ON CONSTRUCTING T CONTROL CHARTS FOR RETROSPECTIVE EXAMINATION. Gunabushanam Nedumaran Oracle Corporation 1133 Esters Road #602 Irving, TX 75061

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1 ON CONSTRUCTING T CONTROL CHARTS FOR RETROSPECTIVE EXAMINATION Gunabushanam Nedumaran Oracle Corporation 33 Esters Road #60 Irving, TX 7506 Joseph J. Pignatiello, Jr. FAMU-FSU College of Engineering Florida State University Florida A&M University Tallahassee, FL Key Words: statistical process control; multivariate process; χ control charts; Hotelling s T ; Tˆ MAX,D adjustment; Monte Carlo simulation. -type statistic; false alarm; Type I error; Bonferroni ABSTRACT In this paper we consider the issue of constructing retrospective T control chart limits so as to control the overall probability of a false alarm at a specified value. We describe an exact method for constructing the control limits for retrospective examination. We then consider Bonferroni-adjustments to Alt s control limit and to the standard χ control limit as alternatives to the exact limit since it is computationally cumbersome to find the exact limit. We present the results of some simulation experiments that are carried out to compare the performance of these control limits. The results indicate that the Bonferroni-adjusted Alt s control limit performs better than the Bonferroni-adjusted χ control limit. Furthermore, it appears that the Bonferroni-

2 adjusted Alt s control limit is more than adequate for controlling the overall false alarm probability at a specified value..0 INTRODUCTION Statistical process control (SPC) charts are tools that are used to monitor the state of a process by distinguishing between common causes and special causes of variability. When several characteristics of a manufactured component are to be monitored simultaneously, multivariate Shewhart-type χ or T control charts can be used (Montgomery 996, pp ). As long as the points plotted on the χ or T control chart fall below the upper control limit (UCL) of the chart, the process is assumed to operate under a stable system of common causes, and hence, in a state of control. When one or more points exceed the UCL, the process is deemed out of control due to one or more special causes and an investigation is carried out to detect these special causes. When the in-control values of the process mean vector and covariance matrix are known, these known parameter values are used for computing the statistic plotted on a χ control chart. The UCL of this chart is based on the chi-square distribution. When initially examining the process, however, these parameter values are not known. To establish a preliminary retrospective control chart, the process parameters are estimated from some m initial subgroups of size n taken when the process is believed to be stable. These parameter estimates are used to calculate Hotelling s statistic for each subgroup, and the subgroup statistic is plotted on a T T control chart.

3 The UCL of this chart is based on the F distribution. This chart is used to test retrospectively whether the process was in control when the m initial subgroups were drawn. The upper control limits of χ and T controls chart are selected such that if the process is in control, only rarely will a false alarm be given. A control chart issues a false alarm if a subgroup statistic exceeds the UCL when the process is actually in control. The usual practice for constructing a retrospective control chart is to design the chart for a specified false alarm probability for each subgroup plotted. We let α denote the false alarm probability for each subgroup plotted on the chart. In many control chart applications, the user might wish to control the overall probability of a false alarm at some desired level for all m subgroups rather than that of individual subgroups. The overall probability of a false alarm during retrospective testing is considerably greater than α since m points are plotted all at once. In this paper, we consider the issue of constructing T control chart limits so as to control the overall probability of a false alarm during retrospective testing at a specified value. We first consider an exact method for constructing the control limit. We then analyze the performance of Alt s (976) control limit and the standard χ control limit with the appropriate Bonferroni-type adjustments. This paper is organized as follows. In the next section we describe the multivariate process model and review some pertinent literature. In the following section, we describe an exact method for constructing T control chart limits so as to control the overall probability of a false alarm during retrospective testing at a specified 3

4 value. We then present the results of some simulation experiments that compare the performance of Alt s (976) control limit and the standard χ limit with the appropriate Bonferroni-type adjustments. Finally, we make some recommendations for constructing retrospective control charts based on the simulation study..0 MULTIVARIATE PROCESS MODEL AND LITERATURE REVIEW We let X ij = ( X ij X ij X ijp),,..., denote a p vector that represents the p characteristics of the jth observation in the ith subgroup, i =,,,... and j =,,..., n. We assume that the X ij s are independent and identically distributed normal random variables with mean m and covariance matrix Σ when the process is in control. That is, we assume that the X ij s are iid N p (m, S) when the process is in control. We let X i denote the average vector for the ith subgroup, and we let S i denote the unbiased estimate of the covariance matrix for the ith subgroup. That is, X i = n n j= X ij and n S = i Xij X i X ij X i n. j= 4

5 When the in-control process parameter values are known, the statistic plotted on the χ control chart for the ith subgroup is X m S X m 0. () χ i = n i 0 0 i When the process is in control this statistic has a chi-square distribution with p degrees of freedom (Montgomery 996, p. 364). It is plotted on the χ control chart with an upper control limit (UCL) given by χ p, α () where χ p, α is the ( α ) th percentile point of the chi-square distribution with p degrees of freedom and α is the probability of a false alarm for each subgroup plotted on the control chart. This stage in control charting process is referred to as Phase II by Alt (985). If the process parameter values are not known, data from m initial subgroups are collected when the process is believed to be in control. Then, pooling data from these m subgroups and assuming that the process was in control, unbiased estimates of the mean vector and the covariance matrix are given by X = m m i= X i m and S = S m i= i (3) respectively. A T control chart is then constructed using these estimated parameters. The control chart is first used to test retrospectively whether the process was in control when the m initial subgroups were drawn. Alt (985) refers to this stage as Phase I, Stage. After the initial control has been established, the control chart can be 5

6 used to monitor the process on-line, i.e., the subgroup averages are plotted one-at-atime on the chart as each new subgroup is obtained. Alt (985) refers to this stage as Phase I, Stage. We consider Phase I, Stage in this paper. Nedumaran and Pignatiello (999) discuss multivariate control charting during Phase I, Stage. Stage ) is The statistic plotted on the T T control chart for each initial subgroup (Phase I, = n X X S X X. (4) i i i Alt (976) gives the UCL of this control chart as where ( m, n, p), mn m UCL (5) = C F T p p + ( m, n, p) p( m )( n ) ( mn m p + ) C =, F ν, ν, α is the ( α ) th percentile point of the F distribution with ν and ν degrees of freedom, and α is the desired false alarm probability for each subgroup plotted on the T control chart. If the process parameters are estimated from a reasonably large number of initial subgroups, the usual practice for constructing retrospective T control charts is to use UCL instead of the exact UCL χ T. However, according to Montgomery (996, p. 367) we must be careful when following this practice. In this paper, we show that Bonferroni-adjusted χ control limits issue relatively large number of false alarms even when the process parameters are estimated from 50 or more subgroups 6

7 In the next section, we describe an exact method for constructing retrospective T control chart limits in which the overall false alarm probability for all m initial subgroups is controlled at a specified value. We then study and compare the performances of Bonferroni-type adjustments made on both Alt s control limit and the standard χ control limit. 3.0 CONSTRUCTING RETROSPECTIVE CONTROL CHARTS We suppose that data from m initial in-control subgroups are available. We let X i and X j ( i, j < m ) denote two initial subgroups. Then, it can be shown by direct evaluation that and E [ X i X ] = 0 m [ X ] = S 0 Var i X mn The ( i X, X j X ) = S 0 Cov X. mn T statistic plotted for the ith subgroup is given by T = n X X S X X. i i i We let φ denote the specified overall false alarm probability and we let UCL E denote the exact upper control limit. Then, Pr [ T UCL, i =,,..., m] = φ i E. 7

8 That is [ ] = φ Pr T max UCL E where T max = max n i ( X i X ) S ( X i X ), i =,,..., m. Here, T max is a Tˆ MAX,D -type statistic discussed by Siotani (959) with Siotani s ( m ) mn γ = and δ = mn. Thus, E point of the distribution of Tˆ MAX,D statistic. UCL is the upper ( φ) th percentile Siotani (959) pointed out that the sampling distribution of Tˆ MAX,D -type statistic is extremely difficult to find. Siotani (959) suggested a two-stage procedure for finding approximations to the upper percentile points of the distribution of Tˆ MAX,D, which was investigated further by Seo and Siotani (993). However, this two-stage procedure involves extensive computations and gives only approximate percentile points. As an alternative, we recommend a much simpler Bonferroniadjustment to Alt s control limit based on the following simulation study. We carried out Monte Carlo simulation experiments to compare the performance of Alt s control limit and the standard χ control limit. Using Bonferroni-adjustment the false alarm probability for each subgroup was set at α = φ m, where φ is the overall false alarm probability for all m initial subgroups. Then, Alt s control limit is given by 8

9 UCL ( m, n, p), = C F T p mn m p + and the standard χ control limit is given by χ p,α. The performance measure considered was the overall probability of a false alarm. We considered three process dimensions of p = 3, 6 and 0, two overall false alarm probabilities of φ = and 0.05, and several values of m. For each combination of p, m and φ, we generated m initial subgroups of size n = 5 from a stable in-control normal distribution with m = 0 and Σ = I using IMSL STAT/LIBRARY (987) FORTRAN routines. The in-control process parameter values were estimated based on these m subgroups. The T statistic for each subgroup was plotted on a chart with a Bonferroni-adjusted Alt s control limit and a chart with a Bonferroni-adjusted standard χ control limit. If a chart issued one or more false alarms, a counter for that chart was increased by one. This procedure was replicated 0,000 times. The overall false alarm probability for each chart was then estimated by dividing the number of replications in which the control chart issued at least one false alarm by the total number of replications. The results are shown in Tables I III. The estimated overall false alarm probability for the Alt s control limit and the standard χ control limit are given as FA-ALT and FA-CHI, respectively. 9

10 TABLE I. Estimated Overall False Alarm Probability for p = 3 and n = 5 m φ = φ = 0.05 FA-ALT FA-CHI FA-ALT FA-CHI TABLE II. Estimated Overall False Alarm Probability for p = 6 and n = 5 m φ = φ = 0.05 FA-ALT FA-CHI FA-ALT FA-CHI

11 Table III. Estimated Overall False Alarm Probability for p = 0 and n = 5 m φ = φ = 0.05 FA-ALT FA-CHI FA-ALT FA-CHI Results in Tables I III show that the Bonferroni-adjusted Alt s control limit performs better than the Bonferroni-adjusted standard χ control limit. The estimated overall false alarm probability for the Bonferroni-adjusted Alt s control limit is relatively close to the specified level for all p, m and φ combinations considered. The simulation study suggests that the use of computationally cumbersome Siotani s twostage procedure may not be warranted in retrospective control charting since Alt s control limits with the appropriate Bonferroni-type adjustments provide overall false alarm probability values that are sufficiently close to the specified value. By contrast, the estimated overall false alarm probability is larger than the specified value for the Bonferroni-adjusted χ control limit. For small p (p = 3), the

12 estimated overall false alarm probabilities get reasonably close to the specified value for m > 30. But, for larger p (p = 6 and 0) the estimated overall false alarm probabilities are much larger than the specified value even for large m. Thus, the Bonferroni-adjusted χ control limit appears to issue more false alarms than one might anticipate. Hence, we recommend that Bonferroni-adjustment to Alt s control limit be used for constructing retrospective control charts when controlling the overall probability of false alarm is of primary concern. 4.0 SUMMARY In this paper we considered the issue of constructing retrospective T control chart limits so as to control the overall probability of a false alarm at a specified value. We described an exact method for constructing the control limits. We considered Bonferroni-adjustments to Alt s control limit and to the standard χ control limit as alternatives to the exact limit since it is cumbersome to find the exact limit. We then presented the results of some simulation experiments that were carried out to compare the performances of these control limits. The results indicate that the Bonferroniadjusted Alt s control limit performs better than the Bonferroni-adjusted χ control limit. Furthermore, it appears that the Bonferroni-adjusted Alt s control limit is more than adequate for controlling the overall false alarm probability at a specified value.

13 BIOGRAPHICAL FOOTNOTE Dr. Gunabushanam Nedumaran is a consultant with Oracle Corporation. His research interests are in the area of applied statistics and statistical process control. He is an ASQ certified Quality Engineer. His address is gnedumar@cs.com. Dr. Joseph J. Pignatiello, Jr. is an Associate Professor in the Department of Industrial Engineering in the FAMU-FSU College of Engineering at Florida State University and Florida A&M University. Dr. Pignatiello's interests are in quality engineering and include statistical process control, process capability, design and analysis of experiments, robust design and engineering statistics. Dr. Pignatiello won the 994 Shewell Award from ASQC's Chemical and Process Industries Division and the 994 Craig Award from ASQC's Automotive Division for his research on process capability. In 990 and 99 he won awards from the Ellis R. Ott Foundation for his research on SPC and the methods of Taguchi. He serves on the editorial boards of IIE Transactions, Journal of Quality Technology and Quality Engineering. His address is pigna@eng.fsu.edu. ACKNOWLEDGEMENTS This material is based in part upon the work supported by the Texas Advanced Research Program under Grant No

14 BIBLIOGRAPHY Alt, F. B. (976). Small Sample Probability Limits for the Mean of a Multivariate Normal Process. ASQC Technical Conference Transactions, pp Alt, F. B. (985). Multivariate Quality Control in Encyclopedia of Statistical Sciences 6, edited by S. Kotz and N. L. Johnson, John Wiley & Sons, New York, NY. IMSL STAT/LIBRARY (987). IMSL STAT/LIBRARY User s Manual, Vol. -3, IMSL Inc., Houston, TX. Montgomery, D. C. (996). Introduction to Statistical Quality Control. John Wiley & Sons, New York, NY. Nedumaran, G. and Pignatiello, J. J., Jr. (999). On Constructing T Control Charts for On-line Process Monitoring. IIE Transactions 3, pp Seo, T. and Siotani, S. (993). Approximations to the Upper Percentiles of Tmax - type Statistics in Statistical Science & Data Analysis, edited by K. Matsusita et al., VSP, Tokyo. Siotani, M. (959). The Exact Value of the Generalized Distances of the Individual Points in the Multivariate Normal Sample. Annals of the Institute of Statistical Mathematics 0, pp