On ARL-unbiased c charts for i.i.d. and INAR(1) Poisson counts

Transcription

1 On ARL-unbiased c charts for i.i.d. and INAR(1) Poisson counts Sofia Raquel Bastos Benvindo Paulino Thesis to obtain the Master of Science Degree in Matemática e Aplicações Supervisor: Prof. Manuel João Cabral Morais Examination Committee Chairperson: Prof. António Manuel Pacheco Pires Supervisor: Prof. Manuel João Cabral Morais Member of the committee: Prof. Patrícia Alexandra de Azevedo Carvalho Ferreira e Pereira Ramos April 2015

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3 To my family i

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5 Acknowledgments I would like to express my gratitude to my supervisor Professor Manuel Cabral Morais for his guidance, constructive criticism and encouragement throughout the preparation of this thesis. Thank you for your patience and support, and for making every work session as much fun and thought-provoking as possible. I want to also say thanks to my close family for their unconditional love and support, and for giving me the strength to carry on, even in the darkest hours. This is all for you. iii

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7 Resumo Em controlo estatístico de processos (Statistical Process Control, SPC) é comum assumir que os processos de contagem têm distribuição marginal de Poisson. O carácter não-negativo, discreto e assimétrico de uma estatística de controlo com tal distribuição e o valor esperado alvo podem impedir o utilizador da carta c de lidar com: um valor sob controlo do número esperado de amostras recolhidas até sinal (average run length, ARL) pré-especificado; um limite de controlo inferior positivo; a capacidade de controlar não apenas aumentos mas também diminuições no valor esperado desses processos de contagem. Mais, tanto quanto pudemos investigar, as cartas c propostas na literatura de SPC tendem a ser enviesadas no que respeita à função ARL, na medida em que alguns valores de ARL fora de controlo são superiores ao valor de ARL sob controlo, i.e., demora-se, em média, mais tempo a detectar alguns shifts no valor esperado do processo do que a emitir um falso alarme. Nesta tese, exploramos as noções de testes não enviesados, aleatorizados e uniformemente mais potentes não enviesados (uniformly most powerful unbiased, UMPU) para corrigir o enviesamento da função ARL da carta c. Utilizamos o software estatístico R para adiantar ilustrações instrutivas de: cartas c não enviesadas em termos de ARL para processos de contagem i.i.d. com distribuição marginal de Poisson; cartas c quase não enviesadas em termos de ARL para o valor esperado de processos inteiros auto-regressivos de primeira ordem (INAR(1)) com distribuição marginal de Poisson. Palavras-chave: controlo estatístico de processos; processos INAR(1) com marginais de Poisson; average run length; testes não enviesados, aleatorizados e UMPU; software estatístico R. v

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9 Abstract In Statistical Process Control (SPC) it is usual to assume that counts have a Poisson distribution. The non-negative, discrete and asymmetrical character of a control statistic with such distribution and the value of its target mean may prevent the quality control practitioner to deal with a c chart with: a pre-specified in-control average run length (ARL); a positive lower control limit; the ability to control not only increases but also decreases in the mean of those counts. Furthermore, as far as we have investigated, the c charts proposed in the SPC literature tend to be ARL-biased, in the sense that some out-of-control ARL values are larger than the in-control ARL, i.e., it takes longer, in average, to detect some shifts in the process mean than to trigger a false alarm. In this thesis, we explore the notions of unbiased, randomized and uniformly most powerful unbiased (UMPU) tests to correct the bias of the ARL function of the c chart. We use the R statistical software to provide instructive illustrations of: ARL-unbiased c charts for i.i.d. Poisson counts; quasi ARLunbiased c charts for the mean of first-order integer-valued autoregressive (INAR(1)) Poisson counts. Keywords: statistical process control; Poisson INAR(1) processes; average run length; unbiased, randomized and UMPU tests; R statistical software. vii

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11 Contents Acknowledgement iii Resumo vii Abstract ix List of Tables xi List of Figures xiv Glossary xv Acronyms xvii 1 Introduction Causes of variation Quality control charts Controlling count processes Aim and overview of the thesis Control charts for the mean of i.i.d. Poisson counts Revisiting the c chart Criticism on the c chart with 3 σ limits Variants of the c chart with 3 σ limits Towards an ARL-unbiased c chart Randomizing the emission of a signal Eliminating the bias of the ARL function Illustrations INAR(1) Poisson counts Time series, stationarity and real-valued time series Integer-valued autoregressive time series models Model building Illustrations ix

12 4 Control charts for the mean of a Poisson INAR(1) process The impact of falsely assuming i.i.d. Poisson counts in the performance of the c chart c chart for the mean of a Poisson INAR(1) process Criticism on the c chart with 3 σ limits Towards a quasi ARL-unbiased c chart Decreasing the bias of the overall ARL function Randomizing the emission of a signal Illustrations Conclusions Bibliography Appendix R code Main functions I.i.d. case INAR case Auxiliary functions I.i.d. case INAR case x

13 List of Tables 2.1 List of control statistics and limits of charts for defects LCL, UCL, supremum of ARL, bias of the ARL function and in-control ARL, for λ 0 = 2, 8, 19 listed in order corresponding to the 3 σ, Ryan & Schwertman and quantile based control limits LCL, UCL, m, pair of randomization probabilities; supremum, bias and in-control value of ARL/SDRL/CVRL/Median RL, for λ 0 = 2, 8, 19 listed in order corresponding to the c chart with quantile based control limits, the randomized c chart and the ARL-unbiased c chart Sunspot groups Yule-Walker (top row) and CLS (bottom row) estimates of the parameters β 1,..., β p, µ ɛ, σɛ 2 and associated value of AICC(p), for p = 1,..., Claims related to soft tissue injuries Yule-Walker (top row) and CLS (bottom row) estimates of the parameters β 1,..., β p, µ ɛ, σɛ 2 and associated value of AICC(p), for p = 1,..., In-control and out-of-control ARL, SDRL, CVRL and Median RL values of the c chart with 3 σ control limits in the i.i.d. case (β = 0), for λ 0 = 5, Estimates of the in-control and out-of-control ARL, SDRL, CVRL and Median RL values of the c chart with 3 σ control limits for the i.i.d. case (β = 0), for λ 0 = 5, Exact values of the overall ARL of the c chart with 3 σ limits for the mean of a Poisson INAR(1) process LCL, UCL, pair of randomization probabilities, in-control ARL/SDRL/CVRL/Median RL, for (λ 0, β 0 ) = (0.4, 0.8), (3, 0.6), (10, 0.5) listed in order corresponding to the c chart with 3 σ control limits, the unrandomized c chart and the randomized c chart List of functions xi

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15 List of Figures 2.1 c charts with unrevised 3 σ trial control limits for the three original sets of numbers of defects (rectangular steel plates, top; telephone cable, center; printed circuit boards, bottom) ARL(δ) curves, for 3 σ (curves in black), Ryan & Schwertman (curves in orange) and quantile based control limits (curves in violet) λ 0 = 2 (top), 8 (center), 19 (bottom) ARL(δ) and SDRL(δ) curves (left and right, resp.), for the c chart with quantile based control limits (curves in violet), the randomized c chart (curves in blue) and the ARLunbiased c chart (curves in green) λ 0 = 2 (top), 8 (center), 19 (bottom) CV RL(δ) and M edianrl(δ) curves (left and right, resp.), for the c chart with quantile based control limits (curves in violet), the randomized c chart (curves in blue) and the ARL-unbiased c chart (curves in green) λ 0 = 2 (top), 8 (center), 19 (bottom) Daily number of sunspot groups as observed by the Space Environment Laboratory in ACF of the daily number of sunspot groups PACF of the daily number of sunspot groups Diagnostics for the CLS estimated INAR(1) fit of the daily number of sunspot groups Monthly number of workers claims related to soft tissue injuries from 1985 to ACF of the monthly number of claims related to soft tissue injuries PACF of the monthly number of claims related to soft tissue injuries Diagnostics for the CLS estimated INAR(1) fit of the monthly number of claims related to soft tissue injuries Estimates of in-control and out-of-control ARL, SDRL, CVRL and Median RL λ 0 = 5 (left) and δ λ = 0, 1 (black and orange curves, resp.); λ 0 = 20 (right) and δ λ = 1, 0, 1 (violet, black and orange curves, resp.) Exact ARL(λ, β 0 ) and ARL(λ 0, β) of the c chart with 3 σ limits for the mean of a Poisson INAR(1) process (λ 0, β 0 ) = (0.4, 0.8), (3, 0.6), (10, 0.5) (top, center, bottom, resp.) ARL(λ, β 0, γ LCL, γ UCL ) and ARL(λ 0, β, γ LCL, γ UCL ) vs. estimated overall ARL (λ 0, β 0 ) = (5, 0.5), LCL = 2, UCL = 21, γ LCL = and γ UCL = Overall ARL, SDRL, CVRL and Median RL as functions of λ (left) and β (right), for the c chart with 3 σ control limits (curves in black), the unrandomized c chart (curves in violet) and the randomized c chart (curves in blue) (λ 0, β 0 ) = (0.4, 0.8) Overall ARL, SDRL, CVRL and Median RL as functions of λ (left) and β (right), for the c chart with 3 σ control limits (curves in black), the unrandomized c chart (curves in violet) and the randomized c chart (curves in blue) (λ 0, β 0 ) = (3, 0.6) xiii

16 4.6 Overall ARL, SDRL, CVRL and Median RL as functions of λ (left) and β (right), for the c chart with 3 σ control limits (curves in black), the unrandomized c chart (curves in violet) and the randomized c chart (curves in blue) (λ 0, β 0 ) = (10, 0.5) xiv

17 Glossary 1 (resp. 0) vector of ones (resp. zeros). a vector. a transpose of a. A = [a ij ] l i,j=k matrix with entries a ij, i, j = k, k + 1,..., l. e u (u + 1)-th vector of the orthogonal basis for R x+1, u {0, 1,..., x and x N. I identity matrix. r.v. random variable. i.i.d. independent and identically distributed. n sample size. t sample index. xv

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19 Acronyms ARL average run length. CVRL coefficient of variation of the run length. INAR(1) first-order integer-valued autoregressive. LCL lower control limit. RL run length. SDRL standard deviation of the run length. SPC statistical process control. UCL upper control limit. UMPU uniformly most powerful unbiased (test). xvii

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21 Chapter 1 Introduction Many have tried to establish the meaning of quality and provided several definitions for this essential and distinctive attribute. For instance, Shewhart (1931, p. 37) mentions that there has been some tendency to conceive quality as the goodness of an object since at least the time of Aristotle. Ramos (2013, p. 1) adds that fitness for use and conformance to requirements are probably the shortest and the two most accurate and consensual definitions of quality and that these definitions are due to Joseph Juran (Juran and Godfrey, 1999, p. 27) and Philip Crosby (Crosby, 1979, p. 17), respectively. Moreover, no matter what the definition of quality may be, the measure of quality is a (random) variable (Duncan, 1959, p. 37) we shall term from now on quality characteristic. Even though quality control is old as industry itself (Duncan, 1959, p. 1), statistical process control (SPC) and the allied techniques of sampling inspection and quality control were only developed in 1920s (Wetherill and Brown, 1991, p. 1). SPC can be seen as an objective statistical analysis of process variation and its causes (Wetherill and Brown, 1991, p. 6) and has an advantage over sampling inspection: it focuses on early detection and prevention of problems, instead of concentrating on their correction after they occur (Wikipedia, 2015b). Unsurprisingly, SPC is now recognized as having a very important role to play in modern industry (Wetherill and Brown, 1991, p. xiii). 1.1 Causes of variation When the founder of SPC, Walter A. Shewhart ( ), joined the Bell Telephone Laboratories, he realized that every manufacturing process results in some random variation in the items it produces (Ross, 1987, p. 406). Shewhart named the sources responsible for this kind of variation as chance causes (Wieringa, 1999, pp ), also known as common causes of variation. They are inherent to the nature of the process itself (Woodall, 2000), and little can be done for no major part of these random variations can be traced to a single chance cause (Duncan, 1959, p. 315). The presence of unusual disruptions leads to unacceptable variability due to what is usually termed as special causes or assignable causes (Woodall, 2000). These relatively large variations can be attributable to assignable causes consisting for the most part of differences among machines, workers and materials, of differences in each of these factors over time or in their relationships to one another (Duncan, 1959, p. 315). When a process is only affected by chance causes of variation, it is said to be in-control; it is considered to be out-of-control when assignable causes are also present. According to Wieringa (1999, p. 20), improving quality by reducing variability requires what is called statistical thinking in quality improvement (Snee, 1990). The reduction of variation can be accomplished by performing a careful analysis of the process. That, together with the knowledge of the process in question, makes it possible to identify the causes of variation that are present and adapt the solving strategy to the nature of the problems. This is essential, since the reduction of the effect of common causes can only be achieved by altering the process itself, while the detection and removal of assignable causes should be done by the people who work with the process (Wieringa, 1999, p. 21). 1

22 1.2 Quality control charts At the basis of the theory of control charts is the differentiation of the causes of variation on quality (Duncan, 1959, p. 315). Quality control charts are, thus, statistical devices not only used to detect the presence of assignable causes in repetitive processes, but also to describe in concrete terms what statistical control is, and for attaining statistical control and for judging whether statistical control has been attained (Duncan, 1959, p. 316). They were introduced by Walter A. Shewhart in a memorandum sent to his superiors in 1924 (Montgomery, 2009, p. 12). A quality control chart is a graphical display where the observed values of a control statistic, usually an estimator of a parameter of a quality characteristic, are regularly plotted over time and compared with the target value of such parameter, the center line (CL), and most importantly with a pair of horizontal lines corresponding to the lower and upper control limits (LCL and UCL, respectively). These control limits are defined in order that nearly all of the observed values of the control statistic will belong to the interval [LCL, UCL] when the process is in-control. Thus, when a value falls within the limits, we consider that the process is in-control and no action needs to be taken (Montgomery, 2009, p. 182). If a point is beyond those control limits, a signal is triggered because we interpret this occurrence as an evidence of the potential presence of assignable causes and deem the process out-of-control. An investigation should follow to bring the process to the state of statistical control if assignable causes are identified in this case, the chart triggered a valid signal; if no assignable causes have been detected, a false alarm has been emitted. The occurrence of an assignable cause can lead to an alteration in a parameter of the quality characteristic. When this parameter suffers an abrupt and sustained change, we say that a shift has occurred. In quality control, the run length (RL) corresponds to the number of samples taken until a signal is given (Wetherill and Brown, 1991, p. 100), and is frequently used to assess the performance of control charts. The most popular RL-related performance measure is the average run length (ARL). When the process is in-control, it is desirable to deal with large ARL values (corresponding to infrequent false alarms), whereas when the process is out-of-control, we should favour charts with small ARL values (corresponding to quick emissions of valid signals). 1.3 Controlling count processes Each specification that is not satisfied by a unit of a product is considered a defect, demerit, error or, more generally, a nonconformity (Montgomery, 2009, p. 308; White et al., 1997). Processes of counts of nonconformities arise frequently in SPC. The marginal distribution of those counts is usually assumed to be Poisson. This distribution will arise if both the following are true (Wetherill and Brown, 1991, p. 215): the defects occur in an independent way within and between inspection units; when the process is on target the average rate of occurrence of a defect is constant. Montgomery (2009, p. 309) expertly adds that count data is well modelled by the Poisson distribution if: the number of opportunities or potential locations for nonconformities be infinitely large; the inspection unit must be the same for each sample and always represent an identical area of opportunity for the occurrence of defects. 2

23 As put by Montgomery (2009, p. 308), although charts for count data are usually less informative than charts for variables, they have important applications since they arise in several settings where the quality characteristics are not easily measured on a numerical scale. The c chart enjoys widespread popularity as the chart of choice to control the mean of the number of nonconformities (White et al., 1997) in a constant-size sample. Since the standard deviation of the Poisson distribution is the square root of its parameter λ, the control limits of the standard c chart tend to be set at λ 0 ± k λ 0, where λ 0 denotes the target mean and k is a positive constant (usually equal to 3). Dealing with a nonnegative, discrete and asymmetrical (e.g. right-skewed) distribution such as Poisson can prevent us from: setting a c chart with a pre-specified in-control ARL; obtaining a positive lower control limit for this chart and therefore to be able to quickly detect small and moderate decreases in the process mean; define an ARL-unbiased control chart (Pignatiello et al., 1995, Acosta-Mejía, 1999), in the sense that all out-of-control ARL values are smaller than the in-control ARL, i.e., it takes longer, in average, to trigger a false alarm than to detect any shifts in the process mean. 1 Since c charts may indeed behave poorly when it comes to the detection of decreases in λ, several authors pursued other alternatives and tried to define charts that would be more sensitive to downward shifts in λ. Aebtarm and Bouguila (2011) listed many variants of the c chart that were established over the years and were obtained by making use of one of the three following approaches: transforming data, standardizing data or optimizing control limits. However, all these charts still have low responsiveness of the lower control limit when dealing with small process means. Furthermore, they tend to be ARL-biased control charts in the sense that some out-of-control ARL values are larger than the in-control ARL. A few authors have proposed ARL-unbiased charts for parameters of absolutely continuous quality characteristics. As pointed out by Knoth and Morais (2013): Uhlmann (1982, pp ) seems to be the only Statistical Quality Control textbook to propose an ARL-unbiased S 2 chart for the variance σ 2 of a normally distributed quality characteristic; Pignatiello et al. (1995), certainly inspired by Ramachandran (1958), proposed alternative control limits for the S 2 chart in order to achieve an ARL-unbiased chart for σ 2 ; 2 Ramalhoto and Morais (1995) and Ramalhoto and Morais (1999) proposed ARL-unbiased charts for the scale parameter of a Weibull quality characteristic. Another important issue is related to the fact that count processes are often autocorrelated but we falsely assume serial independence, especially while planning a control chart to monitor the process mean. Unsurprisingly, using λ 0 ± 3 λ 0 control limits has a severe impact on ARL and this calls for the use of charts accounting for the autocorrelation structure such as the ones proposed by Weiss (2009a, Chapter 20). Regretfully, these charts are also ARL-biased. As far as we have investigated, no ARL-unbiased chart has been proposed for count data, regardless of the fact that the detection of decreases in λ plays a crucial role in the effective implementation of any quality improvement program. This was the starting point of this thesis. 1 Pignatiello et al. (1995) invented the term ARL-unbiased and used to coin any control chart with such a desirable characteristic. This behavior of the ARL function means that the chart satisfies what Ramalhoto and Morais (1995) and Ramalhoto and Morais (1999) called the primordial criterion (Knoth and Morais, 2013). 2 According to Knoth and Morais (2013), those control limits depend on critical values satisfying conditions which can be traced back to Neyman and Pearson (1936), Fertig and Proehl (1937), Ramachandran (1958), Kendall and Stuart (1979, p. 219) or which have been equivalently rewritten by Uhlmann (1982, p. 213) and Krumbholz and Zöller (1995). 3

24 1.4 Aim and overview of the thesis The aim of this thesis is to propose and illustrate control charts for the mean of count processes that allow us to: pre-specify an in-control ARL value, say ARL ; control in a timely fashion not only increases, but also decreases in the process mean; deal with an unbiased (or quasi unbiased) ARL function. In order to achieve this, whether we are in the presence of i.i.d. or autocorrelated Poisson counts, the thesis is organized as follows. In Chapter 2, we tackle i.i.d. Poisson counts. The c chart is presented, along with some variants introduced by other authors. Then, we revisit the c chart with quantile based control limits and, inspired by randomized tests (rarely used in practice in SPC), propose another variant of the c chart; this variant is associated with quantile based control limits and randomization probabilities of triggering a signal when the value of the control statistic is equal to LCL and UCL, to bring the size of the test exactly to level α = (ARL ) 1. Furthermore, influenced by uniformly most powerful unbiased (UMPU) tests and capitalizing on our last variant of the c chart, we obtain an ARL-unbiased c chart with in-control ARL exactly equal to ARL. We present some striking and instructive examples that show that the performance of this last variant of the c chart supersedes the previous ones. In Chapter 3, we revise concepts such as time series, autocorrelation, stationarity and autoregressive processes. Then, we motivate and recall the definition of binomial thinning and first-order integer-valued autoregressive (INAR(1)) process with Poisson distributed marginals. Brief descriptions of two estimation methods and of a model selection criterion for this particular process are followed by illustrations with two data sets. We start Chapter 4 by illustrating the impact of falsely assuming i.i.d. Poisson counts in the performance of the standard c chart with 3 σ control limits. We revisit the modified c chart introduced by Weiss (2007) to monitor the mean of a Poisson INAR(1) process. Two variants of this chart are introduced; one of them comprises randomization probabilities. We go on to compare the performance of all these charts. In Chapter 5 we can find the major contributions of this thesis and a few recommendations for future work. Several programs for the R statistical software (R Core Team, 2013) have been written to produce all the results, tables and graphs in this thesis. All of them can be found in the Appendix. 4

25 Chapter 2 Control charts for the mean of i.i.d. Poisson counts 2.1 Revisiting the c chart The c chart is one of the most popular procedures to monitor the expected number of defects (λ) in a sample of n units. The control statistic of this chart is X t, the total number of defects in the t th sample (t N). We shall assume that the random variables (r.v.) X t, t N, are independent and identically distributed (i.i.d.) to the r.v. X Poisson(λ). Let λ = λ 0 + δ be the process mean, where: λ 0 denotes the target mean and is frequently assumed to be known; δ represents the magnitude of the shift in the process mean. The process is said to be in-control (resp. out-of-control) if δ = 0 (resp. δ ( λ 0, 0) (0, + )). If δ ( λ 0, 0) (resp. δ (0, + )) then a downward (resp. upward) shift has occurred and the practitioner ought to be alerted to this change. The c chart triggers an alarm at sample t if X t is beyond the control limits. The c chart makes use of the k σ control limits, defined for this integer-valued control statistic as the following ceiling and floor functions of λ 0 : LCL = UCL = max{0, λ 0 k λ 0 (2.1) λ 0 + k λ 0, (2.2) where k is a positive constant, usually selected so that false alarms are rather infrequent and is often taken as equal to 3. The performance of this chart is generally evaluated using the number of samples taken until an alarm is triggered, the run length (RL) or, more specifically, the expected value of this number, the average run length (ARL). Ideally, it is desirable that the emission of false alarms occurs rarely (corresponding to a large in-control ARL) and the emission of valid signals occur as quickly as possible (corresponding to a small out-of-control ARL). The RL of this Shewhart-type chart has a geometric distribution and depends on the magnitude of the shift in the process mean, δ ( λ 0, + ): RL(δ) Geometric(ξ(δ)), (2.3) where ξ(δ) = P (emission of a signal δ) = P (X [LCL, UCL] δ). (2.4) Hence ARL(δ) = 1 ξ(δ). (2.5) 5

26 Besides the ARL, we should also consider other RL-related performance measures, namely its standard deviation (SDRL), its coefficient of variation (CVRL) and its quantiles of order p: SDRL(δ) = [1 ξ(δ)] 1 2 ξ(δ) CV RL(δ) = SDRL(δ) ARL(δ) F 1 ; (2.6) = [1 ξ(δ)] 1 2 ; (2.7) RL(δ) (p) = inf{m R : F RL(δ)(m) p ln(1 p) =, 0 < p < 1. (2.8) ln[1 ξ(δ)] Interestingly enough, these three functions and ARL(δ) share a maximum value at the same point, say δ, as stated in the next proposition. Proposition 2.1. Consider a c chart with positive lower control limit. 1 Then, for δ ( λ 0, + ), [ ] [ ] d ARL(δ) d SDRL(δ) sign = sign d δ d δ [ ] d CV RL(δ) = sign d δ [ d F 1 RL(δ) = sign (p) ], 0 < p < 1 d δ [ ] ξ(δ) = sign. (2.9) d δ Consequently, if ξ(δ) takes a minimum value for some δ ( λ 0, + ) then argmax ARL(δ) = argmax SDRL(δ) δ ( λ 0,+ ) δ ( λ 0,+ ) = argmax CV RL(δ) δ ( λ 0,+ ) = argmax F 1 RL(δ) (p), 0 < p < 1 δ ( λ 0,+ ) = argmin ξ(δ). (2.10) δ ( λ 0,+ ) Proof. The first set of equalities is obvious if we note that ξ(δ) is a probability, p (0, 1) and [ ] { d ARL(δ) d [ξ(δ)] 1 sign = sign d δ d δ [ ] d ξ(δ) d δ = sign [ξ(δ)] 2 [ ] ξ(δ) = sign d δ 1 If LCL = 0, the supremum point of ARL(δ), SDRL(δ), CV RL(δ) and F 1 RL(δ) (p), for 0 < p < 1, is δ = λ 0. 6

27 [ ] d SDRL(δ) sign = sign d δ d [1 ξ(δ)] 1 2 ξ(δ) d δ ( d ξ(δ) = sign d δ = sign [ ] d CV RL(δ) sign = sign d δ [ = sign sign [ d F 1 RL(δ) (p) d δ [ ξ(δ) d δ { ] { d [1 ξ(δ)] 1 2 d δ 1 2 [1 ξ(δ)] 1 2 ξ(δ) [1 ξ(δ)] 1 2 ] d ξ(δ) d δ 1 2 [1 ξ(δ)] 1 2 [ ] d ξ(δ) = sign d δ ] d ln(1 p) ln[1 ξ(δ)] = sign d δ ( d ln[1 ξ(δ)] ) d δ ln(1 p) = sign {ln[1 ξ(δ)] 2 { d [1 ξ(δ)] d δ = sign 1 ξ(δ) { d [1 ξ(δ)] d δ = sign 1 ξ(δ) [ d ξ(δ) = sign d δ ]. 1 ) [ξ(δ)] 2 The second set follows in a straightforward manner if we capitalize on the first set of equalities and on the assumption that ξ(δ) takes a minimum value for some δ ( λ 0, + ). 2.2 Criticism on the c chart with 3 σ limits The c chart associated with three standard deviation limits has been historically used, implicitly assuming that the normal approximation to the Poisson distribution is adequate (Ryan and Schwertman, 1997). Adopting 3 σ limits, as Shewhart did, corresponds to a probability of a false alarm of approximately [1 Φ(3)] (Ramos, 2013, p. 3) if that approximation is reasonable. However, since the Poisson distribution is skewed to the right, 2 the lower (resp. upper) tail area tends to be smaller (resp. larger) than (Ryan and Schwertman, 1997), in particular for small values of the target mean λ 0 and in the absence of assignable causes, leading to an in-control ARL that can substantially differ from ( ) In addition, let us remind the reader that the performance of the c chart with 3 σ limits is determined by the reciprocal of the sum of those two tail areas (Ryan and Schwertman, 1997). Furthermore, when the Poisson mean is smaller than three times its square root, the LCL of this c chart is set to zero and the probability of triggering a signal is further distorted (White et al., 1997). We might add that when we deal with a c chart with 3 σ control limits, LCL > 0 iff λ 0 > 9; in case λ 0 9, we are leading with an upper-one sided chart, which triggers a valid signal in the presence of any decreases in λ less 2 Pearson s moment coefficient of skewness is equal to E[(X E(X)] 3 /[V (X)] 3/2 = λ 1/2, for X Poisson(λ) (Wikipedia, 2015a). 7

28 frequently than a false alarm. Unsurprisingly, Woodall (1997) added that the c chart with 3 σ limits may have an unacceptable performance. In conclusion, there is plenty of room for improvement, namely because it is very important to quickly detect if λ has decreased and, thus, use the c chart as a tool for continuous improvement. 2.3 Variants of the c chart with 3 σ limits As we mentioned earlier, the c chart with 3 σ limits may behave poorly, namely when it comes to the detection of decreases in λ. Expectedly, many researchers looked for ways to improve the performance of this Shewhart-type chart. What follows is a brief account of ten alternative charts, thoroughly described by Aebtarm and Bouguila (2011), to monitor the process mean of i.i.d. Poisson counts. These authors essentially reviewed and divided them in three categories, according to the adopted approach: transforming data; standardizing data; optimizing control limits. In the definition, set up and performance assessment of these ten charts and the c chart with 3 σ limits, the target value of the process mean was assumed unknown (in most cases), unlike what we have done so far in this thesis. Indeed λ 0 was replaced by its estimate c in the expression of the control limits of this latter chart, where c represents the mean number of defects in n preliminary sampling inspection units of product 3 taken from the output of the process when it was in-control. All the control statistics and limits have been summarized in Table 2.1, which was inspired by Table 5 of Aebtarm and Bouguila (2011). For a more detailed account of these charts, namely the theoretical ideas behind each chart, the reader is referred to Aebtarm and Bouguila (2011, Sec. 4). The transforming data approach the rationale behind the four charts that belong to this first category is transforming asymmetrical distributions into symmetric ones. The first three charts can be found in Ryan (1989, p. 197) and are associated with data transformations originally introduced by Bartlett (1936a,b), Anscombe (1948) and Freeman and Tukey (1950). The transformations (or observed values of the control statistics) and estimated control limits can be found in Table 2.1 and are: y = 2 c, y = 2 c and y = c + c + 1 (respectively), where c represents the number of defects in a sample; ȳ±3, with ȳ denoting the mean of the transformed number of defects referring to n preliminary samples taken from the output of the process in the absence of assignable causes. The fourth chart in Aebtarm and Bouguila (2011) follows the same lines as a proposal by Tsai et al. (2006) for an alternative chart meant to monitor the sample percentage of defective items. It is associated with: the improved square root transformation (ISRT), y = c; the control limits c ( ) and 1 c c ( Let us remind the reader that a (sampling) inspection unit of product is simply an entity; it can be a single unit of product, or a group of 5, 10, etc. units of product (Montgomery, 2009, p. 309). 1 c ). 8

29 The standardizing data approach is obviously related to the previous approach (after all we are once again transforming the data). However, the three charts Aebtarm and Bouguila (2011) included in the second category share the same control limits ±3, regardless of the control statistic we use or the fact that we adopt fixed or varying sample sizes. To accommodate varying sample sizes, let us assume now that: λ 0 represents the known target value of the expected number of defects in a single unit of product; the sample comprises n units of product (n may vary from sample to sample); c denotes the number of defects in that sample. As reported by Aebtarm and Bouguila (2011), these three charts can be found in Quesenberry (1991a). They were termed the Z, W and Q charts and the observed values of their control statistics were taken from this last reference: z = c nλ0 nλ0 ; w = 2 c 2 nλ 0 ; q = Φ 1 (u), where u = F P oi(nλ0)(c). The first two are standardizations of the observed values of the control statistics of the c chart and the chart based on the Bartlett transformation, considering λ 0 known. When the target value is unknown, the control statistics are obtained by: replacing λ 0 by an estimate, in case Z and W charts are used; and considering a different expression for u, 4 if we adopt the Q chart. 5 Aebtarm and Bouguila (2011) refer some points to be considered if we decide to adopt any of these charts, namely: the transformation data approach requires time and sophisticated inspectors (we beg to differ!); and the transformed charts cannot illustrate the exact level of defects in a process (since we are no longer dealing with the original data). The optimizing control limits approach the three alternative charts in this final category are easy to construct and to use, according to Aebtarm and Bouguila (2011), and their interpretation is straightforward. The first chart follows the work of Ryan and Schwertman (1997), and the remaining two were proposed by Winterbottom (1993) and Kittlitz (2003, 2006). 6 The observed value of their control statistics and limits and eventually a brief comment on each chart follow: the number of defects, c, in a sample of fixed size is plotted against the control limits LCL = c c and UCL = c c, and the optimal control limits for the c chart were obtained by Aebtarm and Bouguila (2011) by linear regression based on a table of the best c chart limits for several values of λ 0 considering a certain loss function; 7 the number of defects c is confronted with the Winterbottom control limits c ± 3 c ;8 4 The interested reader is referred to Quesenberry (1991a, p. 299). 5 We ought to note that Aebtarm and Bouguila (2011) failed to consider the correct control statistic of the Q chart for λ unknown. 6 According to Ryan (1989, p. 200), Kittlitz (1979) proposed modified limits that could be used rather than employing any transformations. 7 The control limits found in Ryan and Schwertman (1997) differ negligibly: LCL = λ λ 0 ; UCL = λ λ 0. Besides the fact that λ 0 was considered instead of its estimate, these control limits were obtained considering a related but slightly different loss function and set of target values. 8 Let us just refer that Winterbottom s chart is the only one of this third category that can be used when we are dealing with variable sample sizes. For more details, please refer to Aebtarm and Bouguila (2011). 9

30 the last c chart in this third category makes use of a transformation of the number of defects c inspired by previous work by Haldane (1938) and Read and Cressie (1988), y = (c ) 2 3, and is associated with the so called (almost) exact control limits (ECL) LCL = [( c ) 2 3 3( 2 3 )( c) 1 6 ] and UCL = [( c ) ( 2 3 )( c) 1 6 ] As rightly remarked by Aebtarm and Bouguila (2011), the ten alternative charts in Table 2.1 were compared by categories and these charts were assessed in terms of three criteria: lowest mean such that the LCL is positive in order to ensure some sensitivity to decreases in the process mean; lowest loss in the in-control state, where the loss function reflects the difference between the reciprocal of the tail areas and ; highest sensitivity in the presence of a shift in the process mean. The comparison study relied on extensive Monte Carlo simulations and on data sets generated with different means from 1 to 50 and in the presence of four mean shifts, and it led to the conclusion that the optimal control limits for the c chart showed outstanding results and is the best defect chart to replace the c chart with 3 σ limits. Nonetheless, Aebtarm and Bouguila (2011) do not claim that it is the only defect chart that should be applied to monitor the process mean of i.i.d. Poisson counts. Regretfully, Aebtarm and Bouguila (2011) did not make any reference to: how biased are the ARL functions of any of the eleven charts; 9 any chart with a control statistic with a recursive character, such as the Poisson CUSUM (cumulative sum) scheme proposed by White et al. (1997). Before being acquainted with the alternative charts for defect counts in Aebtarm and Bouguila (2011), it occurred to us that control limits based on quantiles of the Poisson distribution could be an appropriate alternative to the 3 σ limits, 10 and their adoption a possible remedy for the ARL-biased character of this standard chart. Moreover, since we are aware that the discrete character of this distribution makes it difficult to achieve a pre-specified probability of false alarm, say α (0, 1), we suggest the use of the following control limits termed, from now on, quantile based control limits. Definition 2.2. Quantile based control limits Requiring that P (X [LCL, UCL] δ = 0) α and setting α = α LCL + α UCL leads to the obtention of a LCL and a UCL such that: P (X < LCL δ = 0) α LCL ; (2.11) P (X > UCL δ = 0) α UCL. (2.12) Thus, the quantile based LCL (resp. UCL) is the largest (resp. smallest) integer satisfying (2.11) (resp. (2.12)). 9 The performance analysis of these charts was not made with that criterion in mind after all. 10 Coincidentally, Wetherill and Brown (1991, p. 215) also recommended the use of quantiles, and go on to suggest that if the practitioner wants (a non negative integer) x such that, for example, F P oi(λ) (x) = 1 F χ 2 2(x+1) (2λ) 0.999, then he/she must have λ 1 2 F 1 χ 2 ( ). 2(x+1) 10

31 Table 2.1: List of control statistics and limits of charts for defects. Chart Control statistic LCL UCL c chart c c 3 c c + 3 c Bartlett y = 2 c ȳ 3 ȳ + 3 Anscombe y = 2 c ȳ 3 ȳ + 3 Freeman & Tukey y = c + c + 1 ȳ 3 ȳ + 3 ISRT y = c c ( 1 c ) c Z z = c nλ0 nλ0 3 3 W w = 2 c 2 nλ0 3 3 Q q = Φ 1 [ F P oi(nλ0) (c) ] 3 3 Ryan & Schwertman (AB) c c c c c Winterbottom c c 3 c c + 3 c ECL y = (c ) 2 3 [( c ) 2 3 3( 2 3 )( c) 1 6 ] UCL = [( c ) ( 2 3 )( c) 1 6 ] ( 1 c ) 11

32 We now compare the ARL functions of the charts associated with: the 3 σ control limits (ARL curves in black); the Ryan & Schwertman control limits, as defined by Aebtarm and Bouguila (2011) (ARL curves in orange); the quantile based control limits with α = and α LCL = α UCL (ARL curves in violet). The comparisons between the ARL functions of these three charts shall be done with great care, because the charts are bound to have distinct in-control ARL values, which are in turn different from the desired in-control ARL value α The illustrations ahead have the sole purpose of giving us an idea of how the adoption of quantile based control limits can lead to less biased ARL functions than the two other pairs of control limits, for the following small, medium and large target values: λ 0 = 2 is the integer part of the target mean of the data set taken from Exercise 7.36 of Montgomery (2009, p. 339, Table 7E.10), corresponding to the number of surface defects on 25 rectangular steel plates; 11 λ 0 = 8 this value is larger than the previous one, but the Poisson distribution associated with it is still considerably skewed to the right; it is the integer part of the target value associated with the data set from Exercise 7.41 of Montgomery (2009, p. 339, Table 7E.13) referring to the number of defects per 1000 meters in telephone cable (22 samples); 12 λ 0 = 19 now the target mean is large enough for us to be dealing with an almost symmetric Poisson distribution; this value of λ 0 is the integer part of the target mean of the number of defects present in 26 samples of 100 printed circuit boards from Example 7.3 of Montgomery (2009, pp , Table 7.7). 13 In Figure 2.1 we can find three c charts with unrevised 3 σ trial control limits; these charts refer to the three original sets of numbers of defects described above and were obtained using the qcc package (Scrucca, 2004) for the R statistical environment and the following commands: qcc(dataset1, type= c ) qcc(dataset2, type= c ) qcc(dataset3, type= c ) Please note that the observations beyond the unrevised 3 σ trial control limits are in red. Before we proceed, let us recall that a control chart for Poisson counts is said to be ARL-unbiased if its ARL function decreases as the process mean λ tends away from its in-control value λ 0. Expectedly, the bias of the ARL function can be defined as argmax ARL(δ). (2.13) δ ( λ 0,+ ) If LCL = 0 the bias is equal to λ 0 because the supremum of ARL is at δ = λ 0. In Table 2.2 we listed several quantities corresponding to the charts with: the 3 σ control limits; the Ryan & Schwertman control limits, as defined by Aebtarm and Bouguila (2011); the quantile based control limits with α = and α LCL = α UCL. Besides the lower and upper control limits, Table 2.2 displays the supremum of ARL, the bias of the ARL function and in-control ARL of those three charts, for 11 Sample 13 should be excluded and the estimate of λ 0 and the trial control limits should be revised. 12 Samples 10, 11 and 22 should be excluded; the estimate of λ 0 and the trial control limits should be revised. 13 Montgomery (2009, pp , Table 7.7) alerted the reader to the fact that samples 6 and 20 were excluded and trial control limits revised. 12

33 the target values λ 0 = 2, 8, 19; these quantities were computed using functions k lim, exact lim, ryan lim, ARL and delta written by us for the R package. Figure 2.1: c charts with unrevised 3 σ trial control limits for the three original sets of numbers of defects (rectangular steel plates, top; telephone cable, center; printed circuit boards, bottom). 13

34 The ARL curves of these three charts and target values are depicted in Figure 2.2; they were also obtained for the three target values of λ using the R package. Table 2.2: LCL, UCL, supremum of ARL, bias of the ARL function and in-control ARL, for λ 0 = 2, 8, 19 listed in order corresponding to the 3 σ, Ryan & Schwertman and quantile based control limits. λ 0 [LCL, UCL] Supremum of ARL Bias of ARL In-control ARL 2 [0, 6] λ [0, 6] λ [0, 7] λ [0, 16] λ [1, 17] [1, 18] [6, 32] [7, 32] [7, 33] Firstly, λ 0 = 2 is not large enough to guarantee a positive LCL. Hence, all the charts are upper one-sided, therefore less likely to detect any decrease in the process mean than to trigger a false alarm, an extremely undesirable property. Unsurprisingly, the supremum of ARL is in any case and the bias of ARL equal to λ 0. Furthermore, since the quality practitioner should favour control charts with large in-control ARL and associated with a fast detection of out-of-control situations, he/she should: not be tempted to choose the chart with the largest in-control ARL, the one with quantile based control limits, because the out-of-control ARL values are fairly large for increases up to 50% of the target value; choose instead the chart associated with the Ryan & Schwertman control limits or the chart with 3 σ control limits 14 because they are the ones with the in-control ARL value closer to α and reasonable out-of-control ARL values, as shown by Table 2.2 and Figure 2.2; Secondly, when λ 0 = 8, we compare an upper one-sided chart with 3 σ control limits 15 with two charts with positive LCL. In this case we can state that: the adoption of the Ryan & Schwertman and quantile based control limits results in an improved ARL function not only the in-control ARL values are larger, but also the detection of increases and most decreases in λ requires in average a number of samples under the in-control ARL value; the quantile based control limits lead to an ARL function not as biased as the one obtained when the Ryan & Schwertman control limits are used, as portrayed by Table 2.2 and Figure 2.2; even though the chart with 3 σ control limits should not be used if we anticipate decreases in λ, there are increases in the process mean that are detected (in average) more quickly by this chart because it has smaller (yet fairly large) in-control ARL than the two other alternative charts. 14 Since the limits associated with both these charts coincide, their ARL functions are equal and, therefore, the curve for the chart with 3 σ control limits is overlapped in the first plot of Figure Recall that if we deal with 3 σ control limits then λ 0 9 implies LCL = 0 and no sensitivity to downward shifts in λ. 14

35 Figure 2.2: ARL(δ) curves, for 3 σ (curves in black), Ryan & Schwertman (curves in orange) and quantile based control limits (curves in violet) λ 0 = 2 (top), 8 (center), 19 (bottom). 15

36 Thirdly, in case λ 0 = 19, the quantile based control limits lead to a rather large in-control ARL and a practically ARL-unbiased c chart judging by the results from Table 2.2 and the plots at the bottom of Figure 2.2. Since the major purpose of this thesis is to find ARL-unbiased charts this is the chart we should favour, despite the fact that: the chart with quantile based control limits compares unfavorably to the two other charts (resp. chart with Ryan & Schwertman control limits) when it comes to the detection of increases (resp. most decreases) in the process mean; the chart with Ryan & Schwertman control limits is the one with the in-control ARL closer to the desired value of α Finally, the results attest to the inexistence of an optimal pair of control limits for all the target values. For this reason, we must choose the chart with the ARL characteristics consonant with our goals. 2.4 Towards an ARL-unbiased c chart Woodall (2000) alerted to the importance of distinguishing between the use of a control chart on a set of historical data to determine whether or not a process has been in-control (Phase 1) and its prospective use with samples taken sequentially over time to detect changes in a process (Phase 2). This author also noted that, since in Phase 2 the form of the distribution is assumed to be known along with the target values and the process is considered to be in-control if the observed value of the control statistic falls within the control limits (and out-of-control otherwise), there is a close (yet controversial) resemblance between control charting and repeated hypothesis testing over time. The purpose of this section is to capitalize on hypothesis testing theory to achieve some of the goals of this thesis, namely a c chart with a pre-specified in-control ARL and an unbiased ARL function Randomizing the emission of a signal Inspired by the somewhat promising results of the quantile based control limits and by randomized tests (in a discrete distribution setting), we decided to combine the use of quantile based control limits LCL and UCL and randomization probabilities, γ LCL and γ UCL, of triggering a signal when the value of the control statistic is equal to LCL and UCL, in order to bring the in-control ARL exactly to a pre-specified value α 1 and to decrease the bias of the ARL function. With this purpose in mind, we start by considering X Poisson(λ) and a two-sided hypothesis test H 0 : λ = λ 0 vs. H 1 : λ λ 0, (2.14) 16