Published online: 17 May 2012.

Transcription

1 This article was downloaded by: [Central University of Rajasthan] On: 03 December 014, At: 3: Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Communications in Statistics - Simulation and Computation Publication details, including instructions for authors and subscription information: On Confidence Intervals for Process Capability Indices in a One-Way Random Model K. K. Jose a & Jane A. Luke b a Department of Statistics, St. Thomas College, Kerala, India b Department of Statistics, Newman College, Kerala, India Published online: 17 May 01. To cite this article: K. K. Jose & Jane A. Luke (01) On Confidence Intervals for Process Capability Indices in a One-Way Random Model, Communications in Statistics - Simulation and Computation, 41:10, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 Communications in Statistics Simulation and Computation, 41: , 01 Copyright Taylor & Francis Group, LLC ISSN: print/ online DOI: / On Confidence Intervals for Process Capability Indices in a One-Way Random Model Downloaded by [Central University of Rajasthan] at 3: 03 December 014 K. K. JOSE 1 AND JANE A. LUKE 1 Department of Statistics, St. Thomas College, Kerala, India Department of Statistics, Newman College, Kerala, India 1. Introduction In this article, we investigated the bootstrap calibrated generalized confidence limits for process capability indices C pk for the one-way random effect model. Also, we derived Bissell s approximation formula for the lower confidence limit using Satterthwaite s method and calculated its coverage probabilities and expected values. Then we compared it with standard bootstrap (SB) method and generalized confidence interval method. The simulation results indicate that the confidence limit obtained offers satisfactory coverage probabilities. The proposed method is illustrated with the help of simulation studies and data sets. Keywords Bissell s formula; Bootstrap calibration; Generalized confidence interval; One-way random effect model; Process capability index; Satterthwaite s method. Mathematics Subject Classification Primary 6P30; Secondary 6F5, 6F40. The need for manufacturing defect-free products and continuously monitoring the production processes against quality barometers have led to the development of various process capability indices (PCIs) by industrial statisticians. Many Statistical Process Control (SPC) techniques have been proven useful in quality and productivity improvement of products and processes. Process capability indices are widely used statistical tools to assess the performance of manufacturing processes. As a numerical measure, PCI uses both the process variability and the process specifications to determine whether the process is capable. PCIs have received substantial attention in the statistical literature. For a review of the work on PCIs, refer to Kotz and Johnson (1993, 00), Kotz and Lovelace (1998), and Pearn and Liao (006). The formal practice of constructing confidence intervals for PCIs began with Chou et al. (1990). Since then, there has been numerous attempts to propose confidence intervals for PCIs; see Bissell (1990), Franklin and Wasserman (1991, 199), Pearn et al. (004), Wu and Pearn (005) and Lin et al. (005). Received January, 011; Accepted September 5, 011 Address correspondence to K. K. Jose, Department of Statistics, St. Thomas college, Pala, Kerala, India; kkjstc@gmail.com 1805

3 1806 Jose and Luke Downloaded by [Central University of Rajasthan] at 3: 03 December 014 Bootstrapping is a nonparametric, but computer-intensive statistical technique, introduced by Efron (1979) that plays an increasingly important role in modern statistical analysis and applications. Bootstrap iteration has been particularly useful in improving the accuracy of confidence intervals. Franklin and Wasserman (1991) introduced the use of bootstrap sampling procedures for deriving nonparametric confidence intervals for the process capability index C pk. Assuming normal distribution, Mathew et al. (007) derived the generalized confidence interval (GCI) methodology for constructing confidence limits for process capability indices. Under the one-way random model, Kurian et al. (008) developed the generalized confidence interval methodology for constructing confidence limits for process capability indices. Their methodology results in accurate confidence intervals in many cases, the coverages can be unsatisfactory for certain sample size and parameter combinations. In this work, we propose to improve the accuracy of the generalized confidence interval using bootstrap calibration. Even though the resulting solution is computationally somewhat intensive, numerical results show improved accuracy. In this article, we intend to improve the accuracy of the generalized lower confidence limit for PCIs using bootstrap calibration in the one-way random effect model with balanced data. The main objectives of the present study is to improve the accuracy of the generalized lower confidence limit for PCIs using bootstrap calibration and derive Bissell s approximation formula in the context of one-way random effect model with balanced set up. In Sec., we consider generalized lower confidence limits for C pk and obtain Bissell s approximation formula for one-way random effect model. Bootstrap calibration is used to improve the performance of the approximate lower confidence limits and the procedure is explained in Sec. 3. In Sec. 4, we present simulation studies on coverage probabilities and expected values of the lower confidence limits in different situations using various methods. An example is also provided in Sec. 5 to illustrate the efficiency of the estimates. The conclusions are given in Sec. 6. Let U and L be the upper and lower specification limits, d = U L and M = U+L and T be the target value. Every process capability index is a non-negative function of these values and the parameters of the process, namely its mean () and standard deviation () ofx. Of the numerous indices proposed to date, the most commonly used PCIs are C pk introduced by Kane (1986), C pmk by Pearn et al. (004), and Pearn and Chen (1998) proposed C pk are defined as follows: U L d M C pk = min = 3 3 U L C pmk = min 3 d M = + T 3 + T C pk = d A 3 where d = minu T T L, A = max { d T U T Note that C pk = C pmk = C } d T T L. pk when = T Then C pk is utilized to evaluate the capability of centered processes, whereas the other two are for off-centered processes.

4 Confidence Intervals for Process Capability 1807 Downloaded by [Central University of Rajasthan] at 3: 03 December 014. Generalized Confidence Intervals The concept of generalized confidence interval was introduced by Weerahandi (1993). For a detailed discussion and numerous applications, we refer the reader to Weerahandi (1995, 004). The approach of generalized confidence interval will be valuable whenever standard pivotal quantities are either non existent or difficult to obtain based on the conventional method. Let X be random variable whose probability distribution is fx, where is the parameter of interest and is a nuisance parameter, where could be a vector. Let x denote the observed value of X. To define a generalized confidence interval for, we construct a generalized pivotal quantity TX x, satisfying the following two properties. Property (i). The probability distribution of TX x is free of any unknown parameters. Property (ii). The observed value of TX x, i.e., Tx x is free of the nuisance parameter Given a generalized pivotal quantity T = TX x and the desired confidence coefficient, let T denote 1001 th percentile of T. Then, T T is a one-sided confidence interval for. The quantity T is the generalized lower confidence limit for..1. Generalized Confidence Intervals for One-Way Random Model with Balanced Data The mathematical model that describes the relationship between response and treatment for the one-way ANOVA is given by y ij = + i + e ij (1) where y ij represents the jth observation (j = 1n on the ith treatment (i = 1b levels), is the common effect for the whole experiment, i represents the ith treatment effect, and e ij represents the random error in the jth observation on the ith treatment. It is assumed that i s and e ij s are all independent having the distributions i N0 and e ij N0e. Then Vary ij = = + e, i = 1bj = 1n. We use the following notations: n y i = y ij ȳ i = y i n y = SSB = n j=1 b i=1 j=1 n y ij ȳ = y () n b ȳ i ȳ SSE = i=1 b i=1 j=1 n y ij ȳ i Then it can be shown that ( ȳ N n + ) e bn so that Z = ȳ n + e bn N0 1

5 1808 Jose and Luke U = SSB n + e b 1 and V = SSE e bn 1 (3) where e = MSE = SSE/bn 1, MSB = SSB/b 1 and = 1 MSB MSE n When batch variation is taken into account, the definition of the capability index, in the one-way random model is min LSL USL C pk = 3 d M + = e 3 + (4) e Downloaded by [Central University of Rajasthan] at 3: 03 December 014 In order to derive a generalized lower confidence limit for C pk defined in (4), we have to construct a generalized pivotal quantity satisfying the above Properties (i) and (ii). Let ȳ, SSB, and SSE denote the observed values of ȳ, SSB, and SSE, respectively. The generalized pivotal quantity will then be a function of ȳ, SSB, SSE, ȳ, SSB, and SSE, and possibly the unknown parameters,, and e.to construct a GPQ for C pk, let us define and ȳ / T = y T = 1 n n + e SSB bn bn SSB n + e [ SSB U =ȳ Z U + n 1 SSE ] V SSB bn where Z U, and V are defined earlier. T and T are and + e, respectively, and the conditional distribution of T T is free of any unknown parameters. It may be verified that the observed values of T and T satisfy the Properties (i) and (ii) defined earlier. On substituting T and T defined above in the place of and + e, in the expression of the capability index C pk in Eq. (4), we get a GPQ for C pk given by T Cpk = d T M 3 (5) T Also, T Cpk satisfies the Properties (i) and (ii) for a GPQ as defined earlier. Then, a 1001 % generalized lower confidence limit of C pk is given by T Cpk, the 1001 th percentile of T Cpk, such that PT Cpk T Cpk =. For more details, see Kurian et al. (008)... Bissell s Approximate Confidence Intervals for One-Way Random Effect Model In the context of processes following N, Bissell (1990) derived an approximate expression for the variance of Ĉ pk given by VarĈ pk = 1 + C pk. Based on this 9n n expression, Bissell (1990) proposed a 1001 % lower confidence limit for C pk as 1 B pk = Ĉ pk Z 1 9n + Ĉ pk n where Z 1 is the 1 percentile value of the standard normal distribution.

6 Confidence Intervals for Process Capability 1809 The above derivation due to Bissell is applicable only for a normally distributed process. In order to apply the procedure to the one-way random model with balanced data, let ˆ = ȳ, and let ˆ and ˆ e denote unbiased estimators of and e based on the sum of squares defined earlier. These are given by ˆ = 1 n [ SSB b 1 SSE bn 1 ] ˆ e = SSE (6) bn 1 By replacing, and e in (4) with ˆ, ˆ, and ˆ e, respectively an estimate of C pk, namely Ĉ pk, can be obtained as Downloaded by [Central University of Rajasthan] at 3: 03 December 014 d ˆ M Ĉ pk = 3 ˆ (7) +ˆ e where ˆ and ˆ e are defined in Eq. (6). We will now use the Satterthwaite s approximation in order to obtain an approximate chisquare variable associated with ˆ +ˆ e ; for details refer to Montgomery (009). The approximation states that ˆ +ˆ e a + e 1, with a degrees of freedom where a 1 and a are calculated by matching means and variances. Note that ( ) ˆ E +ˆ e + = 1 (8) e and ( ) [ ˆ V +ˆ e 1 nˆ + = +ˆ e e + + n ] 1ˆ4 e = c (say) (9) e n b 1 b But since ˆ +ˆ e a + e 1, approximately, with degrees of freedom = a,weget ( ) ˆ E +ˆ e + = a 1 a (10) e and ( ) ˆ V +ˆ e + = a 1 a (11) e Equating the respective means and variances, using Eqs. (8) (11), we get a 1 a =1, and a 1 a = c where c is defined in Eq. (9). Hence, a 1 = c, and a = 1 = 1.An a 1 c estimate of c is obtained as follows: [ 1 nˆ ĉ = +ˆ e + n ] 1ˆ4 e (1) n ˆ +ˆ e b 1 b From Bissell (1990), an approximate confidence limit for C pk is given by Ĉ pk Z 1 Ĉ pk ĈV Ĉ pk where ĈV Ĉ pk is the estimated coefficient of variation {CVd } { ( )} of Ĉ pk. Since Ĉ pk is a ratio, CVĈ pk = ˆ M + CV 3 ˆ +ˆ e approximately. For more details, see Stuart and Ord (1987).

7 1810 Jose and Luke Using ˆ = y, Downloaded by [Central University of Rajasthan] at 3: 03 December 014 { ( )} CV d ˆ M = CV ˆ M Vy = Ed ˆ M 1 9C pk = = Vy d ˆ M approximately n + e bn d ˆ M = 1 ( ) n + e 9C pk bn + e (13) ( ) Since 1 ˆ +ˆ e, approximately, with c degrees of freedom, where c is c ˆ +ˆ e defined in Eq. (9), we get { ( )} CV 3 ˆ +ˆ e = c (14) ( ) n Then, CVĈ pk = + e + c, using Eqs. (13) and (14). Hence, bn + e ĈV Ĉ pk = 1 9Ĉ pk ( ) nˆ +ˆ e bnˆ +ˆ e + ĉ The approximate lower confidence limit for C pk is thus Ĉ pk Z 1 Ĉ pk [ 1 9Ĉ pk ( ) nˆ +ˆ e bnˆ +ˆ e + ĉ ].3. Standard Bootstrap Confidence Interval Franklin and Wasserman (1991, 199) investigated the bootstrap methodology for computing confidence limits for C pk for a normally distributed process. For the oneway random model ȳ N ( ) n + e bn, SSB n + e b 1, and SSE e bn 1. Consider B bootstrap samples ȳi SSB i SSE i such that ȳ i N (ˆ ) nˆ +ˆ e bn, U i = SSBi nˆ +ˆ e b 1, and V i = SSE i ˆ e bn 1 Here, r denotes a central chisquare distribution with r degrees of freedom. Let Ĉ pk i denote the estimate of C pk obtained from the ith bootstrap sample. Now calculate the sample average of the bootstrap estimates B Ĉ pk i Ĉ pk = B and the standard deviation of the bootstrap estimates S Cpk = B Ĉ pk i Ĉ pk B 1 i=1 i=1

8 Confidence Intervals for Process Capability 1811 An approximate lower confidence limit for C pk is given by Ĉ pk Z 1 S Cpk Note that the interval is centered at the value of Ĉ pk derived from the original data and the bootstrap method is only used to estimate the standard deviation. Downloaded by [Central University of Rajasthan] at 3: 03 December Bootstrap Calibration The idea of bootstrap calibration for improving the performance of a confidence interval is described in Efron and Tibshirani (1986). In the context of computing a lower confidence limit for C pk, the methodology can be described as follows. Suppose Ĉ pk is a 100(1-% approximate lower confidence limit for C pk, obtained, for example, using any of the methods given in the previous section. Let p = 1 ProbĈ pk C pk We certainly want p =, so that the confidence limit is exact. If there is significant difference between p and, there may exist different from satisfying p =. However, such a could depend on unknown parameters, and this is where the bootstrap comes in to provide an estimate of. Such an estimate is obtained by following the steps given below. 1. Let Ĉ pk denote the estimate of C pk obtained from the given data. Generate B bootstrap samples ȳi SSB i SSE i, i = 1 B, as explained for the standard bootstrap confidence interval mentioned in the previous section.. From each bootstrap sample, compute a 1001 % lower confidence limit, say Ĉ pk i from the ith bootstrap sample, for a grid of values of where, is determined such that ˆp =. 3. For each compute ˆp = #Ĉ pk i Ĉ pk /B. 4. Find the value of satisfying ˆp =. Let ˆ denote the solution. 5. Now go back to the original sample and compute a 1001 ˆ% lower confidence limit for C pk. The 1001 ˆ% lower confidence limit for C pk so obtained is the bootstrap calibrated lower confidence limit. We note that above calibration can be carried out for any lower confidence limit for C pk. This procedure (in particular, the computation of ˆ) is carried out for the example in Sec Simulation Study The estimation of unknown process parameters such as process mean and process standard deviation appearing in the expression of C pk is an important point in the confidence interval estimation of these indices. The process must be stable in order to produce reliable estimates of and and hence a reliable estimate Ĉ pk. The current practice is to compare this estimated index with a recommended minimum value and if the estimated index is greater than or equal to the minimum value, then the process is considered capable. The coverage probabilities of approximate lower confidence limits without bootstrap calibration and approximate lower confidence limits with bootstrap

9 181 Jose and Luke Table 1 Before and after calibration coverage probabilities and expected values of the generalized lower confidence limits for process capability indices of C pk in one-way random effect model for a 95% nominal level Downloaded by [Central University of Rajasthan] at 3: 03 December 014 C pk values e b n Coverage probability (With calibration) (Without calibration) Expected values (With calibration) (Without calibration) calibration and corresponding expected values of the confidence limits are computed using MATLAB software using a series of simulation studies. For the simulation we used LSL = 7, U = 14, = 10, and various values of that will provide C pk = 1, 1.5,,.5, and 3. Coverage probabilities and expected values have been calculated for various values parameter combinations. Before and after calibration coverage probabilities and expected values are calculated and displayed in the Table 1. It reveals that the coverage probabilities with bootstrap calibration achieves nearest to 95% accuracy than coverage probability without bootstrap calibration while holding various values of b and n. From the numerical results in Table, it is clear that Bissell s approximation method achieves 95% confidence limit, which is our target. Between these two, the Bissell s approximation method is to be preferred. Of course, we can calibrate the standard bootstrap method, but we are not attempting due to the heavy computation involved. 5. An Illustration Montgomery (009) provided a set of 15 measurements of the diameters (mm) of piston rings for an automotive engine, produced by a forging process. The quality characteristic is the inside diameter measurements of piston rings. The data have b = 5 batches, each having n = 5 observations. The analysis of X and S charts for this data shows that the process is in statistical control. Here, L = 7395, U = 7405, and C pk = The computations based on the data gave ȳ = , SSB = , and SSE = Using generalized pivotal method, before and after calibration approximate lower confidence limits are obtained as

10 Confidence Intervals for Process Capability 1813 Table Before calibration coverage probabilities and expected values of approximate lower confidence limit for the process capability index C pk in one-way random effect model using Bissell s approximation method and Standard Bootstrap method for a 95% confidence level Downloaded by [Central University of Rajasthan] at 3: 03 December 014 C pk e b n Bissel s approximation method Coverage probability Expected values Standard bootstrap method Coverage probability Expected values and , respectively. For bootstrap calibration, we chose in the interval (0.035, 0.075), and get ˆ = In Standard bootstrap method, lower confidence limit is and in Bissell s approximation method, it is A comparative study is also made. It is clear that bootstrap calibration improves the coverage probability. Also it indicates that bootstrap calibration method is an appropriate method for getting approximate lower confidence limit for the process capability index C pk in one way random effect model. Our numerical study revealed that the method works exceptionally well obtaining lower confidence limits for the process capability index C pk.

11 1814 Jose and Luke Downloaded by [Central University of Rajasthan] at 3: 03 December Conclusions An investigation into the context of providing generalized lower confidence limits via bootstrap calibration for the process capability indices has been made in this article. Also, we have succeeded in deriving Bissell s approximate lower confidence limit for the process capability index C pk for the one-way random effect model. Bootstrap calibration improved performance of the generalized lower confidence limit. The expected value of the generalized lower confidence limit is larger under bootstrap calibration, which is desirable. Also, we compared the performance of coverage probabilities and expected values of confidence limits using standard bootstrap method and Bissell s approximation method. Between these two, the confidence limits using Bissell s approximation method is to be preferred. Acknowledgment The authors are grateful to the referees and the Editor for their comments and suggestions which helped in improving this paper. The second author acknowledges that financial assistance from the University Grants Commission of India for supporting this research under the Teacher Fellowship Scheme. References Bissell, A. F. (1990). How reliable is your capability index?. Applied Statistics 39: Chou, Y. M., Owen, D. B., Borrego, A. S. A. (1990). Lower confidence limits on process capability indices. Journal of Quality Technology :3 9. Efron, B. (1979). Bootstrap methods: Another look at the jackknife. The Annals of Statistics 7:1 6. Efron, B., Tibshirani, R. (1986). Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Statistical Science 1: Franklin, L. A., Wasserman, G. S. (1991). Bootstrap confidence interval estimates of C pk : An introduction. Communications in Statistics Simulation and Computation 0:31 4. Franklin, L. A., Wasserman, G. S. (199). A note on the conservative nature of the tables of lower confidence limits for C pk with a suggested correction. Communications in Statistics Simulation and Computation 1: Kane, V. E. (1986). Process capability indices. Journal of Quality Technology 18:41 5. Kotz, S., Johnson, N. L. (1993). Process Capability Indices. London: Chapman and Hall. Kotz, S., Johnson, N. L. (00). Process capability indices A review, Journal of Quality Technology 34: 19. Kotz, S., Lovelace, C. R. (1998). Process Capability Indices in Theory and Practice. London: Arnold. Kurian, K. M., Mathew, T., Sebastian, G. (008). Generalized confidence intervals for process capability indices in the one-way random model. Metrika 67:83 9. Lin, G. H., Pearn, W. L., Yang, W. S. (005). A Bayesian approach to obtain a lower bound for the C pm capability index. Quality and Reliability Engineering International 1: Mathew, T., Sebastian, G., Kurian, K. M. (007). Generalized confidence intervals for process capability indices. Quality and Reliability Engineering International 3: Montgomery, D. C. (009). Introduction to Statistical Quality Control. Newyork: John Wiley. Pearn, W. L., Chen, K. S. (1998). New generalization of process capability index C pk. Journal of Applied Statistics 5: Pearn, W. L., Liao, M. Y. (006). One sided process capability assessment in the presence of measurement errors. Quality and Reliability Engineering International :

12 Downloaded by [Central University of Rajasthan] at 3: 03 December 014 Confidence Intervals for Process Capability 1815 Pearn, W. L., Lin, P. C., Chen, K. S. (004). The C pk index for asymmetric tolerances implications and inference. Metrika 60: Stuart, A., Ord, J. K. (1987). Kendall s Advanced Theory of Statistics. London: Griffin. Wu, C. W., Pearn, W. L. (005). Capability testing based on C pm with multiple samples. Quality and Reliability Engineering International 1:9 4. Weerahandi, S. (1993). Generalized confidence intervals. Journal of the American Statistical Association 88: Weerahandi, S. (1995). Exact Statistical Methods for Data Analysis. New York: Springer Verlag. Weerahandi, S. (004). Generalized Inference in Repeated Measures. New York: Wiley.