Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts Copyrght 207 by Taylor Enterprses, Inc., All Rghts Reserved. Adjusted Control Lmts for U Charts Dr. Wayne A. Taylor Abstract: U charts are used for count data followng the Posson dstrbuton. However, the U chart has symmetrcal control lmts when the Posson dstrbuton s nonsymmetrcal. As a result, the upper control lmt can have a rate of false detecton as hgh as n.5 ponts plotted. Ths can result n wasted resources nvestgatng false sgnals. Further, the lower control lmt has a rate of false detecton consstently above n 000. Ths makes t slow to detect mprovements n qualty. Adjusted control lmts are provded that correct both these problems. The adjusted control lmts result n a U chart truly based on the assumpton of the Posson dstrbuton..0 Introducton The average run length (ARL) of a control chart s the average number of ponts that are plotted before one goes outsde the control lmts. The ARL vares based on the sze of the shft. When a sgnfcant shft occurs the ARL should be small, approachng. Also of nterest s the ARL when there s no shft. Ths s the tme between false sgnals. It should be large. The ARL when there s no change wll be referred to as the false detecton tme, FDT. /FDT wll be referred to as the false detecton rate, FDR. Control charts based on the normal dstrbuton, such as X and IN charts (Taylor 207b), wth ±3 standard devaton control lmts, have FDRs of: n 740 relatve to the upper control lmt: Φ ( ) = 0.003499 3 = 740.80 n 740 relatve to the lower control lmt: Φ( 3) = 0.003499 = 740.80 n 370 for both control lmts combned: 2 0.003499 = 370. 40 Φ ( z) s the probablty of beng less than z for the normal dstrbuton (µ=0, σ=). U charts assume count data and are based on the Posson dstrbuton. The Posson dstrbuton has one parameter, the average count λ. For the Posson dstrbuton: Average = λ Standard Devaton = λ The standard control lmts for a U chart are: LCLStandard = Average 3 Average UCLStandard = Average + 3 Average As the Posson dstrbuton s not symmetrcal and the standard control lmts are symmetrcal, the FDRs of a U chart wll dffer from those above. 3 standard devaton control lmts are generally robust to the assumpton of normalty as most dstrbutons have a hgh percentage of values wthn three standard devatons of the average. Whle ths may be true most of the tme, for the U chart the false detecton rate can be as hgh as n.5. Ths s an unacceptable rate. Ths artcle documents the false detecton rates for U charts and offers a soluton n the form of an adjustment to the control lmts. Revson 2: October 3, 207 www.varaton.com/techlb/bref2.html
Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts 2.0 False Detecton Rates The U chart works dfferently than an X chart when detectng a worsenng of qualty. For an X chart, both upward and downward shfts can sgnal a shft from the target and a worsenng of qualty. The FDR relatve to a worsenng n qualty s then n 370. For a U chart of complants or nonconformtes, only an ncrease n the counts sgnals a worsenng of qualty. Therefore, t s approprate to have a n 370 FDR assocated wth the UCL by tself. It would be of concern f the FDR dropped below n 200, as ths represents nearly a doublng n the number of false sgnals. The lower control lmt sgnals an mprovement n qualty. The consequence of a false detecton relatve to the LCL s dfferent from that assocated wth the UCL. It s also approprate to have a n 370 FDR assocated wth the LCL by tself. It would be of concern f the FDR dropped below n 200. Fgure shows the FDR for the UCL as a functon of the average count for two dfferent scales. Formulas for the FDR are gven n Secton 4. The FDR s not a smooth curve due to counts beng ntegers. The FDR jumps whenever the control lmt crosses an nteger value. Ths creates an oscllaton pattern. The FDR can be as low as.5. It does not stay above n 200 untl the average count reaches 0. Standard control lmts should not be used due to false rejectons when the average s less than 0. n 370 n 200 n 370 n 200 Fgure : False Detecton Rate for the Standard Upper Control Lmt Revson 2: October 3, 207 2 www.varaton.com/techlb/bref2.html
Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts As the average count exceeds 70, the FDR for the UCL vares between n 400 and n 600. Ths s less frequent than the target n 370. Ths means the U chart s not as effectve as t could be at detectng a worsenng n qualty. Fgure 2 shows the FDR assocated wth the LCL as a functon of the average count. The LCL s zero untl the average counts reaches 9. Ths means the FDR s nfnte. Above 9 the FDR s consstently above n 000 and generally far above ths. Ths means the LCL s not as effectve as t could be at detectng an mprovement n qualty. n 370 n 370 Fgure 2: False Detecton Rate for the Standard Lower Control Lmt As the counts ncrease the Posson dstrbuton more closely resembles the Normal dstrbuton. At the same tme, the counts are less lkely to follow the Posson dstrbuton. STAT-0 Statstcal Technques for Trendng Data n Taylor (207c) states the U chart s generally the best chart for counts less than 25 but that the IN chart (or Laney U chart) s generally the best chart for counts greater than 25. For counts greater than 25 the data tends to be normal but overdspersed, meanng t vares more than the Posson dstrbuton. The nche for the U chart s when counts are less than 25. Ths s when the counts are most lkely to follow the Posson dstrbuton. It s also when the counts are most Revson 2: October 3, 207 3 www.varaton.com/techlb/bref2.html
Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts skewed, so most nonnormal. The U chart s based on Posson property that the standard devaton s the square root of the average. But U charts also use symmetrcal control lmts when the Posson dstrbuton s not symmetrcal. Ths s why adjusted control lmts are needed. 3.0 Adjusted Control Lmts The formula for the adjusted UCL s: UCL = Average + 2.782 Average + Adjusted 2.782 s from the Normal dstrbuton. It corresponds to a n 370 FDR just lke 3 corresponds to n 740 FDR. The adjustment factor, adjusts for both the dscreteness of the values and postve skewness. It was selected so that the mnmum FDR s near n 200, as shown n Fgure 3. The adjustment ( UCL Adjusted - UCL Standard) ncreases the UCL by slghtly less than when the average s or below, reducng the false detecton rate. For larger counts, the correcton lowers the UCL by up to, makng the chart qucker to react. n 370 n 200 n 370 n 200 Fgure 3: False Detecton Rate for the Adjusted Upper Control Lmt Revson 2: October 3, 207 4 www.varaton.com/techlb/bref2.html
Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts The formula for the adjusted LCL s: LCLAdjusted = Average 2.782 Average +. The formula s vald for averages of 5.33 and above. Below that the LCL s zero. As before, 2.782 s from the Normal dstrbuton. The adjustment factor. adjusts for both the dscreteness of the values and postve skewness. It was selected so that the mnmum FDR s near n 200 (Fgure 4). The adjustment ( UCL Adjusted - UCL Standard) ncreases the LCL by. for smaller counts and up to 3.3 for hgher counts, makng the chart qucker to react to an mprovement n qualty. n 370 n 200 n 370 n 200 Fgure 4: False Detecton Rate for the Adjusted Lower Control Lmt An alternatve approach s to use what are sometmes called exact control lmts. It nvolves settng the control lmts as: UCLExact = mn x where -Posson x Average 740 Revson 2: October 3, 207 5 www.varaton.com/techlb/bref2.html
Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts LCLExact = max x where Posson x Average 740 Ths results n the curves lke those n Fgures 3 and 4 beng entrely above 740. It would be better to use /370 than /740 for the reasons gven before. Whle exact control lmts resolve the false detecton ssue, they are not as fast at detectng a change compared to the adjusted control lmts. 4.0 U Chart Formulas It s assumed that each pont s a count C followng the Posson dstrbuton wth parameterλ: ( ) C ~ Posson λ The Posson dstrbuton has the property: Standard Devaton = Average where Average = λ The fact that the average s used to estmate the standard devaton smplfes the chart as a separate estmate of the standard devaton s not needed and makes the chart more powerful for Posson data. However, t makes the chart senstve to the assumpton of the Posson dstrbuton. The Posson dstrbuton occurs when the tems beng counted occur ndependently of each other. When counts are larger, the counts tend to vary more than the Posson due to dependences. Ths s called overdspersed. For example, complants may be grouped together for processng or multple partcles may be ntroduced at the same tme. For counts the standard control lmts are: UCL = Average + 3 Average Standard LCL = Average 3 Average Standard LCL Standard =0 for λ 9. Ths assumes the number of opportuntes or sample sze s constant. Ths s a specal case of a U chart that s commonly called a C chart. The adjusted control lmts are: UCL = Average + 2.782 Average + Adjusted LCLAdjusted = Average 2.782 Average +. = 0 for λ 5.33 = ( ) The value 2.782 s more exactly 2 ( 3) 2 ( 2 3 ) ( 2 ( 3) ) 2 Φ Φ ± Φ Φ 4.4 ( ) Φ Φ = 2.782749668872. It was selected to gve a FDR for just the UCL equal to that of both control lmts for the X and IN charts. Ths s around n 370. As the average count ncreases, the FDR relatve to the UCL converges to around n 370 as shown n Fgures 3 and 4. Revson 2: October 3, 207 6 www.varaton.com/techlb/bref2.html 4.
Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts The false detecton rates are: FDR FDR FDR Lower Upper Both = Posson cel LCL ( ) λ = - Posson floor ( UCL) λ = Posson cel( LCL) λ + - Posson floor ( UCL) ( λ ) Cel() rounds the value up to an nteger. Floor() rounds the value down to an nteger. Posson() s the Posson dstrbuton functon. A control chart of counts s referred to as a C chart. For a U chart, rates R are plotted based on the number of opportuntes O: R C = wth C ~ Posson( λ O) O Then the standard control lmts for the rates R are: UCL Rates-Standard λo + 3 λ O 3 λ O O = = λ + LCL λo 3 λ O 3 λ O Rates-Standard = = λ O = 0 for λ 9. O Smlarly, the adjusted control lmts for the rates R are: UCL Rates-Adjusted λo + 2.782 λo + λ = = λ + 2.782 + O O O λo 2.782 λo +. λ. LC LRates-Adjusted = = λ 2. 782 + O O O = 0 for λ 5.33 O. Revson 2: October 3, 207 7 www.varaton.com/techlb/bref2.html
Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts 5.0 Conclusons U charts are useful for count data followng the Posson dstrbuton, whch s most lkely for counts below 25. For Posson data, the U chart s better than an IN chart because t s based on a better estmate of the standard devaton. However, a U chart s not entrely based on the Posson dstrbuton because the Posson dstrbuton s skewed whle a U chart uses symmetrcal control lmts. For the standard upper control lmt, the rate of false detecton can be as hgh as n.5 for low counts. It does not reman above n 200 untl the average count reaches 0. The standard upper control lmt should not be used due to false rejectons when the average s less than 0. Ths can result n wasted resources nvestgatng false sgnals. The adjusted UCL control lmt fxes ths problem. It mantans the false detecton rate above n 200 and averages around n 370. The adjusted UCL also mproves the detecton of a worsenng of qualty for larger counts, as long as the assumpton of the Posson contnues to hold. For the standard lower control lmt, the rate of false detecton s consstently above n 000. Ths makes t slow to detect mprovements n qualty. Ths can result n mssng mprovements and the assocated lessons learned. The adjusted LCL makes the chart much qucker at detectng mprovements n qualty. The adjusted control lmts result n a U chart truly based on the assumpton of the Posson dstrbuton. The same approach can be appled to P charts as descrbed n Taylor (207a). 6.0 References Adjusted Control Lmts for P Charts (207a), Dr Wayne A. Taylor, Taylor Enterprses, Inc. (www.varaton.com/techlb/bref3.html). Normalzed Indvduals (IN) Chart (207b), Dr Wayne A. Taylor, Taylor Enterprses, Inc. (www.varaton.com/techlb/bref.html). Statstcal Procedures for the Medcal Devce Industry (207c), Dr Wayne A. Taylor, Taylor Enterprses, Inc. (http://www.varaton.com/procedures). Revson 2: October 3, 207 8 www.varaton.com/techlb/bref2.html