Quality Control & Statistical Process Control (SPC)

Transcription

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

2 Agenda Some Terminology & Background SPC Methods & Philosophy Univariate Control Charts More Advanced Charts: EWMA & CUSUM A Bit on Multivariate SPC Some Illustrative Advanced Applications 2

3 Some Terminology & Background 2018 DATAWORKS CONFERENCE

4 Traditional Definition of Quality Quality means fitness for use Two aspects: Quality of design Quality of conformance Idea: Product must meet requirements of user/customer 4

5 Traditional Approach: Inspection Model INPUTS PROCESS INSPECT Pass OUTPUTS Fail Yes REWORK? No SCRAP 5

6 Modern Definition of Quality Quality is inversely proportional to variability It s about consistency If variability is high, quality is low If variability is low, quality is high 6 Idea: Low variability results in a consistent product

7 Modern Approach: Prevention Model INPUTS PROCESS OUTPUTS Improve Collect Data Analyze 7

8 Quality Characteristics & Data 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) 8

9 Quality Specifications 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 9

10 Specification Limits 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 10

11 In a picture: ü All items produced are conforming X Some items are not conforming nonconforming conforming conforming nonconforming LSL Target Value USL 11

12 Definition of Quality Improvement Quality Improvement is the reduction of variability in processes and products Idea: Quality improvement is: Waste reduction More consistent product or process 12

13 In a picture: Quality Improvement Quality Improvement nonconforming conforming conforming nonconforming ideal A quality characteristic 13

14 SPC Methods & Philosophy 2018 DATAWORKS CONFERENCE

15 Statistical Process Control (SPC) 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 15

16 Causes of Variation 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 16

17 Goals of SPC 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 17

18 Detecting Assignable Causes 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 18

19 In a picture: not capable USL Quality Characteristic observations over time capable Goal: detect a shift before not 19capable LSL 19

20 Statistical Basis of Control Charts 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 20

21 Setting Control 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 21

22 Setting Up a Control Chart: Phase I & II 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 22

23 Univariate Control Charts 2018 DATAWORKS CONFERENCE

24 Types of Control Charts Variables control charts For continuous data Examples: shaft diameter, motor speed Attributes control charts For discrete data Examples: number of defects in unit 24

25 Control Charts for Variables 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 25

26 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 26

27 Shewhart ( X-bar ) Charts Observations follow an in-control distribution f 0 (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 h 1 and h 2 such that ò { x: x³ h or x h } 1 2 f ( x) dx = p 0 Sequentially observe values of x i ; stop and conclude the mean may have shifted at time i if x h or x h i ³ 1 i 2 27

28 x Example of a Shewhart Chart 28 Montgomery, D.C. (2009). Introduction to Statistical Quality Control, John Wiley & Sons, p. 401.

29 Average Run Length (ARL) 29 ARL is a measure of chart performance In-control ARL or ARL 0 is expected number of observations between false signals Assuming f 0 (x) known, time between false signals is geometrically distributed, so ARL = 1 p Larger ARL 0 are preferred 0 Out-of-control ARL or ARL 1 is expected number of observations until a true signal for a given out-of-control condition é ù For a one-sided test and a particular f 1 (x), ARL 1 = ê ò f1( x) dxú êë{ xx : ³ h} úû -1

30 Example: Monitoring a Process with X i ~N(µ,s 2 ) With 3s control limits, when in-control, probability an observation is outside the control limits is p = , so ARL = = 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 = , so ARL = » 44 For a 2s shift, Etc. ARL = »

31 Shewhart Charts, continued If only interested in detecting increases in the mean, can use a one-sided chart Sequentially observe values of x i ; stop and conclude the mean may have shifted at time i if x > h Can also use Shewhart charts to monitor process variation along with mean In industrial SPC, called S-charts or R-charts i 31

32 <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> <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> Why Control Variability? Want at/near process target value X<latexit sha1_base64="qefw5qx4b/i2hyuf3b1thghtdta=">aaab7xicbvbns8naej3ur1q/qh69lbbbu0leqn6kxjxwmlbqhjlzbtqlm03y3qgl9ed48adi1f/jzx/jts1bwx8mpn6bywzemaqujet+o6w19y3nrfj2zwd3b/+genj0qjnmuebtrcsqe6jmgkvmg24e66skyrwk1g7htzo//csu5ol8mjoubteojy84rwoldi9elxem/wrnrbtzkfxifaqgbvr96ldvknasztjqgvp3ptc1qy7kccrytnllneurjnhiupzkjjko8vm5u3jmlqgjemvlgjjxf0/kggs9iupbgamz6wvvjv7ndtmtxqu5l2lmmkslrvemieni7hcy4ipriyawifxc3krocbvsyxoq2bc85zdxix9rv65795e15k2rrhlo4btowymgnoeowuadhte8wyu8oanz4rw7h4vwklpmhmmfoj8/30cpdq==</latexit> 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 X<latexit sha1_base64="qefw5qx4b/i2hyuf3b1thghtdta=">aaab7xicbvbns8naej3ur1q/qh69lbbbu0leqn6kxjxwmlbqhjlzbtqlm03y3qgl9ed48adi1f/jzx/jts1bwx8mpn6bywzemaqujet+o6w19y3nrfj2zwd3b/+genj0qjnmuebtrcsqe6jmgkvmg24e66skyrwk1g7htzo//csu5ol8mjoubteojy84rwoldi9elxem/wrnrbtzkfxifaqgbvr96ldvknasztjqgvp3ptc1qy7kccrytnllneurjnhiupzkjjko8vm5u3jmlqgjemvlgjjxf0/kggs9iupbgamz6wvvjv7ndtmtxqu5l2lmmkslrvemieni7hcy4ipriyawifxc3krocbvsyxoq2bc85zdxix9rv65795e15k2rrhlo4btowymgnoeowuadhte8wyu8oanz4rw7h4vwklpmhmmfoj8/30cpdq==</latexit> And controlling variability (with mean at target) gives: More consistent process so items are produced at or near target value Fewer nonconforming items 32

33 <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> Process Improvement Goal: keep process mean on target with minimum variability chart will monitor mean X<latexit sha1_base64="qefw5qx4b/i2hyuf3b1thghtdta=">aaab7xicbvbns8naej3ur1q/qh69lbbbu0leqn6kxjxwmlbqhjlzbtqlm03y3qgl9ed48adi1f/jzx/jts1bwx8mpn6bywzemaqujet+o6w19y3nrfj2zwd3b/+genj0qjnmuebtrcsqe6jmgkvmg24e66skyrwk1g7htzo//csu5ol8mjoubteojy84rwoldi9elxem/wrnrbtzkfxifaqgbvr96ldvknasztjqgvp3ptc1qy7kccrytnllneurjnhiupzkjjko8vm5u3jmlqgjemvlgjjxf0/kggs9iupbgamz6wvvjv7ndtmtxqu5l2lmmkslrvemieni7hcy4ipriyawifxc3krocbvsyxoq2bc85zdxix9rv65795e15k2rrhlo4btowymgnoeowuadhte8wyu8oanz4rw7h4vwklpmhmmfoj8/30cpdq==</latexit> 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 33

34 <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> Prospective Monitoring For prospective monitoring, take samples of size n and: For the x-bar chart, plot the sample mean at time i: X i = X 1 + X X n n 34 For the R-chart plot the range: R i = X max - X min Or, for the S-chart, plot the standard deviation: S i = v u t 1 n 1 nx j=1 X j Xi 2

35 Setting Up the Charts: Some Notation Before beginning monitoring, collect m samples of size n Grand mean: X = X 1 + X m X m 35 Calculate the average range: Or average standard deviation: R = R 1 + R R m m S = S 1 + S S m m

36 X-bar and R Charts X-bar control chart (using R): UCL = x + A 2 R Center Line = x LCL = x - A 2 R R-chart: UCL = LCL = D D 4 3 R Center Line = R R Look up constants A 2, D 3 and D 4 in table. E.g., see Montgomery (2009). 36

37 X-bar and S Charts X-bar chart (using S): UCL = x + A 3 S Center Line = x LCL = x - A 3 S S-chart: UCL = LCL = B B 4 3 S Center Line S = S Look up constants A 3, B 3 and B 4 in table. E.g., see Montgomery (2009). 37

38 Control Charts for Attributes 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 38

39 Disadvantages of Attribute Charts 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 39

40 Example: p-chart 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 40

41 Constructing the Chart For p unknown, then the control limits for fraction nonconforming are p(1 - p) UCL = p + 3 n p(1 - p) LCL = p - 3 n where CL = p and p m å D mn m å i = i= 1 = i= 1 m pˆ i 41

42 Example: Silicon Wafer Defects 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: Sample Number Number Defective Fraction Defective Sample Number Number Defective Fraction Defective Sample Number Number Defective Fraction Defective Source:

43 Example: Silicon Wafer Defects 43 Source:

44 Example: np-chart 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 44

45 Constructing the Chart For p unknown, then the control limits for number nonconforming are UCL CL = LCL = np + 3 np = np - 3 np(1 - np(1 - p) p) 45 where as before p m å D mn m å i = i= 1 = i= 1 m pˆ i

46 Example: Silicon Wafer Defects Redux 46 Source:

47 Other Types of Attribute Charts 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 47

48 Choosing Between Control Charts 48 Source:

49 More Advanced Charts: EWMA & CUSUM 2018 DATAWORKS CONFERENCE

50 Pros & Cons 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 50

51 Exponentially Weighted Moving Average (EWMA) Control Chart The EWMA (exponentially weighted moving average) plots or tracks x i is the observation at time i 0 < l 1 is a constant that governs how much weight is put on historical observations l =1: EWMA reduces to the Shewhart Typical values: z = x + (1- ) z l l - i i i 0.1 l 0.3 With appropriate choice of l, can be made to perform similar to Shewhart (or CUSUM) 1 51

52 EWMA Chart Example 52 Montgomery, D.C. (2009). Introduction to Statistical Quality Control, John Wiley & Sons, p. 421.

53 Cumulative Sum (CUSUM) Control Chart The two-sided CUSUM plots two statistics: typically starting with Stop when either C + 0 > h or C - 0 >-h A one-sided test only uses one of the statistics Must choose both k and h k = s /2 ( µ ) { } C = max 0, C + x - -k i i i ( µ ) { } C = min 0, C + x - + k i i i C + C 0 = 0 - = E.g., Setting h =5s and works well for 1s shift in the mean: ARL 0 approximately 465 and ARL 1 =8.4 (Shewhart: 44) 0 53

54 Two-Sided CUSUM Chart Example 54 Montgomery, D.C. (2009). Introduction to Statistical Quality Control, John Wiley & Sons, p. 407.

55 A Bit About Multivariate SPC 2018 DATAWORKS CONFERENCE

56 Multivariate Control Charts They are not used as often More complicated to implement and interpret But can be more sensitive to some shifts Some charts: T 2 chart generalization of the Shewhart x-bar MEWMA multivariate EWMA MCUSUM multivariate CUSUM 56

57 Some Multivariate SPC Methods Hotelling s T 2 (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 E = z S z ³ -1 i i z i Crosier s MCUSUM (1988) Cumulates vectors componentwise As with CUSUM, good at detecting small shifts T ( ) ( 1 ) 1 zi = l x i - µ + - l zi- h 2-1 = ( X - µ ) Σ ( X - µ ) 57

58 An Illustrative Advanced Application 2018 DATAWORKS CONFERENCE

59 Disease Surveillance & Biosurveillance 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,

60 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] 60 [1] Homeland Security Presidential Directive HSPD-21, October 18, 2007.

61 Challenges in Applying SPC 61

62 Deriving Daily Syndrome Counts Examples of chief complaint data: 62

63 Deriving Daily Syndrome Counts 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: 63

64 Determining the Outbreak Periods 64

65 Baseline ILI Definition Results 65

66 The New Status Quo? 66

67 References & Additional Reading 2018 DATAWORKS CONFERENCE

68 Additional Reading Fricker, Jr., R.D., Introduction to Statistical Mehtods for Biosurveillance, Cambridge University Press, Montgomery, D.C., Introduction to Statistical Quality Control, 7 th edition, Wiley, NIST/SEMATECH e-handbook of Statistical Methods,

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