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Quality Control & Statistical Process Control (SPC) DR. RON FRICKER PROFESSOR & HEAD, DEPARTMENT OF STATISTICS DATAWORKS CONFERENCE, MARCH 22, 2018

Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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Page 1: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

Quality Control & Statistical Process Control (SPC)

DR. RON FRICKER

PROFESSOR & HEAD, DEPARTMENT OF STATISTICS

DATAWORKS CONFERENCE, MARCH 22, 2018

Page 2: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 3: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

Some Terminology & Background

2018 DATAWORKS CONFERENCE

Page 4: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 5: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

Traditional Approach: Inspection Model

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PROCESSINPUTS OUTPUTSINSPECT

REWORK?

SCRAP

Pass

Fail

No

Yes

Page 6: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 7: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

Modern Approach: Prevention Model

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PROCESSINPUTS OUTPUTS

Collect Data

Analyze

Improve

Page 8: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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)

Page 9: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 10: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 11: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

11

In a picture:

Target Value

nonconforming

LSL USL

nonconformingconformingconforming

ü All items produced are conforming

X Some items are not

conforming

Page 12: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 13: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

13

A quality characteristic

ideal

Quality Improvement Quality Improvement

In a picture:

conformingconformingnonconforming nonconforming

Page 14: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

SPC Methods & Philosophy

2018 DATAWORKS CONFERENCE

Page 15: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 16: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 17: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 18: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 19: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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:

Page 20: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 21: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

§ 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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Page 22: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 23: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

Univariate Control Charts

2018 DATAWORKS CONFERENCE

Page 24: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 25: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 26: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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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Page 27: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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£

1ix h³ 2ix h£3/22/18

27

Page 28: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

Example of a Shewhart Chart

Montgomery, D.C. (2009). Introduction to Statistical Quality Control, John Wiley & Sons, p. 401.

x

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Page 29: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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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Page 30: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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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Page 31: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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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Page 32: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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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3/22/18

Page 33: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

X̄<latexit sha1_base64="QEfW5qX4B/i2hyuf3b1ThgHTdtA=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqN6KXjxWMLbQhjLZbtqlm03Y3Qgl9Ed48aDi1f/jzX/jts1BWx8MPN6bYWZemAqujet+O6W19Y3NrfJ2ZWd3b/+genj0qJNMUebTRCSqE6JmgkvmG24E66SKYRwK1g7HtzO//cSU5ol8MJOUBTEOJY84RWOldi9ElXem/WrNrbtzkFXiFaQGBVr96ldvkNAsZtJQgVp3PTc1QY7KcCrYtNLLNEuRjnHIupZKjJkO8vm5U3JmlQGJEmVLGjJXf0/kGGs9iUPbGaMZ6WVvJv7ndTMTXQU5l2lmmKSLRVEmiEnI7Hcy4IpRIyaWIFXc3kroCBVSYxOq2BC85ZdXiX9Rv65795e15k2RRhlO4BTOwYMGNOEOWuADhTE8wyu8Oanz4rw7H4vWklPMHMMfOJ8/30CPdQ==</latexit><latexit sha1_base64="QEfW5qX4B/i2hyuf3b1ThgHTdtA=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqN6KXjxWMLbQhjLZbtqlm03Y3Qgl9Ed48aDi1f/jzX/jts1BWx8MPN6bYWZemAqujet+O6W19Y3NrfJ2ZWd3b/+genj0qJNMUebTRCSqE6JmgkvmG24E66SKYRwK1g7HtzO//cSU5ol8MJOUBTEOJY84RWOldi9ElXem/WrNrbtzkFXiFaQGBVr96ldvkNAsZtJQgVp3PTc1QY7KcCrYtNLLNEuRjnHIupZKjJkO8vm5U3JmlQGJEmVLGjJXf0/kGGs9iUPbGaMZ6WVvJv7ndTMTXQU5l2lmmKSLRVEmiEnI7Hcy4IpRIyaWIFXc3kroCBVSYxOq2BC85ZdXiX9Rv65795e15k2RRhlO4BTOwYMGNOEOWuADhTE8wyu8Oanz4rw7H4vWklPMHMMfOJ8/30CPdQ==</latexit><latexit sha1_base64="QEfW5qX4B/i2hyuf3b1ThgHTdtA=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqN6KXjxWMLbQhjLZbtqlm03Y3Qgl9Ed48aDi1f/jzX/jts1BWx8MPN6bYWZemAqujet+O6W19Y3NrfJ2ZWd3b/+genj0qJNMUebTRCSqE6JmgkvmG24E66SKYRwK1g7HtzO//cSU5ol8MJOUBTEOJY84RWOldi9ElXem/WrNrbtzkFXiFaQGBVr96ldvkNAsZtJQgVp3PTc1QY7KcCrYtNLLNEuRjnHIupZKjJkO8vm5U3JmlQGJEmVLGjJXf0/kGGs9iUPbGaMZ6WVvJv7ndTMTXQU5l2lmmKSLRVEmiEnI7Hcy4IpRIyaWIFXc3kroCBVSYxOq2BC85ZdXiX9Rv65795e15k2RRhlO4BTOwYMGNOEOWuADhTE8wyu8Oanz4rw7H4vWklPMHMMfOJ8/30CPdQ==</latexit><latexit sha1_base64="QEfW5qX4B/i2hyuf3b1ThgHTdtA=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqN6KXjxWMLbQhjLZbtqlm03Y3Qgl9Ed48aDi1f/jzX/jts1BWx8MPN6bYWZemAqujet+O6W19Y3NrfJ2ZWd3b/+genj0qJNMUebTRCSqE6JmgkvmG24E66SKYRwK1g7HtzO//cSU5ol8MJOUBTEOJY84RWOldi9ElXem/WrNrbtzkFXiFaQGBVr96ldvkNAsZtJQgVp3PTc1QY7KcCrYtNLLNEuRjnHIupZKjJkO8vm5U3JmlQGJEmVLGjJXf0/kGGs9iUPbGaMZ6WVvJv7ndTMTXQU5l2lmmKSLRVEmiEnI7Hcy4IpRIyaWIFXc3kroCBVSYxOq2BC85ZdXiX9Rv65795e15k2RRhlO4BTOwYMGNOEOWuADhTE8wyu8Oanz4rw7H4vWklPMHMMfOJ8/30CPdQ==</latexit>

3/22/18

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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

<latexit sha1_base64="HoSe3Pi47rFUEAJkyUIq+ji5v5I=">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</latexit><latexit sha1_base64="HoSe3Pi47rFUEAJkyUIq+ji5v5I=">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</latexit><latexit sha1_base64="HoSe3Pi47rFUEAJkyUIq+ji5v5I=">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</latexit><latexit sha1_base64="HoSe3Pi47rFUEAJkyUIq+ji5v5I=">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</latexit>

Page 35: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

m<latexit sha1_base64="8lfwB1bLFbL0QaU77ezZ/DRFEfA=">AAACD3icbVBNS8NAEN3Ur1q/qh69LBZRKJSkCOpBKHrxWIuxhSaEzWbTLt1Nwu5GKCE/wYt/xYsHFa9evflv3LY5aOuDgcd7M8zM8xNGpTLNb6O0tLyyulZer2xsbm3vVHf37mWcCkxsHLNY9HwkCaMRsRVVjPQSQRD3Gen6o+uJ330gQtI4ulPjhLgcDSIaUoyUlrzqseMjkXVyeAmdUCCcdTyr3vGadQcHsZJ12PF4nvHcq9bMhjkFXCRWQWqgQNurfjlBjFNOIoUZkrJvmYlyMyQUxYzkFSeVJEF4hAakr2mEOJFuNn0oh0daCWAYC12RglP190SGuJRj7utOjtRQznsT8T+vn6rw3M1olKSKRHi2KEwZVDGcpAMDKghWbKwJwoLqWyEeIp2L0hlWdAjW/MuLxG42LhrW7WmtdVWkUQYH4BCcAAucgRa4AW1gAwwewTN4BW/Gk/FivBsfs9aSUczsgz8wPn8AQR6brw==</latexit><latexit sha1_base64="8lfwB1bLFbL0QaU77ezZ/DRFEfA=">AAACD3icbVBNS8NAEN3Ur1q/qh69LBZRKJSkCOpBKHrxWIuxhSaEzWbTLt1Nwu5GKCE/wYt/xYsHFa9evflv3LY5aOuDgcd7M8zM8xNGpTLNb6O0tLyyulZer2xsbm3vVHf37mWcCkxsHLNY9HwkCaMRsRVVjPQSQRD3Gen6o+uJ330gQtI4ulPjhLgcDSIaUoyUlrzqseMjkXVyeAmdUCCcdTyr3vGadQcHsZJ12PF4nvHcq9bMhjkFXCRWQWqgQNurfjlBjFNOIoUZkrJvmYlyMyQUxYzkFSeVJEF4hAakr2mEOJFuNn0oh0daCWAYC12RglP190SGuJRj7utOjtRQznsT8T+vn6rw3M1olKSKRHi2KEwZVDGcpAMDKghWbKwJwoLqWyEeIp2L0hlWdAjW/MuLxG42LhrW7WmtdVWkUQYH4BCcAAucgRa4AW1gAwwewTN4BW/Gk/FivBsfs9aSUczsgz8wPn8AQR6brw==</latexit><latexit sha1_base64="8lfwB1bLFbL0QaU77ezZ/DRFEfA=">AAACD3icbVBNS8NAEN3Ur1q/qh69LBZRKJSkCOpBKHrxWIuxhSaEzWbTLt1Nwu5GKCE/wYt/xYsHFa9evflv3LY5aOuDgcd7M8zM8xNGpTLNb6O0tLyyulZer2xsbm3vVHf37mWcCkxsHLNY9HwkCaMRsRVVjPQSQRD3Gen6o+uJ330gQtI4ulPjhLgcDSIaUoyUlrzqseMjkXVyeAmdUCCcdTyr3vGadQcHsZJ12PF4nvHcq9bMhjkFXCRWQWqgQNurfjlBjFNOIoUZkrJvmYlyMyQUxYzkFSeVJEF4hAakr2mEOJFuNn0oh0daCWAYC12RglP190SGuJRj7utOjtRQznsT8T+vn6rw3M1olKSKRHi2KEwZVDGcpAMDKghWbKwJwoLqWyEeIp2L0hlWdAjW/MuLxG42LhrW7WmtdVWkUQYH4BCcAAucgRa4AW1gAwwewTN4BW/Gk/FivBsfs9aSUczsgz8wPn8AQR6brw==</latexit><latexit sha1_base64="8lfwB1bLFbL0QaU77ezZ/DRFEfA=">AAACD3icbVBNS8NAEN3Ur1q/qh69LBZRKJSkCOpBKHrxWIuxhSaEzWbTLt1Nwu5GKCE/wYt/xYsHFa9evflv3LY5aOuDgcd7M8zM8xNGpTLNb6O0tLyyulZer2xsbm3vVHf37mWcCkxsHLNY9HwkCaMRsRVVjPQSQRD3Gen6o+uJ330gQtI4ulPjhLgcDSIaUoyUlrzqseMjkXVyeAmdUCCcdTyr3vGadQcHsZJ12PF4nvHcq9bMhjkFXCRWQWqgQNurfjlBjFNOIoUZkrJvmYlyMyQUxYzkFSeVJEF4hAakr2mEOJFuNn0oh0daCWAYC12RglP190SGuJRj7utOjtRQznsT8T+vn6rw3M1olKSKRHi2KEwZVDGcpAMDKghWbKwJwoLqWyEeIp2L0hlWdAjW/MuLxG42LhrW7WmtdVWkUQYH4BCcAAucgRa4AW1gAwwewTN4BW/Gk/FivBsfs9aSUczsgz8wPn8AQR6brw==</latexit>

Page 36: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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).

Page 37: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

=

=

=

3/22/18

Look up constants A3, B3 and B4 in table.

E.g., see Montgomery (2009).

Page 38: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 39: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

Disadvantages of Attribute Charts

3/22/18

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

Page 40: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

Example: p-Chart

3/22/18

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

Page 41: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 42: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

Example: Silicon Wafer Defects

3/22/18

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

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Example: Silicon Wafer Defects

3/22/18

43 Source: www.itl.nist.gov/div898/handbook/pmc/section3/pmc332.htm

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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

Page 45: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

--=

=

-+=

Page 46: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

Example: Silicon Wafer Defects Redux

3/22/18

46 Source: www.itl.nist.gov/div898/handbook/pmc/section3/pmc332.htm

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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

Page 48: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

Choosing Between Control Charts

3/22/18

48 Source: www.cqeacademy.com/cqe-body-of-knowledge/continuous-improvement/quality-control-tools/control-charts/

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More Advanced Charts: EWMA & CUSUM

2018 DATAWORKS CONFERENCE

Page 50: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 51: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

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EWMA Chart Example

Montgomery, D.C. (2009). Introduction to Statistical Quality Control, John Wiley & Sons, p. 421.

3/22/18

52

Page 53: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

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

Page 54: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

Two-Sided CUSUM Chart Example

Montgomery, D.C. (2009). Introduction to Statistical Quality Control, John Wiley & Sons, p. 407.

3/22/18

54

Page 55: Quality Control & Statistical Process Control (SPC) · 2018-03-27 · Statistical Process Control (SPC) 3/22/18 15 §A collection of analytical tools §When used can result in process

A Bit About Multivariate SPC

2018 DATAWORKS CONFERENCE

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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

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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

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An Illustrative Advanced Application

2018 DATAWORKS CONFERENCE

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“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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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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Challenges in Applying SPC

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Deriving Daily Syndrome Counts

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§ Examples of “chief complaint” data:

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Deriving Daily Syndrome Counts

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§ 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:

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Determining the Outbreak Periods

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Baseline ILI Definition Results

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The New Status Quo?

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References & Additional Reading

2018 DATAWORKS CONFERENCE

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Additional Reading

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§ 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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Copyright 2017 • Virginia Tech • All Rights Reserved69

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