# Risk-Adjusted Control Charts for Health Care Monitoring

Authors: Willem Albers
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

Attribute data from high-quality processes can be monitored effectively by deciding on whether or not to stop at each time wherer≥1 failures have occurred. The smaller the degree of change in failure rate during out of control one wants to be optimally protected against, the larger the r should be. Under homogeneity,
the distribution involved is negative binomial. However, in health care monitoring, (groups of) patients will often belong to different risk categories. In the present paper, we will show how information about category membership can be used to adjust the basic negative binomial charts to the actual risk incurred. Attention is also devoted to comparing such conditional charts to their unconditional counterparts. The latter do take possible heterogeneity into account but refrain from risk-adjustment. Note that in the risk adjusted case several parameters are involved, which will all be typically unknown. Hence, the potentially considerable estimation effects of the new charts will be investigated as well.


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## Body

## 1. Introduction

We are interested in processes which exhibit only a (very) small proportion of defectives. Due to ever increasing efforts and standards, such high-quality processes become more and more common in industrial setups. Moreover, for the quite different field of health care monitoring, they are in fact the rule: errors, such as malfunctioning equipment, fatally delayed help, or surgical failures, should occur only (very) rarely. Now review papers on health care monitoring (see, e.g., [1–3] and Sonesson and Bock [4]) strongly suggest to apply SPC methods, in particular control charts, and we shall follow that line here.The common starting point for monitoring such attribute data is to watch the number of failures in a series of given sampling intervals. However, for high-quality processes, thisp-chart may not be the best choice. A first improvement is to switch to a “time-between-events” or “geometric” chart, which uses the number of successes between failures to judge whether the process has remained in control (IC). See, for example, Liu et al. [5], Yang et al. [6], Xie et al. [7], Ohta et al. [8] and Wu et al. [9], for details. When the process goes out of control (OoC), such geometric charts quickly react to substantial increases of the failure rate p from IC but are admittedly rather slow in detecting moderate changes. Especially in health care applications, this is undesirable, and several of the authors mentioned above have suggested a second improvement step.Here a decision whether or not to stop is no longer made after each failure but instead only afterr>1 failures have occurred. Typically, the smaller the increase of failure rate during OoC one wants to have optimal detection power against, the larger the r should be. This negative binomial chart is analyzed in some detail in Albers [10]. In particular, a simple rule of thumb is presented for selecting r, and the resulting chart is both easy to understand and to apply. However, as subsequently pointed out in Albers [11], a serious complication arises if the underlying homogeneity assumption is not warranted. In industrial applications, it may often—but by no means always—be reasonable to indeed assume one and the same IC failure probability p for each item inspected. But, in medical settings, patients tend to exhibit quite a bit of heterogeneity, and we will regularly have to take such variation between subjects into account.In Albers [11] the basic situation is considered where in fact all we know is that such heterogeneity does occur. It can, for example, stem from the occurrence of different groups, each with its own IC probability of failure, but we lack further information. The only way, in which it becomes visible, is through an increase of variance, as compared to the homogeneous case. For a discussion of this overdispersion phenomenon, see, for example, Poortema [12] for a general review and Christensen et al. [13] and Fang [14] for earlier applications concerning attribute control charts. In Albers [11] it is demonstrated how the negative binomial charts can be generalized to cover the present overdispersion situation. Essentially the ill-fitting single parameter homogeneous model is widened there into a two-parameter model. In addition to the failure rate p, a second parameter is added, in order to capture the degree of overdispersion. In view of the lack of knowledge about the precise underlying mechanism of the process, this wider family still remains an approximation of reality. But, as demonstrated in Albers [11], the results under overdispersion are far better than those provided by the homogeneous approach.As was already pointed out in Albers [11], quite a different situation occurs when we do have information about the underlying structure. For example, suppose a number of risk categories can be distinguished, each with its own pj during IC, and for each incoming patient we register to which class he/she belongs. First of all, such detailed knowledge about the process in principle allows a more accurate analysis. But probably even more important is the fact that it opens the way to applying so-called risk adjustment methods (see, e.g., [15, 16] for an overview, and [17] for a risk-adjusted version of the sets method introduced by Chen [18]). Here the baseline risk of each patient is taken into account in deciding whether the process is still IC. If, for example, a surgeon's performance decreases over time, an OoC signal may nevertheless not be justified if we observe that meanwhile his/her patients are gradually shifting to higher risk categories.Clearly this is an interesting area, giving rise to quite a few questions, both from a practical and a technical point of view. For example, in practice, one can wonder under what circumstances one should adjust the risk and when one should ignore this possibility and stick to a rigid overall chart. A more technical issue is the following. In risk adjustment several parameters are involved (cf. thepj above), which typically are unknown and need to be estimated. Now estimation effects for control charts tend to be conveniently ignored in practice. But if these are studied, they typically turn out to be substantial; for example, Chen et al. [19], mention a 30–90% increase in false alarm rate for a 10% bias in the estimator for p. This estimation topic was studied more systematically in Albers and Kallenberg [20, 21]. There it has been amply demonstrated that the small probabilities involved, such as p, invariably produce large relative errors, and; thus, corrections are called for. In the present situation, we have not one, but several parameters, and hence the effect is likely to be even more serious. But, as Woodall [1] remarks, little work has been done on the effect of the estimation error on the performance of risk-adjusted charts. Consequently, the purpose of the present paper is to remedy this by studying how the negative binomial charts from the simple homogeneous case can be adapted to the situation where risk adjustment is desirable.As concerns the relation of the methodology proposed here to the existing methods as described in, for example, Steiner et al. [15] and Grigg and Farewell [16, 17], the following remarks are in order. The latter category are of CUSUM-type and as such may be slightly more efficient. In passing note that this actually is a rather subtle matter, as it also seems to depend on the type of performance criterion used (e.g., steady state or not). Nevertheless, using the negative binomial type of approach implies some aggregation of the data over time, which could indeed mean some loss of information compared to the stepwise CUSUM approach. However, precisely this aggregation effect makes the resulting structure less complicated, thus, allowing a detailed analysis of estimation effects, as well as corrections of these. For the CUSUM case (so far), this seems intractable. Hence, the issue in comparing the two types of approach actually is robustness. On the one hand, we have procedures which (maybe) are superior if optimal conditions hold. In the present case that means known parameters, which is almost never realistic. In addition, the impact of just plugging in estimates for these parameters is known to be huge (e.g., causing an average run length during IC which is systematically substantially lower than prescribed). Under such circumstances, it seems an attractive alternative to pay a small insurance premium (in the form of a small loss in detection power) in order to obtain robust procedures which allow control of the validity through adequate corrections for the charts. Incidentally, note that this robustness issue is by no means typical for the application at hand, but is of a rather general nature.The paper is organized as follows. As far as possible, the technicalities involved are relegated to the appendix section, while the body of the paper provides the main ideas. Section2 is devoted to introducing briefly the negative binomial chart from Albers [10], which forms our starting point. In Section 3 these charts are subsequently adapted to situations where risk adjustment is called for. The estimation aspect will be the subject of Section 4. For illustrative purposes, throughout the paper examples are presented. Moreover, at the end of the paper, we summarize the application of the proposed chart in a simple set of steps.


## 2. The Homogeneous Case

Here we briefly introduce the homogeneous case (see [10] for a detailed description). The process being monitored can be characterized through a sequence of independent identically distributed (i.i.d.) random variables (r.v.'s)D1,D2,…, with P(D1=1)=1-P(D1=0)=p in the IC stage. During OoC, p becomes θp for some θ>1, and we should stop as quickly as possible. As discussed in the Section 1, the option to stop arises each time r failures have been assembled, for some r≥1. Let Xi,i=1,2,…, be the successive numbers of Ds involved, then these Xi are i.i.d. as well, and moreover, distributed as a negative binomial r.v. Xr,p=X (indices will be suppressed unless confusion might arise); that is, for k=r,r+1,…, we have(2.1)P(Xr,p=k)=(k-1r-1)pr(1-p)k-r.
A stopping signal should occur the first time an Xi≤n, with the lower limit n=nr,p selected such that the false alarm rate (FAR) Fr,p(n)=P(Xr,p≤n) equals rα, for some small α>0. Then the average run length (ARL) in terms of number of failures during IC equals r/(rα)=1/α for all r, thus aligning the various negative binomial charts for r≥1 in a proper way. Consequently, n=Fr,p-1(rα), the rαth quantile of Fr,p, which can be obtained numerically.For the purpose of analyzing the behavior of the lower limitn, a simple and transparent approximation is most useful. This can in its turn also be used for finding an approximation for the ARL during OoC; that is, when p has turned into θp. As this ARL is a function of α,θ, and r, it is in particular interesting to figure out which choice of r is the best for given α and θ. This task has been carried out in Albers [10], to which we refer for a complete description. However, to facilitate independent reading, we do present some of the details in the Appendix section. At this point we just mention that n≈λ/p, where λ solves P(Zλ≥r)=rα, in which the r.v. Zλ is Poisson’s with parameter λ. Moreover, by considering a table of numerically obtained optimal values of r (i.e., resulting in the lowest ARL) for the various α and θ of interest and fitting to these r′s a simple approximation in terms of α, θ, and αθ, the following easy rule of thumb has been obtained:(2.2)r̃opt=1α(2.6θ+2)+0.01(4θ-3).
As can be seen from Table  3 in Albers [10], this simple rule works remarkably well. Quite often the solution from (2.2) is truncated at 5: most of the improvement over the geometric chart (where r=1) has already been achieved at r=5. Moreover, using really large values of r may be considered undesirable in practice anyhow.As already observed in the Section1, the underlying parameters typically need to be estimated in practice. In the simple homogeneous case, we merely have to worry about the IC failure rate p. To this end, a Phase I is added, during which the process is observed until m failures have occurred, leading to m geometric X1,p's (cf. (2.1)) and, thus, to the obvious estimator p̂=1/X¯, where X¯=m-1Σi=1mXi. Then we simply replace p by p̂ in n (or ñ), after which the actual monitoring can start using the new n̂ (or ñ̂). Consequently, application of the chart is almost as easy as that in the known parameter case. However, it does remain to investigate, and possibly to correct, the resulting estimation effects. What happens is that performance characteristics such as FAR (or ARL) are no longer fixed at rα (or 1/α) but instead will be random in X¯. Noting that U=pX¯-1 satisfies EU=0 and EU2=(1-p)/m≈1/m, it follows that the Taylor expansion in terms of U allows adequate appraisal of errors such as FAR̂-FAR. This can be done in terms of bias by taking expectations or in terms of exceedance probabilities by, for example, looking at P(FAR̂>FAR(1+ɛ)). Next, if desired, suitable corrections can be derived. Then slightly lower limits n̂c=n̂(1-c) are used in which the small positive c is selected such that either the bias is removed or the exceedance probability will stay below a suitably small upper bound. To illustrate the simple application of the homogeneous chart, we conclude the present section with an explicit example.Example 2.
Suppose we are monitoring some large population w.r.t. some unpleasant type of event (“a failure”). As long as matters areIC, its incidence rate is p=0.001. Stopping during IC should be rare, so we decide that on average a false alarm once every 200 failures is acceptable, implying that the IC-ARL=200 and α=0.005. Deciding about stopping or continuing at each failure (the geometric chart, with r=1) is known to be not very efficient, so we increase r somewhat, and, for example, take r=3. (Background: the rule of thumb (2.2) tells us that this is optimal at the given α for θ=6, that is, for an increase during OoC to p=0.006; someone interested in optimality for lower θ should take an even somewhat larger r; for example, r=5 is the best for θ=4). During IC, each third failure will on average arrive after 3000 observations, and, hence, the chart should signal if this arrival happens much sooner. What “much sooner” should mean under the given conditions has been derived above: the exact lower bound here equals n=F3,0.001-1(0.015)=509. Using λ which solves P(Zλ≥3)=0.015 produces λ=0.509, and hence n=λ/p=509 as well, while the further approximation through (A.2) gives the quite close value 506. Monitoring is now completely straightforward: watch the sequence of realized “third-failure-times” and stop as soon as a number at or below 509 (or 506) is observed. If the IC value of p is in fact unknown, monitoring is preceded by a Phase I sample. First wait till, for example, m=100 failures have been obtained and replace p by the outcome p̂=100/Σi=1100Xi. If p̂ happens to equal 0.001, the above remains as it was; for any other value, the computation is adapted in a straightforward way.


## 3. The Heterogeneous Case

Our starting point is the homogeneous case described in Section2: an underlying sequence of i.i.d. Bernoulli r.v.'s Di, giving rise to negative binomial r.v.'s for controlling the process. To this situation we now add heterogeneity, and subsequently we investigate how to accommodate this complication, both in the unconditional case, where the underlying information is ignored, and the conditional case, where it is used. Of course, the emphasis will be on the latter situation, where risk adjustment is applied. In fact, the unconditional case has already been dealt with in Albers [11]; we only review it here to provide the proper perspective.In line with the common notation for bivariate situations, we will useY's for the main r.v.'s, on which the control decisions are based and X's for the r.v.'s supplying additional information. To be more precise, we will use X as a risk category indicator: P(X=j)=πj, j=1,…,k, for some k≥2. If this is done individually, that is, for each Di separately, the unconditional situation remains simple. In fact, ignoring or not having the information from the X's means that effectively we still are in the homogeneous case. Each incoming patient simply has the overall failure probability p=Σπjpj, with pj=P(D=1∣X=j) during IC, and thus the negative binomial charts from Section 2 are still appropriate.However, matters change if patients arrive in groups, and heterogeneity does lead to nonnegligible overdispersion. In the Appendix section, this is demonstrated in some detail for the relatively simple setting of a fixed number of groups, each of fixed size, to which routinely a standardp-chart would be applied. Moreover, in this context, the difference in behavior between unconditional and conditional charts is analyzed. An excursion to the situation of continuous rather than attribute data provides additional clarity, as matters for the normal case are somewhat more transparent. The conclusion is that, whenever Y goes OoC, two situations should be distinguished. In the first, the p.d.f. of X (and, thus, the πj) remains unchanged. Then the two charts react essentially in the same way, with the conditional chart being somewhat more efficient, as it takes the additional information into account. However, once the p.d.f. of X also changes during OoC, the behavior of the charts will diverge. Which of the two provides the right answer depends on whether these changes in X should be ignored or taken into account. (cf. the surgeon from Section 1 who is faced with increasingly more risky patients).Our present setup is of course even more complicated: group sizes will typically neither be fixed nor equal. As a consequence, the p.d.f. of the waiting times till therth failure is not only no longer negative binomial but in fact rather intractable in general. For this reason Albers [11] proposes by way of approximation a parametric model containing an additional parameter to cover the overdispersion aspect (c.f. the references mentioned in the Appendix section for accommodating tail length in case of the normal mixture model). In this way an adequate analysis of the unconditional chart indeed becomes possible; see Albers [11] for details.Having set the background and explained the interpretation, we can now fill in the details for the risk-adjusted chart in a straightforward way. Just as in the homogeneous case, define a new sequenceY1,Y2,… on the basis of the Di's. Here the Yi are i.i.d. copies of Y=Yr,p, the number of Di to be observed until the rth failure arrives. We decide to stop as soon as a Yi≤n, for some suitable lower limit n=nr,p. In line with Section 2, “suitable” will be interpreted to mean that P(Yr,p≤n)=rα for some small, given α. To obtain n in the present case, note that while we are waiting for the realization of an Yi, at each time t, we now have at our disposal the information that t=Σgj, where gj=gj,t is the number of patients from category j, j=1,…,k. Arguing along the same lines as in Section 2, we obtain (once more see the Appendix section for some of the details) that the simple approximation from the homogeneous case n=λ/p, with λ solving P(Zλ≥r)=rα, is updated into (cf. (C.2)):(3.1)n=Σj=1kgj,withthegj=gj,nsuchthatλ=Σj=1kgjpj.
Note that n from (3.1) can obviously be written as n=λ/(Σωjpj), with weights ωj=gj/(Σgj). Hence, if we are in the homogeneous case after all; that is, pj≡p, we get back n=λ/p exactly. Moreover, during IC, the ωj will tend to be close to the underlying probabilities πj, and n=λ/p will hold approximately for arbitrary pj as well.Nevertheless, we should observe that, unlike in the homogeneous case, we no longer have a single fixedn. Each of the realizations Yi from the above-mentioned sequence Y1,Y2,… has its own ni (and; thus, writing n=nr,p,i would be appropriate now). In fact, we do not even need to obtain all of these ni explicitly. For example, in those cases where a signal occurs, we have Yi≤ni, which means that in this situation at time t=Yi we still have Σgj,tpj≤λ. Evaluating the actual ni would imply continuing with the D's corresponding to Yi+1. That would not be the right thing to do; fortunately, it is also superfluous, as the information suffices that the event {Yi≤ni} has occurred.Example 3 (continued).
Suppose that for the situation considered in Section2 additional information, and hence the possibility for risk adjustment, has become available. To keep matters again as simple as possible, we distinguish just two risk categories: “mild” and “severe.” Suppose 10% of the population is severe and their risk is 11 times as high as that of the mild cases. Hence,k=2,π2=0.1,p2=11p1, and thus Σπjpj=2p1=p, resulting in p1=0.0005 and p2=0.0055. After (3.1) it was observed that the risk-adjusted chart replaces the homogeneous choice n=λ/p by n=λ/(Σωjpj), in this way, taking the actually observed weights into account. Here this means using n=(λ/p){2/(1+10ω2)}, which boils down to 509{2/(1+10ω2)}. During IC, ω2 will be close to π2=0.1, and hence n will be close to 509, as in the homogeneous case. Nevertheless, monitoring becomes slightly less trivial than before: now not only the sequence of realized third failure times but also the corresponding fractions of severe cases ω2 need to be recorded. For each data pair (Yi,ω2i), it is checked whether Yi≤509{2/(1+10ω2i)}. For example, an outcome (498,0.15) produces a signal in the homogeneous case, (“498≤509”), but not in the risk-adjusted situation (“498≥472=509/{2/(1+1.5)}”). In the latter case the occurrence of the third failure at a very early moment is deemed acceptable after all in view of the somewhat larger than expected presence of severe cases.The step in (3.1) is essentially all that is needed to deal with the risk adjusted version of the chart. In the Appendix section, it is demonstrated how the approximation steps for, for example, λ and the ARL carry over. Of course, if the process goes uniformly OoC, in the sense that all pj are replaced by θpj, matters are most straightforward. But even if each pj has its own θj, it remains easy to adapt the previously obtained expressions (cf. (C.3)). It is also explicitly demonstrated how the above-mentioned weights ωj=gj/(Σgj) influence matters. If these remain close to the πj, the risk-adjusted chart behaves in the same manner as its unconditional counterpart. It only is slightly more efficient. However, if the ωj’s are quite different, the behaviour will diverge. To illustrate this, just as in the normal example from the fixed case, an explicit example is given in the Appendix section of an OoC situation for Y which is completely due to the change in X. The risk-adjusted chart then indeed continues to consider the process as being IC. To illustrate matters, we conclude the present section with α numerical example.Example 3 (continued).
In the setup considered before, let us next consider theOoC situation. First let θ=2 uniformly; that is, the mild category gets θp1=0.001 and the severe one θp2=0.011. From, for example, Albers [11], we obtain that the homogeneous chart has an ARL of 36 for this case, and, according to the above, this will continue to hold here as well. Next consider a nonuniform example: let θ1=7/9, and let θ2=3, then Σπjθjpj=2p, and; hence, once more θ=2. If during OoC the p.d.f. of X remains the same, the πi’s do not change, and ω2 will still be close to π2=0.1. Then the risk-adjusted chart will continue to behave like the homogeneous one, that is, with an ARL of about 36 at this θ=2. However, if instead the pj’s remain the same and the θj’s are used to transform (π1,π2)=(0.9,0.1) into (θ1π1,θ2π2)=(0.7,0.3), the chart will shift towards using a lower limit n close to λ/(2p)=509/2. Moreover, as θ*=1 in (C.3) for this particular choice, its ARL will remain at about the IC value 200. Clearly, from a risk adjustment perspective, this is precisely what should happen. The mild patients still have p1=0.0005, and the severe ones still have p2=0.0055. What has changed is that the latter category has shifted from 10% to 30% of the total. Hence, not the quality of the performance has deteriorated, but rather that of the incoming patients (cf. the surgeon example in Section 1).


## 4. Estimation Effects

Typically the underlying parameters of control charts are not known in practice. Here this means that we will have to estimate not just the overallp=Σπjpj, but the individual pj as well. As concerns p, in Albers [10] a Phase I sample of “size” m was used for this purpose, meaning that we observe D1,D2,… until m failures have arrived. Note that m, and hence the length Ym,p of this sample as well, is independent of r, implying that the comparison between charts for different r remains fair also w.r.t. estimation. Next, p was simply estimated by p̂=m/Ym,p. In the present context we can use this same sample, but in a more detailed way, as follows. Let Hj be its number of patients from category j (i.e., Σj=1kHj=Ym,p), and in addition let Dji, i=1,…,Hj denote the corresponding D's. Then we have as a straightforward choice for estimating the pj:(4.1)p̂j=Σi=1HjDjiHj,j=1,…,k.Of course, formally there is a problem in (4.1), as each Hj can be 0 with positive probability. Using a slightly modified definition could remedy this. However, we shall not bother to do so, as the probabilities involved are exponentially small. Moreover, if too small hj’s are observed, this anyhow indicates that the design may not be right, and additional effort is required before monitoring can begin. Given Hj=hj, we have Ep̂j=pj and var(p̂j)=pj(1-pj)/hj. As EHj=πjEYm,p=mπj/p, it follows that, ignoring terms involving pj2,(4.2)p̂jpj-1≈AN(0,pmπjpj),
with “AN” denoting asymptotic normality. Hence, if the Phase I sample is chosen in this way, the estimators p̂j are indeed well behaved, in the usual sense of having a relative error which is OP(m-1/2). Only if the contribution πjpj of a certain category j is really small compared to p=Σπipi, the coefficient involved will be large.In fact, the above is all that is needed to transform the chart into its estimated version: just replace in (3.1) the pj by their estimated counterparts p̂j from (4.1). To be precise, we use a lower limit n̂ defined by(4.3)n̂=Σj=1kĝj,withtheĝj=ĝj,nsuchthatλ=Σj=1kĝjp̂j,
where once again λ is such that P(Zλ≥r)=rα. The alternative notation then becomes n̂=λ/(Σω̂jp̂j), with weights ω̂j=ĝj/(Σĝj). Once this lower limit n̂ has been obtained, the actual monitoring can start: each time wait till the rth failure, and if this occurs at or before n̂, a signal results. Hence, straightforward application of the estimated chart remains easy.To investigate the effects of the estimation step, we proceed as follows. As remarked in Section2, FAR will no longer be fixed at some given value rα nor will ARL be precisely equal 1/α. These performance characteristics have now become the random variables FAR̂ and ARL̂, respectively. They depend on the estimators p̂j and consequently fluctuate around the intended values. A rather detailed analysis of the consequences for the homogeneous case can be found in Albers [10]. To avoid repetition, we shall be much more brief here and mainly focus on the additional complication caused by the fact that the estimation step is now split into k categories. See once more the Appendix section for the details. It is evaluated how large the size m of the Phase I sample should be in order to ensure that the exceedance probabilities discussed at the end of Section 2 will fall below a prescribed small quantity δ. If the resulting m is too large for use in practice, a small correction c is calculated, such that for a given m the desired δ can be met after all by using the slightly more strict n̂c=n̂(1-c) rather than n̂ itself. A complication here in comparison to the homogeneous case is the presence of an additional quantity τ≥1 (see (D.5)), which represents the unbalance caused by (possible) differences between the ideal πj and the actually occurring ωj.To illustrate matters we again present an explicit example.Example 4 (continued).
Once again, we add a complication to our ongoing example: now the values ofp1 and p2 are no longer known. Suppose, therefore, that we first wait till m=100 failures have occurred and then use the resulting Phase I sample to estimate these pj (see (4.1)). Just as in the example of Section 2, the application of the chart remains straightforward: simply plug in the p̂j to replace the unknown pj. However, if we also want to study the impact of the estimation step, and possibly correct for it, a bit more effort is needed (we shall refer to the Appendix section for the required formulae). The point is that, in the subsequent monitoring phase, we may still aim at an ARL of 200, but we have to accept that the actual ARL̂ may differ. In particular, this can substantially be smaller than 200, leading to a lot more frequent false alarms than anticipated. To monitor not only the process but this effect as well, we can, for example, look at the probability of a deviation of more than 20%, that is, of the ARL̂ falling below 160. In terms of (D.4), this is precisely PExc with 1/(1+ɛ)=0.8, and; thus, ɛ=0.25. As moreover r=3, it then follows from (D.6) that PExc≈1-Φ(2.5/(3γτ)). Since γ is close to, but smaller than, 1, a close upper bound for PExc is 1-Φ(2.5/(3τ)). As long as X is IC (regardless of whether the same holds for Y or not), τ will be close to 1 as well, in which case this upper bound approximately boils down to 1-Φ(2.5/3)=0.20. If such a 20% probability for a more than 20% too low ARL is acceptable, we can continue to the monitoring phase. Otherwise m should be larger than 100; to be precise, m≥(12uδ)2 is needed to get PExc≤δ. Or, alternatively, for m fixed at 100, this can be achieved by using n̂c with c=(100)1/2uδ-0.25/3 (cf. (D.7)). For, for example, δ=0.10, we have uδ=1.28 and m≥236 and c=0.045 result. If in fact X is OoC, τ will be larger than 1, and we need to use m≥(12τuδ)2 instead. For example, consider once more the situation from the previous example, where π1=0.9,p1=p/2,π2=0.1,p2=11p/2, and subsequently ω1=0.7,ω2=0.3. Then in addition to Σπjpj=p,Σωjpj=2p, we obtain Σ[ωj2/πj]pj=5.22p, which leads through (D.5) to τ2=1.31. Hence, to still get δ=0.20, m should now be 131, rather than just 100. This increase is still rather mild, but it is easy to see that higher increases of m can be necessary. For example, slightly generalize our example into π1=1-q,π2=q for some small q>0, while otherwise keeping p2/p1=11 and (ω1,ω2)=(0.7,0.3). In that case p1=p/(1+10q), producing Σπjpj=p,Σωjpj=4p/(1+10q) and Σ[ωj2/πj]pj={0.49/(1-q)+0.99/q}p/(1+10q). This leads to τ2≈(1+10.5q)/(16q), which, for example, for q=0.05 equals 1.9 and for q=0.02 already 3.8. Of course, the latter case is rather extreme, as the frequency of severe cases has increased by a factor 15.To conclude this section, for convenience, we briefly summarize the steps involved in applying the new chart (see Algorithm1).Algorithm 1
(1) Before the monitoring phase starts, take the following preliminary steps:(a) Select a desiredARL=1/α and a degree of change θ during OoC thatshould be optimally protected against.(b) Apply rule of thumb (2.2) to obtain r (typically truncate at 5 in practice).(c) Findλ such that P(Zλ≥r)=rα, where Zλ is Poission, or simply use itsapproximationλ̃ from (A.2).(d) Wait tillm failures have occurred. Take for example, m=100 (or use Section 4(e.g., see (D.6)) to make a more elaborate choice).(e) From this Phase I sample, evaluate the fraction of failuresp̂j for each of thecategoriesj=1,…,k.(2) Now wait tillY1, the moment at which the rth failure occurs.(3) Obtain the corresponding numbersgj from category j (i.e., Σj=1kgj=Y1).(4) Give a signal ifΣj=1kgjp̂j≤λ; otherwise go back to Step  2, leading to Y2,Y3,….



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*Source: 895273-2011-10-30.xml*