In the retrospective (preliminary) analysis of individual observations collected over time, the in-control parameters of the process are unknown, and there is the possibility of a shift in the mean or standard deviation at any point in the observations. In many cases a special cause of variation will produce a single, sustained shift in the mean, standard deviation, or both. Presented here is a FORTRAN program for detecting such shifts, developed from a likelihood ratio test under the assumption of normality. The advantages over use of the standard X- chart and moving range chart are much higher probabilities of detection of sustained shifts and much improved diagnostic information. When a shift is detected, the most likely location is given, as well as whether it is primarily due to a shift in the mean, standard deviation, or both. Multiple shifts can often be detected by recursive application of the algorithm under the user's direction. The program calculates an approximate upper control limit, using an expression that is determined from simulation by Sullivan and Woodall (1996). Alternatively, the user can provide the desired upper control limit.

Keywords: Change Point, Maximum Likelihood, Moving Range Control Charts, X Control Charts.

*by* **Tcleveland D. Turner, JR, American Management
Systems, Inc., Birmingham, Alabama 35244, Joe H. Sullivan,
Mississippi State University, Mississippi State, MS 39762,
Robert G. Batson, The University of Alabama Tuscaloosa, Alabama
35487-0288** and **William H. Woodall, Virginia Polytechnic
Institute and State University, Blacksburg, VA 24061-0439
**

*INTRODUCTION*

Control charting applications involve detecting assignable causes that are revealed by shifts from the in-control statistical parameters of a process. In Stage 1, referred to as "retrospective" or "preliminary" analysis, historical observations are analyzed to decide if the process was in statistical control and to estimate the in-control parameters of the process. The difficulty is that the parameter estimates may be affected by special causes, possibly masking their presence. In the prospective Stage 2, control charts are used to detect departures from the statistical parameters estimated in Stage 1.

The in-control statistical parameters of a new or modified process are not known, so there is often a need to establish that a process is statistically in-control and estimate the parameters. It is especially important to detect assignable causes, if present, in Stage 1, since this leads to a better understanding of the process. The nature of the process may suggest rational subgroups within which the quality measurements would be homogeneous. On the other hand, there are many situations where it is reasonable to analyze individual observations, possibly because measurement is automated and every unit is measured, because the rate of production is slow, or for other situations in which a special cause can occur between any observations. See, for example, Montgomery (1996, pp. 221-222) for other situations in which a rational subgroup of size one is appropriate.

The common recommendation for this problem is the use of preliminary control charts for individual observations and moving ranges (X and MR control charts) to detect a shift in the process mean and variance. See, for example, Nelson (1982), Montgomery (1996), Ryan (1989), or Wadsworth, Stephens, and Godfrey (1986). Recently, it has been noted by Roes, Does, and Schurink (1993), Rigdon, Cruthis, and Champ (1994), and Sullivan and Woodall (1996) that the MR chart has very little added value, and the use of the X-chart alone has been recommended. The X-chart and the MR chart have the advantage of simple manual construction; however, with the increasing use of computers for quality control applications, it is reasonable to consider other approaches that are more powerful in discovering out-of-control conditions. The program described here uses a procedure considerably more powerful than an X-chart (alone, or combined with a MR chart) in detecting sustained shifts in the mean, standard deviation, or combinations thereof for a normal distribution. For a more complete description of the procedure see Sullivan and Woodall (1996). A generalization to the case of multivariate observations is given in Sullivan and Woodall (2000). This procedure, the LRT control chart, consists of the calculation of a likelihood ratio test statistic for all possible partitions of the historical data into two groups. When the plotted statistic exceeds an upper control limit, an out-of-control condition is indicated, and the maximum likelihood location for a single shift is the boundary of the partition that maximizes the statistic. Furthermore, for diagnostic purposes, an out-of-control signal can be attributed to a shift in mean or standard deviation or both. The process is most sensitive to a single sustained shift, but multiple shifts can often be detected by recursive use of the chart process. The FORTRAN program described below implements the LRT chart process, with recursive application that is either automatic or manually controlled by the user. This program should be used in conjunction with other commonly available software, such as Excel, for the graphical display of the data.

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