Classification of Distribution-Free Quality Control Charts

Proceedings of the Annual Meeting of the American Statistical Association, August 5-9, 2001

CLASSIFICATION OF DISTRIBUTION-FREE QUALITY CONTROL CHARTS Saad T. Bakir College of Business Administration, Alabama State University, P O Box 271, Montgomery, AL 36101. E-mail: [email protected] Key Words: ARL, Bootstrap, Nonparametric, Ranks, Sequential ranks, Sign-test

1. INTRODUCTION Since their inception by W. A. Shewhart (1926, 1927, and 1931), statistical quality control charts have been increasingly applied in various disciplines. Most control charts are distribution-based procedures in the sense that the process output is assumed to follow a specified probability distribution (usually, normal for continuous measurements and binomial or Poisson for attribute data). Recent literature witnessed the development of a number of distribution-free (or nonparametric) control charts where no underlying probability distribution is assumed on the process output (often continuity is assumed). While the distribution-based control charts are well known and well documented in the statistical literature, the distribution-free control charts are less known and rarely referred to even in textbooks specialized in control charts. This is despite the fact that distribution-free control charts possess many advantages and in some cases are even more efficient than their distribution-based counterparts. Recently, Chakraborti, Laan and Bakir (2001) gave an overview of several nonparametric control charts. Woodall and Montgomery (1999) presented recent issues and ideas for control charts in general. In this paper we classify the distribution-free control charts according to the driving nonparametric ideas behind them. 2. STATISTICAL QUALITY CONTROL CHARTS A quality control chart (CC) is a statistical scheme (usually allowing graphical implementation) devised for the purpose of checking and then monitoring the statistical stability of a process. A universally accepted principle (e.g., Deming 1986, 1993) is that the output of every process exhibits some variation. A process is said to be stable (in-control) if the sources of its variation are only due to common causes that are random and inherent in the

process itself day in and day out. A process is said to be unstable (out-of-control) if the sources of its variation are due to common causes plus additional special (assignable causes). Special causes of variation are not naturally inherent but are extraneous to the process. Control charts are designed to detect the existence of special causes as quickly as possible after their occurrence. The number of sampling instances before a CC gives an out-of-control signal is called the run length. The efficiency of a CC depends on the probability distribution of the run length. The most common efficiency criterion is to consider the average run length (ARL), which is the expected value of the run length distribution. In practice, the stability of a process is usually determined relative to one or more of its output characteristics such as the mean and/or the variance. 2.1. TYPES OF CCs Depending on the structure of the control sequence and the nature of the signal rule, CCs can be classified as follows: 1. The Shewhart-type CCs, first introduced by Shewhart, (1926), (1927), and (1931). 2. The cumulative sum-type (CUSUM) CCs, first introduced by Page (1954). 3. The exponentially weighted moving averagetype (EWMA) CCs, first introduced by Roberts (1959). 4. The linear barrier-type (LB) CCs. It is important for us now to state that each of these types may be divided into: A. Distribution-based (parametric) CCs that assume a certain probability distribution for the process output. For a reference on these charts, see Montgomery (2001). B. Distribution-free (nonparametric) CCs that do not assume a particular probability distribution for the process output. For a reference on these charts, see Chakraborti, Laan and Bakir (2001) and Bakir (2002).

3. A BIRD’S EYE-VIEW OF DISTRIBUTION-FREE CONTROL CHARTS Definition: Let ü be a class of underlying probability distributions containing more than one distributional form. A CC is said to be distribution-free (nonparametric) over the class ü if the in-control run length distribution of the CC is the same for every underlying distribution in ü. This definition implies that the in-control ARL of a DFCC is the same for all parent distributions in the class ü. Borrowing ideas form distribution-free (DF) hypothesis testing procedures, researchers proposed several DFCCs based on signs, ranks, signed-ranks and other DF statistics. The sign test statistic is probably the simplest to use although it may not be very efficient. Arnold (1985, 1986) and Amin, Reynolds and Bakir (1995) developed Shewhart and CUSUM DFCCs for grouped data based on the sign test statistic. In their development, a target value for the process center is assumed known. Farnum and Stanton (1986) used counts above and below two known gauge limits to develop a Shewhart DFCC. Because the probability distribution of the sign statistic is obtainable from the binomial distribution, the theoretical properties, such as the ARL, of CCs based on the sign statistic can in general be computed exactly. Amin and Widmaier (2000) developed Shewhart sign charts (for controlling the center) with added features such as variable sampling intervals (VSI), variable sampling size (VSS), and curtailed sampling. These features improve the performance of the sign chart substantially. The effects of curtailed sampling on Shewhart and CUSUM sign charts for center were extensively studied by Amin and Sattler (2001). Shewhart and CUSUM sign charts for variability were developed by Amin and Golz (2001). To enhance efficiency, efforts were made to construct DFCCs using ranks and signed-ranks. However, some inherent difficulties face the implementation of traditional ranking techniques in control charting applications. One difficulty arises from the fact that, in control charting applications, observations are usually sampled in groups. It is not clear how to rank such sequences of grouped data. If single observations are taken sequentially, it is also not clear how to rank long sequences of individual observations. Several

solutions to these difficulties have been suggested as discussed below. Parent (1965) and Reynolds (1975 a) employed the idea of sequential ranking to develop CUSUM and linear barrier DFCCs for individuals to control the process center. In a sequence of observations, the sequential rank of an observation is defined to be its rank with respect to all its predecessors in the sequence. Because Parent (1965) and Reynolds (1975 a) charts actually use sequential signed-ranks, their proposed CCs require the knowledge of a known target value for the process center. The idea of sequential ranking (with no signs) was used by Bhattacharyya and Frierson (1981) to develop a linear barrier DFCC and by McDonald (1990) to develop a CUSUM DFCC without assuming a known target value for the process center. Two difficulties accompany the idea of sequential ranking in control charting. Firstly, one may end up with ranking long sequences of individual observations. Secondly, it is not clear how to apply the idea of sequential ranking to CCs if observations are taken in groups of size larger than one. To alleviate the difficulty that is inherent in sequential ranking, Bakir (1977) and Hackl and Ledolter (1992) suggested “limited sequential ranking”, where an observation is ranked with respect to a fixed number of its immediate predecessors. Bakir (1977) developed linear barrier and CUSUM DFCCs for individuals based on limited sequential signedranks where an in-control target is assumed known for the process center. Hackl and Ledolter (1992) developed EWMA DFCCs for individuals based on limited sequential ranks (with no signs) where no in-control target for the process center is assumed known. To adapt ranking techniques to CCs for grouped observations, Bakir and Reynolds (1979) suggested ranking the observations within groups. This idea of “within-group ranking” was used earlier by Wilcoxon, Rhodes and Bradley (1963) and Van der Laan (1966) to develop DF sequential probability ratio tests for the twosample problem. Bakir and Reynolds (1979) used within-group signed-ranks to develop a CUSUM DFCC that requires the knowledge of an in-control target for the process center. Amin and Searcy (1991) used within-group signedranks to develop an EWMA DFCC (an incontrol target central value is assumed known). Another idea that works for grouped or individual data is to assume that a “reference sample” or a “reference known probability

distribution” is available when the process is incontrol. Instead of within-group ranking, one ranks the observations of the current group (sample) with respect to the reference sample. This is analogous to the ranking scheme of the Wilcoxon rank-sum test. Using this idea of a reference sample, Park and Reynolds (1987) developed Shewhart and CUSUM DFCCs based on Orban’s (1982) placement statistics. Hackl and Ledolter (1991) developed an EWMA DFCC for individuals based on ranking the current observations with respect to a reference sample. One advantage of the idea of a reference sample is that one does not need to assume the availability of an in-control target level for the process. Bakir (1997) employed the idea of “reference sample” to develop two Shewhart DFCCs one based on the two-sample Kolmogorov-Smirnov statistic and the other based on the two-sample Cramer-von Mises statistic. Another implementation of the idea of a “reference sample” was made by Willemain and Runger (1996) to construct Shewhart CCs based on some order statistics obtained from the reference sample. Further, Hackl and Ledolter (1991) and Bakir (1997) developed DFCCs assuming the availability of a known “reference probability distribution” for the process output. Amin and Besenfelder (2000) constructed Shewhart charts based on the sum of squared ranks statistic for the purpose of controlling variability (curtailed sampling enhancement is also considered) in the process. Alloway and Raghavachari (1991) suggested a Shewhart DFCC (for groups) based on the Hodges-Lehmann estimator and the

associated distribution-free confidence intervals based on Walsh averages, a target central value is not assumed known. However, Pappanastos and Adams (1996) demonstrated that the Alloway and Raghvachari (1991) chart is not truly distribution-free because its in-control ARL is not constant across a class of underlying distributions. Bakir (1995a, 1995b, 1996, 2000) suggested Shewhart DFCCs (based on the analysis of means technique) applicable to data on the ordinal scale of measurements. Very few multivariate DFCCs exit. Hayter and Tsui (1994) proposed a Shewharttype multivariate DFCC for individuals to monitor the process location vector parameter. The chart is based on the “M” statistic, which is the maximum of deviations of the observations from their sample means. The calculation of the control limits is based on the empirical distribution of an initial reference sample. Kapatou and Reynolds (1994, 1998) proposed EWMA-type multivariate CCs for groups based on the sign and the signed-rank statistics. Kapatou and Reynolds charts are not truly nonparametric because some elements of the covariance matrix need to be estimated. Liu (1995) combined the idea of a reference sample with the concept of data depth to construct three multivariate DFCCs: A Shewhart-type chart for individuals, a Shewhart-type chart for groups, and a CUSUM-type chart for individuals. Liu’s charts were studied extensively by Stoumbos, et al. (1999) and Stoumbos and Jones (2000). The various DFCCs may be classified as in the Appendix.

4. SUGGESTIONS FOR FUTURE RESEARCH

length distributions and the ARLs of these charts are worthwhile. Third, practical aspects of the implementation of the DFCCs are very much lacking. Many articles on control charts do not provide numerical values for the control limits of the proposed charts. The would be user of these charts has no clue as to what control limits he or she should use to attain a certain in-control ARL or false alarm rate. Accurate tabulations of control limits of the proposed DFCCs need be made available for the potential user. Other issues involve the practicality of the sampling plan, the sample size, and the ranking schemes that are required for the DFCCs. In this respect, Stoumbos et al. (1999) and Stoumbos et al. (2000) studied some multivariate DFCCs based on the simplicial depth of the data. Some success

First, some deep investigations of the currently existing charts are required. Investigations by Pappanastos and Adams (1996) and Jones and Woodall (1998) have shown that some presumably DFCCs are in fact not truly distribution-free. Second, more reliable efficiency comparisons of the DFCCs among themselves and comparisons with parametric competitors are much needed. Previous comparisons resulted mostly from limited simulation runs due to limited computer power. The new Millennium promises vast leaps in computer speed powers. Further, theoretical investigations of the run

stories of actual implementations of the DFCCs in manufacturing or elsewhere need to be publicized. Finally, we certainly need additional DFCCs for monitoring variability and for multivariate situations. Amin, Reynolds and Bakir (1995) and Amin and Golz (2001) suggested Shewhart and CUSUM charts for variability based on the sign statistic. Amin and Besenfelder (2000) developed Shewhart and CUSUM control charts for variability based on the sum of squared ranks statistic. Stromberg (2001) developed a Shewhart DFCC for the range based on bootstrap and jackknife sampling. Multivariate DFCCs have recently been proposed by Kapatou and Reynolds (1994, 1998), Hayter and Tsui (1994) and Liu (1995). REFERENCES Alloway, J. A., Jr., and Raghavachari, M. (1991). “Control Chart Based on the HodgesLehmann Estimator.” J. Of Quality Technology 23, 336-347. Amin, R. and Besenfelder, W. (2000). “A Control Chart for Variability Based on a Sum of Squared Ranks Statistic.” Technical Report # RA 2000-2, Dept. of Mathematics and Statistics, University of West Florida. Amin, R. and Golz, A. (2001). “Shewhart and CUSUM Sign Charts for Variability.” Technical Report # RA 2001-3, Dept. of Mathematics and Statistics, University of West Florida. Amin, R. and Sattler, G. (2001). “Curtailed Sampling Plans Applied to Some Nonparametric Control Charts.” Technical Report # RA 2001-4, Dept. of Mathematics and Statistics, University of West Florida. Amin, R. and Widmaier, O. (2000). “A Simple Non-Parametric Control Chart with Variable Sampling Intervals and Curtailed Sampling.” Technical Report # RA 2000-1, Dept. of Mathematics and Statistics, University of West Florida. Amin, R; Reynolds, M. R., Jr.; and Bakir, S. T. (1995). “Nonparametric Quality Control Charts Based on the Sign Statistic.” Commun. Statist.- Theory Meth. 24(6), 1597-1623. Amin, R. W. and Searcy, A. J. (1991). “A Nonparametric Exponentially Weighted Moving Average Control Scheme.” Commun. Statist.Simula., 20(4). 1049-1072. Arnold, B. (1985). “The Sign Test in Current Control.” Statistische Hefte 26, 253-262. Arnold, B. (1986). “Comparison of the Approximate and Exact Optimum Economic

Design of Control Charts Basing on the Sign Test.” Statistische Hefte 27, 239-241. Bajgier, S. M. (1992). ”The Use of Bootstrapping to Control Limits on Control Charts.” Proceedings of the Decision Science Institute. San Diego, CA, pp 1611-1613. Bakir, S. T. (1977). “Nonparametric Procedures for Process Control.” Ph.D. Dissertation, Dept. of Statistics, Va. Tech., Blacksburg, Va. Bakir, S. T. (1995a). “A Quality Tool for Detecting Out-of-Control Conditions in Education.” 1995 Proceedings of the Section on Statistical Education, American Statistical Association, pp 363-367. Bakir, S. T. (1995b). “A Nonparametric Shewhart Control Chart in the Presence of a Blocking Effect.” 1995 Proceedings of the Section on Quality and Productivity, American Statistical Association, pp 110-114. Bakir, S. T. (1996). “A Quality Tool for Performance Appraisal with Ordinal Ratings.” 1996 Proceedings of the Section on Business and Economics, American Statistical Association, pp 266-271. Bakir, S. T. (1997). "Quality Control Charts for Detecting a General Change in a Process." 1997 Proceedings of the Section on Quality and Productivity, American Statistical Association, pp 53-56. Bakir, S. T. (2000). “A Quality Control Chart for Implementing TQM/Deming’s Principles in HRM.” 2000 Proceedings of the American Society of Business and Behavioral Sciences, Vol. 7, No. 3, pp 15-22. Bakir, S. T. (2002). “Distribution-Free Quality Control Charts: Executive Summaries (1920’s2000).” Technical Report. Personal Collection. Bakir, S. T. and Reynolds, M. R. Jr. (1979). “A Nonparametric Procedure for Process Control Based on Within Group Ranking.” Technometrics 21, 175-183. Bhattacharyya, P. K. and Frierson, D., Jr. (1981). “A Nonparametric Control Chart for Detecting Small Disorders.” Annals of Statistics, 9, 544-554. Chakraborti, S., Van Der Laan, and Bakir, S. T. (2001). “Nonparametric Control Charts: An Overview and Some Results.”, J. of Quality Technology, 33(3), 304-315. Deming, W. E. (1986). Out of the Crisis. Massachusetts Institute of Technology, Center for Advanced Engineering Study, Cambridge, Mass. Deming, W. E. (1993). The New Economics for Industry, Government, and Education. Massachusetts Institute of Technology, Center

for Advanced Engineering Study, Cambridge, Mass. Farnum, N. R. and Stanton, L. W. (1986). “Using Counts to Monitor a Process Mean.” J. of Quality Technology 18, 22-28. Hackl, P. and Ledolter, J. (1991). “A control Chart Based on Ranks.” J. of Quality Technology 23(2), 117-124. Hackl, P. and Ledolter, J. (1992). “A New Nonparametric Quality Control Technique.” Commun. Statist. -Simula. 21(2), 423-443. Hayter, A. J. and Tsui, K. L. (1994). ”Identification and Quantification in Multivariate Quality Control Problems.” J. of Quality Technology. 26, 197-208. Jones, L. A. and Woodall, W. H. (1998). “The performance of Bootstrap Control Charts.” J. of Quality Technology 30(4), 362-375. Kapatou, A. and Reynolds, M. R., Jr. (1994). “Multivariate Nonparametric Control Charts Using Small Samples.” Proceedings of the Section on Quality and Productivity, American Statistical Association, 241-246. Kapatou, A. and Reynolds, M. R., Jr.(1998). “Multivariate Nonparametric Control Charts for the Case of Known 3.” Proceedings of the Section on Quality and Productivity, American Statistical Association, 77-82. Liu, R. Y. (1995). “Control Charts for Multivariate Processes.” JASA 90, 1380-1387. Liu, R. and Tang, J. (1996). “Control Charts for Dependent and Independent Measurements Based on Bootstrap.” JASA 91, 1694-1700. McDonald, D. (1990). “A CUSUM Procedure Based on Sequential Ranks.” Naval Research Logistics. 37, 627-646. McGilchrist, C. A. and Woodyer, K. D. (1975). “ Note on a Distribution-Free CUSUM Technique.” Technometrics 17, 321-325. Montgomery, D. C. (2001). Introduction to Statistical Quality Control, 4rd ed. John Wiley & Sons., Inc., New York, NY. Orban, J. and Wolfe, D. A. (1982). “A Class of Distribution-Free Two-Sample Tests based on Placements.” J. Am. Statist. Assoc. 77, 666-670. Page, E. S. (1954). “Continuous Inspection Schemes.” Biometrika 41, 100-115. Pappanastos, E. A. and Adams, B. M. (1996). “Alternative Designs of the Hodges-Lehmann Control Chart.” J. Quality Technology 28(2), 213-223. Parent, E. A. Jr. (1965). “Sequential Ranking Procedures.” Technical Report # 80, Dept. of Statistics, Stanford University. Park, C. and Reynolds, M. R., Jr. (1987). “Nonparametric Procedures for Monitoring a

Location Parameter Based on Linear Placement Statistics.” Sequential Analysis 6(4), 303-323. Reynolds, M. R., Jr. (1975 a). “A Sequential Signed-Rank Test for Symmetry.” Annals of Statistics. 3, 382-400. Reynolds, M. R., Jr. (1975 b). “Approximations to the Average Run Length in Cumulative Sum Control Charts.” Technometrics 17, 65-71. Roberts, S. W. (1959). “Control Charts Tests Based on Geometric Moving Average.” Technometrics 1, 239-250. Seppala, T.; Moskowitz, H.; Plante, R.; and Tang, J. (1995). “Statistical Process Control with Subgroup Bootstrap.” J. of Quality Technology. 27, 139-153. Shewhart, W. A. (1926). “Quality Control Charts.” Bell Systems Technical Journal, 593603. Shewhart, W. A. (1927). “Quality Control.” Bell Systems Technical Journal, 722-735. Shewhart, W. A. (1931). Economic Control of Quality of Manufactured Product. Princeton: Van Nostrand Reinhold. Shewhart, W. A. (1939). Statistical Methods from the Viewpoint of Quality Control. Republished in 1986 by Dover Publications, New York, N.Y. Stoumbos, Z. G., Jones, L. A., and Woodall, W. H. (1999). “On Nonparametric Multivariate Control Charts Based on Data Depth.” Frontiers in Statistical Quality Control. To appear. Stoumbos, Z. G. and Jones, L. A. (2000). “On the Properties and Design of Individuals Control Charts Based on Simplicial Depth.” Nonlinear Studies. To appear. Stromberg, A. J. (2001). “A New Nonparametric Control Chart for the Range.” A Presentation at the JSM of the American Statistical Association, Atlanta, GA. Van der Laan, P. (1966). “A Sequential Distribution-Free Two-Sample Grouped Test with Three Possible Decisions.” Statistica Neerlandica 20, 31-41. Wilcoxon, F., Rhodes, L. J. and Bradley, R. A. (1963). “Two Sequential Two-Sample Grouped Rank Tests with Applications to Screening Experiments.” Biometrics 19, 58-84. Willemain, T. R. and Runger, G. C. (1996). “Designing Control Charts Using an Empirical Reference Distribution.” J. of Quality Technology 28(1) 31-38. Woodall, W. H. and Montgomery, D. C. (1999). “Research Issues and ideas in Statistical Process Control.” J. of Quality Technology 31(4) 376-386.

Appendix: Univariate and Multivariate DFCCs Classified by Driving Nonparametric Idea and Type*

Driving Idea Signs

I

II

III

Type Shewhart 1. Arnold 86 2. Farnum 86 3. Amin; Reynolds; Bakir 95 4. Amin; Golz 01 5. Amin; Sattler 01 6. Amin; Widmaier 00

Sequential Ranks

NONE

Sequential Signed Ranks

NONE

CUSUM 1. Amin; Reynolds; Bakir 95 2. Amin; Golz 01 3. Amin; Sattler 01 4. Amin; Widmaier 00

EWMA 1. (Multiv) Kapatou Reynolds 94, 98

Linear Barrier NONE

1. McDonald 90

NONE

1. Bhattachryya; Frierson 81

1. Reynolds 75

NONE

1. Parent 65

NONE

NONE

1. Hackl Ledolter 92

NONE

Limited Sequential Signed Ranks Within group Ranks Within Group Signed Ranks

NONE

1. Bakir 77

NONE

1. Bakir 77

NONE

NONE

NONE

NONE

NONE

1. Bakir; Reynolds 79

Reference Sample / Reference Probability Dist

1. Park; Reynolds 87 2. Willemain; Runger 96 3. Bakir 97 4. Amin; Besenfelder 00 1. Bajgier 92 2. Seppala et al. 95 3. Liu; Tang 96 4. Stromberg 01 1. (Multiv) Liu 95 2. (Multiv) Stoumbos et al.99 3. (Multiv) Stoumbos; Jones 00 1. Alloway; Raghavachari 91 2. Bakir 95a, 95b, 96, 00 3. Pappanastos; Adams 96 4. (Multiv) Hayter & Tsui 94

1. Amin; Searcy 91 2. (Multiv) Kapatou Reynolds 94, 98 Hackl; Ledolter 91

2. Reynolds 75 IV

V VI VII

VIII

IX

Limited Sequential Ranks

Bootstrap

Simplicial Depth

X

XI

Others

1. Park; Reynolds 87 2. Amin; Besenfelder 00

NONE

NONE

NONE

NONE

NONE

1. (Multiv) Lui 95 2. (Multiv) Stoumbos et al. 99

NONE

NONE

NONE

NONE

NONE

* Executive summaries of these charts can be found in Bakir (2002).