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1 This article was downloaded by: [Chung Ang University] On: 10 February 2015, At: 23:30 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Communications in Statistics - Theory and Methods Publication details, including instructions for authors and subscription information: Design of and ewma charts in a variance components model Changsoon Park a a Dipartment of Applied Statistics, Chung-Ang Uuiversity, Dongjak-gu, Seoul, Huksukdong, , Korea Published online: 05 Jul To cite this article: Changsoon Park (1998) Design of and ewma charts in a variance components model, Communications in Statistics - Theory and Methods, 27:3, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 COMMUN. STATIST.-THEORY METH., 27(3), (1998) DESIGN OF x AND EWMA CHARTS IN A VARIANCE COMPONENTS MODEL Changsoon Park Department of Applied Statistics Chung-Ang University Dongjak-gu, Huksuk-dong Seoul, , Korea Key Work: average run length; process control; Shewhart model; between-group variation; within-group variation ABSTRACT In statistical process control, the Shewhart model postulates that an individual observation consists of a constant plus a random variation about zero. In processes where group-to-group variability exists, the mean within a given group can be thought of a realization of the between-group variability. For such cases we consider a variance components model where individual observations have between-group variation plus within-group variation. The average run lengths of the standard X and exponentially weighted moving average charts designed for the Shewhart model are calculated in the variance components model. It is shown that the standard procedures are quite misleading. The standard procedures are modified to construct the control limits using the in-control variance of subgroup means in the variance components model. The modification of the procedures improves significantly the performance of the standard procedures. A method of estimating the in-control between-group variance is shown using the observed in-control run lengths from past experiences. Copyright O 1998 by Marcel Dekker, Inc.

3 PARK 1. INTRODUCTION Control charts have been widely used in production processes for monitoring shifts of parameters which specify the quality of products. 'Ihree major types of the control chart are the Shewhart, the cumulative sum and the exponentially weighted moving average (EWMA) charts. Let Xij, i = 1,2,..., j = 1,2,..., n, be the j-th observation from the i-th subgroup taken during the process and let n be the subgroup size. Then the Shewhart model of individual observation for monitoring the mean is usually defined as where is the process mean and e;j is a random variation within the i-th subgroup. This model assumes that e;,'~ are independent and identically distributed (iid) N (0, u2) random variables. In the model (1.1) the state of the process can be determined according to the value of rather than the distribution of observations. Hence the process is determined to be statistically in control if is equal to the target value &,. Consequently the in-control state in the model (1.1) is equivalent that observations taken over time are iid with the mean equal to the target. Suppose that groups of items are produced sequentially in time during a process and a random sample is taken as a subgroup from each group. If each group consists of items produced under the same circumstances, then items within each group are relatively homogeneous compared to items in other groups. Then it is not realistic to expect the mean of observations within each group to be constant over time as in (1.1) even in a well-controlled process. In such cases the simple Shewhart model does not explain the process data adequately and a proper model is essential in making valuable decisions based on statistical theories. It would be more realistic to assume extra variation in the model so that the mean of observations within each group can wander around to give the mean over time as [. The mean of observations within a given group can be thought of the realization of a between-group variability whose mean is [. To be an appropriate model for such cases a between-group component of variability should be added to the model (1.1). The model with this type of variance structure is often called the variance components model and has been studied for statistical process control by Hahn and Cockrum (1987), Laubscher (1996). Wetherill and Brown (1991), and Woodall and Thomas (1995).

4 DESIGN OF X AND EWMA CHARTS 661 In this paper we consider two components of variability: within-group and betweengroup. In section 2 a variance components model with within- and between-group variabilities is defined. In section 3 the properties of the standard X and EWMA charts are obtained in the variance components model. In section 4 we modify the standard control procedures to compensate for the between-group variability and evaluate properties of the modified procedures. An example of estimating the between-group variability is given in section 5. The conclusion and remarks regarding the use of modified procedures are given in section A VARIANCE COMPONENTS MODEL A statistical model for observations including the within- and between-group vari- abilities can be defined as X;, = p, + e;j, i=1,2,..., j =l,2,..., n, (2.1) where p, denotes the mean of observations within the i-th group. We assume that p,'s are iid N ((, r2 (()) random variables and independent from e;, 's. Then ( corresponds to the unconditional mean of the observation and p; corresponds to the conditional mean of the observation given that the observation is taken from the i-th group. Throughout this paper we call < and p, the overall mean and the i-th group mean, respectively. Individual and the subgroup means X, 's observations X,, 's in the model (2.1) are iid N ( El oh (c)) are iid N((, (<)), where (() = 02 + r2(() and 0% (() = 02/n + r2((). In the model (2.1) the process is determined to be statistically in control if ( = (, as in the model (1.1). This indicates that deviations of the group mean from the target are allowed to the in-control state as long as the overall mean is equal to the target. For convenience, we consider only positive shifts of the overall mean from the target. Let 6 be the deviation of the overall mean from the target in unit of the in-control standard deviation of subgroup means, i.e. We assume that the between-group standard deviation increases linearly as the overall mean deviates from the target as follows.

5 662 PARK Then we have the following expressions: 6 = *, (since 6=0 for r2(&)) The coefficient a represents the ratio of the between-group standard deviation to the within-group standard error when the process is in control (6 = 0), while the coefficient b represents the increasing rate of between-group standard deviation per unit change of 5 6 when out of control. If a = b = 0 it indicates that there is no between-group variability. If a > 0 and b = 0 the between-group variability remains the same when 6 changes. 3. PROPERTIES OF THE STANDARD X AND EWMA CHARTS In this section the properties of the standard X and EWMA charts in the model (2.1) are evaluated in terms of the average run length (ARL). The standard procedure of the X chart is to signal at the first i for which u Xi 2 to + c- or Xi 5 E 6 O - CJ;;' The standard procedure of the EWMA chart is to signal at the first i for which where 2; = rxi + (1- r)z,-l, i = 1,2,..., Zo = to, and r is the weight (0 < r 5 1). When the process standard deviation is estimated by the subgroup range or subgroup variance, we are estimating the within-group standard deviation a instead of the true standard deviation a x (6). In both of the standard procedures, hence, " is used in J;; calculating the control limits as the standard deviation of subgroup means instead of the true one (ax (E)). Let ARL~ (a, b, 6, c) and ARL$(a, b, 6, k) be the ARLs of the standard X and EWMAcharts in the model (2.1) with associated parameters a, b, 6, c and k, respectively. a

6 DESIGN OF X AND EWMA CHARTS 663 Also let Lx (A, c) and LE(A, k) be the ARLs of X and EWMA charts in the model (1. I), respectively, for A = (( - &)/(a/&. Then the ARLs of the standard procedures in models (1.1) and (2. I) have the following relations (see Appendix A. 1). and ARL; (a, b, 6, c) = Lx (A*, CS) d7. cs = l+(a+bj) d+ where A* = 6 dttz and ks = From (3.1) and l+(a+b&) (3.2), since A' # A, cs f c and ks # k in general, we see that the ARLs of the standard procedures in the model (2.1) are different from those expected in the model (1.1). The ARL of the X chart is simply the reciprocal of the probability that X, falls outside the control limits. For example, Crowder (1987) developed a method of computing the ARL of the EWMA chart numerically. SAS provides the function EWMAARL to calculate LE(A, k) by the method of Crowder (1987) (see SAS/QC Software:Usage and Reference, 1995). The ARLs of the standard X and EWMA charts in the model (2.1) for c = 3.0, r = 0.2, k = 3.0 are listed in Table I. In the table {a = 0.0, b = 0.0) corresponds to the model (1.1). From the table we see that the ARLs in the model (2.1) are significantly different from those in the model (1.1) for small values of 6. The difference is becoming larger as a or b increases. We see that both of the two coefficients a and b are important to the ARL. False alarm rates (reciprocal of the in-control ARL) of the X chart are , ,0.0339, for a=o, 0.5, 1.0, 2.0. False alarm rates of the EWMA chart are , , , for a=o, 0.5, 1.0, 2.0. As a increases the false alarm rate increases rapidly in both charts. For values of a the standard procedures give false alarms too often to be applied in practice. For a more detailed comparison of a the ARLs in log scale are plotted for b = 0.0 in Figure MODIFICATIONS OF THE x AND EWMA CHARTS The standard X and EWMA procedures are modified using the true in-control stan-

7 PARK TABLE I. ARLs of the standard X and EWMA charts ARL~ (a, b, 6,s) ARL~(~, b, 6,3) (r=0.2) I II 6 I log (ARL) FIG. 1. The ARL curve of the standard X and EWMA charts for b=o

8 DESIGN OF X AND EWMA CHARTS 665 dard deviation of subgroup means ax(&) (=ad=/&) in the variance components model. The x chart is modified to signal at the first i for which The EWMA chart is modified to signal at the first i for which Notice that the modified control limits do not include the coefficient b. Let ARL~(~, b, 6, c) and ARL$$(U, b, 6, k) be the ARLs of the modified X and EWMA charts in the model (2.1) with associated parameters a, b, 6, c and k, respectively. Then the ARLs of the modified procedures can be expressed as (see Appendix A.2) and where A* = 6,,&. CM = C- and km = 1 + (a+bw (4.2), since A* # A, c i # c and km # k in general, we aiso $ee that the ARLs of the modified procedures in the model (2.1) are different from those expected in the model (1.1). However the differences c - CM and k - km are smaller than the corresponding differences c - cs and k - ks in the standard procedures if a > 0. Thus the ARLs of the modified procedures are closer than the standard procedures to those expected in the model (1.1). If 6 = 0 the ARLs of the modified procedures in the two models are the same since A* = 0, c~ = c and k~ = k. The ARLs of the standard and modified procedures are the same if a = 0. The ARLs of the modified X and EWMA charts for c = 3.0, r = 0.2, k = 3.0 are listed in Table 11. The ARLs in log scale for b = 0.5 are plotted in Figure 2. The performance of the modified procedures is significantly improved in the model (2.1) when compared to the standard procedures. If b = 0, the ARLs are the same for

9 PARK TABLE 11. ARLs of the modlfied X and EWMA charts FIG. 2. 7he ARL curve of the modified X and EWMA charts for b=0.5

10 DESIGN OF X AND EWMA CHARTS 667 different values of a. For small values of 6, the modified procedures perform slightly better than the standard ones. The ARLs of the modified procedures are more sensitive to b than a since there is a substantial difference among b values for given a while there is no noticeable difference among a values for given b. In designing the modified procedures the coefficient a is used to obtain the true in-control standard deviation of subgroup means. Suppose that a is estimated as a and the in-c~ntrol standard deviation of subgroup means is used as L J;; J in the ~ modified procedures, then the ARLs of the modified procedures in the model (2.1) can be estimated analogously to (4.1) and (4.2) as and - M ARLx (a, b, 6, c) = Lx(A*, EM) (4.3) - M ARLE (a, b, 6, k) = LE(A*, k ~ ), From (4.5) a d (4.4) we can s& that the estimation error of a affects the control limit only while the amount of shift remains the same. In such cases we are using a constant(jw/ Jm) times of cm and km in calculating the control limits. Thus larger control limits are used if a is overestimated and smaller ones if underestimated. 5. ESTIMATION OF THE IN-CONTROL BETWEEN-GROUP VARIANCE In determining the control limits we need to estimate the within- and between- group variances. The within-group variance u2 can be easily estimated by the subgroup variance or subgroup range. An approach to estimating the two variances is to estimate them directly from the historical data using the one-way analysis of variance technique (see Wetherill and Brown, 199 1). For the construction of the control limts of the modified procedures, it will suffice to estimate variances when the process is in control only. The in-control between- group variance can be estimated from past experiences of the ARLs by estimating the coefficient a. Suppose that either the standard X or EWMA chart has been applied to a process for certain period of time and the average of the observed in-control run lengths

11 668 PARK is known as Ao. If A. is greater than or equal to the in-control ARL in the model (1.I), that is Lx (0, C) or LE(O, k), then we believe that there is no between-group variability. Otherwise we assume the existence of the between-group variability and estimate the coefficient a. In the X chart we let, by (3.1) with A* = 0, cs = c / J m and (3.3), and obtain the estimate of a as In the EWMA chart the coefficient a can not be estimated directly as is done in the x chart. The in-control ARL of the standard EWMA chart in the model (1.1) is plotted in log scale in Figure 3. In Figure 3 we find the k value corresponding to log(ao) as log(arl) from the curve for the given r value and let the value be ko. When r = 0.2 and A. = 300, for example, k corresponding to log(3oo) (= 5.70) is Since LE(O, ks) = A. for ks = k / J m by (3.2), we let and obtain the estimate of a as 6. CONCLUSIONS AND REMARKS The simple Shewhart model, despite its important role in statistical process control, was shown to be inadequate for cases where group-to-group variation exists in the process data. A variance components model was selected in such cases for a better explanation of the data and the properties of the X and EWMA charts are evaluated in the context of the ARL. The standard control procedures designed for the Shewhart model perform

12 DESIGN FIG. 3. The in-control ARL of the standard EWMA chart in the model (1.I): LE(~, k) poorly in the variance components model because they give false alarms too often when only a small variability is allowed in each group. In order to improve the performance of the standard procedures, modified control procedures are proposed using the true in-control variance of subgroup means in con- structing the control limits. The modifications make the false alarm rates much smaller than for the standard procedures. An example of estimating the in-control between-group variance was illustrated in constructing control limits of the modified procedures. Although this paper considered a variance components model in the statistical design only, this feature can be appkied to the economic design of the control chart. Since the ARL is important in designing the economic control scheme, it is expected that the modified control procedures will perform substantially better than the standard ones in a variance components model as they do in the statistical design.

13 PARK APPENDIX A.l Proof of (3.1) and (3.2) Notice that A similar argument would imply - A*). Thus. P(X; > Eo + C-) ARL~ (a, b, 6, c) = u J;; = 1 - A*) - A*) A*) Since (2;- EO)/aX(E) = xj1 + (a + b6)2, the inequality is equivalent to Notice that Thus we have, by (A.l) and (A.2),

14 DESIGN OF X AND EWMA CHARTS A.2 Proof of (4.1) and (4.2) Notice that Thus, similarly to A. 1, we have = (a(-cm - A*). ARL~(~, b, 6, c) = Lx (A*, CM). is equivalent to Therefore we have, by (A.l) and (A.3), ACKNOWLEDGEMENTS This research was partially supported by the Chung-ang University Research Grants in The author would like to thank the referee and the associate editor for their helpful suggestions and insightful critiques on an earlier version of this paper.

15 PARK BIBLIOGRAPHY Crowder, S. V. (1987). Average Run Length of Exponentially Weighted Moving Average Charts, Journal of Quality Technology, 19, Hahn, G. J. and Cockrum, M. B. (1987). Adapting control charts to meet practical needs:a chemical processing application, Journal of Applied Statistics, 14, Laubscher, N. F. (1996). A variance components model for statistical process control, South African Statistical Journal, 30, Wetherill, G. B. and Brown, D. W. (1991). Statistical Process Control: Theory and Practice, Chapman and Hall, London Wocdall, W. H. and Thomas, V. E. (1995) Statistical process control with several components of common cause variability, IIE Transactions 27, SASIQC Software:Usage and Reference, Version 6, First Edition (1995). Cary, NC : SAS Institute Inc. Received January, 1997; Revised October, 1997.