Understanding Control Charting: Techniques and Assumptions

Transcription

1 Undestanding Cntl Chating: Techniques and Assumptins Chales H. Daby, Health Sevices Evaluatin, May Clinic, Rcheste, Minnesta Abstact Evey pcess has sme cmpnent f vaiatin, and by measuing that vaiatin diectly via cntl chats it becmes pssible t detemine when that vaiatin is attibutable t andmness (typically temed cmmn-cause vaiatin), t sme identifiable cause (usually efeed t as special-cause vaiatin). Tw such cntol chats-the pptin nncnfming p-chat and the mean value x-ba chat ae imptant tls f quantitative effts at quality impvement. This pape will addess the cmpnents f cntol chating, the assumptins which fm the basis f these chat's cnstuctin, the classes f cntl chats, techniques f quick access t the SAS-QC Cntl Chating Mdule via SAS-ASSlS, and hw t ecgnize changes in chat pattens which might indicate "special cause" vaiatin. Intductin Cntl chating has becme a widely used technique in quality impvement since its intductin by D. Walte Shewhat f Bell Labaties in the 1920' s. Althugh the techniques exist which ae useful in quality effts. cntol chats ae amng thse which ae mst technically sphisticated. The ealiest uses wee in the mniting f pductin and manufactuing pcesses, hweve in the mst ecent yeas these techniques have been intduced t pactically evey sect f u ecnmy. CntOl chats display a single seies f measuements taken in sequence fm a pcess. By viewing the data in a set f cntol chats, investigats can gain a bette undestanding f vaiatin in a given pcess set f pcesses. Chating Cmpnents At a minimum. each cnstucted cntl chat has the fllwing cmpnents: I. The cente line, 2. Uppe and Lwe Cntl Limits 3. Data pints and line. and 4. Waning Limits (in sme instances). These cmpnents ae highlighted in Figue I belw. Figue 1-- Cntl Chat Cmpnents p 12 c 1 D t 8 f 0 6 P 4 R D P 2 D Su.gup Inde. Subgup Size.: liin n=234 II =654 8 I D [PERIOD) 2,," Limits: UCL P=7.3 lcl 248

2 Cntl Chat Initiatin in SAS-QC P-Chating At the fist SAS-ASSIST sceen. chse Data Analysis, Duality Cntl, and Cntl. Identify the data set cntaining the data values. and select Pgtins f nncnfming unde the Chat f Attibutes listing n Types f Cntl Chats Menu. The system will pmpt f a fixed subgup size f all bsevatins in the and f the subgup vaiable name whse values descibe the cmpnents t be aveaged f each data pint. SAS will als pmpt f the pcess vaiable name (in this case the pptin, vaiable in the p-chat) and ne f the fu methds identified peviusly t btain the cntl limits. If this data begins a new pcess measuement, I wuld ecmmend the fll"st chice--cmpute cntl limits fm the data Optins may be selected n the Additinal Optins sceens. The ptins sceen entitled TNts f Special Causes identifies Zne A. B. and C. These keywds descibe values within and I-sigma distances f the cente line espectively. Once the ptins and custmizatin chices ae cmpleted. click n the B.lm bunn t pepae the chat. If the value f the cente line is "in the neighbhd" f 0 1. and the pcess vaiability is high. limits ae set badly. then the pssibility exits that an uppe lwe cntl limit will exceed ne fall belw ze. The SAS system will autmatically adjust these limits t eflect these limiting values shuld this ccu. This shuld be the nly ccasin in which the p-chat limits ae nn-symmetic with espect t the cente line. X-Ba Chating The pcess f x-ba chating initiatin is vitually identical t that f p-chating. Select Mean and Range Chats unde the Type f Cntl Chat buttn. The ange chat which is paied t the mean (x-ba) chat indicates the diffeence between the minimum value f the pcess vaiable and the maximum value in each subgup. The ange chat is cnsideed t be a subsidiay chat t the x-ba. but can be used t pfile hw vaiability itself changes between identified subgups. Again. nce chat ptins have been selected (if any). chse the Run bunn t pepae the chat. Patten Recgnitin Once statistical cntl has been established. if special-cause vaiatin is pesent. sme type f nn-andm patten will emege fm the sequence f pints which ae fmed fm the subgups. F chats having 2-sigma {3-sigma} cntl limits. anyne f the pattens in Table 2 belw pvides sufficient justificatin t seach f ne me assignable causes. Items A. B. C. and D have will ccu by chance alne appximately I in 20 {I in 300} times f nmally distibuted data. Table 2-Pattems indicative f "special cause" vaiatin f 2-sigma and {3-sigma} cntl limits. A. Any single pint utside a pltted 2-sigma {3- sigma} cntl limit. B. A un f at least 5 {S} points shwing an inceasing tend a deceasing tend. C. Any 3 f 5 {4 f 51 cnsecutive pints falling utside eithe 1-sigma cntl limit (in Zne B futhe). D. Any un f 5 {15} cnsecutive pints n eithe side f the cente line. E. Any unusual nn-andm pattem in the data (such as a cyclical apid swinging pattem) _.. _ _--_.. _---- See Figues 2A-2E f examples f these pattens. F additinal infmatin. see Refeence {5} beginning n page 117. Once chat stability has been veified and a nncyclical tend f data is bseved, cnside using test f hyptheses t veify the tend's significance. If veified. it will be apppiate t ecast the chat n a new cente line and cntl limits f cntinued mniling. 249

3 pcess had attained "statistical cntl" and futue measuements will infm apppiately. All cntl limits (including waning limits) may be set in SAS in ne f fu ways: 1. Cmputed diectly fm the data. 2. Manually enteed based n pcess mean and standad deviatin, 3. Laded fm an existing data set, 4. Manually enteed as LCL, UCL andl aveage with multiple sigma values. The pmpts within SAS-ASSIST guide uses t make ne f these selectins and t allw selectin f the pcessing and display paametes. Chating Assumptins In designing any cntl chat f mniting a pcess, thee assumptins abut the pcess measuement ae made. The fist is that the pcess fm which the data is being taken is stable, ( i.e. that the data ae independent f each the and identically distibuted in each subgup). The secnd assumptin is that ealvalued data is at least appximately nmal in distibutin, in the case f binmial (yesln) data, that the data is well-appximated by the nmal distibutin. If these assumptin ae nt suppted by the fist values taken fm the pcess, additinal steps may be necessay t adjust the data sthat this expectatin is justified. See Refeences [1], [2], and [3] f additinal infmatin. The thid assumptin is that II sampled cmpnent fm the pcess is included in nly ne sampling subgup. If the same cmpnent can be selected s as t be included in me than ne subgup, autcelatin will be intduced int the chat. that the pcess is ut f statistical cntl me ften than it eally is. F example, suppse we cnstucted an x-ba chat with 3-sigma cntl limits. We wuld nmally expect that abut J subgup in 300 wuld fall utside the cntl limits puely by chance. Sevee autcelatin might make this seies f measuements appea "ut f cntl" pehaps thee t five times as ften. F additinal infmatin see Refeences [31. [4].. Cntl Chat Styles Thee ae tw badjy-defined classes f cntl chats. One class, knwn as Vaiable Cntl Chats tack sme eal-valued pcess. Examples might include the length f a manufactued cmpnent the time-t-failue f an electical cicuit. The mst widely used vaiable chat is the X-ba chat. The secnd class f chats is the Attibute Cntl Chat. A cmmn chat in this class, the p-chat plots a subgup pptin f items which cnfm fail t cnfm t a defined standad. Examples wuld include the pptin f pats meeting a cetain numbe f quality chaacteistics, the pptin f patients impving afte a medical pcedue has been pefmed. Attibute chats in geneal lack the sensitivity t pcess change that vaiable chats pvide f equivalent sample sizes. Hweve. in many instances, atti bute chats ae a bene chice due t ecnmic time cnstaints. Setting up vaiable cntl chating sampling may equie additinal expense and time f pttyping data acquisitin methds. F additinal details see Refeence [5]. Autcelatin, depending n the extent t which multiple subgups have the same data. can ange in seveity fm a min pblem, t a maj ne. If pesent. the chat will display geate vaiability than is exists in the pcess. If sevee, it culd lead a use t believe 250

4 -The cente line f a cntl chat can be thught f as descibing the tue estimated pcess mean. Fequently, little is knw abut the pcess t be measued. s a chice is made t estimate the tue pcess mean typically as the weighted aveage f the data cntibuting t the fist 5 t 8 data pints. In the SAS system, each data pint epesents the aveage f the pcess vaiable values in gups defined by values f the subgup vaiable. Table I shws hw SAS detemines subgup values f a p-chan. Table I-SAS System Vaiable Intepetatin f P-Chat Cnstuctin Example. Peid Sample Size Defectives % 19.1% 12.4% 12.6% 19.9% 12.6% 16.8% 14.2% 11.8% Peid descibes the sequential subgup f pltting, Sample size is the value n which cntl limits ae detemined, the ati f defectives t sample size pduces the pecentage which is the pcess vaiable. Once pltted. these data values ae then cnnected by a seies f line segments t fm the data line. When cmpleted t a given measuement, tends in the chat may becme appaent. It is these tends which allw infeence n the effect f a pcess impvement effn. F claity pupses, each pcess vaiable f inteest in a data set is pltted n its wn cntl chat. The Uppc (UCL) and Lwe (LCL) Cntl Limits ae set as theshlds. Shuld data pints fall at beynd eithe limit, "Special Cause" vaiatin is indicated. As seen in Figue I, the UCL and LCL ae paallel and symmetic with espect t the cente line. When the cente line itself is nea an uppe lwe tleance specificatin limit. the cntl limits may nt be equidistant fm the cente line. This cncept will be funbe discussed in the sectin n p chating. Cntl limits geneally need t be chsen with cae. Limits which ae t clse t the cente line will falsely indicate that a pcess is "ut f statistical cntl" and limits which ae set t distant fm the cente line may falsely indicate that a pcess is "in cntl" when it is nt. Typically, new cntl chat limits ae set at sme multiple f the pcess sampling standad e (in the SAS-QC mdule standad es ae dented as sigmas). This multiple is usually based upn the size and fequency f the samples intended t be pltted n the chat. As sample size andl fequency f sampling incease, the sensitivity f a given chat t pcess change als inceases. Small t mdeate sample sizes may be best pltted n chats having cntl limits set at 2-sigmas. TIieesigma chats ae mst useful when the samples ae lage faily fequent (minutes. hus. days). In nging cntl chaning. inceased sensitivity may als be btained by pltting waning limits within the cntl limits. These. ae usually set at 1-. and 2-sigmas fm the tue estimated pcess mean. By selecting a lage sample me infmatin abut a pcess becmes available. s cntl limits can be me pecise. Cntl limits will vay acss subgups in invese ppnin t the sample size. That is. the lage the sample. the clse the cntl limits will be t the cente line. F example. suppse a p-chan cntl limit is set at 2-sigmas. If the subgup size is 30. its cnttollimits will appea me naw than the same chan in which the subgup size is 15. Evey new pcess which is measued via a cntl chat methd equies an initializatin peid. Five t eight data pints ae usually equied as a minimum t establish that the pcess is in "statistical cntl". That is. until a cente line can be detemined and cntl limits set. pcess data tends shuld emain suspect. Additinally. until the patten f data pints appeas t stabilize, n infeence abut changes in pcess shuld be made. Based n visual inspectin f the fist 5-8 subgups, if the patten appeas andm and the data pints d nt exhibit a egula patten identified in the Patten Recgnitin sectin belw. then we can easnably assume the 251

5 Figue 2--Chan Panems descibed in Table , LiatC.s: '. : -6. -C::::::;::::::::::;==:==:;::::::;:::::::::;d """ 7 IIUDtP'OUP tftd.ez ("IOD) 1IUlIq lis.., IUD Mal: n-s09 2A : -j:::::=::;:===::;::=:;:===:;::=:::;:=1 :: liidt..: f 7 f---=..,::.::;, t==-l... s...,aap 1 & fpliltod' 8;1z.. :.un _._509.cL 2B Cnclusin Cntl Chaning f Attibute and Vaiables Data is an endeav which elies n ppe cnsn-uctin and administatin techniques. This pape addessed sme f these techniques: cntl chating elements. chan classes and cncens, sample size effects. and patten ecgnitin. Undestanding hw inceasing sample sizes pmotes impvement in cntl chans ability t detect eal changes and hw data autcelatin can influence chan pefmance is vital t eliable infeence making. Undestanding key pattens in cntl chans impves u abilities t detect change when it des ccu. Infeences made abut a pcess can nly be as eliable as ae the data which suppot that infeence..' : J--; n < f p J- laou 'nrjod) p n f : la J "1., _ _ _-.J : n 7 f 5, audjl'oup 1'" IPDIOD,.1:1": Mh.,...D9 MS lg' liai".z """... 3.c. 2C,.. 2. L'IU.:. ', uc::. ;'7.3 lci. 2D {': 1-2" L.imt.s=. MD... 2E " Refeences [I] [2] [3] (4] [5] A1wan. Layth c. and Rbets. Hay V. The Pblem f Mjslace!i Cntl Limits, Junal f Applied Statistics. V!. 44. N.3. pp Caulcutt. Rland. The Riihts and Wngs f Cntl Chats. Junal f Applied Statistics. V!.44. N.3. pp SAS Institute, Inc. SASlQ Sftwae: Usage and Refeence. Vlumes 1 & 2. Vesin 6. Cay. NC. SAS Institute. Inc Alwan. Layth C. Effects f Autcelatin n Cntl Chan Pefunance, Cmmunicatins in Statistics--They and Methds. V.21. N.4. pp Mntgmey. Duglas C. Intductin t Statistical Quality Cntl. Secnd Editin, Jhn Wiley & Sns. New Yk,

6 Acknwledgements. SAS is a egisteed tademak f SAS Institute. Inc. in the USA and the cunties. indicates USA egistatin. Auth The auth may be cntacted at: Chales H. Daby Health Sevices Evaluatin May Clinic 200 Fist St. SW Rcheste, MN (507) daby@may.edu 253