Systematic Patterns in T2 Charts - ASQ

Systematic Patterns in T2 Charts

Nonrandom or systematic patterns occurring in a univariate Shewhart control chart have often been used as indicators of extraneous sources of process variation. Proper diagnosis of these patterns can lead to process improvement by reducing the overall system variation. Similarly, in multivariate statistical process control, many di.erent systematic patterns may occur in the control charts used to monitor the process. The purpose of this paper is to examine the process conditions that lead to the occurrence of certain nonrandom patterns in a T2 control chart. Examples resulting from cycles, mixtures, trends, process shifts, and autocorrelated data are identi.ed and presented. Results are applicable to a Phase I operation or a Phase II operation where the T2 statistic is based on the most common covariance matrix estimator.

Key Words: Autocorrelation, Multivariate Statistical Process Control, Trends.

By ROBERT L. MASON, Southwest Research Institute, San Antonio, TX 78228-0510
YOUN-MIN CHOU, The University of Texas at San Antonio, San Antonio, TX 78249-0664
JOE H. SULLIVAN, Mississippi State University, MS 39762-9582
ZACHARY G. STOUMBOS, Rutgers, The State University of New Jersey, Piscataway, NJ 08854-8054
JOHN C. YOUNG, McNeese State University, Lake Charles, LA 70609-2340


One of the most popular control chart statistics for monitoring a multivariate normal process is Hotelling's T2. This recognition, in large part, is due to the fact that, among similar statistics, it is the easiest to interpret (e.g., see Mason, Tracy, and Young (1997) or Ryan (2000)). The T2 chart is sometimes referred to as a multivariate Shewhart chart, and several of its properties, such as its performance under varying process conditions, have been examined recently. For example, Stoumbos and Sullivan (2002) show that, for some multivariate non-normal distributions, the T2 chart based on known in-control parameters has an excessive false-alarm rate as well as a reduced probability for detecting shifts in the mean vector. Also, Linna, Woodall, and Busby (2001) study the effect of measurement error on the performance of the T2 chart.

Control charts indicate out-of-control conditions when the value of the corresponding control chart statistic is not contained in the control region. For univariate Shewhart charts, signals occur when the value of the statistic exceeds the upper control limit (UCL), or when the plotted statistic falls below the lower control limit (LCL). Lack of statistical control also may be indicated by a nonrandom pattern in the plotted statistic. These patterns often are detected using supplemental run rules, such as those specified in The Western Electric Handbook (1956), or in many control chart texts (e.g., see Montgomery (2001)). However, application of run rules is known to cause a considerable reduction in the in-control average run length (e.g., see Champ and Woodall (1987)).

The T2 statistic can be expressed as a univariate measure of the squared statistical distance that the p-dimensional multivariate normal observation vector x is from the process center represented by the population mean vector . One form of the T2 statistic is given by

where is the covariance matrix associated with x. When a T2 statistic is used to monitor a multivariate process, out-of-control conditions are generally suggested when a T2 value exceeds its UCL, although cases may exist where small T2 values are a concern so that an LCL would be used (e.g., see Tracy, Young, and Mason (1992)).

In applications, we estimate the unknown values of and using the common unbiased estimators, and , respectively. These estimates are obtained using the historical data set (HDS), x1, x2, ..., xn, under baseline conditions that are indicative of good operational conditions. The form of the T2 statistic under these conditions is given as T2

In-control data taken on a multivariate process reflect the process conditions. In turn, the conditions are reflected in the T2 control chart. For example, data obtained from a fixed-parts industry often exhibit only random variation when the process is in control. A T2 statistic for such data should produce a random signature in the corresponding T2 control chart. Any pattern other than a random signature would indicate lack of statistical control for the process. For such a situation, some form of run rules might be useful for generating signals based on seemingly non-random T2 patterns. However, because the T2 statistic has a non-normal distribution, it is not appropriate to apply the Western Electric run rules to the corresponding T2 control chart. Further, only limited run rules have been studied for T2 control charts. For example, Bozzello (1989) discusses the use of run rules with a T2 control chart in the case where the observation vectors are taken from a bivariate normal distribution.

In contrast to the above example, in-control data for some non-stationary processes, such as occur in the chemical processing industry, may actually contain inherent variation due to ramp changes, step changes, and even weak levels of autocorrelation. A T2 control chart for these data will often contain nonrandom patterns corresponding to the data characteristics. Application of run rules would be inappropriate for this type of process data, however, as the nonrandom patterns occurring in the T2 chart are not indicative of lack of statistical process control, but due to the inherent variation in the data. Since multivariate statistical process control is achieved when the mean vector and covariance matrix remain stable over rational subgroups of observations (e.g., see Wierda (1994)), it becomes most important to recognize the source of the patterns in a T2 chart.

Given this situation, the major purpose of this paper is to aid the T2 control-chart user by identifying process conditions that can lead to nonrandom patterns in a T2 chart. In doing so, we extend the use of the T2 statistic in a Phase I operation and show that certain nonrandom patterns that occur in the T2 chart of a preliminary data set can be used as a diagnostic data tool. This provides the user with an additional means of locating cycles, mixtures, trends, process shifts, and autocorrelation within the data set. These procedures also have applications in a Phase II operation.

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