Impact of censored data on reliability analysis

Impact of Censored Data on
Reliability Analysis
(截尾数据对于可靠性估计和
寿命试验计划的影响)
Rong Pan
©2014 ASQ
http://www.asqrd.org

Rong Pan
Associate Professor
Arizona State University
Email: rong.pan@asu.edu
Impacts of Censored Data on
Reliability Analysis

Outlines
1/11/2014Webinar for ASQ Reliability Division3
 Objectives
 To provide an introduction to the statistical analysis of
failure time data
 To discuss the impact of data censoring on data analysis
 To demonstrate software tools for reliability data analysis
 Organization
 Reliability definition
 Characteristics of reliability data
 Statistical analysis of censored reliability data

Reliability
 Meeker and Escobar (1998) ‒ “Reliability is often
defined as the probability that a system, vehicle,
machine, device, and so on will perform its intended
function under operating conditions, for a specified
period of time.”
 Condra (2001) ‒ “Reliability is quality over time.”
 Leemis (1995) ‒ “The reliability of an item is the
probability that it will adequately perform its specified
purpose for a specified period of time under specified
environmental conditions.

Reliability Function
 The reliability function is the probability that an item
performs its function for a fixed period of time:
 The time at which an item fails to perform its intended
function is called its failure time.
 The failure time of an item is a continuous
nonnegative random variable, often denoted T
𝑹 𝒕 = 𝑷𝒓𝒐𝒃(𝐢𝐭𝐞𝐦 𝐩𝐞𝐫𝐟𝐨𝐫𝐦𝐬 𝐢𝐧𝐭𝐞𝐧𝐝𝐞𝐝 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝐮𝐧𝐝𝐞𝐫
𝐢𝐧𝐭𝐞𝐧𝐝𝐭𝐞𝐝 𝐜𝐨𝐧𝐝𝐢𝐭𝐢𝐨𝐧 𝐟𝐨𝐫 𝐚𝐭 𝐥𝐞𝐚𝐬𝐭 𝐭 𝐭𝐢𝐦𝐞 𝐮𝐧𝐢𝐭𝐬)
)Pr()( tTtR  )Pr()(1)( tTtRtF 

Understanding Hazard Function
 Reliability function
 Define a hazard function
 Instantaneous failure
 Is a function of time
 Only exponential distribution has constant hazard (failure rate)
 Relationships between reliability function and hazard function
t
tTttTt
th t


 
)|(Pr
lim)( 0
)(
)(
)(
tR
tf
th 

t
dxxhtH
0
)()( )(
)( tH
etR 
 )(1)( tRtF 
dt
tdF
tf
)(
)( 
)Pr()( tTtR 

Characteristics of Reliability Data
 Failure time censoring
 Right censoring
 Left censoring
 Interval censoring
 Data from reliability tests
 Type-I censoring (time censoring)
 Type-II censoring (failure censoring)
 Read-outs (multiple censoring)

Right Censoring
 Actual failure time exceeds
observation
 In life tests
 Type-I censoring (time
censoring)
 Type-II censoring (failure
censoring)

Example
Low-cycle fatigue test of nickel super alloy (Meeker &
Escobar (1998), p. 638, attr. Nelson (1990), p. 272)
kCycles Censor
211.626 0
200.027 0
57.923 1
155 0
13.949 0
112.968 1
152.68 0
156.725 0
138.114 1
56.723 0
121.075 0
122.372 1
112.002 0
43.331 0
12.076 0
13.181 0
18.067 0
21.3 0
15.616 0
13.03 0
8.489 0
12.434 0
9.75 0
11.865 0
6.705 0
5.733 0

Left Censoring
 Actual life time less than observation
 May occur when the item is inspected at a fixed
time point
 Less often than
right censoring

Example
In the nickel super alloy example, suppose that the
observation starts only after 10,000 cycles.
kCycles start kCycles end
211.626 211.626
200.027 200.027
57.923 *
155 155
13.949 13.949
112.968 *
152.68 152.68
156.725 156.725
138.114 *
56.723 56.723
121.075 121.075
122.372 *
112.002 112.002
43.331 43.331
12.076 12.076
13.181 13.181
18.067 18.067
21.3 21.3
15.616 15.616
13.03 13.03
8.489 8.489
12.434 12.434
* 10
11.865 11.865
* 10
* 10

Interval Censoring
 Observation gives upper and lower bound on failure
time
 Occurs often with scheduled inspections
 Right censoring and left censoring are special cases
 Read-outs
 Grouped data

Example
In the nickel super alloy example, suppose that the test
units are inspected at 25, 50, 100, 200 kCycles.
kCycles start kCycles end read-outs
0 25 12
25 50 2
50 100 2
100 200 8
200 * 2

Multiple Censored Data
 More than one censoring mechanisms are
employed
 Exact failure times, right censoring times and
interval censoring times are very common in
practice
 May not be easily recognized
 Calendar time vs. lifetime

Example
Adapted from an example in Meeker & Escobar (1998), p. 8.
 Nuclear power plant use heat exchangers to transfer energy from the
reactor to stream turbines. A typical heat exchanger contains
thousands of tubes. With age, heat exchanger tubes develop cracks.
 Suppose there are three plants. Plant 1 had been in operation for 3
years, Plant 2 for 2 years, and Plant 3 for only 1 year. All heat
exchangers are of the same design and operated under similar
conditions. At the beginning, each plant has 100 new tubes. Failed
tubes will be removed from the heat exchanger during operation.
 Plant 1: 1 failure in the first year, 2 failures in the second year, and 2
failures in the third year.
 Plant 2: 2 failures in the first year, 3 failures in the second year.
 Plant 3: 1 failure in the first year.

Example (cont.)
 Year 1: 300 tubes are tested, 4 failed (left censored), 99 removed (right
censored).
 Year 2: 197 tested, 5 failed (interval censored), 95 removed (right
censored).
 Year 3: 97 tested, 2 failed (interval censored), 95 survived (right
censored).
Y1 Y2 Y3
P1
P2
P3
(95 survived)
(95 survived)
(99 survived)

Example (cont.)
initial year 1 year 2 year 3
Plant 1 100 1 2 2
Plant 2 100 2 3
Plant 3 100 1
start year end year at risk removed failed
0 1 300 99 4
1 2 197 95 5
2 3 97 2
start year end year readouts
* 1 4
1 * 99
1 2 5
2 * 95
2 3 2
3 * 95
Data by group Data by age
Data by censoring type

Simulation
 Monte Carlo method
 Assume a probabilistic model
 Generate random numbers
 Compute the statistics of interest
 Repeat it many times
 Estimate confidence intervals
 Useful for evaluating and comparing data
analysis methods
 Useful for evaluating and comparing life test
plans

Simulate Censored Failure
Time
 In MS Excel®
 Many build-in functions for generated random numbers from
specific distribution, such as NORMINV(p, mu, sigma)
 Utilize GAMMAINV(p, alpha, beta) to generate exponentially
distributed failure times
 Set p=rand(), alpha=1, beta=mean failure time
 Utilize NORMAINV(p, mu, sigma) to generate the failure times with
lognormal distribution
 Set p=rand(), mu and sigma are the parameters of the lognormal
distribution
 Compute exp(normal random number)
 No build-in inverse function for Weibull distribution
 Use function [(-ln(1-rand()))/a]^(1/b)
 Where a is the intrinsic failure rate of Weibull distribution, b is a shape
parameter
 Use If() function to create censored failure times

Features of Lifetime
Distribution
Failure data from electrical appliance test (Lawless, p.7.
Attr. Nelson (1970))
Variable: cycles to failure (exact failure time)
 Nonnegative
 Right (positively) skewed
 Some long life observations
Normal distribution may not be a good idea!

Exponential Distribution
 The simplest lifetime distribution
 One parameter
or
 Constant failure rate (constant mean-time-to-
failure, MTTF)
 Memoryless property
 Regardless of past experience, the chance of failure
in future is the same.
 Closure property
 System’s failure time is still exponential, if its
components’ failure times are exponential and they
are in a series configuration.
)exp()|( ttf   )/exp(/1)|(  ttf 

Weibull Distribution
 When the hazard function is a power function of time
 Two common forms
 Two parameters
 Either characteristic life or intrinsic failure rate and shape
parameter
 Relationship with exponential distribution
 When the shape parameter is known
1
)(










 t
th





















 

 tt
tf exp)(
1

















t
tR exp)(
1
)( 
 
tth  
ttR  exp)( 
 tttf  
exp)( 1

 /1

 Rectification
 Plot failure probability on a complementary log-log scale
 Plot time on a log scale
 Some important features on the plot
 Slope is the shape parameter
 Characteristic life can be found at where the failure
probability is (1-1/e)=0.632
 Reliability at a given lifetime depends on distribution
parameters, except at the characteristic life
Weibull Plot
)log(log))](1log(log[   ttF

Weibull Plot of Electrical Appliance
Data

Lognormal Distribution
 From normal to lognormal and vice versa
 If T has a lognormal distribution, then log(T) has a
normal distribution
 If X has a normal distribution, then exp(X) has a
lognormal distribution
 Median failure time
 Log(t50) is a robust estimate of the scale parameter
of lognormal distribution

Parametric Distribution Models
 Maximum likelihood estimation (MLE)
 Likelihood function
 Find the parameter estimate such that the chance of having such failure
time data is maximized
 Contribution from each observation to likelihood function
 Exact failure time
 Failure density function
 Right censored observation
 Left censored observation
 Failure function
 Interval censored observation
 Difference of failure functions
)(tR
)(tF
)()( 
 tFtF
)(tf

Exponential Distribution
 Exact failure times
 Failure density function
 Failure rate estimate
 Type-I censoring
it
i etf 
 
)(
 

n
i itn
n etttL 1
),...,,;( 21



 n
i it
n
1
ˆ
ct
c etR 
)(
c
r
i i trntr
ccr etttttL
)(
21
1
),...,,...,,;(
  




 r
i ci trnt
r
1
)(
ˆ

Exponential Distribution (cont.)
 Type-II censoring
 General formula for Exponential failure times
r
r
i i trntr
rrr etttttL
)(
121
1
),...,,...,,;(


 




 r
i ri trnt
r
1
)(
ˆ
timetestingtotal
failuresofnumber
RateFailure 
failuresofnumber
timetestingtotal
MTTF

Precision of Failure Rate
Estimate
 Sum of exponential distributions becomes gamma
distribution
 Independently identical distributed (i.i.d.)
 Exact confidence intervals for the cases of exact failure
time and type-II censoring
 An approximated confidence interval for the case of type-
I censoring

n
i
i ngammaT
1
),(~ 








TTTTTT
rr
2
,
2
2
2/1,2
2
2/,2  
 )ˆ.(.ˆ),ˆ.(.ˆ
2/2/   eszesz 
r
es
2ˆ
)ˆ.(.

 

Effect of Censoring
 Widened confidence bounds
 Use the electrical appliance data
 Right censoring at 10000, 5000, 4000, 3000, 2000,
1000
# of censored 0 1 4 5 11 23 28
% censoring 0 2.78% 11.11% 13.89% 30.56% 63.89% 77.78%
MTTF 2756.806 2738.343 2645.406 2591.097 2956.92 4336.846 3830.125
lower bound 2038.868 2056.893 1964.68 1916.454 2124.224 2809.555 2262.377
upper bound 3936.114 4095.027 4047.949 3998.769 4863.355 9502.421 12474.54
difference 1897.246 2038.134 2083.27 2082.315 2739.131 6692.866 10212.16

Final Remarks
 Be aware of censoring when analyzing reliability data
 Ignoring censored data will bias failure(reliability) estimates
 Often underestimate reliability
 The amount of information of censored data depends on the
censoring type
 Nonparametric methods are based on ranks
 Often utilize the ratio of number of failures and number of items at risk
 Parametric methods are based on likelihood functions
 Maximum likelihood estimation
 Computation becomes complicated
 Use software
 Simulation is a very useful tool for studying the effect of
sample size or censoring

Impact of censored data on reliability analysis

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Impact of censored data on reliability analysis