A preliminary cumulative sum (CUSUM) chart based on individual observations is developed from the uniformly most powerful test for the detection of linear trends. This CUSUM chart is compared with several of its alternatives which are based on the likelihood ratio test and on transformations of standardized recursive residuals on which, for instance, theQ-chart methodology is based. It turns out that the proposed CUSUM chart is not only superior in the detection of linear trend out-of-control conditions, but also in the detection of other out-of-control situations considered in this paper. Approximate control limits, determined from simulation and an example of its use in practice are given for the proposed CUSUM chart.

Keywords: control charts, CUSUM control charts, statistical process control

*by* **Alex J. Koning, Erasmus University Rotterdam,
NL-3000 DR, Rotterdam, The Netherlands **and **Ronald J.
M. M. Does, University of Amsterdam, NL-1018 TV, Amsterdam,
The Netherlands**

*INTRODUCTION*

Control charts are basic and powerful tools in statistical process control (SPC) and are widely accepted and applied techniques for controlling various industrial processes. Originated by Shewhart in 1924 (see Shewhart (1931)), the effectiveness of control charts is due in part to their simplicity. In fact, these charts are simple graphs with time on the horizontal axis and a quality characteristic such as sample mean or sample range plotted on the vertical axis. Shewhart developed the use of 3-sigma control limits as action limits, that is, if the quality characteristic is outside these limits, then the process is called out-of-control and action is needed to eliminate the special cause.

Another technique to control an industrial process was developed by Page (1954). He proposed the so-called cumulative sum (CUSUM) chart. This technique plots the cumulative sums of deviations of the sample values of a quality characteristic from a target value against time. There are two ways to represent CUSUM´s: the tabular (or algorithmic) CUSUM and the V-mask form of the CUSUM. An excellent overview of CUSUM techniques is given in Hawkins and Olwell (1998).

It is said (see Montgomery (1997)) that Shewhart-type control charts for averages are very effective if the magnitude of the shift is 1.5-sigma to 2-sigma or larger. For smaller shifts, the CUSUM is a good alternative or additional tests for special causes (see Nelson (1984)) are needed to improve the effectiveness of the Shewhart-type control chart.

In the literature two phases of using control charts are distinguished: Phase I and Phase II. Phase II, in which the true distribution is known, we do not consider. Phase I comprises two stages: Stage 1 (retrospective) and Stage 2 (prospective). In Stage 1, historical data are analyzed to decide if the process is in statistical control and to estimate the in-control parameters of the process. The next stage, the prospective Stage 2, is started when the analysis of past data did not reveal any out-of-control signals. It is very important that all special causes are detected in Stage 1 because this leads to a better understanding of the process and it avoids inflation of the estimates of the parameters needed for Stage 2. In the course of time, the preliminary estimates are revised based on available data from Stage 2 and we are again in a Stage 1 situation. This updating and checking of the parameters is a recurrent phenomena, but one hopes to move into a situation where out-of-control signals are fairly rare.

Today´s manufacturing environment hardly resembles the high volume production of the 1920´s to the early 1980´s, in which period SPC charting methods were introduced. Current manufacturing practice is typically characterized by either frequent set-up changes to accommodate a wide range of different products (short-run processes) or intrinsically low production volumes of the same type of product. Printed circuit boards for specialized applications are an example of the former situation, while the production of wafer steppers for semiconductor production is an example of the latter. In the last few years there has been a revival of interest in handling these problems. The increasing use of computers for SPC applications enables industrial statisticians to consider other approaches from mathematical statistics that are more powerful when searching for special causes.

Hawkins (1987) introduced CUSUM*Q*-charts
and Quesenberry (1991, 1995) and Del Castillo and Montgomery
(1994) developed some enhancements and alternative methods.
All these methods require the use of a computer. The present
study was motivated by a recent paper of Sullivan and Woodall
(1996), where a preliminary control chart for individual observations
based on a likelihood ratio test intended for the Stage 1
situation was developed. It is remarkable that some of the
new charts show similarities to well-known papers in econometrics.
For instance, the*Q*-chart methodology can be written
as a transformation of standardized recursive residuals as
proposed in section 2.3 of Brown, Durbin, and Evans (1975).
In addition, the likelihood ratio test (LRT) control chart
in Sullivan and Woodall (1996) is related to Quandt´s
log-likelihood ratio technique (see Quandt (1960)). In this
paper, we consider individual observations and we focus on
the retrospective (Stage 1) situation. After introducing standardized
recursive residuals and*Q*-statistics, we develop a new
CUSUM-type chart for the detection of linear trends based
on the uniformly most powerful test, and we pare this new
chart with the LRT chart of Sullivan and Woodall (1996); the
CUSUM chart of Brown, Durbin, and Evans (1975); and several
charts based on*Q*-statistics. It turns out that the
proposed CUSUM chart has the best properties to detect trends
and shifts in location in the data. However, it should be
noted that the LRT chart is more general than the other charts
considered in this paper; usually one has to trade-off performance
and increased generality.

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