Robustness of Group Runs Control Chart to Non-normality

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1 c Heldermann Verlag Economic Quality Control ISSN Vol 20 (2005), No. 1, Robustness of Group Runs Control Chart to Non-normality M.P. Gadre, R.N. Rattihalli and A.V. Shewade Abstract: Most of the commonly used control charts for variables have been developed under the assumption that the observations are coming from a normal population. One of them is Group Runs GR control chart to detect shifts in the process mean proposed by Gadre and Rattihalli (2004). In this article, we study the robustness of GR chart to violation of the assumption of normality. It is observed that, for moderate to large shifts in the process mean, GR chart is insensitive to the non-normality. It is also illustrated that, the GR chart performs better than the synthetic chart, when the process data are not coming from normal distribution. Key Words: Statistical process control, control chart, synthetic control chart, group runs control chart. 1 Introduction Control charts are effectively used in statistical process control to detect changes in the process level that may affect quality of the product. Starting from Shewhart s X-chart, various control charts have been developed to monitor the process mean. Some of the other well-known control charts are the Exponentially Weighted Moving Average (EWMA) control chart proposed by Roberts (1959) and the Cumulative Sum (CUSUM) chart by Page (1954). Though an X-chart detects large shifts in the process mean effectively, its performance is poor in detecting small or moderate shifts. The EWMA or CUSUM chart can be used to detect small to moderate shifts. For further details of these charts one may refer to Lucus et al. (1990) and Montgomery (1996). Albin et al. (1997) developed an X-EWMA chart by combining X-chart and EWMA chart. The chart has better performance to detect large shifts, but it is inferior to detect small shifts as compared to the EWMA chart. Such combinations of charts have been studied in literature (Ncube (1990), Shamma and Shamma (1992)). To detect small shifts in the process mean, Wu and Spedding (2000) have proposed a synthetic control chart by combining the Shewhart s X-chart and Conforming Run Length (CRL) chart (Bourke, 1991) by considering sample (group) as a unit. Wu and Spedding (2000) have defined CRL as the number of conforming units between two successive nonconforming units including the non-conforming unit at the end. They have shown that the synthetic chart performs better than X-chart and CRL chart. Gadre and Rattihalli

2 82 M.P. Gadre, R.N. Rattihalli and A.V. Shewade (2004) have developed a GR chart, which is an extension of the synthetic chart. They have illustrated that GR chart performs better than the synthetic chart. All these charts are designed and evaluated under the assumption that the observations are independent and identically distributed (i.i.d.) with a normal distribution. The behaviors of these commonly used control charts have been widely studied when some aspects of the assumption do not hold. One violation of the assumption is non-normality, which has been discussed by many authors for independent data. Schilling and Nelson (1976) showed that if the sample size is not less than 4, an X-chart is reasonably robust with respect to the normality assumption. One can also refer to Yourstone and Zimmer (1992) as well as to Amin and Reynolds (1995). By considering various skewed distributions and heavily tailed symmetric non-normal distributions Borror et al. (1999) have illustrated that the in-control ARL of the Shewhart control chart for individual observations is very sensitive, while a well designed EWMA control chart is robust with respect to the normality assumption. For the synthetic control chart, Calzada and Scariano (2001) have observed that in-control ARL values are significantly small for non-normal distributions as compared to those for normal distribution. This is an indication of a high false alarm rate that would be unacceptable for most of practical applications. In this article, we study the sensitivity of GR chart to non-normality. It is observed that for moderate to large shifts in the process mean, GR chart is insensitive to the non-normality, while for small shifts in the process mean, though it is sensitive to nonnormality, its performance is better than in case of normal distribution. Further, it is observed that, GR chart performs better than synthetic chart when the actual process data are coming from non-normal distribution. The remainder of the paper is organised as follows. In the next section we list some notations required to describe the GR chart and take a brief review of GR chart. Robustness of the GR chart to the failure of the assumption of normality is studied in Section 3. Concluding remarks are given in the last section. 2 Group Runs Chart Gadre and Rattihalli (2004) have developed the GR chart to detect shifts in the process mean. The notations required to describe the chart are listed below. 2.1 Notations 1. µ 0 : In control (target) value of the process mean. 2. σ: Process standard deviation.

3 Robustness of Group Runs Control Chart to Non-normality ATS(δ): The average number of units required by GR chart to detect a shift in the process mean from µ 0 to µ 0 ± δσ. 4. ARL(δ): The average number of groups required by GR chart to detect a shift in the process mean from µ 0 to µ 0 ± δσ. 5. δ 1 : Design shift in the mean, the magnitude of which is considered large enough to seriously impair the quality of the product. 6. k: Coefficient used in the control limits of X sub chart; 7. L X/S = µ 0 k σ n and U X/S = µ 0 + k σ n are the control limits of the X sub chart; 8. L: Lower limit of GR chart. 9. Y r : The rth (r =1, 2,...) value of CRL when the groups are treated as units. In other words, it is the number of groups inspected between (r-1)th and rth nonconformed group, including the rth non-conformed group; 10. τ: The minimum required value of ATS(0) and 11. τ 1 : The minimum required value of ARL(0). In GR chart for mean, a X sub chart is used to decide on the status of the group. The sub chart declares the group as non-conforming if the group mean falls outside the control limits L X/S and U X/S of the sub chart. The GR procedure declares the process as out of control if Y 1 L or for some r (> 1), Y r and Y r+1 are not exceeding L for the first time. The quantities µ 0, δ 1 and τ are called the input parameters and optimal values of the quantities n, k, L are called design parameters of the chart. Values of the design parameters of the GR chart are computed so as to Subject to the constraint Minimise ATS(δ 1 ) ATS(0) τ (1) Let P be the probability of the group being non-conformed. Then ) P (δ) = 1 P (L X/S < X<U X/S X N(µ 0 + δσ, σ2 n ) ) = 1 Φ ( k δ n ) +Φ ( k δ n ) (2) As shown in Gadre and Rattihalli (2004), (11) reduces to Minimise Subject to the constraint n P (δ 1 )(1 (1 P (δ 1 )) L ) 2 n P (δ 1 )(1 (1 P (δ 1 )) L ) 2 τ (3)

4 84 M.P. Gadre, R.N. Rattihalli and A.V. Shewade The design parameters of the GR chart are obtained by using a step-wise procedure discussed in Gadre and Rattihalli (2004). While constructing a GR chart for the process mean, it is assumed that the underlying process data are normally distributed. In the next section we study the effect of deviations from normality on the performance of GR chart. 3 Robustness of the GR Chart As the GR chart takes into account the possibility of false alarms before declaring the process as out of control, it may be interesting to study the chart properties. One of the important properties is robustness. Robustness of the GR chart is related to study the performance of GR chart when the actual process data do not follow a normal distribution. For the symmetric case, we have considered different t-distributions and for the other cases, various gamma distributions. To study the effect of sample (group) size on the performance of GR chart under various distributions, instead of considering ATS criterion given in (1), for fixed group size n, we obtain the other design parameters (k, L) ofthe GR chart under ARL criterion described below. Determine the values of (k, L) soasto Minimise ARL(δ 1 ) Subject to the constraint ARL(0) τ 1 (4) Table-1 given below lists the design parameters of GR charts for various values of n (, the group size) and for different input parameters (δ 1, τ 1 = 370) under ARL model. The values of µ 0 and σ are taken as zero and 1, respectively. Table-1: Design Parameters of GR Charts under ARL Model n δ 1 k L n δ 1 k L n δ 1 k L To study the effect of non-normality, we have considered the process data from the t- distribution with increasingly heavier tails (decreasing degrees of freedom (d.f.) 50, 40, 30, 20, 15, 10, 8, 6, 4, respectively) and that from gamma distribution with different values of single shape parameter α, respectively. The values of α considered are 1, 2, 3 and 4. When the process data actually follow a t distribution or a gamma distribution, the control limits L X/S and U X/S of the X sub chart should be adjusted appropriately. We know

5 Robustness of Group Runs Control Chart to Non-normality 85 that, if X has t distribution with v d.f., the probability density function (p.d.f.) of X is given by f X (x v) = Γ ( ) v+1 ( ) v+1 2 Γ ( ) v 1+ x2 2 for <x< (5) 2 vπ v The mean and variance of X having the p.d.f. given in (5) are respectively zero and v This gives, the value of σ as v and, hence, the control limits of a X sub chart should v 2 be taken as L X/S = k v n(v 2) U X/S = k (6) v n(v 2) It is to be noted that, for the existence of the variance, v should be larger than 2. In case of gamma distribution with single shape parameter α, the p.d.f. of X is given by f X (x α) = xα 1 e x for 0 <x< (7) Γ(α) v 2. with both the mean and variance of X having the p.d.f. given in (7) are α. Thus, L X/S = α k α n U X/S = α + k α n (8) Using the adjusted control limits given in (6) and (8) of the X sub chart, we have computed the simulated ARL(0) values for the above GR charts, when the underlying observations actually follow t distribution with various d.f and the gamma distribution with different values of single shape parameter α, respectively. To study the robustness, the GR charts are designed by using the values of the design parameters (n, k, L) from Table-1, which are suitable only when the observations are from normal distribution. The ARL values are computed using 1,00,000 simulations for every non-normal distribution (t-distribution and gamma distribution) for each GR chart in Table-1. Table-2 and Table-3 display the simulation results along with those on the synthetic chart. The simulated ARL values related to the normal distribution are also included in each table. Every cell of Table-2 and Table-3 is divided into two halves. The upper half indicating the corresponding entry of GR chart, whereas the lower half indicating the matched adjusted simulated ARL value of the synthetic chart. The simulated ARL values of the synthetic chart are taken from Calzada and Scariano (2001).

6 86 M.P. Gadre, R.N. Rattihalli and A.V. Shewade Table-2: Simulated (ARL(0)) values of GR and Synthetic chart for normal and various t-distributions n δ Normal t t t t t t t t t Table-3: Simulated (ARL(0)) values of GR and Synthetic chart for normal and various gamma-distributions n δ Normal G(4, 1) G(3, 1) G(2, 1) G(1, 1) From Table-2 and Table-3, for GR chart, we observe the following. 1. The ARL(0) values get closer to 370 as the d.f. of the t distribution or the parameter α of gamma distribution increases. 2. For the t-distribution with more than 8 d.f. the ARL(0) values are between 96% and 104% of those of the normal distribution.

7 Robustness of Group Runs Control Chart to Non-normality Except for the combination (n =4,δ 1 = 1), for smaller d.f., the false alarm rate is much lower than in the normal case. Table-3 also indicates that the false alarm rate decreases rapidly as the value of α decreases. For the synthetic control chart Calzada and Scariano (2001) have observed that ARL(0) values are much less as compared to those of the normal distribution, which is an indication of high false alarm rate that would be unacceptable for most of the practical applications. This fact also indicates the superiority of GR chart over the synthetic chart. As the ARL(0) values for both t and gamma distributions given in Table-2 and Table-3 are different from those for normal distribution, to have a fair comparison, we obtain adjusted ARL(δ) values for non-normal distribution by using the formula Adj[ARL(δ)] = ARL nn (δ) =ARL(δ) ARL n(0) (9) ARL nn (0) where, the index n and nn, respectively, denote the values under normal and nonnormal distribution. Table-4 to Table-6 and Table-7 to Table-9 below contain the adjusted ARL(δ) values for various GR charts when the underlying process data actually follow the above mentioned t and gamma distributions when the shift in the process mean is from 0.25 to 3 with the increment of Every cell of the Table-4(A) and Table-7(A) is divided into two halves, as discussed for Table-2 and Table-3. The following are some important findings based on Table-4 to Table Table-4(A) and Table-7(A) clearly indicate the superiority of GR chart over the synthetic chart. 2. Table-4 indicate that for n =4,δ 0.5 and more than 8 d.f., the adjusted ARL(δ) values of the GR chart are close enough to those for normal distribution. For smaller d.f. and smaller δ values, the power of detecting a signal is less. This indicates that for δ 1 1, the GR chart is insensitive to non-normality. 3. Table-5 and Table-6 indicate that except for (δ = 0.25, d.f. = 4) the adjusted ARL(δ) values are close to those for normal distribution. 4. From Table-7 to Table-9 we observe that as the value of α decreases, the adjusted ARL(δ) values rapidly decreases. This indicates that the power of GR chart to detect a signal increases.

8 88 M.P. Gadre, R.N. Rattihalli and A.V. Shewade Table-4 (A): Simulated Adjusted ARL(δ) Values of GR and Synthetic Chart Related to n =4andδ 1 =1 Normal t t t t t t t t t Table-4 (B): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =4and δ 1 =1.25 Normal t t t t t t t t t

9 Robustness of Group Runs Control Chart to Non-normality 89 Table-4 (C): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =4and δ 1 =1.5 Normal t t t t t t t t t Table-5 (A): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =5and δ 1 =1 Normal t t t t t t t t t Table-5 (B): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =5and δ 1 =1.25 Normal t t t t t t t t t

10 90 M.P. Gadre, R.N. Rattihalli and A.V. Shewade Table-5 (C): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =5and δ 1 =1.5 Normal t t t t t t t t t Table-6 (A): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =6and δ 1 =1 Normal t t t t t t t t t Table-6 (B): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =6and δ 1 =1.25 Normal t t t t t t t t t

11 Robustness of Group Runs Control Chart to Non-normality 91 Table-6 (C): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =6and δ 1 =1.5 Normal t t t t t t t t t Table-7 (A): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =4and δ 1 =1 Normal G(4, 1) G(3, 1) cline G(2, 1) G(1, 1) Table-7 (B): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =4and δ 1 =1.25 Normal G(4, 1) G(3, 1) G(2, 1) G(1, 1)

12 92 M.P. Gadre, R.N. Rattihalli and A.V. Shewade Table-7 (C): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =4and δ 1 =1.5 Normal G(4, 1) G(3, 1) G(2, 1) G(1, 1) Table-8 (A): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =5and δ 1 =1 Normal G(4, 1) G(3, 1) G(2, 1) G(1, 1) Table-8(B): Simulated Adjusted ARL(δ) Values of GR Chart Related to n = 5 and δ 1 =1.25 Normal G(4, 1) G(3, 1) G(2, 1) G(1, 1) Table-8 (C): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =5and δ 1 =1.5 Normal G(4, 1) G(3, 1) G(2, 1) G(1, 1)

13 Robustness of Group Runs Control Chart to Non-normality 93 Table-9 (A): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =6and δ 1 =1 Normal G(4, 1) G(3, 1) G(2, 1) G(1, 1) Table-9 (B): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =6and δ 1 =1.25 Normal G(4, 1) G(3, 1) G(2, 1) G(1, 1) Table-9 (C): Simulated Adjusted ARL(δ) Values of GR Chart Related to n =6and δ 1 =1.5 Normal G(4, 1) G(3, 1) G(2, 1) G(1, 1) Conclusions Gadre and Rattihalli (2004) have already shown that the GR chart performs better than a synthetic control chart under the assumption of normality. In this article we have observed that also under t and gamma distributions, the GR chart performs better than synthetic control chart. The GR is robust to the assumption of normality. When the actual process data follow a t-distribution the ARL(0) values as well as the adjusted ARL(δ) values are comparable with those of normal distribution. If the actual process data has gamma distribution, the adjusted ARL(δ) values are much smaller if δ 0 indicating quickness in detecting the shift, and the larger values for the ARL(0) indicate a low false alarm

14 94 M.P. Gadre, R.N. Rattihalli and A.V. Shewade probability. Thus we conclude that the GR chart is preferable compared to the synthetic control chart. Acknowledgement: We are thankful to the Referee and the Regional Editor for their valuable comments and suggestions, which helped a lot to improve the presentation significantly. The second author is thankful to Professor Maria Calzada for sending the computer program of computing simulated ARL values in case of synthetic control chart. References [1] Albin, S. L.; Kang, L. and Shea, G. (1997): An X and EWMA Chart for Individual Observations. Journal of Quality Technology 29, [2] Amin, R. W. and Reynolds, M. R. Jr. (1995): Non Parametric Quality Control Charts Base on the Sign Test. Communications in Statistics - Theory and Methods 24, [3] Borror, C. M., Montgomery, D. C., and Runger, G. C. (1999): Robustness of the EWMA Control Chart to Non-normality. Journal of Quality Technology 31, [4] Bourke, P. D. (1991): Detecting a Shift in Fraction Nonconforming Using Run- Length Control Charts with 100% Inspection. Journal of Quality Technology [5] Calzada, M. E. and Scariano, S. M. (2001): The Robustness of the Synthetic Control Chart to Non-normality. Communications in Statistics - Simulation and Computation 30, [6] Gadre, M. P. and Rattihalli, R. N. (2004): A Group Runs Control Chart for Detecting Shifts in the Process Mean. Economic Quality Control 19 (1), [7] Lucus, J. M. and Saccucci, M. S. (1990): Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements. Technometrics 32, [8] Montgomery, D. C. (1996): Introduction to Statistical Quality Control. John Wiley & Sons, New York. [9] Ncube, M. M. (1990): An Exponentially Weighted Moving Average Combined Shewhart Cumulative Score Control Procedure. International Journal of Quality and Reliability Management 7, [10] Page, E. S. (1954): Continuous Inspection Schemes. Bimometrika 41, [11] Roberts, S. W. (1959): Control Chart Tests Based on Geometric Moving Averages. Technometrics 1,

15 Robustness of Group Runs Control Chart to Non-normality 95 [12] Shamma, W. E. and Shamma, A. K. (1992): Development and Evaluation of Control Charts Using Double Exponentially Weighted Moving Averages. Journal of Quality and Reliability Management 9, [13] Schilling, E. G. and Nelson, P. R. (1976): The Effect of Non-normality on the Control Limits of Charts. Journal of Quality Technology 8, [14] Wu, Z. and Spedding, T. A. (2000): A Synthetic Control Chart for Detecting Small Shifts in the Process Mean. Journal of Quality Technology 32, [15] Yourstone, S. and Zimmer, W. (1992): Non-normality and the Design of Control Charts for Averages. Decision Sciences 23, Gadre M. P. Department of Statistics Mudhoji College Phaltan India Rattihalli R. N. and Shewade A. V. Department of Statistics Shivaji University Kolhapur India