Moving Slope Clinical Chemicals

7/28/2019 Moving Slope Clinical Chemicals

1/6

CLI N.CHE M. 37/3, 341-346 (1991)

CL I N ICA L CHE M I S T RY , V o l. 37 , No.3 , 1991 341

A na lyzin g Q ua lity-C on tro l T re nd s w ith M ovin g S lo pe C ha rtsS . J ay Sm Ith a nd G ary L Mye rsWe have developedand evaluateda n ew p ro ced ure fo rdetectingrends in quality-contro l m easurem ents andapp lie ditto la bo ra to ry d ata . T he m eth od re qu ire s th e u seof sequentia l or m oving slope estim ates to identifytrends.Formulae are derived o e stim a te th e re gre ss io nerror for the m oving slope directly from the standardd ev ia tio n o f th e a na ly tic al m ea su re me nts o bta in ed d urin gc ha ra cte riz atio n ru ns . C on tro l lim its fo r th e m o vin g s lo pedepend only on th is regress ion error, the span of theslope, and the desired statis tica l leve l of contro l. Them oving slope can be plotted w ith contro l lim its to deter-m ine out-of-contro l po ints. The statistica l pow er of them oving slope is found to be m uch greater than that of anoften-used test for trends. An exam ple of the use of them oving slope is show n for qua lity -contro l m easurem entsfor to ta l cholestero l obta ined over severa l years. W eco nclud e th at th e m ovin g slo pe p ro ce du re h as co nsid er-ab ly m ore statistica l power than trend ru les and that ity ields m ore usefu l in form ation to the ana lyst.AddItionalKeyphrases:t rends ru les compared . cholesterol

De te ct io n o f t rends in qua li ty -cont ro l (QC) charts formeans has b ee n a cc om p lis he d b y m o nito rin g th e le ng th sof certa in types of runs (1-4). Nelson (2 , 3) discusseseight tests for specia l causes. A specia l cause is aperturbing in flu en ce th at c au ses a n ou t-o f-co ntro l sit-uation; it is d istinguished from a com m on cause,w h ic h a ffe cts al l po ints o n th e c ha rt. N elso ns tests arebased on the evaluation of runs. Two of N els on s ru nstests are probably used in clin ica l laboratory Q C m osto ften : T est 2 , t he number o f c on se cu tiv e p oin ts o n e ith erside of the target m ean, an d Test 3, the num ber ofc on se cu tiv e in cr ea sin g o r d ec re as in g p oin ts . S ta te m en tso n th ese ru le s d o n ot a lw ays a gree . F or e xa mp le , a cco rd -ing to D uncan (1 , p. 392), a ru le of thum b is that a ru nof seven or m ore increasing or decreasing values ind i-cates nonrandom influences, w hereas N elsons Test 3requires a run of si x (2). A lthough the tests are based onprobability , N elson ind icates that the probability forgett ing falses ignals should not be considered veryaccurate because of the lack of norm ality and indepen-dence am ong the tests as w ell as overlapping applica-tion.

Trends teststhat evaluatethe lengths of runs aresubject to in terference from sm all random errors inanalytica l m easurem ents. If a genera l trend upward or

D iv is io n o f E nv ir onm en ta l Health L abo rato ry S cien ces, C enterfo r Environmental H ea lth a nd I nju ry C o nt ro l, C en te rs fo r DiseaseC on tro l, P ublic H ealth Service, U .S. D epartm ent of H ealth an dHuma n S er vic es , Atlanta, GA 30333.R ece iv ed M arch 3 0, 1 99 0; accepted D ecem ber 24, 1990.

dow nw ard exists, the runs tests m ay fa il to detect thetrend quickly because of sm all random deviations,w hich contrad ict the special test ru les for strictly con-s ec utive in cre asing or d ecre asin g va lu es. A tren ds p ro -c ed ure w ith th e fo llo win g c hara cte ris tics w ou ld b e m orehelpfu l: (a) it is less subject to in terference from sm all,r an do m e rro rs ; (b) it y ie lds a probability s ta tis tic ; and (c )it a llow s a contro l chart for the statistic w ith it s s ta tis -tica l contro l lim its to be plotted. W e propose a m ovings lo pe ch art th at m eets a ll o f th ese crite ria .

In th is paper w e are not concerned w ith detectinglong-term trends in Q C m easurem ents. Long-termtrends are detected by th e use of cusum charts tom onitor the sum of cum ulative deviations from thetarget m ean (5). W estgard et a l. (6) report combinedS he wh art.-c us um c on tro l c ha rt fo r use in clin ica l c he m-istry laboratories. C em browski e t a l. (7 ) describe aspecia l m ethod for long-term trend analys is in contro ldata, in w hich an exponentia lly w eighted m oving aver-age of the contro l m eans (Triggs tracking signal) isused to tra ck tren ds.Materialsnd Methods

D erivin g form ulae fo r estimating the regression meansquare error an d stat ist ical controllimits fo r th e movings lo pe c ha rt. The span of the m oving slope is defined asth e n um ber o f c on se cutive p oin ts on th e co ntro l ch art fo rm ea ns th at a re u se d fo r e ac h se qu en tia l slo pe e stim ate.The span corresponds to the num ber of analy tica l runsper slope. E ach sequentia l pa ir o f po ints for the regres-s io n co nsists of th e ru n m ean co nce ntra tio n (Y ) and ana ss oc ia te d X va lue , w he re X = 1 , 2 ,. .n. Because the Xvalue for each regress ion point is know n (1 ... n ), th esum , the sum of squares, and the statistica l m eans-corrected sum of squares for X are determ in istic andf ixed fo r e ach slo pe . T he va lu es of th ese sums a re g iv enin Table A-i (Appendix) for se lected values of n. In theAppendix, w e also show that the consecutive slopes canbe estim ated directly from (a) the sum of the cross-products of the run m eans w ith the consecutive Xvalues, and (b ) the sum of the n run m eans w hei thea pp ro pria te fo rm ulae a re u se d.T o co nstru ct co ntro l lim its on the slope, w e need anestim ate of the regression m ean square error. There gre ss io n m e an square error c an b e o bta ine d d ire ctlyfrom the standard deviation of the run m eans of thea na ly tic al m e as ur em e nts m a de d ur in g c ha ra cte riz atio nruns or from som e other estim ate of th is s tandarddeviation. Form ulae for th is approach are also given inth e Appendix. T his d ire ct e stim atio n m eth od d ep end s onan assum ption that the overall s lope is zero during thecharacterizat ion runs. O nce the regress ion m ean squareerrorhas been estim ated, contro l lim its can be deter-


2/6

N els on s te st 3M ov ing s lope

T ab le 1 . S ta tis tIc al P ow er (% ) o f M ov in g S lo pe v sN els on s T es t 3 fo r T re nd s8True slope as a fra ctio n o f th e sta nd ard d evia tio n

0.01 0.05 0 .10 0.20 0.40 0.50 1.00Test3


3/6

Table 4 . S ta tis tic al Power (% ) o f Mo vin g S lo pe a nd N els on s T es t 3 a s a F un ctio n o f th e S ta rtin g Ou t-o f-C on tro lP oin t in S pa n8FS D

0.10 0.20 0.40Start ingpoint Test 3 M595 MS99 Test 3 MS95 MS99 Test 3 MS95 MS99

2


4/6

Movingslope

3 44 C LIN IC ALC H E M IS T R Y, V o l. 3 7 , N o. 3 , 1 99 1

Table 5 . E xample o f C alc ula tio ns fo r th e Mo vin g S lo pew ith a S pa n o f 1 0 R un s: In itia l T ota l C ho le ste ro lM eas ureme nts S hown in F ig . 1Run Sum of XYno., Run mean Mean Cross-X concn, V V products

1 250.752 251.503 250.334 254.255 255.756 254.507 252.758 250.759 251.25

10 253.25 252.5 13896 0.10411 254.00 252.8 13911 0.06812 255.00 253.2 13933 0.09713 253.25 253.5 13 93 4 -0.08914 252.25 253.3 13922 -0.10515 253.75 253.1 13926 0.086Correctedsumso f squaresforx (equation4 in Appendix) i s 8 2 .5 fo r a s p ano f 10 runs .S u m o f X (e qu atio n 2 in Appendix) is55. Movingslopeforr un n o .10 iscalculatedas: [13 89 6 - (55- 252.5)J /82.5=0.104.

with out-of-contro l slopes at the 99% contro l va lue,occurred b etw een r un s 220 and 230. These slopechanges are caused by severa l very low m easurem entsshown inFigure 1 for the associated runs. For runs 98 to1 00 , t he n eg ativ e s lo pe s a re o ut-o f-c on tro l, re fle ctin g th ene ga tive tre nd s in a na lytica l m ea su re me nts after ru nnumber 90.F igure 2 (bot tom) show s a m oving slope chart w ith aspan of five for these sam e data. Reducing the span ofthe chart results in s lope estim ates that are larger andm o re v aria ble , p ar tia lly because the denom inator of thes lope is sm aller as the span decreases (see Appendix).The w idth of the contro l lim its increases as the spand ec re as es . A lth ou gh th e a re as o f c on ce rn re pre se nte d b yextreme slopes are about the sam e on both charts, forspecif ic ran ge s of run numbers th e re la tio ns hip s a mo ngco nse cu tive slo pe p oin ts m ay b e so mew ha t differe nt.Discussion

S ince the early 1950s, statistic ians have recognizedthat Q C charts for m eans lack the pow er to detectre la tiv ely s ma ll d iffe re nc es from expected values (3 , p.233). C usum charts are used to d ete ct th es e d iffe re nc esover long periods. For exam ple, a cusum ch art w ou ldrequire -30 runs to detect a 0.5 standard deviation shiftfrom the target m ean w here alpha is 0.05 and beta is0.20. Trends tests for runs allow m ore rapid detection ofshifts, u su ally in six o r m ore p oin ts o n a ch art, d ep en din go n th e s pe cifie d alpha value and the ru le used. Appar-e ntly , p ro ba bility -b as ed m o vin g s lo pe c ha rts a re n ot usedfo r Q C p urp ose s; h ow eve r, m ovin g s lo pes h ave b ee n us edfo r inte rp ret ing elect rocardiogram a nd o the r da ta (10).The m oving slope m ethod offers several advantagesover Test 3 for trends. F irst, the m oving slope gives theanalyst a statis tic that quantifies the degree of change

of the assay resu lts over severa l runs. The m oving s lopecan be plotted vs statistica l contro l lim its, sim ilar toother standard contro l charts. B ecause the regressionm ean square can be estim ated directly from the esti-m ated run m ean standard deviation of the analyses, thecontrol lim its are fixed as for standard charts . Second,sm all random changes w ill not affect the estim ate of theleast squares slope greatly, w hereas w ith trends tests,such changes m ay cause an increasing or decreasingtrend not to be detected quick ly. Third , the statis tica lpo we r o f t he m ov in g slo pe is m uch g re ate r th an N elso nsTest 3, regard less of the m agnitude of the slope, assh ow n in T ab le 2 . T h is in cre as ed p ow er h old s reg ardle ssof the analy tica l C V. L inear trends can also be detectedby observing them in the Q C m eans charts and by usingthe 2 SD rule. This m ethod m ay have statistica l pow erthat is as great or greater than the m oving s lope.The m oving slope could be m ade m ore robust byspecia l techniques that w ould reduce the influence ofany w ild data points. F irst, a w eighted least-squaresregression could be used, in w hich the w eights dependon the fit o f the ind ividual points . Second, a resistantline-fitting m ethod could be applied, in w hich case th em ean slope would be obtained by dividing the data in tothree different groups on the basis of the independentvariab le as described in Hoaglin et al. (11). T hird, them ed ia n va lu e o f a ll u niq ue pairwise s lo pe e stim a te s (th eT heil estim ator), as recom mended by D ietz (12), couldbe used.All o f these m ethods would be m uch m ore com puta-tionally in tensive than the least-squares m ethod. F it-ting res istant lines from three g ro up s w ou ld requirethree runs per group, for a m in im um span of n ine runsp er fitte d line. An additional d isadvantage w ould bethat estim ation of the regression error and the sta tis ti-cal control lim its for these m ethods would be m oredifficu lt. B ecause of these problem s, and also becausethe least-squares m ethod is inherently som ewhat ro-bust, w e did not use the specia l robust regression tech-n iq ue s in o ur a na ly sis .In a m oving s lope Q C chart, the observed concentra-tio n p oin ts used for estim ating the s lope w ill not fit in alinear fashion for m ost spans. The points m ay be ex-tre me ly sca tte re d, n on lin ear, or n on mo no to nic. T he re -fore, the linear regression m odel m ay fit the pointspoorly. This lack of fit is not critica l for Q C purposesbecause the m oving slope is used p rim arily to in dica tethe direction of change in the contro l data. This situa-tion is d ifferent from one in which the problem is to findthe statistica l m odel that best fits a set of data. Fornoninonotonic data, the m oving slope w ill indicate con-se cu tive u pw ard a nd do wn wa rd tre nd s e xce pt w he re th eslo pe ch ang es du rin g o ne sp an .Because successive m oving slopes have identica ln - 1 points in com m on from the contro l chart form ea ns, th e slop e e stim ate s a re n ot in de pe nd en t. H en ce ,severa l po ints in a row outs ide or near the contro l lim itsdo not have the sign ificance that they w ould have on anord inary contro l chart, on which all po in ts are consid-


5/6

81116681166611611068866111818011 808881101118177777777777 71686681888806690000001I10000000051170000050001I1000151700012 1224517680121234517010 12 123

C LIN IC AL C HEMIS TR Y, V ol. 3 7, N o. 3 , 1 991 34 5

e re d in de pe nd en t. A n ind ivid ua l p oin t, h ow eve r, ha s th esam e sign ificance as a poin t on an ord inary contro lc ha rt. T his s ituation is sim ila r to that for a m ovinga ve ra ge c ha rt (1 , p. 500). T he corre lation betw een thesuccessive m oving slope estim ates increases w ith thesize of the span. S pecifying the span that is consistentw ith the m in im al num ber of points required for least-squares estim ation to m in im ize th is autocorre la tion istherefore appropria te . A m in im al span of three pointscouldbe used. F or com parative purposes, in our analy-sis, w e w anted to use a span that corresponded to aboutth e re qu ire d ru n le ng th fo r sp ecia l te sts. N elso n (13) hasshow n that the m oving average is subject to oscilla tionfrom random variab ility, even w hen cyclica l varia tion isrem oved. As the span increases, the m oving averagegives longer runs on either side of the target m ean. Them o vin g s lo pe b eh av es s im ila rly b ec au se o f th e in cre as edcorre lation w ith the length of the span; th is behavior,th ere fo re , is a no th er re as on w hy th e sp an sh ou ld be sm all.W e ca n co nstruc t a slo pe c ha rt tha t h as n on co rre la te dpoints and in which each span c o rr es p on d s t o d iff er en tc on se cu tiv e g ro up s o f Q C v alu es . This a pp ro ac h w ou ld b eusefu l if the points on a chart w ere grouped naturallyin to specific tim e interva ls. S uch a chart could be calleda n in de pe nde nt-s pan slo pe ch art. A n independent-spans lope chart w ould be especia lly usefu l w ith autom atedc he m is try m e th od s in w hic h s ev era l c on se cu tiv e s am p le so f th e same Q C sp ecim en w ere a na lyz ed d aily. A s lo pe fo re ach d ay s co nse cu tiv e re su lts w ou ld b e d ete rm in ed a ndw ou ld b e p lo tte d a ga in st c on tro l lim its c alc ula te d a cc ord -ing to the num ber of m easurem ents m ade per day.F igure 3 is an exam ple of an independent-span slopechart in w hich the lengths of the spans vary. Thecholestero l m easurem ents g iven in Figure 1 weregrouped by year and m onth from April 1986 throughM arch 1989. The m onths had an average of e ight runseach, w ith a m in im um of three and a m axim um of 11.For each m onth, a slope w as estim ated on w hich X

YEAR I 4 06 10

Fig.3 . independent-span s lope char t for the qua l it y -contro lmeasure -m en ts fo r th e F ig . 1 c ho le ste ro l d ataEachslopew as de te rmined f rom meas uremen ts madedur ing a s ing le mon th .Forregress ion, the Xvalues were consecut ive lyass ignedto runsm adedur ingt ha t m o n th . T h e n u m be r o f r un s p e r m o n th a ve ra g ed e ig h t a n d r an g ed f ro mthreeto 11.Cont ro l l imi ts (U , upper; L, lower)fortheslopeare showna t a 9 5%p ro b ab ili ty le v el . T h e l im i ts v ar y, r ef le ct in g t he n u m be r o f a n al yti ca l runsperformede a ch m o n th

values were ass igned sequentia lly . The 95% contro llimits fo r the s lope w ere estim ated as show n in Materi -a ls and Methods . B ecause of the different num ber ofruns for each m onth, how ever, the contro l lim its som e-tim es varied. O ne point on the chart, the low est concen-tra tion value in Figure 1, is clearly outs ide the 95%control lim it. A n inspection of F igure 3 for like m onthsshow s that negative s lope values tend to occur in June,p os itiv e s lo pe values in February.

A potential prob lem to keep in m ind w hen construct-ing independent-span slope charts is the poss ib ility ofn on mo no to nic d ata w ithin e ach sp an ; s uc h a c ha rt co uldshow near zero s lopes w ithin each span and not reflectthe directional changes in concentration occurringw ith in spans. T hus, w e recom mend that the data w ith ins pa ns b e in sp ec te d fo r n on mo no to nic c on ditio ns .The overa ll chart m ean and the am ong-run m eans tan da rd d eviatio n co uld b e u se d to sta nd ard iz e th e ru nm eans before the m oving slope chart is produced. T hisprocess would have certain advantages. First, the K1values given in Table A-i o f the Appendix can be useddirectly as the contro l lim its in m oving s lope chartsbecause the run m ean square error is taken as unity.S ec on d, w he n ju dg in g in div id ua l s lo pe s o f s ta nd ard iz eddata, the critica l va lue for statistica l sign ificance andthe contro l lim its w ill be constant, regard less of theam ong-run standard deviation of the assay. Third ,s lopes from reference m ateria ls w ith d ifferent concen-tra tio ns c an b e c om p are d d ire ctly .In sum mary, we have dem onstrated a procedure forid en ti1rin g tre nd s in Q C re su lts th at g en erate s a n e asilyinterpreted s ta ti st ic . T h e statistic is p iotted w ith its prob-a bility -b as ed c on tr ol lim its to d ete ct u nu su al increases ordecreases in m easurem ents. W e have show n that this Q Cprocedure has m uch greater s tatistica l pow er than dotre nd s tests a nd th ere fo re c an d ete ct tr en ds m o re ra pid ly .A ls o , w e d e m on s tr at ed that it is u se fu l fo r d ete ctin g t rendsin results, w hich m ay be logica lly categorized into liketim e in te rv als , e .g ., d ay s, weeks , o r m o nth s.

We acknow ledgethe contributions of the reviewer for helpfulsuggestions on statistical pow er analysis and other concepts pre-sented in th is manuscript.References1. Duncan A J. Q uality control and industrial statistics, 4 th ed .H om ewood, IL: R ichard D Irw in, 1974.2. N elson L. Interpreting Shewhart X control charts. J QualT e ch no l 1 98 5; 17 :1 14 -6 .3 . Nelson L. The Shewhart con trol chart-tests fo r sp ec ia l c au se s.J Q ual T echnol 1984;15:233-9.4. Nelson L . Control charts. In : Encyclopedia of statistical sci-ences, V ol 2. New Y ork: John W iley & Sons, 1982:176-82 .5. Goel AL . C umu la tiv e sum c on tr ol c ha rt s. Ibid.:233-41.6. W estgard JO , G roth T, A rronsson T, de Verdier C . CombinedShewhart-cusum contro l chart for im proved quality contro l inclinical chem istry. C lin C hem 1 977 ;23 :1 88 1-7 .7 . Cem browski GS , Westgard JO , Eggert AA , Toren EC . T renddetection in contro l data: optim ization and in terpretation ofT rigg s techn iq ue fo r tren d analysis. C lin C hem 19 75;2 1:1 396 -40 4.8 . Abell LL , Levy BB, B rodie BB , K endall FE . A sim plifiedm ethod for the estimation of total cholestero l in serum . J BiolC hem 1 95 2;1 95 :3 57 .9. SAS Institute, Inc. SAS/QC users guide, 5th ed. Cary, NC: SASIn stitute Inc., 1986.


6/6

10 . Nasipuri M , Basu DK , D atta GR, Seth S. A new m icroproces-sor-based data reduction for m onitoring , com munication, and stor-age of ECG signal. Proc 7th annual sym p on com puter applicationsin medical care. New York, NY 10017: IEEE Com puter SocietyPress, 1984:922-5.11 . H oaglin DC, M osteller F , T uk ey JW , e ds. U nderstan dingrobust and exploratory data analysis. N ew York: John W iley &S ons , 1 9 83 :1 2 9- 63 .12 . D iets E J. Teaching regression in a nonparametric statisticsc ou rs e. A m S ta t 1 98 9;4 3(1 ):3 5-9 .13 . N elson LS . The deceptiveness of m oving averages. Q ual Tech-n o l 1 9 83 ;1 5 :9 9 .AppendixF orm ula e fo r E va lu atin g a L in ea r, L ea st-S qu are sR eg re ss io n M o vin g S lo pe a nd Its C on tro l L im its fo r aQua li ty -Con tro l Char t

T he follow ing form ulae are used to estim ate a m ovingslope for contro l charts w ith each run corresponding to asequentia l X value beginn ing at 1 w ith a span of n runsper estim ate. Upper-case letters denote orig inal sums ofsquares or cross-products values; lower-case letters de-n ote v alu es co rre cte d fo r m ea ns.Estimated slope of regression

XY- XY/nB= (1)x2 _()2/

Corrected su m of squares for sequential X valuesX =n(n+i)/2

X2 = n(n + i)(2n + 1)/6= [3 ] - [2]21n

where x = (X-X), [3 ] re fers to equat ion 3, and so on.Estimated slope (B) by using sequentia l X values

B = [ XY - [ 2 ] Y /n ]/ [4 ]

34 6 C LIN IC AL C HEM IS TR Y, Vo l. 37, N o. 3, 1991

t-te st fo r ze ro slope by using sequential X valuest = B I [S2 . x2]V2 (6)

Suppose o has been estim ated from the run m eans ofa na lytica l m eas ure men ts d urin g ch ara cte riza tio n a na l-yse s. W e w ish to e stim ate o . for use in a t-test o f z eroslop e fo r each sequ ential s lo pe e stim ate o r e qu iv ale ntcontro l lim its . W e also have an estim ate of , th econtro l chart m ean. The true variance about the regres-s io n m ay b e w ritte n

(7 )w here p is the t ru e c or re la ti on .

B ecause the true slope during the characterizationruns is assum ed to be zero, p2 = 0 and

(8 )The upper and low er quality-control lim its for the

m ovin g s lo pe a re t(a, N-i) (S /[4])2 (9)

w here N = n um be r o f c ha ra cte riz atio n ru ns.T he t-va lue is chosen for the desired Type I error rateo r co ntro l le ve l, an d th e d eg re es o f f reedom are taken as

(2) infinity in Table A-i for the m ost conservative interva ls(Z-value).(3) The upper (UCL) and low er (LCL) contro l lim its forthe m oving s lope are U CL = ZSJ[412 and LC L =

(4 ) -ZS)[4]2.Let K1 = Z /[4 ]2 ; th en

UCL = K1 S, an d LC L = -K 1 S. (10)Values of K1 are given in Table A-i for the 95% and

(5) 99% confidence levels.

T ab le A -i. V alu es fo r D ete rm in is tic T erms in Mo vin g S lo pe E stim a tSpan o f mov ing s lope

io n fo r Q ua lity-C on tro l Cha rt sTerm 5 6 7 8 9 10 20 30

10 17.5 28 42 60 82.5 665.0 2247.50.620 0.469 0.370 0.302 0.253 0.216 . 0.076 0.0410.816 0.617 0.487 0.398 0.333 0.284 0.100 0.055n consecutive observat ions are cons idered for each sequentia ls lope es t imate.x : X -va lues c o r rec ted f o r means .

Documents

Moving Slope Clinical Chemicals