Guaranteed In-Control Performance for the Shewhart X and X Control Charts

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1 Guaranteed In-Control Performance for the Shewhart X and X Control Charts ROB GOEDHART, MARIT SCHOONHOVEN, and RONALD J. M. M. DOES University of Amsterdam, Plantage Muidergracht, 08 TV Amsterdam, The Netherlands When in-control arameters are unknown, they have to be estimated using a reference samle. Due to the use of di erent reference samles in hase I, the control chart erformance in hase II will vary across ractitioners. This variation is esecially large for small samle sizes. To revent low in-control average run lengths, new corrections for Shewhart control charts are roosed that guarantee a minimum in-control erformance with a secified robability. However, a minimum in-control erformance guarantee generally lowers the out-of-control erformance. To balance the tradeo between in-control and out-ofcontrol erformance, the minimum erformance threshold and secified robability can be adjusted as desired. The corrections are given in a closed form so that the bootstra method, which has recently been suggested, is no longer required. The erformance of our roosed correction is illustrated by simulating some ractical situations. Furthermore, a comarison is made with tolerance intervals and self-starting control charts. Key Words: Average Run Length; Conditional Distribution; Parameter Estimation; Self-Starting Control Charts; Statistical Process Control; Tolerance Intervals.. Introduction THE SHEWHART X- and X-charts are commonly used to monitor the rocess mean. Because the in-control values of the rocess mean and variance are generally unknown, they have to be estimated using a reference samle. However, due to the use of di erent hase-i data, the estimated control limits vary across ractitioners. Consequently, the erformance of the chart in hase II will vary across ractitioners as well. This may result in a low in-control average run length (ARL) or high false alarm rate (FAR) because of an insu cient amount of hase-i data. The ractitioner- Mr. Goedhart is a PhD student at the Deartment of Oerations Management, Faculty of Economics and Business and a consultant at the Institute for Business and Industrial Statistics (IBIS UvA). His address is r.goedhart@uva.nl. Dr. Schoonhoven is an Associate Professor in the Deartment of Oerations Management, Faculty of Economics and Business and a senior consultant at IBIS UvA. Her address is m.schoonhoven@uva.nl. Dr. Does is a Professor and the Head of the Deartment of Oerations Management and the Director of IBIS UvA. He is a Fellow of the ASQ. His address is r.j.m.m.does@uva.nl. to-ractitioner variability has already been ointed out by multile researchers. See, e.g., Quesenberry (993) and Chen (997), who investigated the average ARL when arameters are estimated, and Saleh et al. (05a) who consider this roblem more generally. An extensive overview of the current literature on the e ects of arameter estimation is given in Jensen et al. (006) and Psarakis et al. (04). In order to take the uncertainty of estimation into account, several researchers (e.g., Nedumaran and Pignatiello (00), Albers and Kallenberg (004a, 004b, 005), Tsai et al. (004, 005) and, more recently, Gandy and Kvaløy (03)) have suggested alying corrections to the control-chart limits. These corrections are intended to make sure that the erformance of the control chart satisfies a minimum requirement, which is relevant for ractitioners. Generally, this erformance is measured in terms of the ARL, FAR, or by matching secific ercentiles of the run-length distribution. Related to these measures, Albers and Kallenberg (004a, 004b) introduced the exceedance robability criterion. This criterion aims to correct the control limits such that a certain value of an in-control erformance characteristic (e.g., FAR, ARL) is guaranteed with a secified robability. In other words, the erformance of a con- Vol. 49, No., Aril

2 56 ROB GOEDHART, MARIT SCHOONHOVEN, AND RONALD J. M. M. DOES trol chart is at least equal to a certain value with a secified robability. This exceedance robability criterion is of great interest in current SPC literature and similar aroaches are recommended by Jones and Steiner (0), Gandy and Kvaløy (03), and Saleh et al. (05a, 05b). While Albers and Kallenberg (005) give relatively simle closed-form correction terms for di erent erformance measures, a bootstra aroach to determine the required correction factors is becoming more common (cf., Jones and Steiner (0), Gandy and Kvaløy (03), and Saleh et al. (05a, 05b)). In this study, we do not use the bootstra method but determine an analytical correction term for the Shewhart X- and X-charts under normal theory in a very general setu. Our roosed term is alicable to both one- and two-sided control charts and can make use of di erent estimators for location and sread. Also, the secifications of the correction can be adjusted by secifying the arameters. This gives the freedom to adat the correction to the circumstances as desired by the ractitioner. It turns out that our corrections erform better than the correction terms of Albers and Kallenberg (005). Next to that, because our correction terms are analytical exressions, the comutation time is negligible. Furthermore, we comare our aroach with the construction of two-sided tolerance intervals for normal oulations. This roblem has a long history of academic interest (cf., Wald and Wolfowitz (946), Weissberg and Beatty (960), Gardiner and Hull (966), Howe (969), and Krishnamoorthy and Mathews (009)). In those ublications, various aroximations and ractical guidelines were develoed along with rigorous justification for the guarantee of tolerance robability. Finally, we comare our limits with self-starting control charts introduced by Hawkins (989) and Quesenberry (99), which also guarantee the conditional and unconditional incontrol erformance in an exact way. In all comarisons, we show that our corrections erform much better and/or are more general than the existing ones. In the next section, we resent our model and aroach, which includes some baseline terminology and derivations. In Section 3, we derive our roosed correction terms, illustrate the erformance, and comare the results with the case of uncorrected limits. In Section 4, we address the out-of-control erformance of the roosed method. In Section 5, we illustrate the erformance and consequences of the roosed correction terms, comared with existing methods. Concluding remarks and recommendations are given in Section 6.. Model and Aroach In hase I, we have m samles of size n, so that samle i, for i =,..., m, consists of the observations X i,..., X in. Let X ij be i.i.d. N(µ, )- distributed random variables. In hase II, the average of a subsamle (X m+, X m+, etc) is used to determine whether the rocess is in-control or not. Note that X m+ is distributed as N(µ, /n). We consider a standard Shewhart X control chart with both an uer control limit (UCL) and a lower control limit (LCL), as well as the inviduals Shewhart X chart for the case n =. If the arameters µ and are known, the control limits equal UCL = µ + K n LCL = µ K n, () where is the desired FAR and K = ( /) (henceforth K = K), where (x) denotes the inverse of the standard normal CDF. In ractice, however, arameters are generally unknown and need to be estimated. Using unbiased estimators ˆµ and ˆ for µ and, resectively, estimated control limits UCL d and LCL d equal ducl = ˆµ + K ˆ n dlcl = ˆµ K ˆ n () and are unbiased estimators of UCL and LCL, resectively. The true robability of a false alarm is a function of these estimated control limits and is thus deendent on K, m, n, ˆµ, and ˆ. Given the estimators ˆµ and ˆ, it is ossible to calculate the true FAR conditional on these estimators. This FAR, denoted as P mn (K; ˆµ, ˆ), is equal to P mn (K; ˆµ, ˆ) = P dlcl <Xm+ < UCL d = P X m+ < ˆµ + K ˆ n + P X m+ < ˆµ K ˆ n Journal of Quality Technology Vol. 49, No., Aril 07

3 GUARANTEED IN-CONTROL PERFORMANCE FOR THE SHEWHART X AND X CONTROL CHARTS 57 = ˆµ µ / n + K ˆ + ˆµ µ / n K ˆ. where (x) denotes the standard normal cumulative distribution function (CDF). P mn (K; ˆµ, ˆ) can be rewritten as P mn (K; Z, W ) Z = m + KW + Z m KW, (3) where Z = (ˆµ µ)/( / mn) and W = ˆ/. Note that Z and W are random variables, of which the distributions deend on the estimators ˆµ and ˆ, resectively. Consequently, the unconditional FAR (i.e., before ˆµ and ˆ are obtained) is a random variable as well. Because ractitioners use di erent hase-i samles, their estimations will vary, resulting in a di erent FAR for their control charts. This variation is known as ractitioner-to-ractitioner variability; cf., Saleh et al. (05a). Quesenberry (993) and Chen (997) already illustrated the e ect of estimated limits for X and X control charts by means of the ARL and the standard deviation of the run length. We consider the ractitioner variability by considering the average ARL (AARL), which is the average ARL over all ractitioners, and its variation. Note that, conditional on the FAR, the run length follows a geometric distribution with arameter FAR, such that the ARL equals /FAR. The AARL then equals the unconditional exectation of the ARL. As illustrated by Quesenberry (993), Chen (997), and others, this unconditional exectation of the ARL using estimated arameters generally does not equal the desired value of /. To deal with this, the use of correction terms for the control limits is suggested; cf., Nedumaran and Pignatiello (00), Albers and Kallenberg (004a, 004b), and Tsai et al. (004, 005). If the distribution of ˆµ is symmetric (such as is the case for ˆµ =X= (/m) P m P i= X i= (/mn) m P n i= j= X ij), the corrections for one-sided control charts with either a UCL or an LCL are identical. However, one cannot simly aly these corrections searately to two-sided control charts. For the equivalence of the one-sided corrections, note that the FAR, when using only a UCL, is equal to the first art of Equation (3) and can be written as Z Z m + KW = m KW. (4) If ˆµ follows a symmetric distribution around µ, consequently this also holds for Z around 0. That means that the distribution of Equation (4) is in that case equal to that of (Z/ m KW ), which is the FAR for a control chart using an LCL only. Therefore, both one-sided charts require the same correction. However, alying this one-sided correction to both sides of a two-sided chart is essentially the same as correcting for twice the FAR of one side, which is distributed as Z m KW = Z m + KW + Z m KW.(5) Note that the exression in Equation (5) is quite di erent from the actual two-sided FAR as defined in Equation (3). The intuition behind it is that, in the two-sided case, an underestimation of the mean increases P (X m+ > UCL), d but at the same time decreases P (X m+ < LCL) d (and vice versa for an overestimation of the mean). This e ect is not taken into account when one simly alies one-sided corrections to a two-sided control chart. For this reason, one-sided and two-sided control charts should be treated searately. We consider a correction term c that is added to K. This changes our estimated control limits of Equation () into ducl = ˆµ + (K + c) ˆ n dlcl = ˆµ (K + c) ˆ. (6) n If we denote K = K + c, then our conditional FAR when using the correction term c is calculated by P mn ( K; ˆµ, ˆ). A common choice is to correct the control-chart limits such that the exectation of a chosen erformance measure (e.g., FAR or ARL) is equal to a desired value. This would ensure that the control chart erforms as it should on average over all ractitioners. However, this does not take the variability between ractitioners into account. We suggest a correction term that is based on the exceedance robability criterion, introduced by Albers and Kallenberg (004b) and which is also used in some recent research (e.g., Gandy and Kvaløy (03)). The aroach aims to correct the controlchart limits such that at least a secified minimum in-control erformance is obtained with a secified robability. More secifically, for the FAR, the correction term aims to obtain a FAR that is equal to (+ ) or smaller with a robability of. Here, Vol. 49, No., Aril 07

4 58 ROB GOEDHART, MARIT SCHOONHOVEN, AND RONALD J. M. M. DOES determines our minimum threshold, roortional to the nominal level, while reresents the robability of obtaining a control-chart erformance lower than the secified minimum. In our view, should be small (e.g., 0.05 or 0.0) and may be slightly larger (e.g., 0. to 0.5). This is because reresents the robability for the ractitioners with which the minimum erformance is satisfied, while determines our minimum erformance threshold. Our choices of and are the same as in recent research (e.g., Albers and Kallenberg (005)) and for = 0 (see also Saleh et al. (05b)). Hence, for the FAR, we derive a correction term such that P P mn ( K; Z, W ) < ( + ) =. (7) Note that, for the ARL in the in-control situation, large ARLs are referred. The criterion would then be P P mn ( K; Z, W ) > ( ) =. (8) The required correction term for the ARL can be obtained through solving the term for the FAR while relacing by = /( ). This is because the lefthand side of Equation (8) is equivalent to P P mn ( K; Z, W ) < = P P mn ( K; Z, W ) < + = P P mn ( K; Z, W ) < ( + ). Because of this equivalence, our further derivations are based on the FAR. In order to determine the correction term c, we require information on the distribution of P mn ( K; Z, W ), as we need to find the value c for which the ercentile of the distribution of P mn ( K; Z, W ) equals ( + ). Although the exact distribution of P mn (K; Z, W ) for an arbitrary K is unknown, it is ossible to calculate its moments using integrals, similar to Chen (997). The first and second moment can be calculated as by and E (P mn (K; Z, W )) = Z Z 0 E(P mn(k; Z, W )) = Z Z 0 P mn (K; z, w)f(z)f(w)dwdz (9) P mn(k; z, w)f(z)f(w)dwdz, (0) resectively. Here, f(z) equals the robability density of Z = (ˆµ µ)/( / mn) and f(w) is the robability density of W = ˆ/. The variance of P mn (K; Z, W ) can be calculated through Var(P mn (K; Z, W )) = E(P mn(k; Z, W )) E (P mn (K; Z, W )). () Using these moments, we aroximate the distribution of P mn (K; Z, W ) by a a b /b distribution (for a detailed motivation, we refer to Aendix A). Because E(a b /b) = a and Var(a b /b) = a /b, we have a = E(P mn (K; Z, W )) () and a b = Var(P mn (K; Z, W )) = E (P mn (K; Z, W )) Var(P mn (K; Z, W )). (3) Note that both the exectation and variance deend on K and, consequently, after imlementing the correction term through K = K + c, also on c. 3. Correction Terms 3.. Two-sided Control Limits As mentioned in the revious section, both a and b, as in Equations () and (3), deend on K. This means that changing K to K = K + c changes the a b /b distribution as well. The correction factor c has to be determined such that, for K = K + c, the ( ) th ercentile of this distribution lies at ( + ). If we denote the exectation and variance of P mn (K; Z, W ) in Equations (9) and () by E and V, resectively, and their derivatives with resect to K by de/ and dv /, resectively, then the correction term can be exressed by where c = Y (K) = 3 ( + ) 3E/3 V ( ) Y (K) Y 0, (4) (K) 3E V V + 3E Y 0 (K) = 3 ( + ) E /3 V de V + 3 de V 3E dv V V 3E dv V 3E /3 V dv (5) 3 de V 9E (6) Journal of Quality Technology Vol. 49, No., Aril 07

5 GUARANTEED IN-CONTROL PERFORMANCE FOR THE SHEWHART X AND X CONTROL CHARTS 59 and where with de = = Z Z 0 Z Z dv = d(e ) d(e ) = 0 Z Z = dp mn (K; z, w) f(z)f(w)dwdz ale z w + Kw m + z m Kw f(z)f(w)dwdz (7) E de 0 Z Z (8) dpmn(k; z, w) f(z)f(w)dwdz 0 wp mn (K; z, w) ale + z + Kw m z m Kw f(z)f(w)dwdz. (9) For a detailed derivation of the correction term, we refer to Aendix B. Note that, instead of a minimum erformance threshold that is relative (through ) to (or / ), one could also chose to secify an absolute minimum erformance threshold. This can be done by setting = FAR 0 = /ARL 0 and = 0, where FAR 0 and ARL 0 are the desired threshold values for the FAR or ARL, resectively. However, in order to make a better comarison between the corrected and uncorrected charts, we consider relative thresholds. The same aroach has been used by other authors (e.g., Albers and Kallenberg (005)). We use simulation to evaluate the erformance of the roosed correction term. In order to do this, we need to define our estimators and determine the corresonding distributions f(z) and f(w) of Z = (ˆµ µ)/( / mn) and W = ˆ/, resectively. We consider ˆµ =X as the estimator for location, in which case f(z) equals the standard normal robability density. For the sread, we consider two estimators. The first estimator, which we use in the case that n > (grous), is based on the ooled standard deviation (s ) and is equal to ˆ = s c 4 (m(n ) + ), (0) where c 4 (m(n )+) is such that ˆ is an unbiased estimator of. For this estimator, W = ˆ/ is distributed as / with = /c 4 (m(n )+) and = m(n ). The robability density of / is then f(w;, ) ( /) / w w = ex () ( /) (see also Chen (997)). The second estimator, which is used in the case n = (individuals), is based on the average moving range MR and is equal to ˆ = MR d (), () where d () = /, which yields that ˆ is an unbiased estimator of. Although the exact distribution of MR is not easy to obtain, the distribution of W = ˆ/ can be aroximated by /, where = Var(W ) + = +. (3) Var(W ) See for examle Roes et al. (993) for this aroximation, or Patnaik (950) for a similar one. The variance of W = ˆ/ is investigated by Cryer and Ryan (990) and can be aroximated by ale ˆ 0.864m.08 Var = (m ). (4) Note that it is ossible to use any other estimator for location and sread by alying their corresonding robability functions f(z) and f(w). In Table and Table, several estimators for µ and, resectively, are listed. For the estimators of µ, the (aroximated) robability density function f(z) is tabulated. For the mean, we have an exact distribution and, for the median, the distribution has been aroximated (cf., Johnson and Kotz (970)). For es- TABLE. Estimators for Location, Including the (Aroximated) Probability Density Function f (z) of Z = (ˆµ µ)/( / mn) Estimator Average (X) Median ( X) f(z) ex z ex z Vol. 49, No., Aril 07

6 60 ROB GOEDHART, MARIT SCHOONHOVEN, AND RONALD J. M. M. DOES TABLE. Estimators for Sread, Including the (Aroximated) Probability Density Function f (w) of W = ˆ/ Estimator and var(w ) Individuals (n = ) Average moving range, ˆ = MR(X) d () Interquartile range ˆ = IQR(X).349 Samle standard deviation ˆ = s c 4 (m) = Var(W ) +, = = Var(W ) +, = = /c 4 (m), = m + Var(W ) + Var(W ) 0.864m.08 (m ).46.80m c 4(m) c 4 (m) Grous (n > ) Average samle standard deviation = Var(W ) +, = ˆ = s c 4 (n) + Var(W ) c 4(n) mc 4 (n) Pooled samle standard deviation = c 4 (m(n ) + ), = m(n ) c 4(m(n ) + ) c 4 (m(n ) + ) s ˆ = c 4 (m(n + )) timators of, the distribution of W = ˆ/ is either exact or aroximated by a / distribution, so that f(w) is given by Equation (). The aroximation is similar to that of MR, by determining the corresonding values of and. The required values of and for the considered estimators are listed in Table. We refer to Roes et al. (993) for the aroximations and to Albers and Kallenberg (005) for exlicit exressions of the listed estimators of. As the ARL is the most commonly used erformance measure of control charts, we evaluate the erformance of our roosed correction terms based on that. The corresonding criterion function is given in Equation (8). To calculate the required correction terms, we use the model as described in the revious section, thus relacing by = /( ) in the correction terms for the FAR. We have calculated the correction terms and simulated their erformance for a wide range of arameter values. For each combination of arameter values,000,000 simulation runs are erformed. The relative standard errors of all reorted AARL values in Tables 3 8 are less than %. Table 3 and Table 4 illustrate, for multile combinations of n and m, the correction term and its erformance comared with an uncorrected chart, each for a di erent set of,, and. The erformance is measured by the exceedance robability as in Equation (8) while, for the comarisons, the AARLs are also given. The results illustrate that, after imlementing our suggested correction term, the exceedance robability is very close to the desired level. Esecially for small samle sizes, the di erences between the corrected and the uncorrected chart are large. Note that this is in agreement with the samle size recommendations by, e.g., Quesenberry (993), which states that about 400/(n ) samles of size n are required in hase I for the X chart to behave roerly on average. Furthermore, Saleh et al. (05a) show that far larger amounts of hase-i data are required when also taking the ractitioner-to-ractitioner variability into account. As more data are available in hase I, the required correction becomes smaller. At some oint, the exceedance robability of the uncorrected chart is already below the desired level, meaning that the correction term becomes negative to increase the exceedance robability. The use of a negative correction term in this case is questionable, as it will de- Journal of Quality Technology Vol. 49, No., Aril 07

7 GUARANTEED IN-CONTROL PERFORMANCE FOR THE SHEWHART X AND X CONTROL CHARTS 6 TABLE 3. Correction Terms c for the X-Chart, Including the Corresonding Exceedance Probability and AARL for the Corrected (Cor) and Uncorrected (Unc) Control Limits. Parameter values are = 0.007, = 0.05, and = 0. m n Correction c Exc. Pr. (cor) Exc. Pr. (unc) AARL (cor) AARL (unc) crease the in-control erformance in that situation. However, the out-of-control erformance im roves while keeing the in-control erformance at a desired level. As mentioned earlier, this correction term is alicable for any estimator. Previously, we have shown the results for the X-chart using an estimator for based on the ooled standard deviation (see Equation (0)). In this section, we also show the results for the individuals X-chart, by using an unbiased version of the average moving range as an estimator of, as in Equation (). Similarly as for the X-chart, Table 5 and Table 6 indicate the correction term and its erformance comared with the uncorrected chart for multile values of m (and n = ), each for a di erent set of,, and. Even though we use an aroximation of the density of ˆ, the correction term still erforms well, with the exceedance robabilities close to the desired level. 3.. One-sided Control Limits A fairly straightforward change in the revious derivations leads to the correction term in case we are dealing with either a UCL or an LCL only (onesided). As the correction term for the UCL and LCL is the same, we consider the case of the UCL. The main di erence lies in the change of Equation (3) to z P kn (u; Z, W ) = k + uw, (5) where u is now equal to ( ). Then this formula should be used to calculate the exectation and variance of P kn (u; Z, W ) in Equations (9) and (). The exressions in Equations (7) and (9) will change Vol. 49, No., Aril 07

8 6 ROB GOEDHART, MARIT SCHOONHOVEN, AND RONALD J. M. M. DOES TABLE 4. Correction Terms c for the X-Chart, Including the Corresonding Exceedance Probability and AARL for the Corrected (Cor) and Uncorrected (Unc) Control Limits. Parameter values are = 0.0, = 0., and = 0.4 m n c Exc. r. (cor) Exc. r. (unc) AARL (cor) AARL (unc) as Z de du = = Z 0 Z Z d(e Z ) du = = 0 0 Z Z dp kn (u; z, w) f(z)f(w)dwdz du z w k + uw f(z)f(w)dwdz Z dpkn 0 (u; z, w) f(z)f(w)dwdz du wp kn (u; z, w) (6) z k + uw f(z)f(w)dwdz, (7) which can be used to obtain the corresonding exression in (8): dv /du = d(e )/du E(dE/du). (8) The corresonding correction term c in Equation (4) for the one-sided control limit can be obtained by imlementing the resulting exressions in Equations (5) and (6). 4. Out-of-Control Performance Correcting the control-chart limits in order to guarantee a minimum erformance clearly has an advantage in the in-control situation. However, this inevitably leads to a deterioration of the out-of-control erformance. In the revious section, more secifically in Tables 3 6, the AARLs of the corrected and uncorrected charts are given. It can already be seen that there is a large di erence between the corrected and uncorrected control limits for small samle sizes. As more information becomes available, and as a consequence the correction term becomes smaller, this di erence becomes smaller. For the out- Journal of Quality Technology Vol. 49, No., Aril 07

9 GUARANTEED IN-CONTROL PERFORMANCE FOR THE SHEWHART X AND X CONTROL CHARTS 63 TABLE 5. Correction Terms c for the Individuals X-Chart, with the Corresonding Exceedance Probability and AARL for the Corrected (Cor) and Uncorrected (Unc) Control Limits. Parameter values are = 0.007, = 0.05, and = 0. m c Exc. r. (cor) Exc. r. (unc) AARL (cor) AARL (unc) ,3, , , , , , , of-control situation, this behavior is very similar. To illustrate this, we have simulated the AARL of the corrected and uncorrected X-charts for di erent sizes of shifts. We consider = 0.05, = 0., and = and have again simulated for a wide range of values for m and n. For the out-of-control situation, we consider a shift of X from a N(µ, /n) to a N(µ+ ( / n), /n) distribution, with equal to 0.5,, or. The results are listed in Table 7. We find that, in this situation, for small samle sizes, the di erences in AARLs between the corrected and uncorrected charts are substantial. Note that we have chosen a rather strict set of arameters, as we guarantee an in-control ARL of at least 96 with a robability of 90%. The out-of-control erformance becomes better as,, and increase. More secifically, increasing the value of and/or results in a lower minimum in-control erformance threshold and, consequently, in a lower AARL. On the other hand, a larger value of means that we allow a larger roortion of the in-control ARLs to be below the minimum erformance threshold. This has the consequence that the AARL will be smaller. Thus, increasing any of the arameters,, and/or leads to lower AARL values, which is beneficial for the out-ofcontrol situation, but of course not for the in-control situation. This tradeo between in-control and outof-control erformance is inherent to control charts. The advantage of our roosed method is that the arameters can be easily adjusted in order to balance the erformance of the chart as desired by the ractitioner. In addition, Table 8 illustrates the corrections, exceedance robabilities, and AARL values of the corrected and uncorrected charts for various combinations of m and n, for = 0.0, = 0., and = Comarison with Existing Methods In order to illustrate the erformance of the roosed correction term, a comarison is made with the existing methods. First, we make a comarison with the methods of Albers and Kallenberg (005) TABLE 6. Correction Terms c for the Individuals X-Chart, with the Corresonding Exceedance Probability and AARL for the Corrected (Cor) and Uncorrected (Unc) Control Limits. Parameter values are = 0.0, = 0., and = 0.4 m c Exc. r. (cor) Exc. r. (unc) AARL (cor) AARL (unc) , Vol. 49, No., Aril 07

10 64 ROB GOEDHART, MARIT SCHOONHOVEN, AND RONALD J. M. M. DOES TABLE 7. Out-of-Control Performance of the X-Chart for Shifts in the Mean of Size / n, for Both the Corrected (Cor) and Uncorrected (Unc) Control Limits. Parameter values are = 0.007, = 0.05, and = 0. Size of shift 0.5 m n AARL (cor) AARL (unc) AARL (cor) AARL (unc) AARL (cor) AARL (unc) and Gandy and Kvaløy (03). Next, we comare the roosed X chart with tolerance intervals for a normal distribution because these use an equivalent criterion for n =. Finally, we comare the roosed control chart with the self-starting Q chart of Quesenberry (993). 5.. Comarison of Shewhart X and X Control Charts We consider the two-sided case, with ˆµ =X and ˆ as in Equation (0) for n > and Equation () for n =. For this situation, Albers and Kallenberg (005) roosed control limits of the form ducl AK =X +K(ˆ/ n)( + c AK ) where dlcl AK =X K(ˆ/ n)( + c AK ), (9) c AK = ( ) mn K, (30) where = lim mn! [mnvar(ˆ/(eˆ))]. For the estimators ˆ as in Equations (0) and (), the value of equals n/[(n )] and 0.86, resectively. Note that their roosed correction (c AK ) only deends on the initial hase-i samle through its size (m and n). This is in line with our roosed correction. For the bootstra aroach of Gandy and Kvaløy (03), this is di erent, as the actual correction deends on the samle estimates. Therefore, in the comarison, Journal of Quality Technology Vol. 49, No., Aril 07

11 GUARANTEED IN-CONTROL PERFORMANCE FOR THE SHEWHART X AND X CONTROL CHARTS 65 TABLE 8. Out-of-Control Performance of the X-Chart for Shifts in the Mean of Size / n, for Both the Corrected (Cor) and Uncorrected (Unc) Control Limits. Parameter values are = 0.0, = 0., and = 0.4. Size of shift 0.5 m n AARL (cor) AARL (unc) AARL (cor) AARL (unc) AARL (cor) AARL (unc) only the realized exceedance robabilities are shown. For the exlicit bootstraing rocedure, we refer to Saleh et al. (05b), who rovide a simlification of the comutations in Gandy and Kvaløy s (03) aroach for the Shewhart control chart. We erformed the bootstra rocedure (based on 00 bootstras) for 0,000 simulated hase-i samles, in order to calculate the exceedance robability. The considered arameter values are = 0., = 0, and = The results of the simulations are listed in Table 9 for di erent values of m and n = 5. Results for other values of n are similar. It is clear that our roosed correction erforms much better than the correction of Albers and Kallenberg (005), as it is closer to the desired level of = 0.. The bootstra rocedure of Gandy and Kvaløy (03) also has good erformance. There is no real di erence with our roosed method in the sense of erformance. Also, the erformance of our roosed method and the bootstra method aears to be less sensitive to the value of, as becomes clear when changing its value. This is shown in Table 0, which indicates the erformance of the three methods when imlementing = 0. while leaving the other arameters as before. To illustrate the erformance and consequences of the roosed methods grahically, the distributions of the ARLs of the di erent methods in the in-control ( = 0) and out-of-control situation ( = ) are shown in Figures and, resectively, for m = 50, Vol. 49, No., Aril 07

12 66 ROB GOEDHART, MARIT SCHOONHOVEN, AND RONALD J. M. M. DOES TABLE 9. Exceedance Probabilities of the Proosed Correction, Albers and Kallenberg (005) (AK) Correction and Gandy and Kvaløy (03) (bootstra) Method for Di erent Values of m. Parameter values are n = 5, = 0.007, = 0.. and = 0 m Proosed AK Bootstra TABLE 0. Exceedance Probabilities of the Proosed Correction, Albers and Kallenberg (005) (AK) Correction and Gandy and Kvaløy (03) (bootstra) Method for Di erent Values of m. Parameter values are n = 5, = 0.007, = 0., and = 0. m Proosed AK Bootstra n = 5, = 0., = 0., and = The vertical line reresents the desired threshold of the 00 th (in this case 0th) ercentile of the in-control ARL distribution. The desired threshold with = and = 0. is equal to 96. As can also be seen in Table 0, our roosed correction and the bootstra method erform best, with the 0th ercentile close to the desired level. It is gratifying to note that, with our correction term, no extensive bootstraing is needed for the Shewhart control charts for location. The tradeo between in-control and out-of-control erformance of the control chart also becomes clear from these figures, if we comare the roosed methods with the uncorrected chart. The corrected charts corresond with better in-control erformance, but have a slower detection in the out-of-control situation. 5. Tolerance Intervals The literature of tolerance intervals considers a criterion that is closely related to the roosed corrections in this aer. From Krishnamoorthy and Mathew (009), we cite a tolerance interval is exected to cature a certain roortion or more of the oulation, with a given confidence. The tolerance intervals are based on the samle average (X) and the samle standard deviation (s) and are in the form of X ±k s, where k is determined such that P (P (X k s ale X alex +k s X, s) ) =. (3) Using the tolerance limits as control limits guarantees an in-control FAR (ARL) that is smaller (larger) than (/ ) with robability. Thus, k is equivalent to our K in the case that = 0 and n =, when FIGURE. In-Control ARL Distribution when m = 50, n = 5, = 0.007, = 0., and = 0.. The boxlots indicate the 5th, 0th, 5th, 50th, 75th, 90th, and 95th ercentiles of the distributions. The vertical line reresents the desired threshold level of the ARL (96). Journal of Quality Technology Vol. 49, No., Aril 07

13 GUARANTEED IN-CONTROL PERFORMANCE FOR THE SHEWHART X AND X CONTROL CHARTS 67 FIGURE. Out-of-Control ARL Distribution for = when m = 50, n = 5, = 0.007, = 0., and = 0.. The boxlots indicate the 5th, 0th, 5th, 50th, 75th, 90th, and 95th ercentiles of the distributions. using X and s as hase-i estimators. Only in this case, the objective is exactly identical to our aroach. Note that the tolerance limit aroach can be alied when n >, by treating X as an individual variable. However, this would mean that standard deviation of X should be used to estimate rather than the standard deviation of X. This means that the within-subgrou variation is ignored in the estimation. The erformance of the toleranceinterval aroach is then equal to the erformance when n =. Because our aroach does consider the within-subgrou variation, it has less uncertainty in arameter estimation, leading to less variation in ARLs for n >. Because of the arguments mentioned above, we have comared our roosed corrections with the aroximated tolerance factors as in Krishnamoorthy and Mathew (009) for the case n =. Although they rovide tolerance factors K KM instead of corrections c KM, we can determine their correction by subtracting K from K KM. We consider X as estimator of µ and s as estimator of. Their roosed correction c KM then equals c KM = (n ) [, ] (/n) [n,] [d,q]! / K, (3) where reresents the q-quantile of a chi-squared distribution with d degrees of freedom, and [d,q] () reresents the q-quantile of a noncentral chi-squared distribution with d degrees of freedom and noncentrality arameter. For di erent combinations of and, we have calculated the corrections and their erformance. The results are shown in Table for = and = 0.05 and in Table for = 0.0 and = 0.. As can be seen, there aears to be no significant di erence in erformance because the resulting exceedance robabilities are close to the desired value for both aroximations. Note that, for other estimators of for n =, like the average moving range, the results of Krishnamoorthy and Mathew (009) are not straightforward. Furthermore, note that, for one-sided control charts, for normally distributed data, exact solutions are available for the tolerance bounds Self-Starting Control Charts There are also other control chart designs that lead to a desirable in-control erformance. In articular, for normally distributed data, self-starting control charts by Hawkins (987) and Quesenberry (99) can guarantee a good in-control erformance TABLE. Corrections and Exceedance Probabilities of the Proosed Correction and the Aroximated Tolerance from Krishnamoorthy and Mathew (009) for Di erent Values of m and n = (individuals). Parameter values are = 0.007, = 0.05, and = 0 Exc. r. Exc. r. m c (c) c KM (c KM ) , Vol. 49, No., Aril 07

14 68 ROB GOEDHART, MARIT SCHOONHOVEN, AND RONALD J. M. M. DOES TABLE. Corrections and Exceedance Probabilities of the Proosed Correction and the Aroximated Tolerance from Krishnamoorthy and Mathew (009) for Di erent Values of m and n = (individuals). Parameter values are = 0.0, = 0., and = 0 Exc. r. Exc. r. m c (c) c KM (c KM ) , in the long run as well. However, the major drawback is that, because of the continuous udating of the control chart limits, there is a risk that out-of-control data influence the rocess estimates. A small change in the rocess mean can therefore slowly change the control limits with it, making the out-of-control situations harder to detect. This can result in larger out-of-control ARLs. To illustrate this, we have simulated ARLs for both the self-starting Q chart in Quesenberry (99) and our roosed corrections for both the in-control and out-of-control situations (with = ). As can be observed in Figure 3, the self-starting Q chart indeed has a long out-of-control ARL, much longer than the roosed corrections. Next to that, although the in-control erformance of the Q chart is very stable, our roosed correction yields a much longer in-control ARL, which is highly beneficial for ractitioners. 6. Concluding Remarks and Recommendations To deal with the e ect of arameter estimation, we roose new correction terms for the Shewhart X and X control charts. As model arameters are generally unknown, they have to be estimated using a reference samle. Because di erent ractitioners use di erent samles, their control chart limits and, consequently, their control-chart erformance, will vary. The newly roosed correction terms are in line with the idea introduced by Albers and Kallenberg (004), which corrects the control limits to guarantee a secified minimum erformance of the control chart with a secified robability. The new correction terms are shown to be very accurate in achieving this. The erformance of the roosed method is much better than Albers and Kallenberg (005) and similar to the bootstra method of Gandy and Kvaløy (03). However, no bootstraing is required, as the roosed correction only deends on the initial hase- I samle through its size rather than its arameter estimates. In this aer, also comarisons are made with tolerance intervals and self-starting control charts. The conclusions are that our corrections behave very well for the individuals Shewhart X control chart and outerform the self-starting control chart in both in- and out-of-control situations. Because of the guarantee of minimum erformance, the corrected chart erforms better than an uncorrected chart in the in-control situation. This inevitably leads to a deterioration of the out-of-control FIGURE 3. In-Control ( = 0) and Out-of-Control ( = ) ARL Distribution of the Self-Starting Q Chart in Quesenberry (99) and Our Proosed Correction when m = 50, n = 5, = 0.007, = 0., and = 0. The boxlots indicate the 5th, 0th, 5th, 50th, 75th, 90th, and 95th ercentiles of the distributions. Journal of Quality Technology Vol. 49, No., Aril 07

15 GUARANTEED IN-CONTROL PERFORMANCE FOR THE SHEWHART X AND X CONTROL CHARTS 69 erformance. However, the strictness of the correction can be easily adated by changing,, and as desired. The choice of arameters should be based on the context. As costs of a false alarm and costs of running a rocess out-of-control are very deendent on the alication, it is recommended to take this into account when making the tradeo between in-control and out-of-control erformance. An advantage of the roosed method is that this tradeo can be taken into account when setting u the control limits rather than being disregarded. The resent aer considers the Shewhart X and X control charts under normal theory. In other settings (e.g., the CUSUM and EWMA control charts or autocorrelated rocesses), alternative methods to guarantee a minimum erformance to ractitioners are the subject for further investigation. Aendix A First note that P mn (K; Z, W ) can be written according to Equation (3). This can be rewritten as P mn (K; ˆµ, ˆ) = (K + (K)) + (K + (K)), (A.) with (x) = (x) and (K) = ˆµ µ ˆ / n + K (A.) and (K) = ˆµ µ ˆ / n + K Hence, for any function g( ), we can write. (A.3) g(p mn (K; ˆµ, ˆ)) = h(k + (K), K + (K)) = h(x, y). (A.4) Using a two-ste Taylor exansion, this is aroximately equal to g(p mn ) = h(k + (K), K + (K)) h(k, K) + h x (K, K) + h y (K, K) (K) + [h xx(k, K) (K) (K) + h xy (K, K) (K) (K) + h yy (K, K) (K)], (A.5) where h x (K, K) and h y (K, K) are the first-order artial derivatives of h(x, y) with resect to x and y, resectively, h xx (K, K) and h yy (K, K) are the second-order artial derivatives of h(x, y) with resect to x and y, resectively, and h xy (K, K) equals the cross artial derivative of h(x, y) with resect to x and y. Note that h x (K, K) = h y (K, K) and h xx (K, K) = h yy (K, K). Taking this into account, we can simlify (A.5) into g(p mn ) h(k, K) + h x (K, K) [ (K) + (K)] + h xx(k, K) (K) + (K) + h xy (K, K) (K) (K). (A.6) Using Equations (A.) and (A.3), we can rewrite this into ˆ g(p mn ) h(k, K) + h x (K, K)K + (h xx (K, K) h xy (K, K)) ˆµ µ / n + (h xx (K, K) + h xy (K, K))K ˆ. (A.7) For X ij i.i.d. N(µ, )-distributed random variables and ˆµ =X, we know that ˆµ µ / mn has a chi-squared distribution. Common estimators of the standard deviation, such as the (ooled) samle standard deviation, follow a scaled chi distribution, while even the distribution of the average moving range can be aroximated as such (cf., Roes et al. (993)). This means that ˆ/ and ˆ/ generally follow a scaled chi and scaled chi-squared distribution, resectively. Hence, we may conclude that P mn (K; Z, W ) is aroximately a combination of scaled chi and chi-squared distributed random variables. Note that the distribution of P mn (K; Z, W ) should be aroximated such that it not only gives an accurate descrition but also such that it is ossible to obtain the required correction term. The difficulty in obtaining this correction term lies in the sense that the correction term c not only changes the exectation but also the variance of the distribution. As the currently obtained aroximation is still of rather comlicated form and, because the scaled chi-squared art aears to be dominant, we aroximate the distribution of P mn (K; Z, W ) by a a b /b distribution, where we use the first two central moments of P mn (K; Z, W ) to identify a and b. Aendix B In order to obtain the required correction term, we use the Wilson Hilferty transformation (Wilson Vol. 49, No., Aril 07

16 70 ROB GOEDHART, MARIT SCHOONHOVEN, AND RONALD J. M. M. DOES and Hilferty (93)), which states that, for X b, we have 3 arox X/b N( /(9b), /(9b)). This transformation is quite accurate, which was shown recently by Inglot (00). Henceforth, we abbreviate P mn (K; Z, W ) as P mn. Then, in our case, when arox P mn a b/b or similarly we obtain r 3 Pmn a b a P arox mn This is equivalent to q 3 P mn a + 9b q b, arox N( /(9b), /(9b)). 9b arox N(0, ). We want to have P (P mn < ( + ) ) = Equation (7)). This is equivalent to P (P mn < ( + ) ) 0 q 3 P mn a + 9b = q < 0 q (+ ) a q 9b 9b q 3 (+ ) a q 9b (cf. + 9b A + 9b A =, (B.) which in turn, by using the inverse of the standard normal CDF (denoted ), leads to the following equation that needs to be solved: q 3 (+ ) a + 9b q = ( ). (B.) 9b Note again that both a and b are functions of K. Given the values of m, n,, and, the left-hand side of Equation (B.) is a function of K only, say Y (K). Using Equations () and (3), we can write Y (K) as Y (K) = 3 ( + ) 3E(P mn) /3 Var(Pmn ) 3E(P mn ) Var(Pmn ) + Var(Pmn ) (B.3) 3E(P mn ) In order to solve Equation (B.,) we need to find c such that, for K = K + c, there holds Y ( K) = ( ). This value of c is found by a linear aroximation of Y ( K) as Y ( K) Y (K) + c(dy (K)/). If we denote the derivatives of E(P mn ) and V (P mn ) with resect to K by de/ and dv /, resectively, and E(P mn ) and V (P mn ) by E and V, resectively, we have Y 0 dy (K) (K) = = E /3 V de V 3 de V 3E dv V V 3E V 3E /3 V dv 3 de V dv + 9E. (B.4) The values of de/ and dv / can be calculated (as can be seen from Equations (9), (), and (0)) as Z de = = Z 0 Z Z 0 dp mn (z, w) f(z)f(w)dwdz ale z w + Kw m z + m f(z)f(w)dwdz Kw (B.5) and dv / = d(e )/ E(dE/), (B.6) where d(e Z ) Z = dpmn(k; z, w) f(z)f(w)dwdz = 0 Z Z wp mn (K; z, w) 0 ale z + Kw m z + m Kw f(z)f(w)dwdz. (B.7) Getting back to the aroximation Y ( K) Y (K) + c(dy (K)/) = Y (K) + cy 0 (K), we thus obtain c = [ ( ) Y (K)]/Y 0 (K). (B.8) Acknowledgments The authors are very grateful for the remarks and suggestions made by the referees and the Editor. They have substantially imroved the aer. Journal of Quality Technology Vol. 49, No., Aril 07

17 GUARANTEED IN-CONTROL PERFORMANCE FOR THE SHEWHART X AND X CONTROL CHARTS 7 References Albers, W. and Kallenberg, W. C. M. (004a). Are Estimated Control Charts in Control? Statistics 38(), Albers, W. and Kallenberg, W. C. M. (004b). Estimation in Shewhart Control Charts: E ects and Corrections. Metrika 59, Albers, W. and Kallenberg, W. C. M. (005). New Corrections for Old Control Charts. Quality Engineering 7, Chen, G. (997). The Mean and Standard Deviation of the Run Length Distribution of X Charts when Control Limits Are Estimated. Statistica Sinica 7(3), Cryer, K. D. and Ryan, T. P. (990). The Estimation of Sigma for an X Chart: MR/d or S/c 4? Journal of Quality Technology, Gandy, A. and Kvaløy, J. T. (03). Guaranteed Conditional Performance of Control Charts via Bootstra Methods. Scandinavian Journal of Statistics 40, Gardiner, D. A. and Hull, N. C. (966). An Aroximation to Two-Sided Tolerance Limits for Normal Poulations. Technometrics 8(),. 5. Hawkins, D. M. (987). Self-Starting CUSUM Charts for Location and Scale. The Statistician, Howe, W. G. (969). Two-Sided Tolerance Limits for Normal Poulations Some Imrovements. Journal of the American Statistical Association 64(36), Inglot, T. (00). Inequalities for Quantiles of the Chi- Square Distribution. Probability and Mathematical Statistics 30(), Jensen, W. A.; Jones-Farmer, L. A.; Cham, C. W.; and Woodall, W. H. (006). E ects of Parameter Estimation on Control Chart Proerties: A Literature Review. Journal of Quality Technology 38(4), Johnson, N. L. and Kotz, S. (970). Distributions in Statistics: Continuous Univariate Distributions, I. New York, NY: Wiley. Jones, M. A. and Steiner, S. H. (0). Assessing the E ect of Estimation Error on the Risk-Adjusted CUSUM Chart Performance. International Journal for Quality in Health Care 4, Krishnamoorthy, K. and Mathew, T. (009). Statistical Tolerance Regions: Theory, Alications, and Comutation, I. New York, NY: Wiley. Nedumaran, G. and Pignatiello, J. J., Jr. (00). On Estimating X Control Chart Limits. Journal of Quality Technology 33(),. 06. Patnaik, P. B. (950). The Use of Mean Range as an Estimator in Statistical Tests. Biometrika 37, Psarakis, S.; Vyniou, A. K.; and Castagliola, P. (04). Some Recent Develoments on the E ects of Parameter Estimation on Control Charts. Quality and Reliability Engineering International 30(8), Quesenberry, C. P. (99). SPC Q Charts for Start-U Processes and Short or Long Runs. Journal of Quality Technology 3(3), Quesenberry, C. P. (993). The E ect of Samle Size on Estimated Limits for X and X Control Charts. Journal of Quality Technology 5(4), Roes, K. C. B.; Does, R. J. M. M.; and Schurink, Y. (993). Shewhart-Tye Control Charts for Individual Observations. Journal of Quality Technology 5(3), Saleh, N. A.; Mahmoud, M. A.; Keefe, M. J.; and Woodall, W. H. (05a). The Di culty in Designing Shewhart X and X Control Charts with Estimated Parameters. Journal of Quality Technology 47(), Saleh, N. A.; Mahmoud, M. A.; Jones-Farmer, L. A.; Zwetsloot, I.; and Woodall, W. H. (05b). Another Look at the EWMA Control Chart with Estimated Parameters. Journal of Quality Technology 47(4), Tsai, T. R.; Wu, S. J.; and Lin, H. C. (004). An Alternative Control Chart Aroach Based on Small Number of Subgrous. International Journal of Information and Management Sciences 5(4), Tsai, T. R.; Lin, J. J.; Wu, S. J.; and Lin, H. C. (005). On Estimating Control Limits of X-Bar Chart when the Number of Subgrous Is Small. International Journal of Advanced Manufacturing Technology 6( ), Wald, A. and Wolfowitz, J. (946). Tolerance Limits for a Normal Distribution. The Annals of Mathematical Statistics Weissberg, A. and Beatty, G. H. (960). Tables of Tolerance-Limit Factors for Normal Distributions. Technometrics (4), Wilson, E. B. and Hilferty, M. M. (93). The Distribution of Chi-Squared. Proceedings of the National Academy of Sciences of the United States of America 7(), s Vol. 49, No., Aril 07