ k ) ⋅∏ ⎡ R ( S j ; θ1 , θ 2 ,..., θ k ) ⎤ N M i =1 ⎣ j =1 ⎦ P ⋅∏ ⎡ F ( IUl ; θ1 , θ 2 ,..., θ k ) − F ( I Ll ; θ1 , θ 2 ,..., θ k ) ⎤ ⎣ ⎦ l =1
  • 23. 21 MLE Parameter Estimation P t E ti ti The logarithmic likelihood function is: Λ = ln L (θ1 , θ 2 ,K , θ k T1 ,K , TN , S1 ,..., S N , IU 1 , I L1 ,..., IUP , I LP ) The maximum likelihood estimators (MLE) of θ1, θ2, … , θk are the simultaneous solutions of k equations such that: q ∂Λ = 0 i = 1, 2,...k 0, 12 ∂θ i
  • 24. 22 EDUCATION Combining Reliability and DOE g y
  • 25. 23 Combining Reliability and DOE: Life-Factor Relationship 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.
  • 26. 24 Life-Factor Relationship Simplify: Life Characteristic Instead of considering the entire scale of the pdf, the life characteristic g p , can be chosen to investigate the effect of potential factors on life. The life characteristic for the 3 commonly used distributions are: Weibull: η Lognormal: μ Exponential: m
  • 27. 25 Lif F t Relationship Life-Factor R l ti hi Using the life characteristic, the model to investigate the effect of factors on life can be expressed as: μ ' = β 0 + β1 x1 + β 2 x2 + ... + β12 x1 x2 + ... where: μ ' = ln(η ) or μ' = μ or μ ' = ln(m) xj : jth factor value Note that a logarithmic transformation is applied to g pp the life characteristics of the Weibull and exponential distributions. This is because η and m can take only positive values.
  • 28. 26 MLE B Based on Lif F t Relationship d Life-Factor R l ti hi Life-Factor Relationship μi' = β0 + β1xi1 + β2 xi 2 + ...+ β12xi1xi 2 + ... N Failure Time Data L f = ∏ f (Ti ; μi′, σ ) i =1 M Suspension Data LS = ∏ R ( S j ; μi′, σ ) j =1 P Interval Data I t lD t LI = ∏ [ F ( IUl ; μi′, σ ) − F ( I Ll ; μi′, σ ) ] l =1 MLE β 0 , β1 , β 2 ,... and σ for lognormal
  • 29. 27 T ti Testing Eff t Si ifi Effect Significance: Lik lih d R ti T t Likelihood Ratio Test Life-factor relationship is μ = β0 + β1xi1 + β2 xi 2 + ...+ β12xi1xi 2 + ... ' i Likelihood ratio test L(effect k removed ) LR (effect k ) = −2 ln L( full Model ) If LR (effect k ) > χ1,α 2 then effect k is significant or active. active
  • 30. 28 Fluorescent Lights R-DOE: D t and D i Fl t Li ht R DOE Data d Design The design is identical to traditional DOE. Data entered includes suspensions and interval data. FluorescentLights.rdoe
  • 31. 29 Fl Fluorescent Lights R-DOE: R t Li ht R DOE Results lt Life is assumed to follow the Weibull distribution.
  • 32. 30 Fluorescent Lights R-DOE: Analyzing Model Fit Fl t Li ht R DOE A l i M d l Residual Probability Plot When using the Weibull distribution for life, the residuals from the life-factor life factor relationship should follow the extreme value distribution with a mean of zero.
  • 33. 31 Fluorescent Lights R-DOE: Analyzing Model Fit g y g (cont’d) Plot of residuals against run order There should be no outliers or pattern.
  • 34. 32 Fl Fluorescent Li ht R DOE I t t Lights R-DOE: Interpreting th R ti the Results lt 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 life, 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 0 0294 D:D -0.2477 E:E 0.1166
  • 35. 33 EDUCATION Traditional DOE Approach pp
  • 36. 34 Traditional DOE Approach: M d l T diti l A h Model Traditional DOE uses ANOVA models. y = β 0 + β1 x1 + β 2 x2 + β12 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 1 -1 1 16~18 20+ 1 1 1 1 1 12~14 14~16 For the first observation: y1 = β0 + β1 × (−1) + β2 × (−1) + β3 × (−1) + β4 × (+1) + β5 × (+1) ˆ …assuming that the interactions are absent assuming absent.
  • 37. 35 Traditional DOE Approach: Effect Significance T diti l A h Eff t Si ifi The ANOVA model is yi = β 0 + β1 xi1 + β 2 xi 2 + ... + β k xik + β12 xi1 xi 2 + ... ˆ F test MS βk F0 ( β k ) = MS E If F0 ( β k ) > f critical then effect k is significant or active. active
  • 38. 36 Fluorescent Lights Example: Traditional DOE g Approach Suspensions are treated as failures. Mid-points are used as failure times for interval data. Life is assumed to follow the normal distribution. distribution
  • 39. 37 Fluorescent Lights Example: Traditional DOE g p Approach Results B and D come out to b significant using traditional DOE approach. d t t be i ifi t i t diti l h 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.
  • 40. 38 Where to Get More Information Wh t G tM I f ti 1. 1 http://www.itl.nist.gov/div898/handbook/ http://www itl nist gov/div898/handbook/ 2. www.Weibull.com 3. http://www.reliawiki.org/index.php/ReliaSoft_Books
  • 41. Worldwide Headquarters (North America) ReliaSoft Corporation 1450 S. Eastside Loop Tucson, AZ 85710-6703, USA Phone: (+1) 520-886-0410 (USA/Canada Toll Free: 1-888-886-0410) Fax: (+1) 520-886-0399 E-mail: Sales@ReliaSoft.com Web site: www.ReliaSoft.com Regional Centers See Web sites for complete contact info info. Europe and Middle East ReliaSoft Corp. Poland Sp. z o.o. Warsaw, Poland Web site: www ReliaSoft eu www.ReliaSoft.eu Asia Pacific ReliaSoft Asia Pte Ltd Singapore Web site: www ReliaSoftAsia com www.ReliaSoftAsia.com South America ReliaSoft Brasil São Paulo, Brasil Web it W b site: www.ReliaSoft.com.br R li S ft b India ReliaSoft India Private Limited Chennai, India Web site: www.ReliaSoftIndia.com 39