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Statistical Methods

by: Kamren McLaughlin

Statistical Methods STAT 571

Kamren McLaughlin
GPA 3.94


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This 33 page Class Notes was uploaded by Kamren McLaughlin on Monday October 26, 2015. The Class Notes belongs to STAT 571 at University of Tennessee - Knoxville taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/229896/stat-571-university-of-tennessee-knoxville in Statistics at University of Tennessee - Knoxville.


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Date Created: 10/26/15
Unit 2 Models Censoring and Likelihood for FailureTime Data Ram on V Leon Notes largely based on Statistical Methods for Reliability Data by WQ Meeker and L A Escobar Wiley 1998 and on their class notes 8252002 Stat 571 Ram n V Le n Unit 2 Objectives Describe models for continuous failuretime processesdistributions Describe models that we will use for the discrete data from these continuous failuretime processes Describe common censoring mechanisms that restrict our ability to observe all of the failure times that might occur in a reliability study Explain the principles of likelihood how it is related to the probability of the observed data and how likelihood ideas can be used to make inferences from reliability data 8252002 Stat 571 Ram n V Le n 2 Typical FailureTime Probability Functions T is a nonnegative continuous random variable describing the failuretime process The distribution for T can be characterized by any of the adjacent functions 8252002 Cumulative Distribution Function Ft PT gt 1 exp r Probability Density Function dFt t f dt Survival Function 50 PT 2 z f ftdt eXp t1 7 Hazard Function 02M f0 so 1 Ft 39 Stat 571 Ram n V Le n 17 gtlt f7 gtlt cXp t1 7 7 gtltt39 Typical FailureTime Probability Functions Ciimuiaiive Disiribmion Function Probabiiiiy Densiiy Funciion Survivai Funciion 06 mm 02 i0 00 05 10 if 20 2 I 5 Hazard Funciion 8252002 30 20 W 10 00 no 05 10 15 20 2 I 5 Stat 571 Ram n V Le n HazaniFunc on or Instantaneous Failure Rate Function The hazard function Mr is de ned by PtltTStAtTgtt At PtltTStAtPTgtt At gt0 PtltT tAt AMT PTgtt 2 m 2 f0 St 1 Ft W 11mm lim lim 8252002 Stat 571 Ram n V Le n CDF in Terms of the Hazard Function HQ hxdx 03 1 Fx Let u Fx gt du fxdx so HQ Kmf 1n1 uOFquot ln1 Ft ln SQ u gt Ft 1 eXp Hl 1 eXp J hxdx 8252002 Stat 571 Ram n V Le n Hazard Function or Instantaneous Failure Rate Function hit describes propensity of failure in the next small interval of time given survival to time t hrxAthrtltT t l AtlTgtt Some reliability engineers think of modeling in terms of Mt 8252002 81515717 Ramon V Leon Engineers Interpretation of the Hazard Function The hazard function can be interpreted as a failure rate if there is a large number of items say nt in operation at time t Then nt gtlt htAt i Expected number of failures in time t t At 2 ht 2 Expected number of failures per unit of time per unit at risk 8252002 Stat 571 Ram n V Le n FIT Rate A FIT rate is defined as the hazard function in units of 1hours multiplied by 109 A FIT is the expected number of failures per billion hours of operation A FIT is the expected number of failures per 1000 hours of operation per one million units at risk 8252002 Stat 571 Ram n V Le n 9 Bathtub Curve Hazard Function Mun Mm mm Rmtom l39mhuc w mum uhur 8252002 Stat5717 Ramon V Leo39n 10 Cumulative Hazard Function and Average Hazard Rate 0 Cumulative hazard function i H t i 39 39 39 lt gt 11 a Notice that Ft 17egtltp 7Ht 1 iexp if h1cl1 0 Average hazard rate in interval 1172 5 Hum mm 7 Hm t AHRtt L 2 itl bitl 8252002 81515717 Ramon V Leo39ri ii Practical Interpretation of Average Hazard Rate Ft2 Ft1 Pt1 s T s 12 2 t1 t2 t1 if F t2 PT S 2 is small say less than 01 AHRUptz In particular WW Ht N Ft PT s r t t t t ifFt PT S t is small say less than 01 AHRt 8252002 Stat 571 Ram n V Le n 12 Derivation t2 d t2 d L fu u S r M du S L fu u Sol a 500 55 gt FUD F01 t2 FUD F01 SUI s hudu H02 H01 302 gt Fltrzgt Fltrlgt S Hag Hal 2 AHRM SFltr2gt Fltr1gt SCI 12 tl 12 tl 55 12 tl So if F01 SF12 is small St2 S Stl is close to 1 gt AHR1112FU2F11Z Pt1 s T 12 12 t1 5 11 8252002 Stat 571 Ram n V Le n 13 Quantile Function Probability Density Function Shaded Area 2 Ft 57 Cumulative Distribution Function 8252002 Stat 571 Ramon V Leon 14 Distribution Quantiles The p quantile of F is the smallest time IF such that PTStPFtPZp where 0ltplt1 t p When Ft is constant over some intervals there can be more than one solution it to the equation Ft 2 p Taking tp equal to the smallest 1 value satisfying Ft 2 p is a standard convention 8252002 Stat 571 Ramon v Latin 15 Simple Quantile Calculation When Fr is strictly increasing there is a unique value rp that satisfies Fr p 1 and we write t1 2 F710 Example 2 20 is the time by which 20 of the population will faili For F0 2 1 exp fl7 p 2 FUN gives 1 log1 p117 and r2 7 log1 7 2117 414 Terminology r is also know ad 8100p eg MO is also known as 810 8252002 Stat 571 Ram n V Le n 16 Models for Discrete Data from a Continuous Time Process All data are discrete Partition 000 into m 1 intervals depending on inspection times and roundofr as follows fro1 1 t1t2 tiniLtmL t771t1711 where to O and tm1 2 cc Observe that the last interval is of infinite length 8252002 Stat 571 Ramon V Latin 17 Partitioning of Time into Non Overlapping Intervals to 0 T1 t1 tmrl tm tilll 00 Times need not be equally spaced 8252002 Stat 571 Ram n V Le n 1B Graphical Interpretations of the Tc s 8252002 qut FUN FHZD Fun In 0 U m o 5 t u 5 1U tn H 1 13 4 Stat 571 Ram n V Le n 19 Nonparametric Parameters Define PVT121 lt T S 1 7 Fm F6721 FO i 1170771 i 1 F ii StH Because the m values are rnLIltinomial probabilities 3 O and 23711173 1 Also pm1 l but the only restriction on 1pm is 0 1913 1 P2 Prti71lt T S Ti l T gt T271 m1 Notice SUM PT gt tH 7 jZi 7239i piStI71 8252002 Stat 571 Ramon V Latin 20 A Important Derivation Sti 1StiFtiFti 17Ti 2191510141 3 1 190510141 2 Sci 2 54 H1 pj i1m1 j1 8252002 Stat 571 Ram n V Le n 21 Nonparametric Parameters Since Ftl1 1LI1 pj i1m1 and H Ftl7ri i1m1 we view 7 7r17rm1 or p p1pm1 as the nonparametric parameters 8252002 Stat 571 Ram n V Le n 22 Example Calculation of the Nonparametric Parameters Probabilities for the Multinomial Failure Time Model Computed from Ft l 7 exp7t17 ti F07 3a 71 P1 1 7P 00 000 1 000 05 265 735 265 265 735 10 632 368 367 500 500 15 864 136 231 629 371 20 961 0388 0976 715 285 6g 1000 000 0388 1000 000 8252002 Stat 571 Ramon V Leon Examples of Censoring Mechanisms Censoring restricts our ability to observe T Some sources of censoring are Fixed time to end test lower bound on T for unfailed units Inspection times upper and lower bounds on T Staggered entry of units into service leads to multiple censonng Multiple failure modes also known as competing risks and resulting in multiple right censoring Independent simple Non independent difficult Simple analysis requires noninformative censoring assumptions 8252002 Stat 571 Ram n V Le n 24 Likelihood Probability of the Data as a Unifying Concept Likelihood provides a general and versatile method of estimation ModelParameters combinations with large likelihood are plausible Allows for censored interval and truncated data Theory is simple in regular models Theory more complicated in nonregular models but concepts are similar Limitation can be computationally intensive still not general software 8252002 Stat 571 Ram n V Le n 25 Determining the Likelihood Probability of the Data The form of the likelihood will depend on Question and focus of the study Assumed model Measurement system form of available data ldentifiabilityparametrization 8252002 Stat 571 Ram n V Le n 26 8252002 Likelihood Contributions for Different Kinds of Censoring Stat 571 Ram n V Le n Likelihood Contributions for Different Kinds of Censoring Example F t 1 eXp t1397 o Interval censored observations I up f mm FM 7 Feel 39 171 If a unit is still operating at t 2 1 0 but has failed at t 15 inspection L F15 7 F10 231 Left censored observations f Mp O m dr Fan 7 Fm Fm If a failure is found at the first inspection time t 5 L F5 265 8252002 Stat 571 Ram n V Le n 28 Likelihood Contributions for Different Kinds of Censoring Example F I 1 eXp t1397 o Right censored observations quotX Mp t m dr Fee 7 Fan 1 e Fm a If a unit has not failed by the last inspection at t 2 L 1 7F2 0388 8252002 Stat 571 Ramon V Leon 29 Likelihood for Life Table Data o For a life table the data are the number of failures 1 right censored and left censored 0 units on each of the nonoveriapping interval ix42 i 1ml f0 O The likelihood probability of the data for a single obser vation data in 23423 is L7T datal FO M 7139 7 F0741 7T Assuming that the censoring is at ti Type of Characteristic Number Likelihood of Censoring of Cases Responses LL07 datal Left at r1 T g t7 6 Ft Interval n4 lt T 3151 ii Ftv 7 Ftv1di Right at in T gt ti 7 1 1 7 Ft7739i39 8252002 Stat 571 Ram n V Le n Likelihood Probability of the Data The total likelihood or joint probability of the DATA for n independent observations is n L739rDATA C LL7rdata7 i 1 7711 c H F6010lFt17Ft171ld l 400quot i1 where n Zyjll I7 7 J and C is a constant depend ing on the sampling inspection scheme but not on 77 So we can take C 1 Want to find 7r so that L7r is large 8252002 Stat 571 Ramon V Latin 31 Likelihood for Arbitrary Censored Data o In general the the ith observation consists of an interval tfr i 1 11 ff lt ti that contains the time event T for the ith individual The intervals GILL may overlap and their union may not cover the entire time line 0x In general if 74 tisl 0 Assuming that the censoring is at if Type of Characteristic Likelihood of a single Censoring Response L17r datal Left at r T 3 ti Fm Interval 15 lt T g t Ft 7 Fa Right at 1rl T gt ti 1 7 Frv 8252002 Stat 571 Ramon V Leon Likelihood for General Reliability Data o The total likelihood for the DATA with n independent ob servations is ii L7r DATA H L17r datal 21 o Some of the observations may have multiple occurrences Let erJJL j 1 be the distinct intervals in the DATA and let it be the frequency of observation of tJLJ Jl Then Aquot L7rDATA Ljrdataju 11 o In this case the nonparametric parameters 7139 correspond to probabilities of a partition of 000 determined by the data Examples later 8252002 Stat 571 Ram n V Le n


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