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Design of and ewma charts in a variance componentsmodelChangsoon Park aa Dipartment of Applied Statistics , Chung-Ang Uuiversity , Dongjak-gu, Seoul, Huksuk-dong, 156-756, KoreaPublished online: 05 Jul 2007.

To cite this article: Changsoon Park (1998) Design of and ewma charts in a variance components model, Communications inStatistics - Theory and Methods, 27:3, 659-672, DOI: 10.1080/03610929808832119

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COMMUN. STATIST.-THEORY METH., 27(3), 659-672 (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, 156-756, Korea

Key Work: average run length; process control; Shewhart model; between-group varia- tion; 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.

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1. INTRODUCTION

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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 compo-

nents 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).

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DESIGN OF X AND EWMA CHARTS 66 1

In this paper we consider two components of variability: within-group and between-

group. 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 6.

2. 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

observations X,, 's in the model (2.1) are iid N ( E l oh ( c ) ) and the subgroup means X, 's

are iid N ( ( , (<)), where (() = 02 + r 2 ( ( ) and 0% (() = 0 2 / n + r 2 ( ( ) . 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.

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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 a X i 2 to + c- or X i 5 E

6 O - CJ;;' The standard procedure of the EWMA chart is to signal at the first i for which

where 2; = r X i + ( 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 ( a X ( E ) ) . Let A R L ~ ( a , b, 6, c ) and A R L $ ( 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.

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DESIGN OF X AND EWMA CHARTS 663

Also let Lx (A , c) and L E ( 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).

ARL; (a, b, 6, c ) = Lx (A*, C S )

and

d7. cs = d+ and ks = where A* = 6 dTTZ From (3.1) and

l+(a+bJ) 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 numer-

ically. SAS provides the function EWMAARL to calculate L E ( 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 0.0027,

0.0073,0.0339,0.1796 for a=O, 0.5, 1.0, 2.0. False alarm rates of the EWMA chart are

0.0018, 0.0044, 0.0173, 0.0822 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 2 0.5 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 1.

4. MODIFICATIONS OF THE x AND EWMA CHARTS

The standard X and EWMA procedures are modified using the true in-control stan-

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PARK

TABLE I. ARLs of the standard X and EWMA charts

A R L ~ (a , b, 6 , s ) A R L ~ ( ~ , 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

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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 A R L ~ ( ~ , 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 ,,&. C M = 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 - C M and k - kM are smaller than the corresponding

differences c - c s 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

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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

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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 ~ in the J;; 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

- M ARLx ( a , b , 6 , c ) = L x ( A * , E M ) ( 4 .3 )

and

- M ARLE ( a , b , 6, k ) = L E ( 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

c o n s t a n t ( J W / Jm) times of c M and k M 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

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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 , c s = 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 2.77.

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

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DESIGN

FIG. 3. The in-control ARL of the standard EWMA chart in the model (1 .I):

L E ( ~ , 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.

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APPENDIX

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A.l Proof of (3.1) and (3.2)

Notice that

= @(-cs - A*).

A similar argument would imply

u P(X; > Eo + C-) = 1 - @(cS - A*)

J;; Thus.

A R L ~ ( a , b , 6 , c) = 1

@(-cs - A*) + 1 - @(cS - A*)

Since (2; - EO)/aX(E) = x J 1 + ( a + b6)2, the inequality

is equivalent to

Notice that

Thus we have, by (A.l) and (A.2),

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DESIGN OF X AND EWMA CHARTS

A.2 Proof of (4.1) and (4.2)

Notice that

= (a(-cM - A * ) .

Thus, similarly to A. 1, we have

A R L ~ ( ~ , b, 6 , c ) = Lx ( A * , C M ) .

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 1996. 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. Dow

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BIBLIOGRAPHY

Crowder, S. V. (1987). Average Run Length of Exponentially Weighted Moving Average

Charts, Journal of Quality Technology, 19, 161-164.

Hahn, G. J. and Cockrum, M. B. (1987). Adapting control charts to meet practical needs:a

chemical processing application, Journal of Applied Statistics, 14, 33-50.

Laubscher, N. F. (1996). A variance components model for statistical process control,

South African Statistical Journal, 30, 27-47.

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 com-

ponents of common cause variability, IIE Transactions 27,757-764.

SASIQC Software:Usage and Reference, Version 6, First Edition (1995). Cary, NC :

SAS Institute Inc.

Received January , 1997; Revised October, 1997.

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