Control Charts for Mean for Non-Normally Correlated Data

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1 Joural of Moder Applied Statistical Methods Volume 16 Issue 1 Article Cotrol Charts for Mea for No-Normally Correlated Data J. R. Sigh Vikram Uiversity, Ujjai, Idia Ab Latif Dar School of Studies i Statistics, Vikram Uiversity, Ujjai, Idia, lateefdar.007@rediffmail.com Follow this ad additioal works at: Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical heory Commos Recommeded Citatio Sigh, J. R. & Dar, A. L. (017). Cotrol charts for mea for o-ormally correlated data. Joural of Moder Applied Statistical Methods, 16(1), doi: 10.37/jmasm/ his Regular Article is brought to you for free ad ope access by the Ope Access Jourals at DigitalCommos@WayeState. It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods by a authorized editor of DigitalCommos@WayeState.

2 Joural of Moder Applied Statistical Methods May 017, Vol. 16, No. 1, doi: 10.37/jmasm/ Copyright 017 JMASM, Ic. ISSN Cotrol Charts for Mea for No-Normally Correlated Data J. R. Sigh Vikram Uiversity Ujjai, Idia Ab Latif Dar Vikram Uiversity Ujjai, Idia raditioally, quality cotrol methodology is based o the assumptio that seriallygeerated data are idepedet ad ormally distributed. O the basis of these assumptios the operatig characteristic (OC) fuctio of the cotrol chart is derived after settig the cotrol limits. But i practice, may of the basic idustrial variables do ot satisfy both the assumptios ad hece oe may doubt the validity of the ifereces draw from the cotrol charts. I this paper the power of the cotrol chart for the mea is examied whe both the assumptios of idepedece ad ormality are ot teable. he OC fuctio is calculated ad compared with the ormal populatio. Keywords: Cotrol chart, correlatio, Edgeworth Series, stadardized cumulats Itroductio he quality cotrol techiques curretly used i idustry are aimed at the detectio of chages i the productio process that result i quality defects. Quality cotrol charts are curretly the most widely-adopted cotrol techique. raditioally, quality cotrol methodology is based o the assumptio that serially-geerated data are idepedet ad ormally distributed. Uder these coditios, appropriate cotrol limits for ca be worked out from the tables available i stadard textbooks o statistical quality cotrol. But i practice, may of the basic idustrial processes do ot satisfy both the assumptios ad hece oe may doubt the validity of the iferece draw from the cotrol charts. Alwa (199) studied the effect of auto-correlatio o cotrol chart performace. Maragah ad Woodall (199) studied the effect of auto-correlatio o the retrospective -chart. Alwa ad Roberts (1995) coducted ivestigatios of cotrol charts whe the assumptios of ormality, idepedece, or both are violated. Dar ad Sigh (015) studied the effect of correlatio o the power of Ab Latif Dar is a Professor i the School of Studies i Statistics. them at: lateefdar.007@rediffmail.com. 45

3 SINGH & DAR the chart. he purpose of this study is to cosider the power of the cotrol chart ad the effect of correlatio o ype-i error ad the OC fuctio, ad also to cosider relaxig the assumptio of ormality ad cosiderig the productio process to follow a o-ormal distributio represeted by the first four terms of a Edgeworth series. Effect of Correlatio o OC Fuctio for Normal Case Suppose that the observatios x 1, x,, x have a multivariate ormal distributio with E(x i ) = μ, V(x i ) = σ ad ρ is the commo correlatio coefficiet betwee ay x i ad x j, i j. he E x Var x 1 1 (1) where 1 1 () he power of the cotrol chart is judged by its OC fuctio. he cotrol chart for the mea is set up by drawig the cetral lie at the process average θ ad the cotrol limits at k where σ is the process stadard deviatio ad is the sample size. he OC fuctio gives the probability that the cotrol chart idicates the value θ as the process average, whe it is actually ot θ, but 453

4 CONROL CHARS FOR MEAN 454 where is as defied i equatio (). he OC fuctio is derived by itegratig the distributio of the mea with θʹ as the process average betwee the limits of the cotrol chart. For the ormal populatio uder correlated data, 1 f (3) he distributio of the sample mea is give by g (4) where 1 exp ad π r r r t d t t t dt he OC fuctio is obtaied after replacig θ i (4) by θʹ ad itegratig it betwee the limits of the cotrol chart as k k N L d (5) k k N L d (6) Makig the trasformatio y

5 SINGH & DAR ad y γ = t sequetially, the above itegral simplifies to L N k k 1 (7) he error of ype I gives the probability of searchig for assigable causes whe i fact there are o such causes. It is give by k 1 k g d (8) After itegratig above as i the case of the OC fuctio we will get k (9) he Effect of No-Normally Correlated Data o OC Fuctio For o-ormal populatios represeted by the first four terms of a Edgeworth series, f (10) where are the stadardized third ad fourth cumulats, respectively. he distributio of the sample mea for correlated data ca be derived, by followig Gaye (195), as ad 455

6 CONROL CHARS FOR MEAN g (11) he OC fuctio is obtaied after replacig θ i equatio (11) by θʹ ad itegratig it betwee the limits of the cotrol chart, i.e. betwee k Itegratig i the similar way as for the ormal case, we get. L L L L (1) N u b where L N is give by equatio (7). he other two terms of the OC fuctio are give by 13 k 3 k 5 k L u (13) 13 k 3 k 5 k L b (14) he ype-i error for the o-ormal populatio works out to be k 1 k g d c (15) where α as defied by equatio (9) is the ype-i error whe the populatio is ormal ad depedet, ad k c k (16) is the correctio for o-ormality ad depedecies i ype-i error. 456

7 SINGH & DAR able 1. Value of ype-i error for ormally ad correlated data K = K = 3 ρ = 0.0 ρ = 0. ρ = 0.5 ρ = 0.8 ρ = 0.0 ρ = 0. ρ = 0.5 ρ = able. Value of OC fuctio for ormally ad correlated data K = K = 3 γ ρ = 0.0 ρ = 0. ρ = 0.5 ρ = 0.8 ρ = 0.0 ρ = 0. ρ = 0.5 ρ = Results ad Coclusio For ormal populatios with correlatio coefficiet ρ = 0, 0., 0.5, ad 0.8, the values of ype-i error have bee computed ad give i able 1 for k =, 3 ad = 5, 10, 15. able 1 clearly idicates that the effect of correlatio o ype-i error is quite substatial as the error icreases with the icrease i ρ. For example, for = 5, k =, ad ρ = 0, 0., 0.5, ad 0.8, the correspodig values of ype-i error are , , 0.481, ad hough the effect goes o decreasig with icreasig k, it still affects the value of ype-i error quite largely. For o-ormal populatios we have a similar result (able 3) as the error goes o icreasig with a icrease i the value of ρ, λ 3, ad λ 4. From able, it is evidet that the value of the OC are affected seriously as the correlatio betwee the observatios icreases. For example, for ρ = 0, k =, = 5, ad γ = ±1, the value 457

8 CONROL CHARS FOR MEAN of the OC is , while for ρ = 0., 0.5, 0.8, k =, ad = 5, the value reduces to , , For other values of = 10, 15, we have a similar results. he values of the OC for o-ormal populatios with k =, = 5 ad for differet values of ρ = 0, 0., 0.5, 0.8 are give i able 4. For ρ = 0 ad (λ 3, λ 4 ) = (0, 0) we get tabulated values of Sigh, Sakle, ad Ahmad (01), which are show i able 4. he effect of correlatio o the OC fuctio remais more or less of the same magitude whe we move from ormal to o-ormal populatios. As is evidet from the able 4, for ρ = 0, λ 3 = 0, λ 4 = 0, ad γ = ±1, the value of the OC fuctio is while for ρ = 0.5, λ 3 = 0.5, λ 4 = 0.5, ad γ = ±1, the correspodig value of the OC fuctio is reduced to O chagig λ 3 (skewess), λ 4 (kurtosis), or both at the same time, the value of the OC is affected. herefore, it may be iferred that the violatio i the assumptios of idepedece ad ormality have a serious effect o the cotrol chart performace ad it is advisable to take ito accout the depedece ad oormality of the paret populatio while desigig cotrol charts. able 3. Values of the ype-i error for o-ormally correlated data K = K = 3 ρ λ3 λ4=0.0 λ4=0.5 λ4=1.0 λ4=.0 λ4=0.0 λ4=0. λ4=0.5 λ4=

9 SINGH & DAR able 3, cotiued. K = K = 3 ρ λ3 λ4=0.0 λ4=0.5 λ4=1.0 λ4=.0 λ4=0.0 λ4=0. λ4=0.5 λ4= able 4. Values of OC fuctio for o-ormally correlated data (λ 3, λ 4 ) ρ γ (0.0,0.0) (0.0,0.5) (0.0,1.0) (0.0,.0) (0.5,0.0) (0.5,0.5) (0.5,1.0) (0.5,.0) Refereces Alwa, L. C. (199). Effects of autocorrelatio o cotrol chart performace. Commuicatios i Statistics heory ad Methods, 1(4), doi: /

10 CONROL CHARS FOR MEAN Alwa, L. C., & Roberts, H. V. (1995). he problem of misplaced cotrol limits. Joural of the Royal Statistical Society. Series C (Applied Statistics), 44(3), doi: / Dar, A. L., & Sigh, J. R. (015). he power of -chart i presece of data correlatio. Joural of Reliability ad Statistical Studies, 8(1), Retrieved from Gaye, A. K. (195). O settig up cotrol charts for o-ormal samples. Idia Society for Quality Cotrol Bulleti, 53, Maragah, H. D, & Woodall, W. H. (199). he effect of autocorrelatio o the retrospective -chart. Joural of Statistical Computatio ad Simulatio, 40(1-), 9-4. doi: / Sigh, J. R., Sakle, R., & Ahmad, M. (01). Cotrol charts for mea uder correlated data. Joural of Rajastha Statistical Associatio, 1(1),