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
3. 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
4. Reliability
1/11/2014Webinar for ASQ Reliability Division4
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.
5. Reliability Function
1/11/2014Webinar for ASQ Reliability Division5
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
6. Understanding Hazard Function
1/11/2014Webinar for ASQ Reliability Division6
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
7. Characteristics of Reliability Data
1/11/2014Webinar for ASQ Reliability Division7
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)
8. Right Censoring
1/11/2014Webinar for ASQ Reliability Division8
Actual failure time exceeds
observation
In life tests
Type-I censoring (time
censoring)
Type-II censoring (failure
censoring)
9. Example
1/11/2014Webinar for ASQ Reliability Division9
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
10. Left Censoring
1/11/2014Webinar for ASQ Reliability Division10
Actual life time less than observation
May occur when the item is inspected at a fixed
time point
Less often than
right censoring
11. Example
1/11/2014Webinar for ASQ Reliability Division11
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
12. Interval Censoring
1/11/2014Webinar for ASQ Reliability Division12
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
13. Example
1/11/2014Webinar for ASQ Reliability Division13
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
14. Multiple Censored Data
1/11/2014Webinar for ASQ Reliability Division14
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
15. Example
1/11/2014Webinar for ASQ Reliability Division15
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.
17. Example (cont.)
1/11/2014Webinar for ASQ Reliability Division17
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
18. Simulation
1/11/2014Webinar for ASQ Reliability Division18
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
19. Simulate Censored Failure
Time
1/11/2014Webinar for ASQ Reliability Division19
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
20. Features of Lifetime
Distribution
1/11/2014Webinar for ASQ Reliability Division20
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!
21. Exponential Distribution
1/11/2014Webinar for ASQ Reliability Division21
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
22. Weibull Distribution
1/11/2014Webinar for ASQ Reliability Division22
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
23. 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
1/11/2014Webinar for ASQ Reliability Division23
Weibull Plot
)log(log))](1log(log[ ttF
25. Lognormal Distribution
1/11/2014Webinar for ASQ Reliability Division25
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
26. Parametric Distribution Models
1/11/2014Webinar for ASQ Reliability Division26
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
Reliability function
Left censored observation
Failure function
Interval censored observation
Difference of failure functions
)(tR
)(tF
)()(
tFtF
)(tf
27. Exponential Distribution
1/11/2014Webinar for ASQ Reliability Division27
Exact failure times
Failure density function
Likelihood function
Failure rate estimate
Type-I censoring
Reliability function
Likelihood function
Failure rate estimate
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
28. Exponential Distribution (cont.)
1/11/2014Webinar for ASQ Reliability Division28
Type-II censoring
Likelihood function
Failure rate estimate
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
29. Precision of Failure Rate
Estimate
1/11/2014Webinar for ASQ Reliability Division29
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
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)ˆ.(.ˆ),ˆ.(.ˆ
2/2/ eszesz
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es
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)ˆ.(.
31. Final Remarks
1/11/2014Webinar for ASQ Reliability Division31
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