The Efficiency of the 4-out-of-5 Runs Rules Scheme for monitoring the Ratio of Population Means of a Bivariate Normal distribution

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1 Proceedings of the 22nd ISSAT International Conference on Reliability and Quality in Design August 4-6, Los Angeles, California, U.S.A. The Efficiency of the 4-out-of-5 Runs Rules Scheme for monitoring the Ratio of Population Means of a Bivariate Normal distribution Kim Phuc Tran (1), Philippe Castagliola (2) Giovanni Celano (3). (1) LUNAM Université, Université de Nantes & IRCCyN UMR CNRS 6597, Nantes, France, kimphuc.tran@univ-nantes.fr. (2) LUNAM Université, Université de Nantes & IRCCyN UMR CNRS 6597, Nantes, France, philippe.castagliola@univ-nantes.fr. (3) Department of Industrial Engineering, Università di Catania, Catania, Italy, gcelano@dii.unict.it. Keywords: Ratio distribution, Markov chain, Run rules. Abstract In order to monitor the ratio of population means of a bivariate normal distribution, Tran et al. [12] have recently suggested a procedure based on Runs Rules, called RRRZ control charts, and showed how their proposed approach efficiently detects small or moderate process shifts. The goal of this paper is to investigate the RRRZ control charts with the 4-out-of-5 Runs Rules by using a Markov chain methodology. We will also compare the performance of these rules using their Expected Average Run Length. 1 Introduction Quality has become one of the most important consumer decision factors in the selection among competing products and services. Statistical Process Control (SPC) is a set of statistical techniques to monitor, control, and improve a process. Among SPC tools, control charts are very useful tools for the detection and the elimination of assignable causes shifting the process. Shewhart control charts have been widely investigated as easy and efficient means to monitor the mean and/or the dispersion of a single quality characteristic. Usually, the sample mean X or the sample median X is used to monitor on-line the mean of the process while, the sample range R, the sample standard deviation S or the sample variance S 2 is used to monitor the process dispersion, see Montgomery [8]. In specific manufacturing environments, we need to monitor the the ratio Z of p = 2 random variables X and Y instead of the (p 1) process mean vector μ and/or the (p p) variance-covariance matrix Σ stability. In the production area, Celano and Castagliola [1] discussed that there are typically two main situations where the ratio of two quantities should be used: when quality practitioners are mainly interested in the relative comparison of the same property for two components in production processes where several components should be blended together to get a product composition; when the ratio is dimensionless and represents the change of a product quality characteristic measured before and after a chemical or a physical reaction. Previously, control charts monitoring the ratio of two random variables has been discussed by Spisak [11] and Davis and Woodall [3], who investigated an example coming from an unemployment insurance quality control program. Öksoy et al. [10] proposed a set of guidelines for the implementation of Shewhart control charts to perform on-line monitoring in the glass industry. Very recently, Celano et al. [2] discussed the statistical properties of a Shewhart chart for individual measurements, the Shewhart-RZ control chart (denoted as SHRZ ), monitoring the ratio of two normal variables. Celano and Castagliola [1] extended this work by assuming subgroups consisting of n > 1 sample units. It is well known that the Shewhart type control charts are rather slow in the detection of small or moderate process shifts. The statistical sensitivity of a Shewhart control chart can be improved by using supplementary Run Rules. For this reason, Tran et al. [12] introduced one-sided Run Rules Shewhart-type control charts monitoring the Ratio of Population means of a bivariate normal distribution (denoted as RRRZ control charts, that is Runs Rules Ratio Z control charts). The goal of this paper is to propose the RRRZ control charts with the 4-out-of-5 Runs Rules. We will also compare the performance of these rules with the performance of the SHRZ control chart using their Expected Average Run Length. 2 The ratio Z distribution Let us consider the case of two normal random variables X and Y such as W =(X,Y ) T N(μ W,Σ W ) i.e W is a bivariate normal random vector with mean vector μx μ W =, (1) μ Y and variance-covariance matrix σ 2 Σ W = X ρσ X σ Y ρσ X σ Y σy 2, (2) where ρ is the correlation coefficient between X and Y. The coefficients of variation of the two random variables X and Y are denoted as γ X = σ X μ X and γ Y = σ Y μ Y, respectively, and their standard-deviation ratio as ω = σ X σ Y. We denote as Z = X Y the page 148

2 ratio of X to Y. Very rencently, Celano and Castagliola [1] approximated the c.d.f. F Z (z γ X,γ Y,ω,ρ), the p.d.f. (probability density function) f Z (z γ X,γ Y,ω,ρ) and the i.d.f. (inverse distribution function) FZ 1 (p γ X,γ Y,ω,ρ) of Z by following an approach similar to the one suggested by Geary [4], Hayya et al. [5]. In this case, an approximation for F Z (z γ X,γ Y,ω,ρ) can be written A F Z (z γ X,γ Y,ω,ρ) Φ, (3) B where Φ(.) is the c.d.f. of the standard normal distribution and where A and B are functions of z, γ X, γ Y, ω and ρ. Inasimilar way, Celano and Castagliola [1] presented an approximation for the i.d.f. FZ 1 (p γ X,γ Y,ω,ρ) of Z such as C 2 C FZ C 1C 3 (p γ 2C X,γ Y,ω,ρ) 1 if p (0,0.5], C 2 C2 24C 1C 3 2C 1 if p [0.5,1), (4) where C 1, C 2 and C 3 are functions of p, γ X, γ Y, ω and ρ. 3 The RRRZ control charts Let us suppose that the two random variables X and Y of interest are correlated with an in-control coefficient of correlation ρ 0. For the quality characteristic Z = Y X, we collect a sample of n independent couples {W i,1,w i,2,...,w i,n } at each sampling period i = 1,2,..., where each W i, j =(X i, j,y i, j ) T N(μ W,i,Σ W,i ), j = 1,...,n, is a bivariate normal random vector with mean vector. μx,i μ W,i =, (5) μ Y,i and variance-covariance matrix ( σx,i 2 Σ W,i = ρ 0 σ X,i σ Y,i ρ 0 σ X,i σ Y,i σ 2 Y,i ). (6) From one subgroup to another, the values of μ W,i and Σ W,i may change. We assume that each independent couples {W i,1,w i,2,...,w i,n } has a linear relationship σ X,i = γ X μ X,i and σ Y,i = γ Y μ Y,i where γ X and γ Y are the assumed known and constant coefficients of variation. In this paper, we suggest to monitor is Ẑ i = ˆμ X,i = X i = n j=1 X i, j ˆμ Y,i Ȳ i n j=1 Y,i = 1,2,... (7) i, j At each inspection i, the standard-deviations ratio ω i = σ X,i σ Y,i is equal to ω i = μ X,i γ X, i = 1,2,... (8) μ Y,i γ Y When the process runs in-control we assume the ratio μ X,i μ Y,i = z 0, i = 1,2,...,wherez 0 is its in-control value. Therefore, the standard-deviation ratio is ω i = ω 0 = z 0 γ X γ Y, i = 1,2,... (9) where ω 0 is the in-control standard-deviation ratio. As a consequence, the c.d.f. FẐi (z n,γ X,γ Y,z 0,ρ 0 ) and the i.d.f. F 1 (p n,γ Ẑ X,γ Y,z 0,ρ 0 ) of Ẑ i can be deduced from the c.d.f. i F Z (z γ X,γ Y,ω,ρ) in (3) and the i.d.f. F 1 Z (p γ X,γ Y,ω,ρ) in (4), seemoreintranetal.[12]. We suggest to define the Run Rules strategies to monitor Ẑ i a lower-sided r-out-of-s Run Rules control chart (denoted as RRRZ r,s) aiming at detecting a decrease in Ẑ i with a single Lower Control Limit LCL RRRZ (i.e. the Upper Control Limit UCL RRRZ = ). an upper-sided r-out-of-s Run Rules control chart (denoted as RRRZ r,s) aiming at detecting an increase in Ẑ i with a single Upper Control Limit UCL RRRZ (i.e. the Lower Control Limit LCL RRRZ = 0). It is important to note that, like in Klein [6], we only focus on pure Run Rules type charts which only require a single couple of limits (LCL RRRZ,UCL RRRZ ) and assume that an out-of-control condition must be signaled only if the selected Run Rules pattern occurs. In this paper, we will only focus on the 2-out-of-3 and 3-out-of-4 RRRZ control charts in Tran et al. [12]. We also investigate the new 4-out-of-5 RRRZ control charts which correspond to the case (r = 4,s = 5). Concerning the RRRZ2,3 and RRRZ 2,3 control charts, the sequence of points plotted on these charts can be modelled as a stochastic process. The statistical properties of these control charts can be obtained by using the following Markov Chain matrix P, where the 4th state corresponds to the absorbing state: P = Q r = 0 T p 1 p p p 0 1 p p , (10) where P(Ẑ i LCL RRRZ ) (P(Ẑ i UCL RRRZ )) for the RRRZ2,3 (RRRZ 2,3 )chart,0 =(0,0,...,0)T, Q is the (3,3) matrix of transient probabilities, the (3, 1) vector r satisfies r = 1 Q1 (i.e. row probabilities must sum to 1), with 1 =(1,1,1) T. The corresponding (3,1) vector q of initial probabilities associated with the transient states is equal to q =(0,0,1) T (i.e. the initial state is the 3rd one). We assume that the occurrence of an out-of-control condition shifts the in-control ratio z 0 to z 1 = τ z 0,whereτ > 0 is the shift size. Values of τ (0,1) correspond to a decrease of the nominal ratio z 0, while values of τ > 1 correspond to an increase of the nominal ratio z 0. We also consider that when the process shifts to the out-of-control condition the coefficient of correlation can shift from ρ = ρ 0 to ρ = ρ 1. Therefore, the probability p in (10) is equal to For the RRRZ 2,3 chart: p = 1 FẐi (LCL RRRZ n,γ X,γ Y,z 1,ρ 1 ) (11) For the RRRZ 2,3 chart: p = FẐi (UCL RRRZ n,γ X,γ Y,z 1,ρ 1 ). (12) page 149

3 The proposed Markov chain model for RRRZ control charts can be extended to longer Run Rules like, for instance, the 3-out-of-4 Run Rules and the 4-out-of-5 Run Rules but the matrix Q for them will not be presented here. For the RRRZ3,4 and RRRZ 3,4 charts size of matrix Q is (7 7) and vector q = (0,0,0,0,0,0,1) T (i.e. the initial state is the 7th one). Concerning the the RRRZ4,5 and RRRZ 4,5 charts, the size of matrix Q is (15 15) and vector q =(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1) T (i.e. the initial state is the 15th one). Since the number of steps RL until the process reaches the absorbing state is a Discrete PHase-type (or DPH) random variable of parameters (Q,q), (see [9]or[7]), the mean (ARL or Average Run Length) and the standard-deviation (SDRL or Standard Deviation Run Length) of the run length RL of the one-sided RRRZ 2,3, RRRZ 3,4 and RRRZ 4,5 charts are equal to with 4 Numerical analysis ARL = ν 1, (13) SDRL = μ 2, (14) ν 1 = q T (I Q) 1 1, (15) ν 2 = 2q T (I Q) 2 Q1, (16) μ 2 = ν 2 ν 2 1 ν 1. (17) In this paper, the investigated statistical measure of performance is the zero-state ARL, defined as the average number of samples before a control chart signals an out-of-control condition. As for the RRRZ control charts in Tran et al. [12], the onesided control limits LCL RRRZ and UCL RRRZ of the RRRZ r,s and RRRZ r,s control charts can be rewritten as LCL RRRZ = K L z 0 and UCL RRRZ = K U z 0 where K L and K U are constants that depend on ARL 0, n, γ X, γ Y, ρ 0 and z 0 = 1. These constants K L and K U have been obtained numerically for ARL 0 = 200, n {1,5,15}, γ X {0.01,0.2}, γ Y {0.01,0.2} and ρ 0 {0.8,0.4,0,0.4,0.8} and are displayed in Table 1. γ X γ Y ρ 0 n = 1 n = 5 n = (0.9852,1.0151) (0.9933,1.0067) (0.9961,1.0039) (0.9869,1.0133) (0.9941,1.0059) (0.9966,1.0034) (0.9889,1.0112) (0.9950,1.0050) (0.9971,1.0029) (0.9914,1.0087) (0.9961,1.0039) (0.9978,1.0022) (0.9950,1.0050) (0.9978,1.0022) (0.9987,1.0013) (0.7396,1.3521) (0.8746,1.1433) (0.9257,1.0803) (0.7662,1.3051) (0.8886,1.1254) (0.9341,1.0705) (0.7983,1.2526) (0.9049,1.1050) (0.9440,1.0593) (0.8398,1.1908) (0.9255,1.0805) (0.9564,1.0456) (0.9040,1.1062) (0.9563,1.0457) (0.9746,1.0261) (0.8583,1.1946) (0.9315,1.0789) (0.9593,1.0441) (0.8610,1.1910) (0.9328,1.0774) (0.9601,1.0433) (0.8637,1.1873) (0.9341,1.0759) (0.9608,1.0425) (0.8664,1.1835) (0.9354,1.0744) (0.9616,1.0416) (0.8692,1.1797) (0.9368,1.0728) (0.9624,1.0407) (0.8371,1.1650) (0.9269,1.0735) (0.9577,1.0424) (0.8396,1.1615) (0.9282,1.0721) (0.9585,1.0416) (0.8423,1.1579) (0.9294,1.0706) (0.9593,1.0407) (0.8449,1.1542) (0.9308,1.0691) (0.9601,1.0399) (0.8477,1.1504) (0.9321,1.0675) (0.9609,1.0390) Table 1: Values of (K L,K U ) for the RRRZ 4,5, for z 0 = 1, γ X {0.01,0.2}, γ Y {0.01,0.2}, ρ 0 {0.8,0.4,0,0.4,0.8}, n {1,5,15} and ARL 0 = 200 The conclusions for the values of (K L,K U ) of RRRZ 4,5 shown in Table 1 as follows Given (γ X,γ Y ), the values of K L and K U change with n and ρ 0. In particular K L increases and K U decreases as n and ρ 0 increase. For example, when (γ X,γ Y )= (0.01,0.01), wehavek L = and K U = if n = 1andρ 0 = 0.4 and we have K L = and K U = if n = 15 and ρ 0 = 0.4. Given n and ρ 0, the values of K L and K U change with (γ X,γ Y ). In particular, K L decreases and K U increases as (γ X,γ Y ) increase. For example, when n = 5andρ 0 = 0.4, we have K L = and K U = when (γ X,γ Y )= (0.01,0.01) and we have K L = and K U = when (γ X,γ Y )=(0.2,0.2). (γ X = 0.01,γ Y = 0.01) (γ X = 0.2,γ Y = 0.2) τ n = 1 n = 5 n = 15 n = 1 n = 5 n = Table 2: ARL of the RRRZ 4,5 charts for γ X {0.01,0.2}, γ Y {0.01,0.2}, γ X = γ Y, ρ 0 {0.8,0.4,0,0.4,0.8}, ρ 0 = ρ 1, τ {0.90,0.95,0.98,0.99,1.01,1.02,1.05,1.10}, n {1,5,15} and ARL 0 = 200 For the sake of brevity, Tables 2 only presents the out-of-control ARL 1 values of the RRRZ 4,5 control charts when the process shifts from the in-control to the out-of-control condition without a change in the correlation between X and Y (ρ 0 = ρ 1 = ρ) page 150

4 and γ X = γ Y. Similar tables for the other investigated scenarios are not presented here but are available upon request from authors. Discussion of the results shown in Tables 2 can be summarized as follows, (we also give some comments about the SDRL values, which are not shown in the Tables 2): The performance of the one-sided RRRZ 4,5 control charts is strongly influenced by (γ X,γ Y )andρ 0. The smaller the coefficients of variation (γ X,γ Y ), the faster the control charts in the detection of the out-of-control condition, because the interval [K L,K U ] is tighter. For example, when ρ 0 = 0.4, n = 15 and τ = 0.99, we get ARL 1 = 4.0 and SDRL 1 = 0.1 fortherrrz4,5 chart if (γ X,γ Y )= (0.01,0.01);wegetARL 1 = 93.4 andsdrl 1 = 89.9 for the RRRZ4,5 chart if (γ X,γ Y )=(0.2,0.2). Occurrence of negative correlation (ρ 0 < 0), between the random variables X and Y reduces the chart sensitivity vs. positive correlation, (ρ 0 > 0). For instance, if (γ X,γ Y )= (0.2,0.2), τ = 0.99, n = 15 and ρ 0 = 0.4 wehave ARL 1 = 119.9andSDRL 1 = 116.3fortheRRRZ4,5 chart, compared to ARL 1 = 93.4andSDRL 1 = 89.9ifρ 0 = 0.4. When (γ X,γ Y )=(0.01,0.01) and n increases (say n 5) the RRRZ 4,5 charts are very sensitive to a process shift, i.e. we have ARL 1 4andSDRL 1 0, that is the lowest possible for the RRRZ 4,5 charts when τ < 0.99 or τ > 1.01, see for instance Table 2 for n = 15. The statistical performance of the RRRZ 4,5 charts is not identical for the same absolute value of the ratio percentage variation Δ Z = 100 τ 1. The performance difference is a function of (γ X,γ Y ). For example, when (γ X,γ Y )=(0.01,0.01), n = 1, ρ 0 = 0.4 and τ = 0.98(1.02), i.e. Δ Z = 2% we have ARL 1 = 4.9 and SDRL 1 = 1.6(ARL 1 = 5.0 andsdrl 1 = 1.7) for the RRRZ4,5 (RRRZ 4,5 ) chart, see Table 2. In practice, the size of a shift to the out-of-control are are unknown. Similar to Celano and Castagliola [1], when a quality practitioner has an interest to detect a range of shifts Ω =[a,b], but no preference for any particular size of the process shift, then a uniform distribution for the shift size τ can be considered over Ω and the Expected Average Run Length (EARL) value can be computed by the expected value of the ARL when integrated over the distribution function f τ (τ): EARL = ARL f τ (τ)dτ, (18) Ω where ARL is defined as in (13)and f τ (τ)= 1 ba for τ Ω. The values of EARL for the RRRZ 4,5 are not presented here and are available upon request from authors. In order to have an overall evaluation of RRRZ and SHRZ charts, Table 3 summarizes which chart has the smaller EARL values for Ω =[0.9, 1), for ρ 0 = ρ 1 and ρ 0 ρ 1. (γ X = 0.01,γ Y = 0.01) (γ X = 0.2,γ Y = 0.2) Ω n = 1 n = 5 n = 15 n = 1 n = 5 n = 15 (D) RRRZ2,3 RRRZ2,3 SHRZ RRRZ4,5 RRRZ4,5 RRRZ4,5 (I) RRRZ 2,3 RRRZ 2,3 SHRZ RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 (D) RRRZ2,3 SHRZ SHRZ RRRZ4,5 RRRZ4,5 RRRZ4,5 (I) RRRZ 2,3 RRRZ 2,3 SHRZ RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 (D) RRRZ2,3 SHRZ SHRZ RRRZ4,5 RRRZ4,5 RRRZ4,5 (I) RRRZ 2,3 SHRZ SHRZ RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 (D) RRRZ2,3 SHRZ SHRZ RRRZ4,5 RRRZ4,5 RRRZ4,5 (I) RRRZ 2,3 SHRZ SHRZ RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 (D) SHRZ SHRZ SHRZ RRRZ4,5 RRRZ4,5 RRRZ4,5 (I) SHRZ SHRZ SHRZ RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 (γ X = 0.01,γ Y = 0.2) (γ X = 0.2,γ Y = 0.01) Ω n = 1 n = 5 n = 15 n = 1 n = 5 n = 15 (D) RRRZ4,5 RRRZ4,5 RRRZ4,5 RRRZ4,5 RRRZ4,5 RRRZ4,5 (I) RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 (D) RRRZ3,4 RRRZ4,5 RRRZ4,5 RRRZ4,5 RRRZ4,5 RRRZ4,5 (I) RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 (D) RRRZ3,4 RRRZ4,5 RRRZ4,5 RRRZ4,5 RRRZ4,5 RRRZ4,5 (I) RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 (D) RRRZ3,4 RRRZ4,5 RRRZ4,5 RRRZ4,5 RRRZ4,5 RRRZ4,5 (I) RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 (D) RRRZ3,4 RRRZ4,5 RRRZ4,5 RRRZ4,5 RRRZ4,5 RRRZ4,5 (I) RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 RRRZ 4,5 Table 3: Best control charts for γ X {0.01,0.2}, γ Y {0.01,0.2}, ρ 0 {0.8,0.4,0,0.4,0.8}, ρ 0 = ρ 1, n {1,5,15}, ARL 0 = 200, Ω =[0.9,1), i.e. (D)ecreasing case and Ω =[1,1.1) i.e. (I)ncreasing case. A comparison with the ARL 1 values for the RRRZ 2,3, and RRRZ 3,4 control charts obtained in Tran et al. [12] andthe SHRZ control charts obtained in Celano and Castagliola [1] shows that, when γ X = γ Y = 0.01 and ρ 0 = ρ 1, the best charts are SHRZ and RRRZ 2,3 ; for ρ 0 = ρ 1,whenγ X = γ Y = 0.2 or γ X γ Y the best charts are the RRRZ 4,5. As expected, in most cases, the RRRZ the control charts, especially, the control charts with the 4-out-of-5 Runs Rules are more sensitive than the SHRZ chart to small shifts of the nominal ratio z 0. 5 Conclusions The statistical properties of control charts monitoring the ratio of population means of a bivariate normal distribution using Run Rules have been studied when subgroups of n > 1sample units are collected. To meet the production mix demand and allow for more flexibility of implementation, the size of each sample unit in the process can change from one subgroup to another.a Markov chain methodology coupled to an efficient normal approximation of the distribution of Z has been used to evaluate the statistical performance of the RRRZ control charts for different values of the sample size n. We proposed the RRRZ control charts with the 4-out-of-5 Runs Rules. A comparison with the SHRZ chart highlighted the following recommendations: when γ X = γ Y = 0.01 and ρ 0 = ρ 1, the best charts are SHRZ and RRRZ 2,3 ;forρ 0 = ρ 1,whenγ X = γ Y = 0.2orγ X γ Y the best charts are RRRZ 4,5. Finally, possible enhancements and page 151

5 future work about control charts monitoring the ratio of random normal variables include the investigation of their Phase I implementation. References [1] G. Celano and P. Castagliola. Design of a phase II Control Chart for Monitoring the Ratio of two Normal Variables. Quality and Reliability Engineering International, 32(1): , [2] G. Celano, P. Castagliola, A. Faraz, and S. Fichera. Statistical Performance of a Control Chart for Individual Observations Monitoring the Ratio of two Normal Variables. Quality and Reliability Engineering International, 30(8): , [3] R.B. Davis and W.H. Woodall. Evaluation of Control Charts for Ratios. In 22nd Annual Pittsburgh Conference on Modeling and Simulation, [4] R.C. Geary. The Frequency Distribution of the Quotient of Two Normal Variates. Journal of the Royal Statistical Society, 93(3): , [5] J. Hayya, D. Armstrong, and N. Gressis. A Note on the Ratio of Two Normally Distributed Variables. Management Science, 21 (11): , [6] M. Klein. Two Alternatives to the Shewhart X Control Chart. Journal of Quality Technology, 32: , [7] G. Latouche and V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modelling. ASA-SIAM, [8] D.C. Montgomery. Statistical quality control: a modern introduction. Wiley, New York, [9] M.F. Neuts. Matrix-Geometric Solutions in Stochastic Models: an Algorithmic Approach. Dover Publications Inc, [10] D. Öksoy, E. Boulos, and L.D. Pye. Statistical Process Control by the Quotient of two Correlated Normal Variables. Quality Engineering, 6(2): , [11] A.W. Spisak. A Control Chart for Ratios. Journal of Quality Technology, 22(1):34 37, [12] K. P. Tran, P. Castagliola, and G. Celano. Monitoring the Ratio of two Normal Variables using Run Rules Type Control Charts. International Journal of Production Research, 54(6): , page 152