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Quality Control & Statistical Process Control (SPC)
DR. RON FRICKER
PROFESSOR & HEAD, DEPARTMENT OF STATISTICS
DATAWORKS CONFERENCE, MARCH 22, 2018
Agenda
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§ Some Terminology & Background§ SPC Methods & Philosophy§ Univariate Control Charts§ More Advanced Charts: EWMA & CUSUM§ A Bit on Multivariate SPC§ Some Illustrative Advanced Applications
Some Terminology & Background
2018 DATAWORKS CONFERENCE
Traditional Definition of Quality
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Quality means fitness for use• Two aspects:
• Quality of design• Quality of conformance
• Idea: Product must meet requirements of user/customer
Traditional Approach: Inspection Model
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PROCESSINPUTS OUTPUTSINSPECT
REWORK?
SCRAP
Pass
Fail
No
Yes
Modern Definition of Quality
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Quality is inversely proportional to variability
• It’s about consistency• If variability is high,
quality is low• If variability is low,
quality is high
• Idea: Low variability results in a consistent product
Modern Approach: Prevention Model
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PROCESSINPUTS OUTPUTS
Collect Data
Analyze
Improve
Quality Characteristics & Data
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§ Quality characteristics can be:§ Physical (length, weight, viscosity)§ Sensory (taste, color, appearance)§ Time oriented (reliability, durability)
§ Two types of data:§ Attributes - discrete data, often counts§ Variables - continuous measurements (length, weight, etc)
Quality Specifications
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§ Quality characteristics are evaluated against specifications
Specifications are the desired measurements for the quality
characteristics§ The target value is the ideal level of a quality
characteristic
Specification Limits
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§ The upper specification limit (USL) is the largest allowable value for the quality characteristic
§ The lower specification limit (LSL) is the smallest allowable value for the quality characteristic
§ Items that do not meet one or more specifications are called noncomforming
§ A nonconforming item is defective if the safety or the effective use of the product is degraded
11
In a picture:
Target Value
nonconforming
LSL USL
nonconformingconformingconforming
ü All items produced are conforming
X Some items are not
conforming
Definition of Quality Improvement
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Quality Improvement is the reductionof variability in processes and products
• Idea: Quality improvement is:• Waste reduction• More consistent product or process
13
A quality characteristic
ideal
Quality Improvement Quality Improvement
In a picture:
conformingconformingnonconforming nonconforming
SPC Methods & Philosophy
2018 DATAWORKS CONFERENCE
Statistical Process Control (SPC)
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§ A collection of analytical tools § When used can result in process stability and variance reduction§ These days also referred to as Statistical Process Monitoring
§ Most common tool: control chart§ Basics of control charts in this module§ Not covering the rest of the “magnificent seven”
• Histograms • Pareto analysis • Cause and effect diagrams
• Check sheets• Scatter plots • Stratification
Causes of Variation
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§ Chance (or common) causes of variation come from random events§ Natural variability that cannot be controlled§ “Background noise”
§ Assignable (or special) causes of variation come from events that can be controlled or corrected§ Improperly adjusted machines§ Operator errors§ Defective raw material
Goals of SPC
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§ Control chart is a tool to detect assignable causes § Identify and eliminate assignable causes of variation
§ Minimize variation due to common causes§ Structured way to improve process
§ Result: Improved process/product consistency§ Advantages
§ Graphical display of performance§ Accounts for natural randomness§ Removes subjective decision making
Detecting Assignable Causes
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§ A process operating with only chance causes of variation is said to be in (statistical) control§ “In control” does not mean process produces all conforming items
§ A process operating with assignable causes is said to be out-of-control
§ Control chart only detects (possible) assignable causes§ Alarm is an indication of problem – not proof§ Action required by management, operator, engineering to identify
and eliminate assignable causes
19
19
Qua
lity
Cha
ract
eris
tic
observations over time
LSL
USL
capable
not capable
Goal: detect a shift before not capable
In a picture:
Statistical Basis of Control Charts
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§ Choose control limits to guide actions§ If points fall within control limits
§ Assume process in control § No action required
§ If points fall outside control limits§ Evidence process is out of control§ Stop and look for assignable causes
§ Control limits are based on natural process variability§ They are not related to specification limits
§ Setting control limits involve making a trade off between competing requirements§When in control, desire small chance of point falling
outside control limits: low false alarm rate§ Minimize error: concluding the process is out of control
when it is really in control§When out of control, desire high chance of falling out
of control limits§ Want to detect out-of-control condition quickly: high sensitivity
Setting Control Limits
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Setting Up a Control Chart: Phase I & II
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§ Phase I: Retrospective analysis on existing (historical) data to establish appropriate control limits
§ Phase II: Prospective monitoring of the process§ Basic idea in Phase I
§ Gather historical data and use various tools to identify periods with assignable causes
§ Perhaps fix assignable causes, but for purposes of Phase II, eliminate that data for purpose of defining control limits
§ May be an iterative process
Univariate Control Charts
2018 DATAWORKS CONFERENCE
Types of Control Charts
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§ Variables control charts§ For continuous data§ Examples: shaft diameter, motor speed
§ Attributes control charts§ For discrete data§ Examples: number of defects in unit
Control Charts for Variables
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§ Advantage: Provides more process information§ Process mean and variance § Approximate time of process change§ Can indicate impending problems; is a leading indicator
§ Disadvantages§ Perhaps more expensive to take measurements§ If multiple quality characteristics, need multiple charts or
multivariate charts§ Could be more effort and more complicated
Univariate Control Charts
§ Shewhart (1931)§ Stop when observation (or statistic) exceeds pre-defined threshold§ Better for detecting large shifts/changes
§ EWMA (Roberts, 1959)§ Stop when weighted average of observations exceeds threshold§ Very similar in performance to CUSUM
§ CUSUM (Page, 1954)§ Stop when cumulative sum of observations exceeds threshold§ Better for detecting small shifts/changes
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Shewhart (“X-bar”) Charts
§ Observations follow an in-control distribution f0(x), for which we often want to monitor the mean of the distribution
§ If interested in detecting both increases and decreases in the mean, choose thresholds h1 and h2 such that
§ Sequentially observe values of xi; stop and conclude the mean may have shifted at time i if or
12
0{ : or }
( )x x hx h
f x dx p³£
=ò
1ix h³ 2ix h£3/22/18
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Example of a Shewhart Chart
Montgomery, D.C. (2009). Introduction to Statistical Quality Control, John Wiley & Sons, p. 401.
x
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Average Run Length (ARL)
§ ARL is a measure of chart performance§ In-control ARL or ARL0 is expected number of observations
between false signals§ Assuming f0(x) known, time between false signals is geometrically
distributed, so§ Larger ARL0 are preferred
§ Out-of-control ARL or ARL1 is expected number of observations until a true signal for a given out-of-control condition§ For a one-sided test and a particular f1(x),
0ARL 1 p=
1
1 1{ : }
ARL ( )x x h
f x dx-
³
é ù= ê úê úë ûò
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Example: Monitoring a Process with Xi~N(µ,s 2)
§ With 3s control limits, when in-control, probability an observation is outside the control limits is p = 0.0027, so§ If sampling at fixed times, says will get a false signal on
average once every 370 time periods
§ For out-of-control condition where mean shifts up or down 1s, probability an observation is outside the control limits is p = 0.0227, so§ For a 2s shift, § Etc.
0ARL 1 0.0027 370= =
1ARL 1 0.0227 44= »1ARL 1 0.1814 5.5= »
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Shewhart Charts, continued
§ If only interested in detecting increases in the mean, can use a one-sided chart§ Sequentially observe values of xi; stop and conclude the
mean may have shifted at time i if
§ Can also use Shewhart charts to monitor process variation along with mean§ In industrial SPC, called S-charts or R-charts
ix h>
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Why Control Variability?
32
§ Want at/near process target value§ The target (center line) represents best possible quality§ Ideally, would like all items to be exactly at target value§ When is on target, chance of nonconforming items
minimized
§ And controlling variability (with mean at target) gives:§ More consistent process so items are produced at or near
target value§ Fewer nonconforming items
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Process Improvement
33
§ Goal: keep process mean on target with minimum variability§ chart will monitor mean§ R- or S-chart will monitor variability§ Apply in continuous process improvement to make process better
§ Continuously work to reduce variability
§ R charts sometimes preferred to S because R is easy to calculate§ For example, charts might be manually plotted§ Before computers and calculators, S-charts were too hard§ For some users, also easier to understand the range
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3/22/18
Prospective Monitoring
34
§ For prospective monitoring, take samples of size n and:§ For the x-bar chart, plot the sample mean at time i:
§ For the R-chart plot the range: Ri = Xmax - Xmin
§ Or, for the S-chart, plot the standard deviation:
nXXX
X ni
+++=
...21
3/22/18
Si =
vuut 1
n� 1
nX
j=1
�Xj � X̄i
�2
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Setting Up the Charts: Some Notation
35
§ Before beginning monitoring, collect m samples of size n
§ Grand mean:
§ Calculate the average range:
§ Or average standard deviation:
mXXX
X m+++=
...21
mSSS
S m+++=
...213/22/18
R̄ =R1 +R2 + · · ·+Rm
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X-bar and R Charts
36
§ X-bar control chart (using R):
§ R-chart:
Rx
x
Rx
2
2
ALCLLineCenter
AUCL
-=
=
+=
RR
R
3
4
DLCLLineCenter DUCL
=
=
=
3/22/18
Look up constants A2, D3 and D4 in table.
E.g., see Montgomery (2009).
X-bar and S Charts
37
§ X-bar chart (using S):
§ S-chart:
Sx
x
Sx
3
3
ALCLLineCenter
AUCL
-=
=
+=
SS
S
3
4
BLCLLineCenter BUCL
=
=
=
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Look up constants A3, B3 and B4 in table.
E.g., see Montgomery (2009).
Control Charts for Attributes
3/22/18
38
§ Attributes data § Data that can be classified into one of several categories or
classifications§ Classifications such as conforming and nonconforming are
commonly used in quality control
§ Advantages§ Several quality characteristics can be measured at once§ Unit only classified conforming or not§ Classification usually requires less measurement effort§ Applies to non-numeric as well as numeric quality characteristics
Disadvantages of Attribute Charts
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39
§ Less information about the process§ Not a good for quality improvement
§ Has little information about process variability§ Lags process changes, so only find out about problems after
the fact
§ Generally requires larger sample sizes
Example: p-Chart
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40
§ Control chart for fraction nonconforming § Fraction nonconforming: ratio of the number of
nonconforming items to the total number of items
§ Notation§ n = number of units in the sample § D = number of nonconforming units in the sample§ p = (usually unknown) probability of selecting a
nonconforming unit from the sample
Constructing the Chart
3/22/18
41
§ For p unknown, then the control limits for fraction nonconforming are
where and
npppLCL
npppUCL
)1(3
)1(3
--=
-+=
pCL =m
p
mn
Dp
m
ii
m
ii åå
== == 11ˆ
Example: Silicon Wafer Defects
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42
§ The location of 50 chips is measured on 30 silicon wafers§ A defective is a misregistration, in terms of horizontal and/or vertical distances
from the center
§ Results:
Source: www.itl.nist.gov/div898/handbook/pmc/section3/pmc332.htm
Sample Number
Number Defective
Fraction Defective
Sample Number
Number Defective
Fraction Defective
Sample Number
Number Defective
Fraction Defective
1 12 0.24 11 5 0.10 21 20 0.40
2 15 0.30 12 6 0.12 22 18 0.36
3 8 0.16 13 17 0.34 23 24 0.48
4 10 0.20 14 12 0.24 24 15 0.30
5 4 0.08 15 22 0.44 25 9 0.18
6 7 0.14 16 8 0.16 26 12 0.24
7 16 0.32 17 10 0.20 27 7 0.14
8 9 0.18 18 5 0.10 28 13 0.26
9 14 0.28 19 13 0.26 29 9 0.18
10 10 0.20 20 11 0.22 30 6 0.12
Example: Silicon Wafer Defects
3/22/18
43 Source: www.itl.nist.gov/div898/handbook/pmc/section3/pmc332.htm
Example: np-Chart
3/22/18
44
§ Control chart for number nonconforming § Number nonconforming: just the count of nonconforming
items
§ Notation§ n = number of units in the sample § D = number of nonconforming units in the sample§ p = (usually unknown) probability of selecting a
nonconforming unit from the sample
Constructing the Chart
3/22/18
45
§ For p unknown, then the control limits for number nonconforming are
where as beforem
p
mn
Dp
m
ii
m
ii åå
== == 11ˆ
)1(3
)1(3
pnpnpLCL
npCLpnpnpUCL
--=
=
-+=
Example: Silicon Wafer Defects Redux
3/22/18
46 Source: www.itl.nist.gov/div898/handbook/pmc/section3/pmc332.htm
Other Types of Attribute Charts
3/22/18
47
§ c-chart for the number of defects per sample§ May be more than one per unit!§ Do not confuse with np-chart
§ u-chart for the average number of defects per inspection unit§ Do not confuse with p-chart
Choosing Between Control Charts
3/22/18
48 Source: www.cqeacademy.com/cqe-body-of-knowledge/continuous-improvement/quality-control-tools/control-charts/
More Advanced Charts: EWMA & CUSUM
2018 DATAWORKS CONFERENCE
Pros & Cons
3/22/18
50
§ All charts up to now are Shewhart-type control charts§ Characterized by control limits at target value plus or minus
multiples of the statistic standard deviation§ These types of charts have both strengths and weaknesses
§ Shewhart-type chart strengths§ Simple to implement§ Quickly detect large mean shifts
§ Weaknesses§ Insensitive to small shifts§ Sensitizing rules help, but detract from chart simplicity
Exponentially Weighted Moving Average (EWMA) Control Chart
§ The EWMA (exponentially weighted moving average) plots or tracks
§ xi is the observation at time i§ is a constant that governs how much weight is put on
historical observations§ l =1: EWMA reduces to the Shewhart§ Typical values:
§ With appropriate choice of l, can be made to perform similar to Shewhart (or CUSUM)
1(1 )i i iz x zl l -= + -
10 £< l
0.1 0.3l£ £
3/22/18
51
EWMA Chart Example
Montgomery, D.C. (2009). Introduction to Statistical Quality Control, John Wiley & Sons, p. 421.
3/22/18
52
Cumulative Sum (CUSUM) Control Chart
§ The two-sided CUSUM plots two statistics:
typically starting with § Stop when either§ A one-sided test only uses one of the statistics
§ Must choose both k and h§ E.g., Setting h =5s and works well for 1s shift in the mean:
§ ARL0 approximately 465 and ARL1=8.4 (Shewhart: 44)
( ){ }( ){ }
1 0
1 0
max 0,
min 0,
i i i
i i i
C C x k
C C x k
µ
µ
+ +-
- --
= + - -
= + - +
000 == -+ CC0 0 or C h C h+ -> > -
/ 2k s=
3/22/18
53
Two-Sided CUSUM Chart Example
Montgomery, D.C. (2009). Introduction to Statistical Quality Control, John Wiley & Sons, p. 407.
3/22/18
54
A Bit About Multivariate SPC
2018 DATAWORKS CONFERENCE
Multivariate Control Charts
3/22/18
56
§ They are not used as often § More complicated to implement and interpret§ But can be more sensitive to some shifts
§ Some charts:§ T2 chart – generalization of the Shewhart x-bar§ MEWMA – multivariate EWMA§ MCUSUM – multivariate CUSUM
Some Multivariate SPC Methods
§ Hotelling’s T2 (1947)§ Stop when statistical distance to observation
exceeds threshold h§ Like Shewhart, good at detecting large shifts
§ Lowry et al.’s MEWMA (1992)§ Multivariate generalization of univariate EWMA
§ At each time, calculate
§ Stop when
§ Crosier’s MCUSUM (1988)§ Cumulates vectors componentwise
§ As with CUSUM, good at detecting small shifts
)()( 12 µXΣµX -¢-= -T
( ) ( ) 11i i il l -= - + -z x µ z1
i i iE h¥
-¢= S ³zz z
3/22/18
57
An Illustrative Advanced Application
2018 DATAWORKS CONFERENCE
“That’s how it’s gonna be, a little test tube with a-a rubber cap that’s deteriorating... A guy steps out of Times Square Station. Pshht... Smashes it on the sidewalk... There is a world war right there.”
“Josh” West Wing, 1999
Disease Surveillance & Biosurveillance
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59
What is Biosurveillance?
“The term ‘biosurveillance’ means the process of active data-gathering … of biosphere data … in order to achieve early warning of health threats, early detection of health events, and overall situational awareness of disease activity.” [1]
[1] Homeland Security Presidential Directive HSPD-21, October 18, 2007.
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60
61
Challenges in Applying SPC
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Deriving Daily Syndrome Counts
62
§ Examples of “chief complaint” data:
3/22/18
Deriving Daily Syndrome Counts
63
§ Text matching searches for terms in the data to derive symptoms§ E.g., existence of word “flu” results in classifying an individual with the flu
symptom
§ Symptoms then used to determine whether to classify an individual with a syndrome
§ MCHD has used three definitions for ILI syndrome:
3/22/18
Determining the Outbreak Periods
64
3/22/18
Baseline ILI Definition Results
65
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The New Status Quo?
66
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References & Additional Reading
2018 DATAWORKS CONFERENCE
Additional Reading
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68
§ Fricker, Jr., R.D., Introduction to Statistical Mehtodsfor Biosurveillance, Cambridge University Press, 2013.
§ Montgomery, D.C., Introduction to Statistical Quality Control, 7th edition, Wiley, 2012.
§ NIST/SEMATECH e-Handbook of Statistical Methods, https://www.itl.nist.gov/div898/handbook/index.htm, 2012.
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3/22/18