Unit 2: Models, Censoring, and Likelihood for Failure-Time ...web.utk.edu/~leon/rel/Fall04pdfs/567Unit2.pdfUnit 2: Models, Censoring, and Likelihood for Failure-Time Data Notes largely

Unit 2: Models, Censoring, and Unit 2: Models, Censoring, and Likelihood for FailureLikelihood for Failure--Time Data Time Data Likelihood for FailureLikelihood for Failure--Time Data Time Data

Notes largely based on “Statistical Methods for Reliability Data” by W.Q. y y QMeeker 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 ObjectivesUnit 2 ObjectivesUnit 2 ObjectivesUnit 2 Objectives Describe models for continuous

f il ti /di t ib tifailure-time processes/distributions◦ Time to failure

Describe models that we will use 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 censoring, i.e. failure times that one knows that they felt in a interval Rounding


Unit 2 ObjectivesUnit 2 ObjectivesUnit 2 ObjectivesUnit 2 Objectives Describe common censoring mechanisms that

restrict our ability to observe all of the failure 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

h f il ld h d i l that a failure would have occurred past some survival (censoring) time.◦ Left censoring: one only knows that the failure

d b f ( i )tioccurred before some (censoring)time◦ Interval censoring: one only knows that the failure

occurred in an interval and not the exact time to failurefailure


Unit 2 ObjectivesUnit 2 ObjectivesD b h h Describe common censoring mechanisms that restrict our ability to observe all of the failure times that might occur in a reliability studyg y y


Typical FailureTypical Failure--Time Probability FunctionsTime Probability Functions

1.7( ) ( ) 1 exp( )F t P T t t

Cumulative Distribution FunctionT is a ( ) ( ) 1 exp( )

( )

F t P T t t

dF t

Probability Density Function

nonnegative, continuous random variable d ibi th.7 1.7( )( ) 1.7 exp( )dF tf t t t

dt

Survival Function

describing the failure-time process. The distribution for T

1.7( ) ( ) ( ) exp( )t

S t P T t f t dt t

Survival Function distribution for T

can be characterized by any of the

( ) ( )( ) f t f th t

Hazard Function

.71.7 t

any of the adjacent functions


( )( ) 1

h tS t F

1.7( )

tt

Typical FailureTypical Failure--Time Time Probability Probability FunctionsFunctions


Hazard Function Hazard Function or or I I F il R F iF il R F iInstantaneous Instantaneous Failure Rate FunctionFailure Rate Function

The hazard function is defined byh(t) y( )

0( | )( ) lim t

P t T t t T th tt

0( ) ( )lim t

tP t T t t P T t

t

0( )lim

( )

tP t T t t

tP T t

( )

( ) ( )( ) 1 ( )

P T tf t f tS t F t


( ) ( )

CDF in Terms of the CDF in Terms of the Hazard Hazard Function Function

( )f0 0

( )( ) ( )1 ( )

Let ( ) ( ) so

t t f xH t h x dx dxF x

u F x du f x dx

( ) ( )

00

Let ( ) ( ) , so

( ) ln(1 ) ln 1 ( ) ln ( )1

F t F t

u F x du f x dxduH t u F t S t

u

0( ) 1 exp ( ) 1 exp ( )

tF t H t h x dx

0


Hazard Function Hazard Function or or Instantaneous Instantaneous Failure Rate FunctionFailure Rate Function


Engineers Interpretation of the Engineers Interpretation of the H d H d FFHazard Hazard FunctionFunction

The hazard function can be interpreted as a failure prate if there is a large number of items, say n(t), in operation at time t. Then

( ) ( ) Expected number of f il i ti ( )

n t h t t failures in time ( , )t t t

( ) Expected number of

failures per unit of time per unit at riskh t


failures per unit of time per unit at risk

FIT RateFIT RateFIT RateFIT Rate

A FIT rate is defined as the h d f f 1/h hazard function in units of 1/hours multiplied by 109multiplied by 10


Example 2 5 Example 2 5 Example 2.5 Example 2.5

165,000 copies of a componentp p Hazard rate constant over time at 15

FITs.s.◦ 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 from this component in 1 year (8760 hours) of operation is ◦ 15x10-9 x 165,000 x 8760 = 21 715x10 x 165,000 x 8760 21.7


Bathtub Curve Hazard FunctionBathtub Curve Hazard FunctionBathtub Curve Hazard FunctionBathtub Curve Hazard Function


Cumulative Hazard Function and Cumulative Hazard Function and A H d RA H d RAverage Hazard RateAverage Hazard Rate


Practical Interpretation of Average Practical Interpretation of Average H d RH d RHazard RateHazard Rate

2 1 1 2( ) ( ) ( )( ) F t F t P t T tAHR 2 1 1 21 2

2 1 2 1

( ) ( ) ( )( , )

if ( ) ( ) is small say less than 0 1

F t F t P t T tAHR t tt t t t

F t P T t

2 2if ( ) ( ) is small, say less than 0.1

I ti l

F t P T t

0

In particular,

( ) ( ) ( ) ( )th u du H t F t P T t0 ( ) ( ) ( ) ( )( ) = =

if ( ) ( ) is small say less than 0 1

H t F t P T tAHR tt t t t

F t P T t


if ( ) ( ) is small, say less than 0.1F t P T t

DerivationDerivation2 2

21 1

( ) ( )( )t t

tt tf u du f u duf u du

1

2

1 2

2 1 2 1

( ) ( ) ( )

( ) ( ) ( ) ( )

t

t

duS t S u S t

F t F t F t F t

2

1

2 1 2 12 1

1 2

( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )t

F t F t F t F th u du H t H tS t S t

2 1 2 1 2 11 2

1 2 1 2 1 2 2 1

( ) ( ) ( ) ( ) ( ) ( )1 1( , )( ) ( )

So if

F t F t H t H t F t F tAHR t tS t t t t t S t t t

F

1 2 2 1( ) ( ) is small ( ) ( ) is close to 1t F t S t S t So if F 1 2 2 1

2 1 1 2

( ) ( ) is small ( ) ( ) is close to 1

( ) ( ) ( )( )

t F t S t S t

F t F t P t T tAHR t t


1 22 1 2 1

( , )AHR t tt t t t

Quantile FunctionQuantile Function


Distribution QuantilesDistribution QuantilesDistribution QuantilesDistribution QuantilesThe quantile of is the smallest time such that pp F t

( ) ( ) , where 0 1p pP T t F t p p

p

tp

Stat 567: Unit 2 - Ramón V. León 18

Simple Quantile CalculationSimple Quantile CalculationSimple Quantile CalculationSimple Quantile Calculation

Example:

Terminology:Terminology:


Models for Discrete Data from a Models for Discrete Data from a Models for Discrete Data from a Models for Discrete Data from a Continuous Time ProcessContinuous Time Process


Partitioning of Time into Partitioning of Time into NonNon--Overlapping Overlapping IntervalsIntervals


Graphical Interpretations of the Graphical Interpretations of the ’s’sGraphical Interpretations of the Graphical Interpretations of the ss


Nonparametric ParametersNonparametric ParametersNonparametric ParametersNonparametric Parameters

1( )i

iS t

1( )iS t

1m

Notice:

1 1

i 1

( ) ( )

( )

i i jj i

i i

S t P T t

p S t


i 1( )i ip S t

A Important DerivationA Important DerivationA Important DerivationA Important Derivation

( ) ( ) ( ) ( ) ( )S t S t F t F t p S t 1 1 1( ) ( ) ( ) ( ) ( )i i i i i i iS t S t F t F t p S t

1(1 ) ( ) ( )by induction

i i ip S t S t

( ) (1 ), 1,..., 1i

i jS t p i m 1j


Nonparametric ParametersNonparametric Parameters

Sincei

1

( ) 1 (1 ), 1,..., 1i

i jj

F t p i m

and

( )i

1

( ) , 1,..., 1

we view ( ) or ( )

i ij

F t i m

p p

p1 1 1 1we view ( ,..., ) or ( ,..., )

as the nonparametric parameters.m mp p p


Example Calculation of the Example Calculation of the Nonparametric ParametersNonparametric ParametersNonparametric ParametersNonparametric Parameters

i


i

1( )ii

pS t

Examples of Censoring MechanismsExamples of Censoring Mechanismsp gp g

Censoring restricts our ability to observe T. Some sources of censoring are:

Fixed time to end test (lower bound on T for unfailed units)

Some sources of censoring are:

) Inspection times (upper and lower bounds on T) Staggered entry of units into service leads to multiple

censoringcensoring Multiple failure modes (also known as competing risks,

and resulting in multiple right censoring):◦ Independent failure modes (simple)Independent failure modes (simple)◦ Non independent failure modes (difficult)

Simple analysis requires i f ti i ti


non-informative censoring assumptions

Likelihood (Probability of the Data) as Likelihood (Probability of the Data) as Likelihood (Probability of the Data) as Likelihood (Probability of the Data) as a Unifying Concepta 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 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)g )


Determining the Likelihood Determining the Likelihood (P b b l f h D )(P b b l f h D )(Probability of the Data)(Probability of the Data)

The form of the likelihood will depend onThe form of the likelihood will depend on Question and focus of the study Assumed model Assumed model Measurement system (form of available data) Identifiability/parametrization Identifiability/parametrization


Likelihood Contributions for Different Likelihood Contributions for Different K d f CK d f CKinds of CensoringKinds of Censoring



1.7Example: ( ) 1 exp( )F t t Example: ( ) 1 exp( )F t t



1.7Example: ( ) 1 exp( )F t tExample: ( ) 1 exp( )F t t


Likelihood for Life DataLikelihood for Life DataLikelihood for Life DataLikelihood for Life Data

di = 2 = number ofdi 2 number of observations interval censored in ti-1 and ti

l 3 b fli = 3 = number of observations left-censored at ti

ri =2 number of observations right-censored at ticensored at ti


Likelihood for Life Table DataLikelihood for Life Table Data


Likelihood: Probability of the DataLikelihood: Probability of the DataLikelihood: Probability of the DataLikelihood: Probability of the Data


Likelihood for Arbitrary Censored DataLikelihood for Arbitrary Censored Data


Likelihood for General Reliability DataLikelihood for General Reliability Datayy


Other Topics in Chapter 2Other Topics in Chapter 2Other Topics in Chapter 2Other Topics in Chapter 2

Random censoring Overlapping censoring intervals Likelihood with censoring in the intervals How to determine C How to determine C


Unit 2: Models, Censoring, and Likelihood for Failure-Time ...web.utk.edu/~leon/rel/Fall04pdfs/567Unit2.pdfUnit 2: Models, Censoring, and Likelihood for Failure-Time Data Notes largely

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