Chapter 2. Models, Censoring, and Likelihood for Failure-Time Data

Transcription

1 Chaper 2 Models, Censoring, and Likelihood for Failure-Time Daa William Q. Meeker and Luis A. Escobar Iowa Sae Universiy and Louisiana Sae Universiy Copyrigh W. Q. Meeker and L. A. Escobar. Based on he auhors ex Saisical Mehods for Reliabiliy Daa, John Wiley & Sons Inc January 13, h 41min 2-1

2 Chaper 2 Models, Censoring, and Likelihood for Failure-Time Daa Objecives Describe models for coninuous failure-ime processes. Describe some reliabiliy merics. Describe models ha we will use for he discree daa from hese coninuous failure-ime processes. Describe common censoring mechanisms ha resric our abiliy o observe all of he failure imes ha migh occur in a reliabiliy sudy. Explain he principles of likelihood, how i is relaed o he probabiliy of he observed daa, and how likelihood ideas can be used o make inferences from reliabiliy daa. 2-2

3 Typical Failure-ime cdf, pdf, hf, and sf F() = 1 exp( 1.7 ); f() = exp( 1.7 ) S() = exp( 1.7 ); h() = Cumulaive Disribuion Funcion Probabiliy Densiy Funcion F().5 f() Survival Funcion Hazard Funcion S().5 h()

4 Models for Coninuous Failure-Time Processes T is a nonnegaive, coninuous random variable describing he failure-ime process. The disribuion of T can be characerized by any of he following funcions: The cumulaive disribuion funcion (cdf): F() = Pr(T ). Example, F() = 1 exp( 1.7 ). The probabiliy densiy funcion (pdf): f() = df()/d. Example, f() = exp( 1.7 ). Survival funcion (or reliabiliy funcion): S() = Pr(T > ) = 1 F() = Example, S() = exp( 1.7 ). f(x)dx. The hazard funcion: h() = f()/[1 F()]. Example, h() =

5 Hazard Funcion or Insananeous Failure Rae Funcion The hazard funcion h() is defined by h() = lim 0 Pr( < T + T > ) = f() 1 F(). Noes: F() = 1 exp[ 0 h(x)dx], ec. h() describes propensiy of failure in he nex small inerval of ime given survival o ime h() Pr( < T + T > ). Some reliabiliy engineers hink of modeling in erms of h(). 2-5

6 Bahub Curve Hazard Funcion h() Infan Moraliy Random Failures Wearou Failures 2-6

7 Cumulaive Hazard Funcion and Average Hazard Rae Cumulaive hazard funcion: H() = 0 h(x)dx. Noice ha, F() = 1 exp[ H()] = 1 exp [ 0 h(x)dx ]. Average hazard rae in inerval ( 1, 2 ]: AHR( 1, 2 ) = 2 1 h(u)du 2 1 = H( 2) H( 1 ) 2 1. If F( 2 ) F( 1 ) is small (say less han.1), hen AHR( 1, 2 ) F( 2) F( 1 ) ( 2 1 )S( 1 ). An imporan special case arises when 1 = 0, 0 h(u)du AHR() = = H() F(). Approximaion is good for small F(), say F() <

8 Plos Showing ha he Quanile Funcion is he Inverse of he cdf Probabiliy Densiy Funcion Cumulaive Disribuion Funcion f() 0.4 F() Shaded Area =

9 Disribuion Quaniles The p quanile of F is he smalles ime p such ha Pr(T p ) = F( p ) p, where 0 < p < is he ime by which 20% of he populaion will fail. For, F() = 1 exp( 1.7 ), p = F( p ) gives p = [ log(1 p)] 1/1.7 and.2 = [ log(1.2)] 1/1.7 =.414. When F() is sricly increasing here is a unique value p ha saisfies F( p ) = p, and we wrie p = F 1 (p). When F() is consan over some inervals, here can be more han one soluion o he equaion F() p. Taking p equal o he smalles value saisfying F() p is a sandard convenion. p is also know ad B100p (e.g.,.10 is also known as B10). 2-9

10 Pariioning of Time ino Non-Overlapping Inervals π 1 π 2 π 3... π m-1 π m π m = m-1 m m+1 = Times need no be equally spaced. 2-10

11 Graphical Inerpreaion of he π s F( ) F( 4 3 ) F( 2 ) π π π 2 F( 1 ) 0.2 π

12 Models for Discree Daa from a Coninuous Time Processes All daa are discree! Pariion (0, ) ino m+1 inervals depending on inspecion imes and roundoff as follows: ( 0, 1 ], ( 1, 2 ],..., ( m 1, m ], ( m, m+1 ) where 0 = 0 and m+1 =. Observe ha he las inerval is of infinie lengh. Define, π i = Pr( i 1 < T i ) = F( i ) F( i 1 ) p i = Pr( i 1 < T i T > i 1 ) = F( i) F( i 1 ) 1 F( i 1 ) Because he π i values are mulinomial probabiliies, π i 0 and m+1 j=1 π j = 1. Also, p m+1 = 1 bu he only resricion on p 1,...,p m is 0 p i

13 Models for Discree Daa from a Coninuous Time Processes Coninued I follows ha, S( i 1 ) = Pr(T > i 1 ) = m+1 j=i π j π i = p i S( i 1 ) S( i ) = i j=1 ( 1 pj ), i = 1,...,m+1 We view π = (π 1,...,π m+1 ) or p = (p 1,...,p m ) as he nonparameric parameers. 2-13

14 Probabiliies for he Mulinomial Failure Time Model Compued from F() = 1 exp( 1.7 ) i F( i ) S( i ) π i p i 1 p i

15 Examples of Censoring Mechanisms Censoring resrics our abiliy o observe T. Some sources of censoring are: Fixed ime o end es (lower bound on T for unfailed unis). Inspecions imes (upper and lower bounds on T). Saggered enry of unis ino service leads o muliple censoring. Muliple failure modes (also known as compeing risks, and resuling in muliple righ censoring): independen (simple). non independen (difficul). Simple analysis requires non-informaive censoring assumpion. 2-15

16 Likelihood (Probabiliy of he Daa) as a Unifying Concep Likelihood provides a general and versaile mehod of esimaion. Model/Parameers combinaions wih relaively large likelihood are plausible. Allows for censored, inerval, and runcaed daa. Theory is simple in regular models. Theory more complicaed in non-regular models (bu conceps are similar). Limiaion: can be compuaionally inensive (sill no general sofware). 2-16

17 Deermining he Likelihood (Probabiliy of he Daa) The form of he likelihood will depend on: Quesion/focus of sudy. Assumed model. Measuremen sysem (form of available daa). Idenifiabiliy/parameerizaion. 2-17

18 Likelihood (Probabiliy of he Daa) Conribuions for Differen Kinds of Censoring Pr(Daa) = n i=1 Pr(daa i ) = Pr(daa 1 ) Pr(daa n ) Lef censoring f() Inerval censoring Righ censoring

19 Likelihood Conribuions for Differen Kinds of Censoring wih F() = 1 exp( 1.7 ) Inerval-censored observaions: L i (p) = i i 1 f()d = F( i ) F( i 1 ). If a uni is sill operaing a = 1.0 bu has failed a = 1.5 inspecion, L i = F(1.5) F(1.0) =.231. Lef-censored observaions: i L i (p) = f()d = F( i) F(0) = F( i ). 0 If a failure is found a he firs inspecion ime =.5, L i = F(.5) =.265. Righ-censored observaions: L i (p) = i f()d = F( ) F( i ) = 1 F( i ). If a uni has no failed by he las inspecion a = 2, L i = 1 F(2) =

20 Likelihood for Life Table Daa For a life able he daa are: he number of failures (d i ), righ censored (r i ), and lef censored (l i ) unis on each of he nonoverlapping inerval ( i 1, i ], i = 1,...,m+1, 0 = 0. The likelihood (probabiliy of he daa) for a single observaion, daa i, in ( i 1, i ] is L i (π;daa i ) = F( i ;π) F( i 1 ;π). Assuming ha he censoring is a i Type of Characerisic Number Likelihood of Censoring of Cases Responses L i (π;daa i ) Lef a i T i l i [F( i )] l i Inerval i 1 < T i d i [ F(i ) F( i 1 ) ] d i Righ a i T > i r i [1 F( i )] r i 2-20

21 Likelihood: Probabiliy of he Daa The oal likelihood, or join probabiliy of he DATA, for n independen observaions is L(π;DATA) = C = C n i=1 m+1 i=1 L i (π;daa i ) [F( i )] l i [ F( i ) F( i 1 ) ] d i [1 F( i )] r i where n = ( ) m+1 j=1 dj +r j +l j and C is a consan depending on he sampling inspecion scheme bu no on π. So we can ake C = 1. Wan o find π so ha L(π) is large. 2-21

22 Likelihood for Arbirary Censored Daa In general, he ih observaion consiss of an inerval ( L i, i], i = 1,...,n ( L i < i ) ha conains he ime even T for he ih individual. The inervals ( L i, i] may overlap and heir union may no cover he enire ime line (0, ). In general L i i 1. Assuming ha he censoring is a i Type of Characerisic Likelihood of a single Censoring Response L i (π;daa i ) Lef a i T i F( i ) Inerval L i < T i F( i ) F( L i ) Righ a i T > i 1 F( i ) 2-22

23 Likelihood for General Reliabiliy Daa The oal likelihood for he DATA wih n independen observaions is L(π;DATA) = n i=1 L i (π;daa i ). Some of he observaions may have muliple occurrences. Le ( L j, j], j = 1,...,k be he disinc inervals in he DATA and le w j be he frequency of observaion of ( L j, j]. Then L(π;DATA) = k j=1 [ Lj (π;daa j ) ] w j. In his case he nonparameric parameers π correspond o probabiliies of a pariion of (0, ) deermined by he daa (Examples laer). 2-23

24 Oher Topics in Chaper 2 Random censoring. Overlapping censoring inervals. Likelihood wih censoring in he inervals. How o deermine C. 2-24