Statistical Quality Control Charts to Monitor Processes Under ...

Intro EWMA CUSUM Comparison Example Future Work References
Statistical Quality Control Charts to
Monitor Processes Under Competing
Risks
Lin Fang
Department of Mathematics and Statistics
McMaster University
( Supervised by Prof. Román Viveros-Aguilera )
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Intro EWMA CUSUM Comparison Example Future Work References
Outline
1 Introduction
2 Exponentially Weighted Moving Average (EWMA)
Control Chart
3 Cumulative Sum (CUSUM) Control Chart
4 Comparing the CUSUM Control Charts with the EWMA
Control Charts
5 Example: A Manufacturing Process
6 Future Work
7 References
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1. Introduction
Processes under Competing Risks
Assumptions
System fails due to an observed cause.
Failure times are statistically independent with continuous
distributions.
Notations
n (≥ 1) systems sampled, each system is subjected to
p (≥ 2) competing risks.
T ij is the time to failure associated with the j th risk
component of system i ( i = 1, . . . , n , j = 1, . . . , p).
T ij has a pdf. f j (t | θ j ) and a survival function ¯F j (t | θ j ).
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Intro EWMA CUSUM Comparison Example Future Work References
Notations (continued)
Observed data: {(Y 1 , C 1 ), (Y 2 , C 2 ), . . . , (Y n , C n )}.
Y i = min{T 1i , . . . , T pi } and C i = j.
WLOG, the 1 st competing risk component is assumed to
be the one of interest.
Terminology
In-control and out-of-control parameter values: θ (0)
1
and θ (1)
1
Run length (LR) and average run length (ARL)
In-control and out-of-control ARLs: ARL 0 and ARL 1
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Intro EWMA CUSUM Comparison Example Future Work References
2. EWMA Control Chart
Work of Steiner and MacKay (2001)
Conditional Expected Value (CEV) weight
W i = E(T 1i | µ (0)
1 , σ 1, Y i , C i ) =
{
Yi if C i = 1,
R ∞
Y i
if C i ≠ 1 .
t 1i f 1 (t 1i |θ 1 )dt 1i
¯F 1 (Y i |θ 1 )
(1)
EWMA chart statistic is defined as
U l = λ ¯W l + (1 − λ)U l−1 , l = 1, 2, . . . , (2)
where U 0 = µ (0)
1
and smoothing parameter 0 < λ ≤ 1.
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Control limits: LCL and UCL.
Performance of EWMA control charts.
Table: The out-of-control ARLs of some selected EWMA charts under exponential
competing risks when n = 1, µ (0)
1 = 1 3 and µ 2 = 1
µ (1)
1 /µ(0) 1
EWMA 0.02 EWMA 0.05 EWMA 0.08 EWMA 0.10 EWMA 0.15 EWMA 0.20
0.25 48.21 50.57 46.94 52.14 91.23 404.90
0.50 481.15 1520.83 1935.13 2967.16 7789.23 34822.51
0.75 129907.48 50874.29 22229.49 16106.06 9704.83 7190.29
1.00 500.46 500.47 500.57 499.47 499.58 500.25
1.25 93.35 92.60 95.35 99.73 110.70 121.01
1.50 49.73 43.97 42.43 43.37 47.10 51.66
1.75 34.64 28.83 26.58 26.53 27.79 29.99
Problem arising due to sample size and/or underlying
distribution of competing risks.
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3. CUSUM Control Chart
Form of the CUSUM control charts
Upper-sided CUSUM chart
Z + 0
= 0
Z +
l
= max(0, Z +
l−1 + X l + k), l = 1, 2, . . . , (3)
Lower-sided CUSUM chart
Z − 0
= 0
Z −
l
= min(0, Z −
l−1 + X l + k), l = 1, 2, . . . . (4)
Reference value: k (Hawkins and Olwell (1998))
Control limit: h + and h − 7/23

Intro EWMA CUSUM Comparison Example Future Work References
Two candidates for the CUSUM score X l
Likelihood function
L(Θ) =
nY
i=1
»
f ci (y i | θ ci )
pY
–
¯Fk (y i | θ k ) . (5)
k≠c i
Score function
∂
8> <
log f
s i (θ (0)
∂θ 1 (y i | θ 1 ) ˛ if c i = 1,
1 ˛θ1
1 ) = =θ (0)
1
∂
> : log ¯F (6)
∂θ 1 (y i | θ 1 ) ˛ if c i ≠ 1 .
1 ˛θ1 =θ (0)
1
Log-likelihood ratio (LLR)
j (1) L(θ 1
LLR = log
) ff
L(θ (0)
1 )
nX
„
«
=
log f 1 (y i | θ (1)
1 ) − log f 1(y i | θ (0)
1 )
{i:c i =1}
nX
„
+ log ¯F 1 (y i | θ (1)
1 ) − log ¯F 1 (y i | θ (0)
1
«. ) (7)
{i:c i ≠1}
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Table: Control parameters (ARL 0 = 500) and out-of-control ARLs for score and LLR
CUSUM charts to monitor processes under normal competing risks with µ (0)
1 = 0 and
σ 1 = µ 2 = σ 2 = 1.
Score CUSUM
LLR CUSUM
shift in µ 1 k h ARL1 k h ARL1
n = 1
-2.5σ 1 -1.247 -1.84 2.17 -0.036 -1.78 2.17
-2.0σ 1 -0.994 -2.34 3.09 -0.029 -2.26 3.08
-1.5σ 1 -0.739 -3.10 4.87 -0.021 -3.00 4.87
-1.0σ 1 -0.486 -4.40 9.30 -0.013 -4.28 9.28
-0.5σ 1 -0.237 -7.23 26.74 -0.004 -7.12 26.74
0.5σ 1 0.214 7.16 29.31 -0.006 7.30 29.29
1.0σ 1 0.398 4.46 11.46 -0.029 4.67 11.34
1.5σ 1 0.542 3.34 6.87 -0.073 3.58 6.68
2.0σ 1 0.646 2.78 5.08 -0.141 3.08 4.86
2.5σ 1 0.715 2.48 4.27 -0.231 2.88 4.03
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Intro EWMA CUSUM Comparison Example Future Work References
Table: Control parameters (ARL 0 = 500) and out-of-control ARLs for score and LLR
CUSUM charts to monitor processes under exponential competing risks with µ (0)
1 = 1 3
and µ 2 = 1.
n = 1
Score CUSUM
LLR CUSUM
µ (1)
1 /µ(0) 1
k h ARL1 k h ARL1
0.5 -0.6429 -15.26 22.83 0.0108 -1.55 22.60
0.6 -0.5000 -19.28 34.30 0.0061 -2.00 34.17
0.7 -0.3649 -23.44 53.62 0.0031 -2.53 53.58
0.8 -0.2368 -30.38 92.92 0.0012 -3.29 92.72
0.9 -0.1154 -45.31 187.89 0.0003 -4.96 188.14
1.1 0.1098 41.31 187.16 0.0002 4.63 186.36
1.2 0.2143 35.10 106.58 0.0009 3.96 106.33
1.3 0.3140 30.47 70.34 0.0018 3.48 70.12
1.4 0.4091 27.64 51.69 0.0031 3.17 51.49
1.5 0.5000 25.05 40.41 0.0045 2.91 40.22
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Table: Control parameters (ARL 0 = 500) and out-of-control ARLs for score and LLR
CUSUM charts to monitor processes under Weibull competing risks with µ (0)
1 = 4.43,
µ 2 = 5.32 and β 1 = β 2 = 2.
n = 1
Score CUSUM
LLR CUSUM
µ (1)
1 /µ(0) 1
k h ARL1 k h ARL1
0.5 -0.144 -1.36 9.77 1.339 -43.12 9.33
0.6 -0.116 -1.72 14.54 0.774 -58.48 14.10
0.7 -0.086 -2.23 23.83 0.389 -82.26 23.48
0.8 -0.056 -3.02 44.54 0.162 -120.50 44.32
0.9 -0.027 -4.54 107.78 0.036 -189.26 107.79
1.1 0.026 4.71 119.01 0.000 212.65 118.94
1.2 0.050 3.75 58.01 0.111 172.15 57.65
1.3 0.072 3.12 36.74 0.196 150.30 36.34
1.4 0.092 2.75 26.68 0.274 136.65 26.08
1.5 0.110 2.51 21.11 0.356 127.31 20.51
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4. Comparison
n = 1
ARL1
0 100 200 300 400 500
LLR CUSUM (−1 sd)
EWMA r = 0.05
EWMA r = 0.10
EWMA r = 0.20
−2.5 −2.0 −1.5 −1.0 −0.5 0.0
µ1
Figure: ARL curves for detecting mean decreases under normal competing risks.
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Intro EWMA CUSUM Comparison Example Future Work References
n = 1
ARL1
0 10 20 30 40
LLR CUSUM (−1 sd)
EWMA r = 0.05
EWMA r = 0.10
EWMA r = 0.20
−1.4 −1.2 −1.0 −0.8 −0.6
µ1
Figure: (Continued) ARL curves for detecting mean decreases under normal
competing risks.
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Intro EWMA CUSUM Comparison Example Future Work References
n = 5
ARL1
0 200 400 600 800 1000
LLR CUSUM (0.5)
EWMA r = 0.05
EWMA r = 0.10
EWMA r = 0.20
0.4 0.6 0.8 1.0
mean ratio
Figure: ARL curves for detecting mean decreases under exponential competing
risks.
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Intro EWMA CUSUM Comparison Example Future Work References
n = 5
ARL1
0 10 20 30 40
LLR CUSUM (0.5)
EWMA r = 0.05
EWMA r = 0.10
EWMA r = 0.20
0.3 0.4 0.5 0.6 0.7
mean ratio
Figure: (Continued) ARL curves for detecting mean decreases under exponential
competing risks.
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Intro EWMA CUSUM Comparison Example Future Work References
n = 1
ARL1
0 200 400 600 800 1000
LLR CUSUM (0.7)
EWMA r = 0.05
EWMA r = 0.10
EWMA r = 0.20
0.4 0.6 0.8 1.0
mean ratio
Figure: ARL curves for detecting mean decreases under Weibull competing risks.
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n = 1
ARL1
0 50 100 150 200
LLR CUSUM (0.7)
EWMA r = 0.05
EWMA r = 0.10
EWMA r = 0.20
0.55 0.60 0.65 0.70 0.75 0.80 0.85
mean ratio
Figure: (Continued) ARL curves for detecting mean decreases under Weibull
competing risks.
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Intro EWMA CUSUM Comparison Example Future Work References
5. Example
Problem from Nelson (1982): Production process stops
because of shave die or other causes.
Objective: monitor whether the shave dies exhibit shorter
lifetimes.
Failure times observed: 11 (shave die) vs. 26 (other
causes)
Weibull distribution is fitted for both groups.
Estimated in-control MLEs
β die = 2.76 and η die = 78.35 (i.e. µ die = 69.73)
β other = 2.44 and η other = 57.21 (i.e. µ other = 50.73)
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Intro EWMA CUSUM Comparison Example Future Work References
Survival Probability s(t)
0.0 0.2 0.4 0.6 0.8 1.0
Shave Die
Other Causes
0 20 40 60
Time (t)
Figure: Kaplan-Meier curves and estimated survival functions by using Weibull
MLEs.
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Intro EWMA CUSUM Comparison Example Future Work References
CUSUM Chart for Shave Die
CUSUM Chart for Other Causes
LLR CUSUM
−200000 −100000 0
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0 5 10 15 20 25
LLR CUSUM
−150000 −100000 −50000 0
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0 5 10 15 20 25
Sample
Sample
(a)
(b)
Figure: An example that mean operating hours of shave die decreases after sample
20.
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Intro EWMA CUSUM Comparison Example Future Work References
CUSUM Chart for Shave Die
CUSUM Chart for Other Causes
LLR CUSUM
−120000 −80000 −40000 0
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15 0 5 10 20 25 30 35 40 45
LLR CUSUM
−140000 −100000 −60000 −20000
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0 5 10 15 20 25 30 35 40 45
Sample
Sample
(a)
(b)
Figure: An example that mean operating hours of other causes decreases after
sample 20.
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Intro EWMA CUSUM Comparison Example Future Work References
Future Work
Masked failure problem: the cause(s) of failure could not
be fully identified.
Dependent risks: If the assumption of independence is no
valid, how to extend the methods to take into account
correlation among the competing risks?
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Intro EWMA CUSUM Comparison Example Future Work References
References
Hawkins, D. M. and Olwell, D. H. (1998), Cumulative Sum
Charts and Charting for Quality Improvement, New York:
Springer-Verlag.
Nelson, W. B. (1982), Applied Life Data Analysis, New
Jersey: John Wiley & Sons.
Stainer, S. H. and MacKay, R. J. (2001), Monitoring
Processes with Data Censored Owing to Competing Risks
by Using Exponentially Weighted Moving Average Control
Charts. Applied Statistics, 50, part 3, pp. 293-302.
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