Statistical quality control charts originated in the late 1920’s by Shewhart (1926, 1931 and 1939). Their applications in various disciplines have been ever-increasing. Although most control charts are distribution-based, recent literature witnessed the development of a considerable number of distribution-free or nonparametric control charts. The purpose of this thesis is to present the concepts and introduce the researcher to the essentials of univariate nonparametric control charts. Various properties of nonparametric control charts are comprehensively discussed and concepts are clearly explained. Proofs and detailed calculations have been given to help the reader to study and understand the subject more thoroughly. This text contains a wide variety of illustrative examples to give an overall picture of how nonparametric control charts are used. Both simulated and real data examples have been integrated throughout the text. Since most practical problems are too large to be solved using hand calculations, some type of statistical software package is required to solve these problems. There are several excellent statistical packages available and in this thesis we make use of Microsoft Excel, SAS, Minitab, Mathcad and Mathematica to construct (almost all) the tables in this thesis. We point out that a number of Mathematica programs are provided by Chakraborti and Van de Wiel (2003) by means of the website www.win.tue.nl/~markvdw. The aim throughout is to convey the concepts of univariate nonparametric control charts in a way that readers will find attractive and interesting. Since the majority of nonparametric procedures, to be distribution-free, require a continuous population, only variables control charts are covered. We only consider control charts for monitoring the location of a process, since very few nonparametric charts are available for monitoring the spread. In this thesis we consider the three main classes of control charts: the Shewhart, CUSUM and EWMA control charts and their refinements. The text is divided into several chapters. An introduction to nonparametric control charts is presented in Chapter 1. A discussion of some of the advantages of nonparametric control charts is included while pointing out some of the disadvantages. In Chapter 2 we describe the Shewhart-, CUSUMand EWMA-type sign control charts with (and without) warning limits. In Chapter 3 we describe the Shewhart-, CUSUM- and EWMA-type signed-rank control charts with (and without) runs-type signalling rules. The Shewhart-type sign-like control chart with (and without) signalling rules is considered in Chapter 4. In Chapter 5 we consider the Shewharttype signed-rank-like control chart. Finally, in Chapter 6 we consider the Shewhart- and CUSUM-type Mann-Whitney-Wilcoxon control charts. We considered decision problems under both Phase I and Phase II (see Section 1.5 for a distinction between the two phases). In all the sections of this thesis we considered Phase II process monitoring, except in Section 6.2 where a CUSUM-type control chart for the preliminary Phase I analysis of individual observations based on the Mann-Whitney two-sample test is proposed. In the last chapter we have some concluding remarks along with some ideas for future research. / Dissertation (MSc)--University of Pretoria, 2011. / Statistics / unrestricted

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It is often important to detect an increase in the frequency of some event. Particular attention is given to medical events such as mortality or the incidence of a given disease, infection or birth defect. Observations are regularly taken in which either an incidence occurs or one does not. This dissertation contains the result of an investigation of prospective monitoring techniques in two distinct surveillance situations. In the first situation, the observations are assumed to be the results of independent Bernoulli trials. Some have suggested adapting the scan statistic to monitor such rates and detect a rate increase as soon as possible after it occurs. Other methods could be used in prospective surveillance, such as the Bernoulli cumulative sum (CUSUM) technique. Issues involved in selecting parameters for the scan statistic and CUSUM methods are discussed, and a method for computing the expected number of observations needed for the scan statistic method to signal a rate increase is given. A comparison of these methods shows that the Bernoulli CUSUM method tends to be more effective in detecting increases in the rate. In the second situation, the incidence information is available at multiple locations. In this case the individual sites often report a count of incidences on a regularly scheduled basis. It is assumed that the counts are Poisson random variables which are independent over time, but the counts at any given time are possibly correlated between regions. Multivariate techniques have been suggested for this situation, but many of these approaches have shortcomings which have been demonstrated in the quality control literature. In an attempt to remedy some of these shortcomings, a new control chart is recommended based on a multivariate exponentially weighted moving average. The average run-length performance of this chart is compared with that of the existing methods. / Ph. D.

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This study uses Monte Carlo simulation to examine the performance of a variable frequency sampling cumulative sum control chart scheme for controlling the mean of a normal process. The study compares the performance of the method with that of a standard fixed interval sampling cumulative sum control chart scheme. The results indicate that the variable frequency sampling cumulative sum control chart scheme is superior to the standard cumulative sum control chart scheme in detecting a small to moderate shift in the process mean.

Control charts are designed to monitor population parameters. Selection of a control chart sampling plan involves determination of the frequency of samples, size of each sample, and critical values to determine when the system is sending an out-of-control signal. Since the main use of control charts is in industry, a widely accepted measure of a good sampling plan is one that minimizes the total cost of operating the system per unit time. Methods for selection of control chart sampling plans for economically optimal X charts are well established. These plans focus on single stage sampling at each sampling period. However, some populations naturally call for two stage sampling. Here, the cost of operating a system per unit time is redefined in terms of two stage sampling plans, and computer search techniques are developed to determine the control chart parameters. First the sample sizes and critical values are fixed, and Newton's method is used to determine the optimal time between samples. Then, a Hooke - Jeeves search is used to simultaneously determine the optimal critical value, sample sizes and time between samples. Adjustment to the latter is required whenever any of the other three parameters change. Alternative methods are also discussed. Information from a single sample is usually used to control shifts in both the process mean and variance. With two stage sampling, this means two additional control charts are used, one for each variance component. The computer algorithm developed for selection of parameters for X charts is adapted by expanding the Hooke Jeeves search region to a six dimensional space, now over three critical values, sample sizes for both stages of sampling, and the time between samples. These methods are applied to a real data set that requires two stage sampling. A representative analysis of the sensitivity of the optimal sampling scheme to the input parameters completes the paper. / Graduation date: 1990

Distribution-free (nonparametric) control charts can be useful to the quality practitioner when the underlying distribution is not known. The term nonparametric is not intended to imply that there are no parameters involved, in fact, quite the contrary. While the term distribution-free seems to be a better description of what we expect from these charts, that is, they remain valid for a large class of distributions, nonparametric is perhaps the term more often used. In the statistics literature there is now a rather vast collection of nonparametric tests and confidence intervals and these methods have been shown to perform well compared to their normal theory counterparts. Remarkably, even when the underlying distribution is normal, the efficiency of some nonparametric tests relative to the corresponding (optimal) normal theory methods can be as high as 0.955 (see e.g. Gibbons and Chakraborti (2010) page 218). For some other heavy-tailed and skewed distributions, the efficiency can be 1.0 or even higher. It may be argued that nonparametric methods will be ‘less efficient’ than their parametric counterparts when one has a complete knowledge of the process distribution for which that parametric method was specifically designed. However, the reality is that such information is seldom, if ever, available in practice. Thus it seems natural to develop and use nonparametric methods in statistical process control (SPC) and the quality practitioners will be well advised to have these techniques in their toolkits. In this thesis we only propose univariate nonparametric control charts designed to track the location of a continuous process since very few charts are available for monitoring the scale and simultaneously monitoring the location and scale of a process. Chapter 1 gives a brief introduction to SPC and provides background information regarding the research conducted in this thesis. This will aid in familiarizing the reader with concepts and terminology that are helpful to the following chapters. Details are given regarding the three main classes of control charts, namely the Shewhart chart, the cumulative sum (CUSUM) chart and the exponentially weighted moving average (EWMA) chart. We begin Chapter 2 with a literature overview of Shewhart-type Phase I control charts followed by the design and implementation of these charts. A nonparametric Shewhart-type Phase I control chart for monitoring the location of a continuous variable is proposed. 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. The derivations recognize that in Phase I the signalling events are dependent and that more than one comparison is © University of Pretoria v made against the same estimated limits simultaneously; this leads to working with the joint distribution of a set of dependant random variables. An exact expression for the false alarm probability is given in terms of the multivariate hypergeometric distribution and this is used to provide tables for the control limits. Some approximations are discussed in terms of the univariate hypergeometric and the normal distributions. In Chapter 3 Phase II control charts are introduced and considered for the case when the underlying parameters of the process distribution are known or specified. This is referred to as the ‘standard(s) known’ case and is denoted Case K. Two nonparametric Phase II control charts are considered in this chapter, with the first one being a nonparametric exponentially weighted moving average (NPEWMA)-type control chart based on the sign (SN) statistic. A Markov chain approach (see e.g. Fu and Lou (2003)) is used to determine the run-length distribution of the chart and some associated performance characteristics (such as the average, standard deviation, median and other percentiles). In order to aid practical implementation, tables are provided for the chart’s 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 NPEWMA-SN chart can be a strong contender in many applications where traditional parametric charts are currently used. Secondly, we consider the NPEWMA chart that was introduced by Amin and Searcy (1991) using the Wilcoxon signed-rank statistic (see e.g. Gibbons and Chakraborti (2010) page 195). This is called the nonparametric exponentially weighted moving average signed-rank (NPEWMA-SR) chart. In their article important questions remained unanswered regarding the practical implementation as well as the performance of this chart. In this thesis we address these issues with a more in-depth study of the 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 chart’s 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. Results show that the NPEWMA-SR chart performs just as well as and in some cases better than the competitors. In Chapter 4 Phase II control charts are introduced and considered for the case when the underlying parameters of the process distribution are unknown and need to be estimated. This is referred to as the ‘standard(s) unknown’ case and is denoted Case U. Two nonparametric Phase II control charts are proposed in this chapter. They are a Phase II NPEWMA-type control chart and a nonparametric cumulative sum (NPCUSUM)-type control chart, based on the exceedance statistics, © University of Pretoria vi respectively, for detecting a shift in the location parameter of a continuous distribution. The exceedance statistics can be more efficient than rank-based methods when the underlying distribution is heavy-tailed and / or right-skewed, which may be the case in some applications, particularly with certain lifetime data. Moreover, exceedance statistics can save testing time and resources as they can be applied as soon as a certain order statistic of the reference sample is available. We also investigate the choice of the order statistics (percentile), from the reference (Phase I) sample that defines the exceedance statistic. It is observed that other choices, such as the third quartile, can play an important role in improving the performance of these exceedance charts. It is seen that these exceedance charts perform as well as and, in many cases, better than its competitors and thus can be a useful alternative chart in practice. Chapter 5 wraps up this thesis with a summary of the research carried out and offers concluding remarks concerning unanswered questions and / or future research opportunities. © University / Thesis (PhD)--University of Pretoria, 2013. / gm2013 / Statistics / restricted

abstract: Mobile ad hoc networks (MANETs) have attracted attention for mission critical applications. This dissertation investigates techniques of statistical monitoring and control for overhead reduction in a proactive MANET routing protocol. Proactive protocols transmit overhead periodically. Instead, we propose that the local conditions of a node should determine this transmission decision. While the goal is to minimize overhead, a balance in the amount of overhead transmitted and the performance achieved is required. Statistical monitoring consists of techniques to determine if a characteristic has shifted away from an in-control state. A basic tool for monitoring is a control chart, a time-oriented representation of the characteristic. When a sample deviates outside control limits, a significant change has occurred and corrective actions are required to return to the in-control state. We investigate the use of statistical monitoring of local conditions in the Optimized Link State Routing (OLSR) protocol. Three versions are developed. In A-OLSR, each node uses a Shewhart chart to monitor betweenness of its two-hop neighbourhood. Betweenness is a social network metric that measures a node's influence; betweenness is larger when a node has more influence. Changes in topology are associated with changes in betweenness. We incorporate additional local node conditions including speed, density, packet arrival rate, and number of flows it forwards in A+-OLSR. Response Surface Methodology (RSM) is used to optimize timer values. As well, the Shewhart chart is replaced by an Exponentially Weighted Moving Average (EWMA) chart, which is more sensitive to small changes in the characteristic. It is known that control charts do not work as well in the presence of correlation. Hence, in A*-OLSR the autocorrelation in the time series is removed and an Auto-Regressive Integrated Moving Average (ARIMA) model found; this removes the dependence on node speed. A*-OLSR also extends monitoring to two characteristics concurrently using multivariate cumulative sum (MCUSUM) charts. The protocols are evaluated in simulation, and compared to OLSR and its variants. The techniques for statistical monitoring and control are general and have great potential to be applied to the adaptive control of many network protocols. / Dissertation/Thesis / Ph.D. Computer Science 2012

Healthcare is facing great challenges and psychiatric care is no exception. Extensive attempts to improve quality are made. It is essential to use methods that enable learning from experience, to improve performance. The core feature of Statistical Process Control (SPC), the control charts, are in use in various settings to enable learning and to support quality improvement work, but its use in psychiatric settings are scarce. This master´s thesis explores the usefulness of control charts, in quality improvement work. This was done in a case study at a department of psychosis by addressing two questions related to: a) control chart’s contribution to knowledge on if, when, where and how changes occur, and 2) how usefulness of control charts is perceived at the department. Control charts were applied to important variables and development officer’s and manager’s thoughts on usefulness were analysed using pattern matching. The use of charts shows shifts and differences between wards related to ongoing improvement projects. There is a readiness to start using control charts. The perceived usefulness matches the benefits and challenges identified in literature. Control charts as a tool supporting continuous improvement work in a psychiatric context, has a great potential still awaiting its use.