【正文】 Answers, cont. The probability of surviving 45,000 hours is about 90%. Table of Survival Probabilities % Normal CI Time Probability Lower Upper Reliability Nonparametric Analysis 47 Nonparametric Reliability Analysis When you can’t find a distribution, that fits your reliability data or you don’t want to make assumptions about the distribution you can use ―Nonparametric‖ or ―Distributionfree‖ reliability Analysis. 48 Minitab Followalong: Nonparametric reliability analysis In June 20xx a phone manufacturer produced 7,000 phones using two amplifier variants (―Silver‖ and ―Black‖). The assembled units have an individual ID numbers. From June 20xx to April 20xx data on returned phones have been collected on the life time of the amplifier. The results of the analysis are to be transferred to other products. ? Within what time do 1% of the two variants fail? Data: C:/BoosterData/ 49 Minitab Followalong: Nonparametric reliability analysis, cont. 1. Try to find a distribution fitting the data StatReliability/SurvivalDistribution ID PlotRight Cens… 50 Minitab Followalong: Nonparametric reliability analysis, cont. You can also try the other distributions but there seems to be no distribution fitting the data. We have to continue the analysis nonparametric or ―distributionfree‖ 3 3 0 6 , 3 3 0 6N o rm a l3 3 0 6 , 3 3 0 6Ex p o n e n t i a l3 3 0 6 , 3 3 0 6L o g n o rm a l b a s e e3 3 0 6 , 3 3 0 6W e i b u l lA n d e r s o n D a r l i n g ( a d j )Bl a c kSi l v e r9995908070605040302010 5 3 2 1100001000100101Weibu llPercent99958070605040302010 5 1100000010000010000100010010Log nor m al b ase ePercent9998979590807060503010700006000050000400003000020xx0100000Ex pon ent ialPercent99958070605040302010 5 1150010005000Norm alPercentF ourw ay P robabilit y P lot f or Lif e (day s )M L E s t i m a t e s C e n s o r i n g C o l u m n i n C e n s o r51 Minitab Followalong: Nonparametric reliability analysis, cont. 2. Get an overview plot (tick ―Nonparametric analysis‖) toggle 52 Minitab Followalong: Nonparametric reliability analysis, cont. Since we are not using assumptions about distributions we now don’t get round curves B l a ck S i l ve r 30020xx0001 . 0 00 . 9 90 . 9 8Ka 。 Select ―Lower‖ for Confidence Intervals. Then click ―OK‖ ―OK‖ 33 Minitab Follow along: Parametric Reliability Analysis, cont. Use your notebook to follow along. 34 Minitab Follow along: Parametric Reliability Analysis, cont. Note the Probability Plot amp。 wear out Hazard Function is then the “Bathtub” Curve. Reliability Exponential Model 7 Breakin, infant mortality, manufacturing defects, insufficient burnin Useful life Wearout Degradation Fatigue Corrosion Hazard Rate time Components / units may have all or some or a bination of the three “failure modes” Common hazard function: The “Bathtub” Curve 8 ?Exponential ?Normal ?Lognormal ?Weibull R(t) = exp [ (t/?)**?] ?Extreme Value “the weakest link” of identical parts ?Logistic ?Loglogistic ?Nonparametric In Minitab: We will find which of these models fits the data best and estimate percentiles. Other Reliability Models 9 Weibull R(t) = exp [ (t/?)**?] ?Shape (?) amp。 C Some of the units drop out of the test for unrelated reasons. In the Minitab worksheet, you use a column of censoring indicators to designate which times are actual failures (1) and which are censored units removed from the test before failure (0). Note: For Minitab analysis you need to specify the column containing the censoring information in all the dialogue windows at the ―censor‖ button. Data: C:/BoosterData/ Time: 30 min. 36 Exercise: Answers Find an appropriate distribution ? The distribution ID plots suggest to either use the lognormal or the loglogistic distribution. We selected the loglogistic for continued analysis. ? Note: You must specify the Censor columns L o g n o r m a l b a s e e6 7 . 2 2 , 1 6 . 5 0L o g n o r m a l b a s e 1 06 7 . 2 2 , 1 6 . 5 0E x t r e m e v a lu e6 8 . 5 1 , 1 7 . 8 3L o g lo g i s t i c6 7 . 1 4 , 1 6 . 4 5A n d e r s o n D a r l i n g ( a d j )T e m p 8 0T e m p 1 0 0 1 51 02 03 04 05 06 07 08 09 59 91 0 1 0 0L o g n o r m a l b a s e ePercent 1 51 02 03 04 05 06 07 08 09 59 91 0 1 0 0L o g n o r m a l b a s e 1 0Percent 1 2 3 51 02 03 04 05 06 07 08 09 09 59 9 1 0 0 0 1 0 0E x t r e m e v a l u ePercent 1 51 02 03 04 05 06 07 08 09 59 91 0 1 0 0L o g l o g i s t i cPercentF o u r w a y P r o b a b i l i t y P l o t f o r T e m p 8 0 T e m p 1 0 0M L E s t i m a t e s C e n s o r i n g C o l u m n i n C e n s 8 0 . . . C e n s 1 0 037 Exercise: Answers, cont. Create the Overview plot. 3 4 / 63 7 / 1 3F / C1 6 . 4 56 7 . 1 4A D *0 . 4 1 6 40 . 2 8 1 4S ca l e3 . 6 3 5 94 . 0 7 0 9L o ca t i o nT e m p 1 0 0T e m p 8 0T e m p 8 0T e m p 1 0 0100109995908070605040302010 5 1Log log istic Pro bab ilityPercent25020xx501005001 . 00 . 90 . 80 . 70 . 60 . 50 . 40 . 30 . 20 . 10 . 0Su rviva l Fu nct ionProbability25020xx501005000 . 0 30 . 0 20 . 0 10 . 0 0Haz ard Fu nct ionRate25020xx501005000 . 0 20 . 0 10 . 0 0