Schalk William Human

Phase I Statistical Process Control Charts: An Overview and Some Results

ABSTRACT In practice, Phase I analysis constitutes an integral part of an overall SPC program in which control charts play a crucial role. An overview of the literature on Phase I parametric control charts for univariate variables data is presented. Since the Phase I signaling events are dependent and multiple signaling events are to be dealt with simultaneously in making an in-control or out-of-control decision, the joint distribution of the charting statistics is used to control the false alarm probability, which is defined as the probability of at least one false alarm, while designing the charts. An example is given. Concluding remarks include suggestions regarding future research problems.

Parameter Estimation and Performance of the

Effects of parameter estimation are examined for the well-known p-chart for the fraction nonconforming based on attributes (binary) data. The exact run-length distribution of the chart is obtained for Phase II applications, when the fraction of nonconforming items, p, is unknown, by conditioning on the observed number of nonconformities in a set of reference data (from Phase I) used to estimate p. Numerical illustrations show that the actual performance of the chart can be substantially different from what one would nominally expect, in terms of the false alarm rate and/or the in-control average run-length. Moreover, the performance of the p-chart can be highly degraded in that an exceedingly large number of false alarms are observed, particularly when p is estimated, unless the number of reference observations is substantially large, much larger than what might be commonly used in practice. These results are useful in the study of the reliability of products or systems that involve binary data

p

Nonparametric control charts can provide a robust alternative in practice to the data analyst when there is a lack of knowledge about the underlying distribution. A nonparametric exponentially weighted moving average (NPEWMA) control chart combines the advantages of a nonparametric control chart with the better shift detection properties of a traditional EWMA chart. A NPEWMA chart for the median of a symmetric continuous distribution was introduced by Amin and Searcy (1991) using the Wilcoxon signed-rank statistic (see Gibbons and Chakraborti, 2003). This is called the nonparametric exponentially weighted moving average Signed-Rank (NPEWMA-SR) chart. However, important questions remained unanswered regarding the practical implementation as well as the performance of this chart. In this paper we address these issues with a more in-depth study of the two-sided NPEWMA-SR chart. A Markov chain approach is used to compute the run-length distribution and the associated performance characteristics. Detailed guidelines and recommendations for selecting the charts design parameters for practical implementation are provided along with illustrative examples. An extensive simulation study is done on the performance of the chart including a detailed comparison with a number of existing control charts, including the traditional EWMA chart for subgroup averages and some nonparametric charts i.e. runs-rules enhanced Shewhart-type SR charts and the NPEWMA chart based on signs. Results show that the NPEWMA-SR chart performs just as well as and in some cases better than the competitors. A summary and some concluding remarks are given.

-Chart for Attributes Data

ABSTRACT Nonparametric control charts are useful when there is limited or complete lack of knowledge about the form of the underlying distribution. Though traditional statistical process control (SPC) applications of control charts involve subgrouped data, recent advances have led to more and more instances where individual measurements (data) are collected over time. A two-sided nonparametric exponentially weighted moving average (EWMA) control chart for i.i.d. individual data is proposed based on the sign (SN) statistic. A Markov chain approach is used to determine the run-length distribution of the chart and some associated performance characteristics. An important advantage of the nonparametric EWMA-SN chart is its inherent in-control robustness. In fact, the in-control run-length distribution and hence all of its associated characteristics (e.g., false alarm rate, average, standard deviation, median, etc.) of the chart remain the same for all unknown continuous distributions. In order to aid practical implementation, tables are provided for the charts design parameters. An extensive simulation study shows that on the basis of minimal required assumptions, robustness of the in-control run-length distribution and out-of-control performance, the proposed nonparametric EWMA-SN chart can be a strong contender in many applications where traditional parametric charts are currently used.

A nonparametric exponentially weighted moving average signed-rank chart for monitoring location

Abstract The effects of parameter estimation are examined for the well-known c-chart for attributes data. The exact run length distribution is obtained for Phase II applications, when the true average number of non-conformities, c, is unknown, by conditioning on the observed number of non-conformities in a set of reference data (from Phase I). Expressions for various chart performance characteristics, such as the average run length (ARL), the standard deviation of the run length (SDRL) and the median run length (MDRL) are also obtained. Examples show that the actual performance of the chart, both in terms of the false alarm rate (FAR) and the in-control ARL, can be substantially different from what might be expected when c is known, in that an exceedingly large number of false alarms are observed, unless the number of inspection units (the size of the reference dataset) used to estimate c is very large, much larger than is commonly used or recommended in practice. In addition, the actual FAR and the in-control ARL values can be very different from the nominally expected values such as 0.0027 (or ARL0=370), particularly when c is small, even with large amounts of reference data. A summary and conclusions are offered.

A Nonparametric EWMA Sign Chart for Location Based on Individual Measurements

Nonparametric control charts are considered for the median and other percentiles based on runs of sign statistics above and below the control limits. It is noted that the sign charts are advantageous in certain practical situations. Expressions for the run-length distributions are derived using Markov chain theory; several examples are given. The in-control (IC) and the out-of-control (OOC) performance of these charts are studied and compared to the existing nonparametric Wilcoxon signed-ranked charts of Chakraborti and Eryilmaz (2007) under the normal, the double exponential and the Cauchy distributions, using the average run-length (ARL), the standard deviation of the run-length (SDRL), the false alarm rate (FAR) and some percentiles of the run-length, including the median run-length (MDRL). It is shown that the proposed “runs-rules enhanced” sign charts offer more practically desirable IC ARL (ARL 0) and FAR values and perform better for some heavy-tailed distributions. Some concluding remarks are offered.

Properties and performance of the c-chart for attributes data

A nonparametric Shewhart-type control chart is proposed for monitoring the location of a continuous variable in a Phase I process control setting. The chart is based on the pooled median of the available Phase I samples and the charting statistics are the counts (number of observations) in each sample that are less than the pooled median. An exact expression for the false alarm probability (FAP) is given in terms of the multivariate hypergeometric distribution and this is used to provide tables for the control limits for a specified nominal FAP value (of 0.01, 0.05 and 0.10, respectively) and for some values of the sample size (n) and the number of Phase I samples (m). Some approximations are discussed in terms of the univariate hypergeometric and the normal distributions. A simulation study shows that the proposed chart performs as well as, and in some cases better than, an existing Shewhart-type chart based on the normal distribution. Numerical examples are given to demonstrate the implementation of the new chart.

Nonparametric Shewhart-Type Sign Control Charts Based on Runs

The traditional exponentially weighted moving average (EWMA) chart is one of the most popular control charts used in practice today. The in-control robustness is the key to the proper design and implementation of any control chart, lack of which can render its out-of-control shift detection capability almost meaningless. To this end, Borror et al. [5] studied the performance of the traditional EWMA chart for the mean for i.i.d. data. We use a more extensive simulation study to further investigate the in-control robustness (to non-normality) of the three different EWMA designs studied by Borror et al. [5]. Our study includes a much wider collection of non-normal distributions including light- and heavy-tailed and symmetric and asymmetric bi-modal as well as the contaminated normal, which is particularly useful to study the effects of outliers. Also, we consider two separate cases: (i) when the process mean and standard deviation are both known and (ii) when they are both unknown and estimated from an in-control Phase I sample. In addition, unlike in the study done by Borror et al. [5], the average run-length (ARL) is not used as the sole performance measure in our study, we consider the standard deviation of the run-length (SDRL), the median run-length (MDRL), and the first and the third quartiles as well as the first and the 99th percentiles of the in-control run-length distribution for a better overall assessment of the traditional EWMA charts in-control performance. Our findings sound a cautionary note to the (over) use of the EWMA chart in practice, at least with some types of non-normal data. A summary and recommendations are provided.

A Phase I nonparametric Shewhart-type control chart based on the median

Research of the first author was supported in part by STATOMET at the University of Pretoria, South Africa and National Research Foundation through the SARChI Chair at the University of Pretoria, South Africa.

Robustness of the EWMA control chart for individual observations

Runs-rules are typically incorporated in control charts to increase their sensitivity to detect small process shifts. However, a drawback of this approach is that runs-rules charts are unable to detect large shifts quickly. In this article improved runs-rules are introduced to the nonparametric sign chart to address this limitation. Improved runs-rules are incorporated to maintain sensitivity to small process shifts, while having the added ability to detect large shifts in the process more efficiently. Performance comparisons between sign charts with runs-rules and sign charts with improved runs-rules illustrate that the improved runs-rules are superior in performance for large shifts in the process, while maintaining the same sensitivity in the detection of small shifts.