In a reliability evaluation test, one could end up with one of decisions of the reliability specification being met, not met, or inconclusive. In this presentation, we present a methodology of sample size determination prior to the test based on the probability of reaching these decisions. The specific results are obtained for the cases of Exponential distribution and Weibull distribution with a known shape parameter.
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3. 16th ISSAT International Conference Washington, D.C. USA
Reliability and Quality in Design August 5-7, 2010
5 7,
Life Test Sample Size Determination Based
on Probability of Decisions
Jiliang Zhang
Jili Zh
Hewlett-Packard Co.
jiliang.zhang@hp.com
Cell: 208-9549507
2010 ISSAT – Session 5 – Jiliang Zhang 1
4. General Consideration in Sample Size
Determination
Sample Size = # of Units + Test Duration
Test type and objective
Reliability growth, evaluation, qualification
MTBF versus durability
Reliability requirement/specification
Statistical confidence
Unit variation
Cost, schedule, and availability
Resources – testing, engineering
Phase of product development
p p
Other alternatives or additional data – internal, external
Application
pp c o
2010 ISSAT – Session 5 – Jiliang Zhang 2
5. Effectiveness of Sample Size Increase
The effectiveness of sample size reduces over sample size increase
2010 ISSAT – Session 5 – Jiliang Zhang 3
6. Minimum Sample Size
For exponential distribution
χ 2
1−α ;2 r + 2
T = MTBF
2
Caution: # of failures are unknown prior to the test,
which is related to true performance
hi h i l d f
2010 ISSAT – Session 5 – Jiliang Zhang 4
7. Contents
Problem Statement
General Methodology
Exponential Distribution Case
Weibull Distribution with a Known Shape
Parameter Case
Conclusions
2010 ISSAT – Session 5 – Jiliang Zhang 5
8. Problem Statement
Manager: “What is the sample size should be to determine the
reliability specification being met or not?
not?”
COST!
SCHEDULE!
Planning purpose
Reliability Engineer: “Sample size is related to the true
performance that is unknown prior to the test” (with any given
sample size, one could possibly end up with one of the
following decisions with certain probabilities: “met”, “not
met”, or “unknown”
With today’s economic condition, this becomes more
y
challenging and needs better methods
2010 ISSAT – Session 5 – Jiliang Zhang 6
9. Type of Life Test in Study
Consider one situation: from the life test result, the statistical
conclusion could be one of following:
specification is met if the lower limit ≥ specification at
specified confidence level (“Good);
specification is not met if the upper limit < specification at
p pp p
specified confidence level (“Bad”)
otherwise, we don t know the specification is met or not due
don’t
to the limited sample size in the test (“Inconclusive”)
The probability of “Inclusive” will always be there and is
Inclusive
desired to be small but might be acceptable under certain
circumstances due to limited budget, consequence of
“Inconclusive”, product development phase, or additional
information availabilityy
2010 ISSAT – Session 5 – Jiliang Zhang 7
10. General Methodology
Link the sample size to the probability of reaching conclusions
of “Good”, “Bad”, and “Inconclusive” if the performance is
Good Bad Inconclusive
indeed good enough and/or bad enough.
Probability of Decision Table:
Decision Worse-than- Better-than-
specification
ifi ti specification
ifi ti
performance performance
“Good” p0G Type II Error p1G
“Inconclusive” p0U p1U
“Bad” p0B p1B Type I Error
Note: this is not an acceptance test in which there are only two conclusions
can be drawn: either accept or reject
A sample size can be determined if a resulting probability of
decision table is acceptable
2010 ISSAT – Session 5 – Jiliang Zhang 8
11. Exponential Distribution Case – Criterion
of Decisions
For a time-censored life test,
2T
2T 2T
2T
m L1 = mU 1 =
χ α ;2r +2
2
χ 12− α ; 2 r
Criterion f d i i
C i i of decision with the goal of mG
ih h l f
“Good” or the specification is met if mL1 ≥ mG;
“Bad” or the specification is not met if mU1 < mG;
“Inconclusive” if mL1 < mG ≤ mU1U1.
Define rG be the maximum number of failures that satisfies
mL1 ≥ mG, and rB be the minimum number of failures that
satisfies mU1 < mG
2010 ISSAT – Session 5 – Jiliang Zhang 9
12. Exponential Distribution Case – Probability
of Decision Table Calculation
The number of failures within T follows a Poisson distribution.
So,
The probability of decision of “Good” is T
T r −m
rG ( ) e
piG = P( X ≤ rG | m = mi ) = ∑ m , i = 0, 1
r =0 r!
The probability of decision of “Bad” is
Bad T
T r −m
∞ ( ) e
piB = P( X ≥ rB | m = mi ) = ∑ m , i = 0, 1
r = rB r!
The probability of “Inconclusive” is T
T r −m
rB −1 ( ) e
piU = P (rG < X < rB | m = mi ) = ∑ m , i = 0, 1
r = rG +1 r!
2010 ISSAT – Session 5 – Jiliang Zhang 10
13. Exponential Distribution Case – Example
Assume 1 − α = 90%, m0 = mG/2 and m1 = 2mG. Set T/ mG = 7. We
,
can obtain rG = 3 and rB = 11 from the equations in previous slide 6.
So, the following probability of decisions table can be obtained from
equations in Slide 7:
Decision m0 = mG/2 m1= 2mG
“Good’ 0.1% 53.7%
“Inconclusive” 17.4% 46.2%
“Bad” 82.5% 0.1%
Note: In this example, one may feel the probability of “Inconclusive” when m1 = 2mG
example Inconclusive
is too big. Increasing sample size would reduce this uncertainty. In real case,
one can decide to further monitor the performance since this is in the middle of
p
product development , we have additional information, and the consequence of
“Inconclusive” may not be that significant.
2010 ISSAT – Session 5 – Jiliang Zhang 11
14. Weibull Distribution with a Known Shape
Parameter Case – Conversion
A two-parameter Weibull distribution,
⎛t ⎞
β tβ
−⎜ ⎟
⎜η ⎟ −
ηβ
R(t ) = e ⎝ ⎠
=e
Let s = t β , we h
have
s
−
ηβ
R( s) = e
So, s follows an exponential distribution with m = η β .
Therefore, we can expand the previous methodology in
exponential case to the Weibull with a known shape parameter
case
2010 ISSAT – Session 5 – Jiliang Zhang 12
15. Weibull Distribution Case – Criterion of
Decisions
Let S = k (t c ) β
k is the initial number of units
tc is predetermined censoring time
(k, i
(k tc) is considered as a sample size.
id d l i
For a time-censored life test,
2S β 2S
β
η L1 = 2 ηU 1 = 2
χ α ;2r +2 χ 1−α ; 2 r
Define rG be the maximum number of failures that satisfies ηL1
≥ ηG, and rB the minimum number of failures that satisfies ηU1
< ηG
2010 ISSAT – Session 5 – Jiliang Zhang 13
16. Weibull Distribution Case – Probability of
Decision Table Calculation
The probability of decision of “Good” is
S
−
S r ηβ
( β
) e
rG
η
p1G = P ( X ≤ rG | η β = ηiβ ) = ∑ , i = 0, 1
r =0 r!
The probability of decision of “Bad” is
S
−
S ηβ
( β
)r e
∞
η
piB = P( X ≥ rB | η = ηi ) = ∑
β β
, i = 0, 1
r = rB r!
The probability of “Inconclusive” is
S
−
S r ηβ
rB −1 ( χ
) e
η
piU = P(rG < X < rB | η = ηi ) =β β
∑
r = rG +1
+1 r!
, i = 0, 1
2010 ISSAT – Session 5 – Jiliang Zhang 14
17. Weibull Distribution Case – Example
Required reliability for 25,000 hours is 97.5%. Assume 1 − α =
90%. The time-between-failures follows a two-parameter
p
Weibull distribution with a known β = 2.0. The required
reliability can be converted to required ηG = 157,118 hours or
y q ,
β
requiredη G = 24,686,000,000. We further assume η β = η G /2
β
β 0
and η1 = 2η G , which correspond to the reliability of 95.1%
β
and 98.7%, respectively. For tc = 3×25,000 = 75,000 hours and
k = 30, we can obtain rG = 3 and rB = 11 from equations in Slide
10. So, the following probability of decisions table can be
obtained from equations in Slide 11:
Decision η0 β
η1β
“Good” 0% 53.4%
“Inconclusive” 6.6% 46.6%
“Bad” 93.4% 0%
2010 ISSAT – Session 5 – Jiliang Zhang 15
18. Conclusions
Sample size is linked to the probability of decisions or risks of
statistical errors or “inconclusive”
Probability of decision table can be used to determine the
sample size needed
Upper and lower performance as well as values in probability
of decision table depends on specific application. Some
consideration factors could be
Required reliability or failure consequence
q y q
Cost of the test and budget available
Product development p
p phase
Consequence and the possible action items for “Inconclusive”
Additional available information
2010 ISSAT – Session 5 – Jiliang Zhang 16