mss # 1294.tex; art. # 03; 43(4) Design and Analysis of Control Charts for Standard Deviation with Estimated Parameters MARIT SCHOONHOVEN Institute for Business and Industrial Statistics of the University of Amsterdam (IBIS UvA), Plantage Muidergracht 12, 1018 TV Amsterdam, The Netherlands MUHAMMAD RIAZ King Fahad University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Department of Statistics, Quad-i-Azam University, 45320 Islamabad 44000, Pakistan RONALD J. M. M. DOES IBIS UvA, Plantage Muidergracht 12, 1018 TV Amsterdam, The Netherlands This paper concerns the design and analysis of the standard deviation control chart with estimated limits. We consider an extensive range of statistics to estimate the in-control standard deviation (Phase I) and design the control chart for real-time process monitoring (Phase II) by determining the factors for the control limits. The Phase II performance of the design schemes is assessed when the Phase I data are uncontaminated and normally distributed as well as when the Phase I data are contaminated. We propose a robust estimation method based on the mean absolute deviation from the median supplemented with a simple screening method. It turns out that this approach is efficient under normality and performs substantially better than the traditional estimators and several robust proposals when contaminations are present. Key Words: Average Run Length; Mean-Squared Error; Phase I; Phase II; Robust; Shewhart Control Chart; Statistical Process Control. Introduction ters, and an optimal performance requires any change in these parameters to be detected as early as HE PERFORMANCE of a process depends on the possible. To monitor a process with respect to these stability of its location and dispersion parame- T parameters, Shewhart introduced the idea of control charts in the 1920s. The dispersion parameter of the Ms. Schoonhoven is a Consultant in Statistics at IBIS UvA. process is controlled first, followed by the location pa- She is working on her PhD, focusing on control-charting tech- rameter. The present paper focuses on control charts niques. Her email address is [email protected]. for monitoring the process standard deviation. Dr. Riaz is an Assistant Professor in the Department Let Yij, i = 1, 2, 3, . and j = 1, 2, . , n, denote of Mathematics and Statistics of King Fahad University of samples of size n taken in sequence on the process Petroleum and Minerals. His email address is riaz76qau@ variable to be monitored. We assume the Yij’s to be yahoo.com. independent and N(µ, λσ) distributed, where λ is a Dr. Ronald J. M. M. Does is Professor in Industrial Statis- constant. When λ = 1, the standard deviation of the tics at the University of Amsterdam, Managing Director of process is in control; otherwise, the standard devia- IBIS UvA, and fellow of ASQ. His email address is r.j.m.m tion has changed. Let σˆi be an estimate of λσ based [email protected]. on the ith sample Yij, j = 1, 2, . , n. When the in- Vol. 43, No. 4, October 2011 307 www.asq.org mss # 1294.tex; art. # 03; 43(4) 308 MARIT SCHOONHOVEN, MUHAMMAD RIAZ, AND RONALD J. M. M. DOES control σ is known, the process standard deviation over all possible values of the parameter estimates. can be monitored by plotting σˆi on a Shewhart-type The unconditional p is control chart with respective upper and lower control p = E(P (F σˆ)), limits i | UCL = Unσ, LCL = Lnσ, (1) the unconditional average run length is where Un and Ln are factors such that, for a chosen 1 ARL = E . type I error probability ↵, we have P (F σˆ) ✓ i | ◆ P (L σ σˆ U σ) = 1 ↵. n i n − Quesenberry (1993) showed that, for the X and X When σˆi falls within the control limits, the process control charts, the marginal ARL is higher than in is deemed to be in control. We define Ei as the event the σ-known case. Furthermore, a higher in-control that σˆi falls beyond the limits, P (Ei) as the proba- ARL is not necessarily better because the RL distri- bility that sample i falls beyond the limits, and RL bution will reflect an increased number of short RLs as the run length, i.e., the number of samples un- as well as an increased number of long RLs. He con- til the first σˆi falls beyond the limits. When σ is cluded that, if limits are to behave like known limits, known, the events Ei are independent, and there- the number of samples (k) in Phase I should be at fore RL is geometrically distributed with parameter least 400/(n 1) for X control charts and 300 for X − p = P (Ei) = ↵. It follows that the average run length control charts. Chen (1998) studied the marginal RL (ARL) is given by 1/p and that the standard devia- distribution of the standard deviation control chart tion of the run length (SDRL) is given by p1 p/p. under normality. He showed that, if the shift in the − standard deviation in Phase II is large, the impact of In practice, the in-control process parameters are parameter estimation is small. In order to achieve a usually unknown. Therefore, they must be estimated performance comparable with known limits, he rec- from samples taken when the process is assumed to ommended taking at least 30 samples of size 5 and be in control. This stage in the control-charting pro- updating the limits when more samples become avail- cess is called Phase I (cf., Woodall and Montgomery able. For permanent limits, at least 75 samples of (1999), Vining (2009)). The monitoring stage is de- size 5 should be used. Thus, the situation is some- noted by Phase II. Define σˆ as an unbiased estimate what better than for the X control chart with both of σ based on k samples of size n, which are denoted process mean and standard deviation estimated. by Xij, i = 1, 2, . , k. The control limits can be es- timated by Jensen et al. (2006) conducted a literature sur- vey of the e↵ects of parameter estimation on control- UCL = U σˆ, LCL = L σˆ. (2) n n chart properties and identified several issues for fu- These U and L are not necessarily the same as ture research. One of their recommendations is to n d n d in Equation (1) and will be di↵erent if the marginal consider robust or other alternative estimators for probability of signalling is the same. Let Fi denote the location and the standard deviation in Phase I the event that σˆi is above UCL or below LCL. We applications because it seems more appropriate to define P (F σˆ) as the probability that sample i use an estimator that will be robust to outliers and i | generates a signal given σˆ, i.e.,d d step changes in Phase I. Also, the e↵ect of using these robust estimators on Phase II should be assessed P (F σˆ) = P (σˆ < LCL or σˆ > UCL σˆ). i | i i | (Jensen et al. (2006, p. 360)). This recommendation Given σˆ, the distribution of the run length is geo- is the subject of the present paper, i.e., we will study d d metric with parameter P (Fi σˆ). Consequently, the alternative estimators for the standard deviation in conditional ARL is given by | Phase I and we will study the impact of these esti- mators on the Phase II performance of the standard 1 E(RL σˆ) = . deviation control chart. | P (Fi σˆ) | Chen (1998) studied the standard deviation con- In contrast with the conditional RL distribution, trol chart when σ is estimated by the pooled-sample the marginal RL distribution takes into account the standard deviation (S˜), the mean-sample standard random variability introduced into the charting pro- deviation (S), or the mean-sample range (R) un- cedure through parameter estimation. It can be ob- der normality. He showed that the performance of tained by averaging the conditional RL distribution the charts based on S˜ and S is almost identical, Journal of Quality Technology Vol. 43, No. 4, October 2011 mss # 1294.tex; art. # 03; 43(4) DESIGN AND ANALYSIS OF CONTROL CHARTS FOR STANDARD DEVIATION 309 while the performance of the chart based on R is MDM, the ADM, the MAD, and the robust estima- slightly worse. Rocke (1989) proposed robust con- tor of Tatum (1997). Moreover, we investigate the trol charts based on the 25% trimmed mean of the use of a variant of the screening methods proposed sample ranges, the median of the sample ranges, and by Rocke (1989) and Tatum (1997). The performance the mean of the sample interquartile ranges in con- of the estimators is evaluated by assessing the mean- taminated Phase I situations. Moreover, he studied squared error (MSE) of the estimators under normal- the use of a two-stage procedure whereby the initial ity and in the presence of various types of contam- chart is constructed first and then subgroups that inations. Further, we derive the constants that de- seem to be out of control are excluded. Rocke (1992) termine the control limits. We then have the desired gave the practical details for the construction of these marginal probability that the chart will produce a charts. Wu et al. (2002) considered three alternative false signal in Phase II. Finally, we assess the Phase statistics for the sample standard deviation, namely II performance of the control charts by means of a the median of the absolute deviation from the me- simulation study. dian (MDM), the average absolute deviation from the median (ADM), and the median of the average The paper is structured as follows.