Unit 2: Models, Censoring, and Likelihood for Failure-Time Data Notes largely based on Statistical Methods for Reliability Data by W.Q. Meeker and L. A. Escobar, Wiley, 1998 and on their class notes. Ramón V. León 8/21/2010 Stat 567: Unit 2 - Ramón V. León 1
Unit 2 Objectives Describe models for continuous fil failure-time processes/distributions ib ti Time to failure Describe models that we will use for the discrete data from these continuous failure-time processes/distributions Data resulting primarily by interval censoring, i.e. failure times that one knows that they felt in a interval Rounding 8/21/2010 Stat 567: Unit 2 - Ramón V. León 2
Unit 2 Objectives Describe common censoring mechanisms that restrict our ability to observe all of the failure times that might occur in a reliability study Right censoring: survival times, i.e., one only knows that a failure would have occurred past some survival (censoring) time. Left censoring: one only knows that the failure occurred before some (censoring)time Interval censoring: one only knows that the failure occurred in an interval and not the exact time to failure 8/21/2010 Stat 567: Unit 2 - Ramón V. León 3
Unit 2 Objectives Describe common censoring mechanisms that restrict our ability to observe all of the failure times that might occur in a reliability study 8/21/2010 Stat 567: Unit 2 - Ramón V. León 4
Typical Failure-Time Probability Functions Cumulative Distribution Function 1.7 Ft ( ) PT ( t ) 1 exp( t ) Probability Density Function ().7 1.7 f() t df t 1.7t exp( t ) dt Survival Function St PT t f tdt t 1.7 () ( ) () exp( ) t Hazard Function f() t f() t ht () St () 1 F() t 1.7 t.7 T is a nonnegative, continuous random variable describing the failure-time process. The distribution for T can be characterized by any of the adjacent functions 8/21/2010 Stat 567: Unit 2 - Ramón V. León 5
Typical Failure-Time Probability Functions 8/21/2010 Stat 567: Unit 2 - Ramón V. León 6
Hazard Function or Instantaneous Fil Failure Rate Function The hazard function h(t) () is defined by Pt ( Ttt Tt) ht () lim t 0 t P( t T tt) P( T t) lim t 0 t Pt ( Ttt) lim t 0 t PT ( t ) f() t f() t St () 1 Ft () 8/21/2010 Stat 567: Unit 2 - Ramón V. León 7
CDF in Terms of the Hazard Function t t f ( x ) H () t h( x) dx 0 dx 01 F( x) Let u F( x) du f( x) dx, so F() t du F() t H ( t) ln(1 u) ln 1 F( t) ln S( t) 0 0 1 u t Ft ( ) 1 exp Ht ( ) 1 exp hxdx ( ) 0 8/21/2010 Stat 567: Unit 2 - Ramón V. León 8
Hazard Function or Instantaneous Failure Rate Function 8/21/2010 Stat 567: Unit 2 - Ramón V. León 9
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 n(t), in operation at time t. Then nt ( ) ht ( ) texpected number of ht ( ) Expected number of fil failures in time ( t, t t ) failures per unit of time per unitatrisk 8/21/2010 Stat 567: Unit 2 - Ramón V. León 10
FIT Rate A FIT rate is defined as the hazard function in units of 1/hours multiplied by 10 9 8/21/2010 Stat 567: Unit 2 - Ramón V. León 11
Example 2.5 165,000 copies of a component Hazard rate constant over time at 15 FITs. h(t)=15x10-9 failures per unit per hour for all times t measured in units of hours. A prediction for the number of failures from this component in 1 year (8760 hours) of operation is 15x10-9 x 165,000 x 8760 = 21.7 8/21/2010 Stat 567: Unit 2 - Ramón V. León 12
Bathtub Curve Hazard Function 8/21/2010 Stat 567: Unit 2 - Ramón V. León 13
Cumulative Hazard Function and Average Hazard Rate 8/21/2010 Stat 567: Unit 2 - Ramón V. León 14
Practical Interpretation of Average Hazard Rate AHR ( t, t ) 1 2 Ft ( 2 ) Ft ( 1 ) Pt ( 1 T t 2 ) t t t t 2 1 2 1 if Ft ( ) PT ( t) is small, say less than 0.1 2 2 In particular, t hudu ( ) 0 Ht () Ft () PT ( t ) AHR() t = = t t t t if Ft () PT ( t ) is small, say less than 0.1 8/21/2010 Stat 567: Unit 2 - Ramón V. León 15
Derivation t t f( u) du ( ) f( u) du t f u du St ( ) Su ( ) St ( ) 2 2 2 1 t 1 du t 1 1 2 Ft Ft Ft Ft hudu ( ) Ht ( ) Ht ( ) t ( ) ( ) t 2 ( ) ( ) 2 1 2 1 2 1 St ( 1 1) t St ( 2) 1 F( t2) F( t1) H( t2) H( t1) 1 F( t2) F( t1) AHR( t1, t2) S( t1) t2 t1 t2 t1 S( t2) t2 t1 So if F ( t ) F ( t ) is small S ( t ) S ( t ) is close to 1 1 2 2 1 Ft ( 2) Ft ( 1) Pt ( 1 Tt2) AHR ( t1, t2 ) t2 t1 t2 t1 8/21/2010 Stat 567: Unit 2 - Ramón V. León 16
Quantile Function 8/21/2010 Stat 567: Unit 2 - Ramón V. León 17
Distribution Quantiles The p quantile of F is the smallest time t p such that PT ( t ) Ft ( ) p, where 0 p1 p p p t p Stat 567: Unit 2 - Ramón V. León 18
Simple Quantile Calculation Example: Terminology: 8/21/2010 Stat 567: Unit 2 - Ramón V. León 19
Models for Discrete Data from a Continuous Time Process 8/21/2010 Stat 567: Unit 2 - Ramón V. León 20
Partitioning of Time into Non-Overlapping Intervals Stat 567: Unit 2 - Ramón V. León 21
Graphical Interpretations of the s s 8/21/2010 Stat 567: Unit 2 - Ramón V. León 22
Nonparametric Parameters i St ( i 1 ) Notice: St ( ) PT ( t ) m1 i1 i1 j ji ps ( t ) i i i 1 8/21/2010 Stat 567: Unit 2 - Ramón V. León 23
A Important Derivation St ( ) St ( ) Ft ( ) Ft ( ) pst ( ) i1 i i i1 i i i1 (1 p ) St ( ) St ( ) i i1 i by induction i St ( i) (1 pj), i1,..., m1 j1 8/21/2010 Stat 567: Unit 2 - Ramón V. León 24
Nonparametric Parameters Since i Ft ( i) 1 (1 pj), i1,..., m1 and i j1 j1 F ( ti ) i, i 1,..., m1 we view ( 1,..., m 1) or p ( p1,..., pm 1 ) as the nonparametric parameters. Stat 567: Unit 2 - Ramón V. León 25
Example Calculation of the Nonparametric Parameters p i i S( t ) i1 8/21/2010 Stat 567: Unit 2 - Ramón V. León 26
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 censoring Multiple failure modes (also known as competing risks, and resulting in multiple right censoring): Independent failure modes (simple) Non independent failure modes (difficult) Simple analysis requires non-informative censoring assumptions 8/21/2010 Stat 567: Unit 2 - Ramón V. León 27
Likelihood (Probability of the Data) as a Unifying Concept Likelihood provides a general and versatile method of estimation Model/Parameters combinations with large likelihood are plausible Allows for censored, interval, and truncated data Theory is simple in regular models Theory more complicated in non-regular models (but concepts are similar) Limitation: can be computationally intensive (still not general software) 8/21/2010 Stat 567: Unit 2 - Ramón V. León 28
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) Identifiability/parametrization 8/21/2010 Stat 567: Unit 2 - Ramón V. León 29
Likelihood Contributions for Different Kinds of Censoring 8/21/2010 Stat 567: Unit 2 - Ramón V. León 30
Likelihood Contributions for Different Kinds of Censoring 1.7 Example: Ft () 1 exp( t ) 8/21/2010 Stat 567: Unit 2 - Ramón V. León 31
Likelihood Contributions for Different Kinds of Censoring 1.7 Example: Ft ( ) 1 exp( t ) 8/21/2010 Stat 567: Unit 2 - Ramón V. León 32
Likelihood for Life Data d i = 2 = number of observations interval censored in t i-1 and t i l i = 3 = number of observations leftcensored at t i r i =2 number of observations right- censored at t i 8/21/2010 Stat 567: Unit 2 - Ramón V. León 33
Likelihood for Life Table Data 8/21/2010 Stat 567: Unit 2 - Ramón V. León 34
Likelihood: Probability of the Data 8/21/2010 Stat 567: Unit 2 - Ramón V. León 35
Likelihood for Arbitrary Censored Data 8/21/2010 Stat 567: Unit 2 - Ramón V. León 36
Likelihood for General Reliability Data 8/21/2010 Stat 567: Unit 2 - Ramón V. León 37
Other Topics in Chapter 2 Random censoring Overlapping censoring intervals Likelihood with censoring in the intervals How to determine C 8/21/2010 Stat 567: Unit 2 - Ramón V. León 38