IIE Transactions (1998) 30, 545±561

The SPRT chart for monitoring a proportion MARION R. REYNOLDS, JR1 and ZACHARY G. STOUMBOS2 1

Departments of Statistics and Forestry, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0439, USA E-mail: [email protected] 2 Department of Management Science and Information Systems and Rutgers Center for Operations Research (RUTCOR), Rutgers, The State University of New Jersey, Newark, NJ 07102-1895, USA E-mail: [email protected] Received April 1997 and accepted January 1998

A control chart based on applying a sequential probability ratio test (SPRT) at each sampling point is considered for the problem of monitoring a process proportion p. This SPRT chart can be applied in situations in which items are inspected one by one, and the results of each inspection can be conveniently recorded before the next item is inspected. Some corrected di€usion theory approximations are given for the statistical properties of the SPRT and the SPRT chart. These approximations are very accurate and provide a simple method for designing an SPRT chart for practical applications. The sample size for the SPRT chart at a particular sampling time depends on the observations at that time, but the chart can be designed to have a speci®ed average sampling rate when the process is in control. When there is a small shift in p, the average sampling rate per unit time will increase, but for a large shift in p the average sampling rate will decrease. For a given in-control average sampling rate and a given false alarm rate, the SPRT chart will detect changes in p much faster than the standard p-chart, which has traditionally been used for monitoring p. The SPRT chart will also detect changes in p much faster than the CUSUM chart for p. Thus, the SPRT chart can be used in place of traditional control charts to provide faster detection of changes in p or to reduce the sampling e€ort required to provide a given detection capability.

1. Introduction In monitoring a production process, the statistic recorded will frequently be a count of the number of items in a sample that are classi®ed as having a particular attribute such as being defective or failing to conform to some speci®cation. It is usually assumed that the items from the process are independent, with a probability p of having the attribute of interest. Then, the distribution of the number of items with the attribute in a sample of size n is binomial with parameters n and p. For purposes of exposition in this paper, an item having the attribute of interest will usually be referred to as a defective, with the understanding that the results presented here apply more generally. Thus, the problem being considered here is one of detecting changes in the process which are re¯ected by a change in the process proportion defective p from an incontrol value p0 . In most quality control applications, the primary interest would be in detecting an increase in p above p0 . The classical control chart for this problem is the Shewhart p-chart, which plots the proportion of defectives found in each sample. In most cases, in practice, three-sigma control limits are used with this chart. There are several disadvantages to using the p-chart. One is that 0740-817X

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the discreteness of the binomial distribution may result in an undesirable false alarm rate. In particular, if p0 is close to zero and n is not very large then the false alarm rate can be quite di€erent from what would be expected from using three-sigma limits with a normal distribution. Another disadvantage of the p-chart is that it is not e€ective for detecting small changes in p unless n is very large. Runs rules are sometimes used to improve the e€ectiveness of Shewhart charts, but for the p-chart the discreteness of the binomial distribution also causes problems in attempting to apply standard runs rules. There has been some work on developing the CUSUM chart for the problem of monitoring p (see, for example, Gan [1]). The CUSUM chart for p will be considerably more e€ective than the p-chart for detecting small changes in p. In most applications, both the p-chart and the CUSUM chart would take a ®xed sample size (FSS) of n items using a ®xed sampling interval (FSI) between samples. In some cases, in practice, n may vary from one sample to another, but this is usually the unintended result of factors such as variations in the production rate and not the result of a deliberate attempt to vary the sampling rate. In recent years, there has been extensive development of control charts that deliberately vary the rate of sam-

546 pling from the process as a function of the data that are obtained from the process. A variable sampling rate can be achieved by using a variable sampling interval (VSI) chart in which the time interval from the current sample to the next sample depends on the value of the control statistic calculated at the current sample (as is discussed in Reynolds et al. [2], Reynolds et al. [3], Vaughan [4], Reynolds [5], and Stoumbos and Reynolds [6,7]. The next sampling interval will be short if the current value of the control statistic indicates a problem with the process and long if the control statistic does not indicate a problem. If the current value of the control statistic is outside of the control limits, then a VSI chart signals in the same way as a traditional chart. By allowing the sampling interval to vary, the rate of sampling can be increased during times in which there is an indication of a possible change in the process. This increase in the sampling rate when there is an indication of a problem typically results in much faster detection of process changes, as compared to traditional FSI and FSS control charts in which the sampling rate is constant at all times. Another way to vary the sampling rate is to use a variable sample size (VSS) chart in which the sample size to be used at the next sample depends on the value of the control statistic calculated at the current sample as is discussed by Prabhu et al. [8], Costa [9], Park and Reynolds [10], and Stoumbos and Reynolds [7]. A VSS chart would use a large sample size when there is an indication of a problem with the process and a small sample size when there is no such indication. Thus, a VSS chart will increase the sampling rate when there is an indication of a change in the process and, like a VSI chart, will detect most process changes faster than traditional charts. The VSI and VSS features can be used together for improved performance as is discussed by Rendtel [11], Prabhu et al. [12], Costa [13], Park and Reynolds [14], and Stoumbos and Reynolds [7]. A distinguishing feature of the VSI and VSS control charts developed in the references given above is that the sampling interval and/or sample size is varied for the next sampling point as a function of the value of the control statistic at the current sampling point. This approach to varying the sampling rate would be appropriate when it is necessary to know the sample size to use at the next sampling point before reaching this point. A di€erent approach to varying the sampling rate can be used for cases in which the sample size for the next sampling point does not have to be determined in advance, but instead can be determined by the data in the next sample as it is taken. The idea of allowing the size of a sample to depend on the data in the sample is, of course, the basic idea of sequential analysis (see, for example, Wald [15] or Ghosh [16]). This idea was used by Daudin [17] for constructing a control chart for the process mean using two possible sample sizes. This idea was also used by Stoumbos and Reynolds [7,18,19] in developing a control chart for monitoring the process mean and by Stoumbos and

Reynolds and Stoumbos Reynolds [6] in developing a control chart for monitoring a general process parameter. The control charts developed by Stoumbos and Reynolds [6,7,18,19] consist of the application of a sequential probability ratio test (SPRT) at each sampling point. These SPRT charts can be used when observations can be taken sequentially, with the possibility of a decision about the process after each observation. If the time required to take each observation is short enough that it can be neglected relative to the time between samples, then the SPRT chart has the administrative advantage of being an FSI chart. It is a VSS chart but not in the sense of the previously referenced VSS charts which determine the sample size in advance of taking the sample. The advantage of the SPRT chart is that, for a given in-control average sampling rate and a given false alarm rate, the SPRT chart will detect changes in the process much faster than traditional charts such as Shewhart and CUSUM charts. The purpose of this paper is to extend the development of the generalized SPRT chart presented in Stoumbos and Reynolds [6] to the problem of monitoring the process proportion defective p. This SPRT chart for p can be applied in situations in which items are inspected one by one, and the results of each inspection can be recorded before the next item is inspected. It is envisioned that detecting an increase in p will be the primary objective in most applications, and thus this paper concentrates on this problem. The next section gives some background information on the SPRT and shows how the SPRT chart is based on applying an SPRT at each sampling point. Statistical measures of control chart performance are also discussed. Then some corrected di€usion theory approximations are given for the statistical properties of the SPRT chart. These are new approximations which are substantially better in the current problem of monitoring p than those in the problem of monitoring the process mean l considered by Stoumbos and Reynolds [19]. Two simple methods for designing this chart are presented, and an example is used to illustrate the design and application of the SPRT chart. The SPRT chart is compared to the traditional p-chart and to the CUSUM chart for p, and it is shown that the SPRT chart is much more ecient than either of these charts. An SPRT chart can be constructed for the problem of detecting a decrease in p, and this problem is discussed brie¯y. Some exact methods for computing the statistical properties of the SPRT chart are given in an Appendix. The accuracy of the corrected di€usion theory approximations is also discussed in an Appendix.

2. The SPRT and the SPRT chart Before showing how the SPRT can be used to develop a control chart, the SPRT will ®rst be explained in its original context of hypothesis testing. The SPRT is a

The SPRT chart for monitoring a proportion

547

general sequential test developed by Wald [15]. It can be applied to test a simple null hypothesis against a simple alternative hypothesis. For the case of a test involving the proportion defective p, the SPRT can be used to test the null hypothesis H0: p ˆ p0 against the alternative hypothesis H1: p ˆ p1 . In the context of monitoring p, p0 would be the in-control value of p, and p1 would usually be a value above p0 that is the result of a special cause that should be detected quickly. To de®ne the SPRT for testing H0: p ˆ p0 against H1: p ˆ p1 , consider a sequence of items that are being inspected at some sampling time, and let the Bernoulli random variable Xi be de®ned by Xi = 1 if the ith item is defective and Xi = 0 otherwise. The statistic used by the SPRT is de®ned in terms of a log likelihood ratio using the discrete density of f …x; p† ˆ px …1 ÿ p†1ÿx of Xi . After the jth item is inspected, this log likelihood ratio statistic is: Sj ˆ where

j X

Li ;

…1†

iˆ1

f …Xi ; p1 † ˆ r2 Xi ÿ r1 ; Li ˆ ln f …Xi ; p0 † 1 ÿ p1 1 ÿ p0

and r2 ˆ ln

p1 …1 ÿ p0 † : p0 …1 ÿ p1 †

…2†

…3†

where Tj ˆ

j X

Xi :

…4† …5†

iˆ1

Note that Tj is the total number of defective items in the ®rst j items inspected, so Tj is the natural statistic that would be recorded as items are inspected. The SPRT requires the speci®cation of two constants a and b, b < a, and uses the following rules for sampling and making a decision to accept or reject H0 : 1. If b < Sj < a; then continue sampling; 2. If Sj  a; then stop sampling and reject H0 ; 3. If Sj  b; then stop sampling and accept H0 . The inequality: b < Sj < a;

…6†

determines when the SPRT continues sampling and is usually called the critical inequality of the SPRT. In some applications it may be more convenient to carry out the SPRT by substituting (4) into (6) and dividing each term by r2 to obtain an equivalent critical inequality. If p1 > p0 , then this equivalent critical inequality is: g < Tj ÿ cj < h;

…7†

…8†

Thus, after the jth item is inspected, the SPRT is carried out by determining Tj , subtracting cj, and comparing the result to g and h. If (7) holds then inspection is continued, if Tj ÿ cj  g then sampling is stopped and H0 is accepted, and if Tj ÿ cj  h then sampling is stopped and H0 is rejected. When evaluating any statistical test, a critical property of the test is determined by either the probability that the test accepts the null hypothesis or the probability that the test rejects the null hypothesis, expressed as functions of the value of the parameter under consideration. Following the convention in sequential analysis, we will work with the operating characteristic (OC) function, which is the probability of accepting H0 as a function of p, rather than power, which is the probability of rejecting H0 . Let OC… p† be the OC function as a function of p. When p ˆ p0 the probability of a type I error is: …9†

and when p ˆ p1 the probability of a Type II error is: b ˆ OC…p1 †:

Substituting (2) into (1) gives Sj ˆ r2 Tj ÿ r1 j;

c ˆ r1 =r2 :

a ˆ 1 ÿ OC…p0 †;

and the constants r1 and r2 are de®ned by: r1 ˆ ÿln

where g ˆ b=r2 ; h ˆ a=r2 ; and

…10†

For most statistical tests, the sample size is ®xed before the data are taken, but for a sequential test, the sample size, say N , depends on the data and is thus a random variable. Therefore, for a sequential test, the distribution of N must be considered. In most cases, E…N †, called the average sample number (ASN), is used to characterize the distribution of N . Let ASN …p† be the ASN function as a function of p. The SPRT will now be used as a component in developing a control chart for monitoring p. Suppose that a sampling interval of d is used for sampling from the process. At each sampling point, items from the process are inspected one by one and an SPRT is applied, where p0 in the SPRT is the in-control value of p, and p1 is a value of p that should quickly lead to a signal by the control chart. If the SPRT accepts H0: p ˆ p0 , then the decision is that the process is in control, and the process is allowed to continue until the next sampling point, at which time another SPRT is applied. But if the SPRT rejects H0 , then this is taken as a signal that there has been a change in p. The sample size used at each sampling point is determined by the SPRT that is applied at that point. If items can be inspected quickly enough, then the inspection can be done on consecutive items as they come from the production process. For example, if an item is produced every 10 seconds, and the inspection and recording of the result takes no more than 10 seconds, then inspection can be done as the items are produced. On the other hand, if the inspection rate is slower than the rate of production, then inspection could be done on items that have been accumulated after production. Alternately, inspection

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could be done on items as they come from production with some items skipped. For example, if an item is produced every 10 seconds, but inspection requires between 30 and 40 seconds, then every third or fourth item could be inspected during inspection periods. Each SPRT either accepts or rejects H0 , and thus the number of SPRTs until a signal has a geometric distribution with parameter 1 ÿ OC…p†. Because each SPRT corresponds to a sample from the process, the number of SPRTs until a signal equals the number of samples until a signal. The expected number of samples until a signal will be called the average number of samples to signal (ANSS). The ANSS has been called the average run length (ARL) in some contexts, but here we will use ANSS because it is more descriptive. For the SPRT chart, the ANSS for a given p, say ANSS…p†, is thus the mean of the geometric distribution, which is: ANSS… p† ˆ

1 : 1 ÿ OC… p†

…11†

Each item that is inspected corresponds to an observation, so the ASN of the SPRT is the expected number of observations at a sampling point. The expected total number of observations until the SPRT chart signals will be called the average number of observations to signal (ANOS). It is easy to see that the ANOS is the product of the ANSS and the ASN. Thus, if ANOS…p† is the ANOS at a given p, then ANOS…p† ˆ ANSS…p†  ASN …p†:

…12†

When there is a ®xed time interval d between samples, and the time required to take a sample is negligible, then the expected time for a control chart to signal, called the average time to signal (ATS), is the product of d and the ANSS. Thus, the ATS at p, say ATS…p†, is ATS…p† ˆ d  ANSS…p† ˆ

d : 1 ÿ OC…p†

…13†

When p ˆ p0 using (9) gives: d ATS…p0 † ˆ : …14† a When p ˆ p0 , it is desirable that the ATS be large so that the frequency of false alarms is low. When p ˆ p1 using (10) gives: d : …15† ATS…p1 † ˆ 1ÿb If p ˆ p1 , then it is desirable that the ATS be small so that this condition will be detected quickly. The ATS for some p 6ˆ p0 is a measure of the time required to detect a shift to this p, assuming that this shift is present at the time that the control chart is started. However, it is usually desirable to measure the detection time assuming that the shift in p occurs at some random time after the chart has been started. To model this situation it is frequently assumed that the shift occurs after

the chart's control statistic has reached its stationary or steady-state distribution. If it is assumed that the time required to take a sample is negligible relative to the time between samples, then it follows that when a shift occurs, it will occur somewhere between two samples. If it is assumed that the position of the shift in the interval between two samples is uniformly distributed on this interval, then it follows that the expected position of the shift within the interval is the midpoint of the interval. The expected time from the shift to a signal by the control chart computed under these assumptions is usually called the steady-state ATS (SSATS). In this paper, the SSATS will be used as the measure of the time required to detect a shift in p. The SSATS of the SPRT chart can be expressed simply as a function of the ATS because the SPRT applied at one sampling point does not depend on the SPRTs at past sampling points. To explain the expression for the SSATS, suppose that the shift occurs between samples k and k + 1, and the chart signals at sample k 0 , where k 0  k ‡ 1. Then, the time from the shift to the signal at sample k 0 can be expressed as the time from sample k to sample k 0 minus the time from sample k to the shift. The time between samples is d, and thus it follows that the SSATS at p, say SSATS…p†, is: SSATS…p† ˆ ATS…p† ÿ

d d d ˆ ÿ : 2 1 ÿ OC…p† 2

…16†

For the SPRT chart, the steady state ANSS (SSANSS) is the same as the ANSS, and steady state ANOS (SSANOS) is the same as the ANOS. Approximations to the statistical properties of the SPRT chart will be given in the next section.

3. The statistical properties of the SPRT chart The expressions given in the previous section show that the statistical properties of the SPRT chart depend on the statistical properties of the SPRT that is the basis of the SPRT chart. Exact expressions for the OC and ASN functions of the SPRT for p are derived and given in Appendix A. These expressions, however, are relatively complicated, and thus it would be convenient to have simpler expressions that could be used in designing the SPRT chart for practical applications. Some simple approximations to the OC and ASN functions have been developed and will be given in this section. These approximations are based on di€usion corrections to approximations presented by Wald [15] and are thus termed corrected di€usion (CD) approximations. A design procedure for the SPRT will be developed based on these CD approximations. Additional discussions of the derivation of these CD approximations are presented in Appendix B. Before giving the CD approximations a simpli®cation of the structure of the SPRT for p will be discussed. In

The SPRT chart for monitoring a proportion

549

most quality control applications, the values of p0 and p1 that are of interest will be relatively small, and in this case, the value of c ˆ r1 =r2 will also be small. For example, if the null hypothesis is that the process proportion defective is p0 ˆ 0:01, and the alternative hypothesis is that it is p1 ˆ 0:02, then in this case r1 ˆ ÿln…0:98=0:99† ˆ 0:010 15, r2 ˆ lnf‰…0:02†…0:99†Š= ‰…0:01†…0:98†Šg ˆ 0:7033, and c ˆ r1 =r2 ˆ 0:014 44. It will frequently be convenient in situations such as this example to have c ˆ 1=m, where m is a positive integer, so that the SPRT statistic Tj ÿ cj in (7) will take on values which are integer multiples of c. It will usually be possible to make c ˆ 1=m by a slight adjustment of p1 . The value of p1 is the value of p that should be detected quickly by the chart, and its precise speci®cation is usually not critical. In the current example c ˆ 1=69:27, and if p1 is changed slightly to about p1 ˆ 0:020 14, then this will give r1 ˆ 0:010 30, r2 ˆ 0:7105, and c ˆ 1=69, so that m ˆ 69. Changing p1 from 0.02 to 0:020 14 would usually be of no practical consequence. When c ˆ 1=m, where m is a positive integer, the acceptance limit g in (7) can be chosen to be an integer multiple of 1=m, and this will ensure that the SPRT statistic Tj ÿ cj will exactly hit g when the test accepts H0 . In the development of the SPRT and the SPRT chart that follows in this paper, it will be assumed that c ˆ 1=m and that g is an integer multiple of c. If Tj ÿ cj is an integer multiple of c, then it follows that the rejection limit h can also be taken to be an integer multiple of c, although Tj ÿ cj may still overshoot h when the test rejects H0 . The CD approximations to the OC and ASN functions of the SPRT for p are obtained by determining the expected excess of the SPRT statistic over the boundary h. In particular, the CD approximations use: h ˆ h ‡ …1 ÿ 2p0 †=3;

…17†

where …1 ÿ 2p0 †=3 can be interpreted as an adjustment to h to account for the expected excess (see Equation (A20)). The CD approximation to the OC function is: 8 exp…h r2 † ÿ 1 > > if p ˆ p0 , > > exp…h r2 † ÿ exp…gr2 † > > > > > exp…gr2 †…exp…h r2 † ÿ 1† > > > if p ˆ p1 , < exp…h r † ÿ exp…gr † 2 2 OC…p†  …18†  > h r r 2 1 > > if p ˆ , > > h r2 ‡ jgr2 j r2 > > > >  > > exp…n…p†h r2 † ÿ 1 > : ; otherwise. exp…n…p†h r2 † ÿ exp…n…p†gr2 † The CD approximation to the ASN function is: 8  h r2 …jgr2 j† r1 > > if p ˆ , > < r1 …r2 ÿ r1 † r2 …19† ASN …p†   > h r2 …1 ÿ OC…p†† ‡ gr2 OC…p† > > : otherwise. r2 p ÿ r1

In this case n…p† 6ˆ 0 denotes the non-zero solution to the equation:  n…p†   p1 1 ÿ p1 n…p† Ep …expf…r2 X ÿ r1 †n…p†g† ˆ p‡ p0 1 ÿ p0  …1 ÿ p† ˆ 1:

…20†

Clearly, n…p0 † ˆ 1 ˆ ÿn…p1 †. The solution for n…p† for p 6ˆ p0 or p1 can be obtained numerically (see Appendix C). When the SPRT is used for hypothesis testing, it will usually be desirable to choose the constants g and h such that the test has speci®ed values for the error probabilities a and b. This can be achieved by substituting the approximation in (18) for the OC function in (9) and (10) and solving for g and h to give:   1 1ÿb  ; …21† h  ln r2 a and g

  1 b ln : r2 1ÿa

…22†

If nominal values are speci®ed for a and b, then g and h can be determined by using (21) and (22), and then the value of h can be obtained from h using (17). This will give a test with a and b approximately equal to the nominal values. As an example, consider the previous example in which p0 ˆ 0:01, p1 ˆ 0:020 14, r2 ˆ 0:7105, and c ˆ 1=69. Suppose that it is desirable to have a  0:01 and b  0:05. Then using (21) and (17) gives h ˆ ln…0:95=0:01†=0:7105 ˆ 6:4091 and h = 6.0824, and using (22) gives g = ln(0.05/0.99)/0.7105 = ÿ4:2020. The possible values of the SPRT statistic in (7) are all integer multiples of 1/69, so it follows that g and h can be taken to be integer multiples of 1/69. If the g and h values determined above are rounded to the nearest multiple of 1/69, then this will give g ˆ ÿ290=69 ˆ ÿ4:2029 and h ˆ 420=69 ˆ 6:0870. Using these values of g and h will give an SPRT with a type I error probability that is very close to 0.01 (the exact value is 0.009 998 using the exact methods given in Appendix A), and a type II error probability that is very close to 0.05 (the exact value is 0.049 97). The accuracy of the CD approximations to the statistical properties of the SPRT is discussed further in Appendix C. The CD approximation to the ASN at p0 and p1 can be expressed simply in terms of a and b by substituting (9), (10), (21), and (22) into (19). For p ˆ p0 , this gives:    1ÿb ASN …p0 †  aln a   b …23† ‡ …1 ÿ a†ln …r2 p0 ÿ r1 †; 1ÿa and for p ˆ p1 , this gives:

550

Reynolds and Stoumbos    1ÿb ASN …p1 †  …1 ÿ b†ln a   b ‡ bln …r2 p1 ÿ r1 †: 1ÿa

…24†

Thus, for given a and b, evaluating the ASN at p0 and p1 is relatively easy. A possible disadvantage of the SPRT for practical applications is that there is no upper bound on the sample size N that may be required. To avoid the possibility that N takes on an extremely large value, a truncated version of the SPRT is sometimes used. With a truncated SPRT, sampling is stopped at some maximum value, say nmax , and a decision is made to accept or reject H0 . One criterion for choosing nmax is to choose nmax large enough so that P …N > nmax †  0, so that the properties of the SPRT will not be a€ected signi®cantly by the truncation. A rough rule of thumb that is given in sequential analysis texts (see, for example, Wald [15] and Ghosh [16]) is that if nmax is chosen as 3  maxfASN …p0 †; ASN …p1 †g, then P …N > nmax †  0. Some limited numerical results were obtained for this truncation point using the following decision rule. If an SPRT reaches this truncation point without a decision, then accept the null hypothesis that the process is on target and go to the next sampling point to apply another SPRT. This rule will be conservative in the sense that the false alarm rate of the SPRT chart will be no higher than for the case of no truncation. With this rule the time required to detect small shifts in p will increase by a small amount, but the time to detect large shifts will not be a€ected very much.

(e) Using a and b from above, ®nd h from h ˆ …1=r2 †ln……1 ÿ b†=a† ÿ ……1 ÿ 2p0 †=3†; (f) Using a and b from above, ®nd g from g ˆ …1=r2 † ln…b=…1 ÿ a††; (g) Find ASN …p0 † using (23). If this value is too high then change the speci®cations for ATS…p0 † and/or SSATS…p1 † and iterate through the design procedure until a reasonable balance is obtained between sampling costs, the false alarm rate, and the time required to detect the shift of interest. The advantage of this approach is that the false alarm rate of the chart and the time to detect a speci®ed shift are directly controlled. A possible disadvantage of this approach is that the average number of items inspected in each SPRT (the ASN) is not directly speci®ed or controlled. In many applications, it will be desirable to specify the in-control average sampling rate, i.e., specify ASN …p0 †, and design the chart to achieve this value of ASN …p0 †. Thus, a second approach to designing the SPRT chart is to specify d, ATS…p0 †, and ASN …p0 †. The value of a is determined using (14) and the speci®ed values of d and ATS…p0 †. Then, using (23) the value of b can be determined from the speci®ed value of ASN …p0 † and the value of a just determined. The expression in (23) cannot be solved explicitly for b in terms of a and ASN …p0 †, so the solution for b will have to be determined numerically. The steps in this second design approach are as follows.

To design the SPRT chart for practical applications, it is necessary to specify the sampling interval d, specify the value of p1 which determines the chart parameter c, and then determine the constants g and h. Once d and p1 are speci®ed, two approaches to determining g and h will be considered here. The ®rst approach is to specify ATS…p0 † (this will determine the false alarm rate) and specify SSATS…p1 † (this will determine the average time required to detect a shift in p from p0 to p1 ). These speci®cations will determine a and b from (14), (15), and (16), and then g and h can be determined using (21), (22), and (17). The steps in this ®rst design approach are as follows.

Design approach II (a) Determine p0 and specify p1 . Adjust p1 if necessary so that c ˆ 1=m, where m is a positive integer; (b) Specify the sampling interval d; (c) Specify ATS…p0 †. Find a from a ˆ …d=ATS…p0 ††; (d) Specify ASN …p0 †. Use (23) to ®nd b; (e) Using a and b from above, ®nd h from h ˆ …1=r2 †ln……1 ÿ b†=a† ÿ ……1 ÿ 2p0 †=3†; (f) Using a and b from above, ®nd g from g ˆ …1=r2 †ln…b=…1 ÿ a††; (g) Find SSATS…p1 † using SSATS…p1 † ˆ …d=…1 ÿ b†† ÿ …d=2†. If this value is too high then change the speci®cations for ATS(p0 ) and/or ASN(p0 ) and iterate through the design procedure until a reasonable balance is obtained between sampling costs, the false alarm rate, and the time required to detect the shift of interest.

Design approach I (a) Determine p0 and specify p1 . Adjust p1 if necessary so that c ˆ 1=m, where m is a positive integer; (b) Specify the sampling interval d; (c) Specify ATS…p0 †. Find a from a ˆ …d=ATS…p0 ††; Find b from bˆ (d) Specify SSATS…p1 †. …SSATS…p1 † ÿ d=2†=…SSATS…p1 † ‡ d=2†;

The value chosen for p1 in the design approaches above is really just a convenient design device for the SPRT chart, so this value of p would usually not be the only value which should be detected quickly. Thus, when designing an SPRT chart in practice, it would be desirable to use the CD approximation in (18) along with (16) to ®nd the SSATS for a range of values of p around p1 . The accuracy of the CD approximations in the evaluation of

4. The design of the SPRT chart

The SPRT chart for monitoring a proportion the statistical properties and design of SPRT charts is further discussed in Appendix C. If a p-chart is used to monitor p, then a relatively large value of n is required if it is desirable to detect small changes in p. This frequently requires that the sampling interval d must be chosen to be relatively long to maintain a reasonable sampling rate. The SPRT chart does not have this restriction. It will be shown in Sections 6 and 7 that small samples taken at short sampling intervals work just as well for detecting small shifts, and are actually better for detecting large shifts. Thus, the sampling interval for the SPRT chart can be chosen to be a small convenient value.

5. An example of designing an SPRT chart To illustrate the design and application of the SPRT chart, consider a process which produces at a rate of one item every 5 seconds. When the process is stable, the proportion defective has been 1%. Thus, the objective is to use a control chart to detect any special cause that produces an increase in p above the in-control value of p0 ˆ 0:01. The current procedure for monitoring this process is to use a Shewhart p-chart with three-sigma limits. Process engineers are interested in detecting small increases in p, and thus it has been necessary to use a relatively large sample size for the p-chart. The current procedure takes a sample of n ˆ 200 items from the process every d = 4 hours. Items from the process can be tested at the rate of one every 5 seconds (the same as the production rate), so the items can be inspected one by one as they are produced. In this example, the time required to obtain a sample is short relative to the time between samples, so neglecting this time in computations of quantities such as the ATS seems to be reasonable. The three-sigma control limits for the p-chart are determined from 0:01  3‰…0:01†…0:99†=200Š1=2 . This gives an upper control limit of 0.0311, but there is no e€ective lower control limit. A sample proportion defective above the upper control limit is equivalent to seven or more defectives in the sample of 200. When p ˆ p0 , the probability of a signal by the p-chart is 0.004 296, and this corresponds to an ANSS of 232.80 and an ATS of 931.20 hours. Thus, when the process is in control, there will be a false alarm once in every 931.20 hours on average. When p increases from 0.01 to 0.02, the SSATS is 34.85 hours, and thus it will take a relatively long time for this p-chart to detect a doubling of p. If p increases to 0.03, the SSATS is 8.16 hours. There might be two objectives for looking for a more ecient control chart for monitoring this process. One objective would be to obtain faster detection of increases in p; the average of 34.85 hours required to detect a doubling of p might be too high. A second objective would be to decrease sampling costs. The use of the SPRT

551 chart for achieving one or both of these objectives will now be illustrated. To set up the SPRT chart, it is necessary to specify a value of p1 that should be detected quickly. Suppose that process engineers want to reduce the time required to detect small shifts in p and speci®cally decide that it would be desirable to have fast detection when p increases to p1 ˆ 0:02. As in the example from Section 3 which used p0 ˆ 0:01 and p1 ˆ 0:02, it will be convenient to adjust p1 slightly to about 0.020 14 so that c ˆ 1=69. Suppose that it is decided that the SPRT chart should be designed to have approximately the same in-control average sampling rate and approximately the same false alarm rate as the p-chart. Then, d can be taken to be 4, ASN …p0 † can be taken to be 200, and ATS…p0 † can be taken to be 931.20 hours. Thus, the second design approach from Section 4 will be used because it is desirable to have a speci®ed value for ASN …p0 †. Using the speci®ed value for ATS…p0 †, step (c) of the design procedure gives a ˆ 0:004 296. Then, in step (d) solving (23) numerically for b gives b ˆ 0:5138. Then, step (e) gives h ˆ 6:3289, and step (f) gives g ˆ ÿ0:9311. Rounding g and h to the nearest multiple of 1/69 gives g ˆ ÿ64=69 ˆ ÿ0:9275 and h ˆ 437=69 ˆ 6:3333. Thus, the SPRT chart can be applied in this case using the critical inequality: ÿ0:9275 < Tj ÿ j=69 < 6:3333:

…25†

The in-control ASN of this chart should be approximately 200 (the exact value is 199.24), and the in-control ATS should be approximately 931.20 (the exact value is 933.88). Step (g) of the design approach gives a SSATS at p ˆ p1 ˆ 0:020 14 of approximately 6.30 hours (the exact value is 6.25 hours). Thus, compared to the p-chart, the SPRT chart will provide a dramatic reduction in the time required to detect the shift from p0 to p1 . SSATS values for other values of p are given in the next section in the context of a more comprehensive comparison of the performance of the SPRT chart relative to the p-chart. Exact SSATS values are used in this comparison because they are available using the exact methods developed in Appendix A.

6. Comparison of the SPRT chart and the p-chart For the example in the previous section, Table 1 gives OC, ANSS, and ATS values for the p-chart and the SPRT chart. The OC values are the probability of no signal for an individual sample. ASN values for the SPRT chart are also given (the ASN of the p-chart is always n = 200). The columns labeled ATS are ATS when p ˆ p0 and SSATS when p > p0 . For small shifts in p, the SPRT chart has a lower OC and thus a lower ANSS and SSATS than the p-chart. It is interesting to note that for small

552

Reynolds and Stoumbos

Table 1. Exact OC, ASN, ANSS, and ATS and SSATS values for the p-chart and the SPRT chart with p0 = 0.01 p-chart p 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

SPRT chart

OC

ANSS

ATS

OC

ASN

ANSS

ATS

0.9957 0.8914 0.6063 0.3084 0.1237 0.0413 0.0119 0.0030

232.80 9.21 2.54 1.45 1.14 1.04 1.01 1.00

931.20 34.85 8.16 3.78 2.56 2.17 2.05 2.01

0.9957 0.5228 0.2026 0.0893 0.0419 0.0203 0.0099 0.0049

199.24 490.39 332.29 236.50 180.45 144.68 120.23 101.82

233.47 2.09 1.25 1.10 1.04 1.02 1.01 1.00

933.88 6.38 3.02 2.39 2.17 2.08 2.04 2.02

shifts in p the ASN of the SPRT chart is above the incontrol value. This means that the SPRT chart tends to take larger samples, and this makes it easier to detect this shift. For larger shifts in p the ASN of the SPRT chart is lower than the in-control value. Although not shown in Table 1, the ASN is lower than the in-control value when p is below p0 so that an improvement in quality results in a reduction in the sampling rate. Table 2 contains some additional SSATS values for a more complete investigation of the performance of the SPRT chart for this application. Columns 2 and 3 of Table 2 reproduce the SSATS values of the p-chart and SPRT chart from Table 1. Columns 4 and 5 are for two additional SPRT charts with nominal values of p1 of 0.025 and 0.030, respectively, and in-control ASN values of approximately 200. The actual values of p1 used were 0:025 01, to give c ˆ 1=61 (column 4), and 0:029 84, to give c ˆ 1=55 (column 5). In addition to p1 , d, and the incontrol ASN, the column headings give ASN/d which is the in-control average number of observations per hour (labeled obs/h). Columns 3-5 can be used to determine the

e€ect of the choice of p1 on the performance of the SPRT chart. The SSATS values in columns 3±5 show that, as expected, using a small value of p1 gives a chart which is more sensitive to small shifts, while using a larger value of p1 gives a chart that is more sensitive to large shifts. Although p1 was initially chosen as 0.020 in the example, process engineers might very well decide that they would prefer to use the chart with p1 ˆ 0:025 instead of the chart with p1 ˆ 0:020. Choosing the chart with p1 ˆ 0:025 would give up a little performance at small shifts to gain a little at large shifts. The additional numerical results to be presented in this example are for SPRT charts with nominal p1 ˆ 0:025. The fact that the SPRT chart is so much more ecient than the p-chart suggests that the average in-control sampling rate of the SPRT chart could be reduced while still providing reasonably fast detection of shifts in p. The p-chart is taking 200 observations every 4 hours, which corresponds to 50 observations per hour. SSATS values for three SPRT charts with nominal p1 ˆ 0:025 and with an approximate in-control average sampling rate of 25

Table 2. Exact ATS and SSATS values for the p-chart and SPRT charts with p0 = 0.01 p-chart

SPRT chart, obs/h » 50

SPRT chart, obs/h » 25

p1 d ASN obs/h

* 4.0 200 50

0.020 4.0 199.2 49.8

0.025 4.0 200.6 50.1

0.030 4.0 201.1 50.3

p = 0.010 0.015 0.020 0.025 0.030 0.040 0.050 0.070 0.100 0.200

931.20 121.57 34.85 14.95 8.16 3.78 2.56 2.05 2.00 2.00

933.88 24.19 6.38 3.87 3.02 2.39 2.17 2.04 2.00 2.00

924.81 33.65 6.91 3.66 2.77 2.23 2.08 2.01 2.00 2.00

924.37 44.93 8.21 3.78 2.71 2.17 2.05 2.01 2.00 2.00

927.00 54.27 13.10 6.71 4.69 3.22 2.67 2.24 2.06 2.00

936.90 54.33 13.02 6.56 4.46 2.88 2.22 1.63 1.28 1.02

939.15 54.35 13.01 6.52 4.41 2.79 2.10 1.46 1.04 0.62

912.37 83.37 23.78 12.42 8.51 5.50 4.26 3.14 2.49 2.04

952.76 87.34 24.91 12.95 8.80 5.57 4.20 2.91 2.07 1.24

960.43 87.57 24.92 12.93 8.77 5.53 4.14 2.82 1.95 0.99

952.88 87.35 24.90 12.93 8.77 5.52 4.12 2.80 1.91 0.91

* * *

69 437/69 )64/69

61 314/61 )80/61

55 243/55 )92/55

61 286/61 )40/61

61 296/61 )20/61

61 301/61 )10/61

61 252/61 )21/61

61 257/61 )10/61

61 260/61 )5/61

61 261/61 )2/61

m h g

0.025 4.0 98.8 24.7

0.025 2.0 49.5 24.7

0.025 1.0 24.7 24.7

SPRT chart, obs/h » 12.5 0.025 4.0 50.6 12.6

0.025 2.0 24.1 12.1

0.025 1.0 12.1 12.1

0.025 0.4 4.8 12.1

The SPRT chart for monitoring a proportion

553

observations per hour are given in columns 6±8, respectively, of Table 2. These three SPRT charts have sampling intervals of 4.0, 2.0, and 1.0 hours, respectively. The SSATS values in columns 6±8 indicate that, for a given in-control average sampling rate, the sampling interval is not important for detecting small shifts, but for detecting large shifts, it is better to take small samples more frequently. The SPRT charts with sampling intervals of 2 hours and 1 hour both have an SSATS that is uniformly lower than the p-chart. Thus, by using an SPRT chart, it is possible to cut the in-control average sampling rate in half and still detect shifts in p faster than when using the p-chart. If the SPRT chart can provide faster detection with half the sampling rate, then an obvious question is the e€ect of additional reductions in the sampling rate of the SPRT chart. Columns 9±12 of Table 2 contain SSATS values for SPRT charts with an approximate in-control average sampling rate of 12.5 observations per hour (one quarter the value for the p-chart). The sampling intervals for these four SPRT charts are, respectively, 4.0, 2.0, 1.0, and 0.4 hours. The values in columns 9±10 indicate that these SPRT charts are a little better than the p-chart for small shifts, but not quite as good for large shifts. For very large shifts in p, the SPRT charts with short sampling intervals (1.0 and 0.4) are better than the p-chart. Thus, it appears that an SPRT chart with one quarter the in-control sampling rate of the p-chart will have roughly the same detection capability. In contrast to the p-chart, for the SPRT chart there is no need to take large samples with a long sampling interval in order to detect small shifts; small samples at short intervals work just as well for detecting small shifts and better for detecting large shifts.

7. Comparison of the SPRT chart and the CUSUM chart In the case considered in the previous example, the SPRT chart detected shifts in p much faster that the p-chart. It would be expected that a CUSUM chart based on the binomial distribution would perform better than the Shewhart p-chart for detecting all but very large shifts. Thus, an important question is how the SPRT chart would compare to the CUSUM chart. Like the p-chart, the standard CUSUM chart for monitoring p (see, for example, Gan [1]) is based on samples of n items taken at a ®xed sampling interval of d. If Yk is used to represent the total number of defectives in the kth sample, then the statistics Y1 ; Y2 ; . . . are independent binomial random variables. The binomial CUSUM chart uses the control statistic: Ck ˆ max…0; Ckÿ1 † ‡ …Yk ÿ cC †;

k ˆ 1; 2; . . . ;

…26†

and signals at sample k if Ck  hC , where C0 is the starting value and cC is the reference value. Gan [1] has discussed Markov chain methods for evaluating the properties of the binomial CUSUM chart. In the binomial CUSUM chart, the appropriate choice of cC is nr1 =r2 to detect a shift to p1 . Consider the example in the previous sections in which the in-control ATS of the p-chart was 931.20 hours. For the case of n = 200, d = 4.0, and nominal p1 ˆ 0:025, using cC ˆ 200=61 and hC ˆ 237=61 in the CUSUM chart gives an in-control ATS of 870.65 hours (using hC ˆ 238=61 would give an in-control ATS of 948.19). SSATS values for this binomial CUSUM chart are given in column 4 of Table 3. For purposes of comparison, SSATS values for the p-chart and the SPRT chart are

Table 3. Exact ATS and SSATS values for the p-chart, and SPRT charts, and CUSUM charts with p0 = 0.01 and p1 = 0.025 obs/h » 50

obs/h » 25

p-chart

SPRT chart

d ASN obs/h

4.0 200 50

4.0 200.6 50.1

4.0 200 50

p = 0.010 0.015 0.020 0.025 0.030 0.040 0.050 0.070 0.100 0.200

931.20 121.57 34.85 14.95 8.16 3.78 2.56 2.05 2.00 2.00

924.81 33.65 6.91 3.66 2.77 2.23 2.08 2.01 2.00 2.00

870.65 63.81 19.07 10.02 6.64 3.90 2.78 2.09 2.00 2.00

927.00 54.27 13.10 6.71 4.69 3.22 2.67 2.24 2.06 2.00

* *

314/61 )80/61

237/61 *

286/61 )40/61

h and hC g

CUSUM chart

SPRT chart 4.0 98.8 24.7

CUSUM chart

obs/h » 12.5

SPRT chart

CUSUM chart

SPRT chart

CUSUM chart

2.0 49.5 24.7

2.0 50 25

2.0 24.1 12.1

2.0 25 12.5

859.82 93.88 32.04 17.57 11.88 7.12 5.01 3.10 2.20 2.00

936.90 54.33 13.02 6.56 4.46 2.88 2.22 1.63 1.28 1.02

917.02 98.68 33.38 18.29 12.32 7.30 5.12 3.20 2.06 1.06

952.76 87.34 24.91 12.95 8.80 5.57 4.20 2.91 2.07 1.24

926.41 141.33 53.88 30.62 20.96 12.69 9.07 5.77 3.72 1.67

232/61 *

296/61 )20/61

261/61 *

257/61 )10/61

241/61 *

4.0 100 25

554

Reynolds and Stoumbos

given in columns 2 and 3, respectively. From the SSATS values in columns 2±4, it can be concluded that the CUSUM chart is much better than the p-chart for detecting small shifts in p, but that the SPRT chart is uniformly better than the CUSUM chart. In fact, the SPRT chart is dramatically better than the CUSUM chart for detecting small shifts in p. If the sample size in the CUSUM chart is changed to n ˆ 100, then using d ˆ 4:0, nominal p1 ˆ 0:025, cC ˆ 100=61, and hC ˆ 232=61 will give an in-control ATS of 859.82 hours. SSATS values for this CUSUM chart are given in column 6 of Table 3. SSATS values for the corresponding SPRT chart from Table 2 are given in column 5. The SPRT chart is also uniformly better than the CUSUM chart in this case. SSATS values for two more pairs of CUSUM and SPRT charts, each with nominal p1 ˆ 0:025, are given in Table 3. The CUSUM chart in the ®rst pair has n ˆ 50 and d ˆ 2:0, and the CUSUM chart in the second pair has n ˆ 25 and d ˆ 2:0. In both of these pairs, the SPRT chart is uniformly better than the CUSUM chart. In the discussion of Table 1, it was noted that a shift in p may produce either an increase or a decrease in the ASN of the SPRT chart, depending on the size of the shift. In particular, a small shift in p will result in an increase in the average sampling rate per unit time. However, the time required to detect the shift is much lower for the SPRT chart than for other charts, so it is not clear how this will a€ect the total number of observations required by the charts to detect the shift. To investigate this question, SSANOS values for the charts from Table 3 were computed and are given in Table 4 (the in-control values are ANOS values). Examining Table 4 shows that the SSANOS values of the SPRT charts are uniformly lower than the SSANOS values of the p-chart. For example, consider the SPRT chart with d ˆ 4:0 and in-control ASN of 200.6. If there is a shift to p ˆ 0:02, then the ASN of this SPRT chart is 482.2, the ANSS is

2.23, and the ANOS is (2.227)(482.2) = 1074. The ANOS of the p-chart is (9.211)(200) = 1842. This means that, on average, this SPRT chart requires both less time and fewer observations to detect a shift to 0.02. Therefore, from a statistical point of view, the SPRT chart's higher sampling rate per unit time is not really a disadvantage because it reduces both the out-of-control time and the out-of-control sampling cost. Comparing the SSANOS values of the pairs of SPRT and CUSUM charts in Table 4 shows that the SPRT chart has a slightly higher SSANOS for small shifts, but a lower SSANOS for large shifts. Thus, the SPRT chart will detect small shifts much faster than the CUSUM chart while, on average, requiring roughly the same number of observations for detection. For large shifts the detection times of the two charts are roughly the same, but the SPRT chart requires fewer observations. The comparisons between the SPRT chart, the p-chart and the CUSUM chart that have been given here are for a particular case of p0 ˆ 0:01. We have looked at many other cases with p0 ranging from 0.001 to 0.5 and found similar results on the relative performance of the three charts. It is not possible to present all of these results here, but one more set of comparisons will be presented to illustrate the relative performance of the charts for a larger value of p0 . Consider the case in which the target value for p is p0 ˆ 0:05 and samples of size n ˆ 100 will be taken every d=4 hours. Suppose that the in-control ANSS should be approximately 250 so that the in-control ATS should be approximately 1000 hours. A p-chart using three-sigma limits has an upper control limit of 0.1154 and this corresponds to signaling if there are 12 or more defectives in the sample of 100. The in-control ATS of this chart is 935.85. Some SSATS values for this p-chart are given in the column 2 of Table 5. Consider now the design of an SPRT chart with incontrol ASN  100 and in-control ATS  1000. Suppose

Table 4. Exact ANOS and SSANOS values for the charts in Table 3

d ASN p = 0.010 0.015 0.020 0.025 0.030 0.040 0.050 0.070 0.100 0.200

p-chart

SPRT chart

CUSUM chart

SPRT chart

CUSUM chart

SPRT chart

CUSUM chart

SPRT chart

CUSUM chart

4.0 200

4.0 200.6

4.0 200

4.0 98.8

4.0 100

2.0 49.5

2.0 50

2.0 24.1

2.0 25

46 560 6 178 1 842 848 508 289 228 202 200 200

46 370 3 524 1 074 575 386 231 164 103 66 30

43 533 3 291 1 054 601 432 295 239 205 200 200

22 887 2 557 888 495 338 206 148 94 61 28

21 496 2 397 851 489 347 228 175 128 105 100

23 167 2 564 891 498 341 208 150 96 63 30

22 926 2 492 860 482 333 208 153 105 77 52

11 484 1 828 717 416 289 178 129 83 54 25

11 580 1 779 686 395 275 171 126 85 59 33

The SPRT chart for monitoring a proportion

555

Table 5. Exact ATS and SSATS values for p-charts, SPRT charts, and CUSUM charts with p0 = 0.05 and d = 4.0 p-chart

SPRT chart

p1 ASN

* 100

0.077 101.2

p = 0.050 0.055 0.058 0.060 0.065 0.070 0.075 0.080 0.090 0.100 0.150 0.200

935.85 447.80 302.30 236.77 135.75 83.29 54.10 36.89 19.32 11.47 2.78 2.05

977.68 198.78 88.74 55.48 21.83 11.62 7.64 5.73 3.99 3.21 2.20 2.04

* *

160/16 )21/16

h and hC g

0.098 98.2

CUSUM chart

p-chart

SPRT chart

0.077 100

0.098 100

* 51

975.83 289.17 149.48 99.19 39.71 18.75 10.44 6.74 3.90 2.93 2.07 2.01

974.95 279.86 154.57 110.19 56.13 34.48 23.97 18.07 11.86 8.72 3.58 2.24

985.26 363.70 216.59 158.08 79.69 45.88 29.50 20.68 12.15 8.28 2.96 2.09

1105.51 624.21 458.05 377.11 240.69 160.92 111.94 80.57 45.45 28.10 5.92 2.84

971.50 252.90 127.94 85.97 38.08 21.35 14.21 10.58 7.11 5.49 3.01 2.41

1001.40 340.15 190.18 132.66 59.46 30.65 18.04 11.93 6.77 4.77 2.50 2.14

981.64 340.38 206.10 154.34 85.68 55.21 39.46 30.27 20.35 15.20 6.39 3.90

984.13 414.97 265.53 202.45 112.04 69.22 46.83 34.04 20.94 14.68 5.36 3.09

98/14 )30/14

132/16 *

78/14 *

* *

142/16 )11/16

92/14 )16/14

120/16 *

75/14 *

that p1 is initially taken to be 0.075. Using the second design procedure from Section 4 results in p1 being adjusted to about 0.0768 to give c ˆ 1=16. The remaining part of the design procedure gives g ˆ ÿ21=16 and h ˆ 160=16. SSATS values for this SPRT chart are given in column 3 of Table 5. To investigate the e€ect of the choice of p1 on the SPRT chart, consider the choice of p1 = 0.10. Repeating the design procedure using this value of p1 results in p1 being adjusted to 0.0977 with c ˆ 1=14, g ˆ ÿ30=14, and h ˆ 98=14. SSATS values for this second SPRT chart are given in column 4 of Table 5. Comparing columns 3 and 4 shows that the ®rst SPRT chart with p1 ˆ 0:0768 is better for detecting small increases in p, while the second SPRT chart with p1 ˆ 0:0977 is a little better for detecting large increases in p. Both SPRT charts are much better than the p-chart. To determine how the CUSUM chart would perform in this example, two CUSUM charts were constructed using n ˆ 100 and the same values of p1 as the two SPRT charts in columns 3 and 4. SSATS values for these two CUSUM charts are given in columns 5 and 6 of Table 5. The SPRT charts in columns 3 and 4 have much lower SSATS values than the corresponding CUSUM charts in columns 5 and 6. The ®ve charts in columns 2±6 of Table 5 have an approximate sample size of 100. To investigate the e€ect of reducing the sample size, ®ve more charts were constructed with an approximate sample size of 51. A pchart with n ˆ 51 and three-sigma limits has a upper control limit of 0.1416 (signal if the number of defectives in a sample is 8 or more). SSATS values for this p-chart are given in column 7 of Table 5. SSATS values for two SPRT charts with an in-control ASN of 51.8 are given in columns 8 and 9, and SSATS values for two CUSUM charts with n ˆ 51 are given in columns 10 and 11. The

0.077 51.8

0.098 51.8

CUSUM chart 0.077 51

0.098 51

SPRT and CUSUM charts in columns 8±11 have the same values of p1 as the corresponding charts in columns 3±6. The general e€ect of reducing the sample size is of course to increase the time required by all of the charts to detect a speci®c shift. But the conclusions about the relative performance of the charts are the same for n ˆ 51 as for n ˆ 100; the CUSUM chart is much better than the p-chart for detecting all but large shifts, and the SPRT chart is much better than the CUSUM chart. SSANOS values were computed for all of the charts in Table 5. The conclusions about the relative SSANOS performance of the charts are similar to the conclusions obtained for the charts in Table 4, so the results are not given here. The CUSUM chart is known to be the optimal ®xed sampling rate control chart for detecting a speci®c shift in a process parameter. In Tables 3 and 5 the superiority of the SPRT chart over the CUSUM chart is dramatic. When the SPRT chart and the CUSUM chart have approximately the same in-control average sampling rate and approximately the same false alarm rate, the SPRT chart will on average require less than half the time of the CUSUM chart to detect the shift at which the CUSUM is optimal. An alternate way to look at the comparison of the SPRT chart and the CUSUM chart is to note that the SPRT chart will provide a given shift detection capability with a much smaller average in-control sample size. If the CUSUM charts in Tables 3 and 5 are compared with SPRT charts with half the in-control sampling rate, the SPRT charts will provide faster detection except at large shifts. For example, compare the SPRT chart in column 8 of Table 5 with an in-control ASN of 51.8 to the CUSUM chart in column 5 with n = 100. The SPRT chart will provide much faster detection of all shifts except for p ˆ 0:200.

556

Reynolds and Stoumbos

8. The SPRT chart for detecting a decrease in p In most quality control applications the primary interest would be in detecting an increase in p, but detecting a decrease in p might be of interest if e€orts were being made to improve the process. Unfortunately, the standard p-chart with three-sigma limits does not have a lower control limit for many of the values of n and p0 that are commonly used in applications. An SPRT chart, however, can easily be designed to detect a decrease in p. If p1 < p0 , then both r1 and r2 will be negative and the critical inequality in (6) of the SPRT is equivalent to: h < Tj ÿ cj < g;

…27†

where g ˆ b=r2 and h ˆ a=r2 . The null hypothesis H0 : p ˆ p0 is rejected if Tj ÿ cj  h and accepted if Tj ÿ cj  g. The CD approximations given in Section 3 for the case p1 > p0 can be applied to the current case of p1 < p0 by replacing h with the original h and replacing g with: g ˆ g ‡ …1 ÿ 2p0 †=3;

…28†

in (18) and (19). In quality control applications in which detecting a decrease in p is of interest it would also be desirable to detect an increase in p. Unfortunately, it is not a simple matter to develop a two-sided SPRT chart by just running two one-sided SPRT charts simultaneously. The two one-sided charts would typically stop sampling at di€erent points, and thus it is not completely clear how to de®ne the decision rules to be used. Because the development of the decision rules for the two-sided SPRT chart is not straightforward, this development will be pursued in a separate paper.

9. Conclusions and discussion The SPRT chart has been presented here for the problem of monitoring the process proportion defective p. Using the highly accurate CD approximations to the OC and ASN functions, it is easy to design an SPRT chart for use in an application. The SPRT chart is much more ecient in detecting shifts in p than either the traditional Shewhart p-chart or the CUSUM chart. This means that the SPRT chart can be used to give much faster detection of process changes for a given in-control average sampling rate and false alarm rate, or it can be used to give a required detection capability at a much lower in-control average sampling rate. When there is a large shift in p, the average sampling rate per unit time for the SPRT chart will be lower than the in-control sampling rate, but for a small shift in p the average sampling rate will be higher. Although the SPRT chart has a higher sampling rate for small shifts, the expected number of observations required to detect this shift will be lower than the expected

number for the p-chart, and roughly the same as the expected number for the CUSUM chart. The statistical properties of the SPRT chart have been developed here for the case in which the time required to obtain an individual observation is short enough that it can be neglected, relative to the time between samples. In some applications, however, the time between observations will be longer, and it will not be reasonable to neglect this time. The SPRT chart can still be used in these applications, and the design methods given is this paper would usually provide a reasonable design. But, if a precise evaluation of the statistical properties of the SPRT chart are required in this case, then some of the results given in this paper would need to be modi®ed. For example, if the time required to take a sample cannot be neglected, then the SSATS computation would have to account for the possibility of p shifting during the time that a sample is being taken. This issue arises in the development and the evaluation of the statistical properties of all control charts, including FSS charts such as the standard p-chart and the CUSUM chart. It is an issue, however, which is usually ignored. In the extreme case in which the time required to inspect an item is very long, it is possible that sampling for one SPRT would not be ®nished before the time at which the next SPRT should be started. The application and evaluation of SPRT charts in situations of theses types can be handled in several ways as is discussed in Stoumbos and Reynolds [6]. In some applications, the rate of production may be slow, and/or the in-control value p0 may be very low, and this may result in a relatively high proportion of items being inspected. This is another example of a situation in which it would not be reasonable to neglect the time required to take a sample. The application of SPRT charts in this type of situation is considered by Stoumbos and Reynolds [6]. It is not uncommon for all items from a process to be inspected. For example, 100% inspection would be reasonable if the inspection process is automated. With 100% inspection, the sampling rate could not be varied, and in this case, a CUSUM chart would be appropriate as is discussed in Reynolds and Stoumbos [20]. In general, when using a variable sampling rate control chart, an additional issue that needs to be considered is the possibility of increased administrative cost associated with the varying sampling rate. With the SPRT chart, the sample size to be used at a sampling point is not known until sampling stops at this point. In addition, there is no upper bound to this sample size. In applications of the SPRT, it is common to truncate the test so that there is a reasonable upper bound to the sample size, so this could be done for the SPRT chart. In some cases, it may be more administratively convenient to take observations in groups. For example, if it is desirable to have an in-control ASN of about 50, then observations could be taken in groups of 10, and the results for each group of 10 items could be used in the SPRT at each sampling point. In this

The SPRT chart for monitoring a proportion case, the statistic used in the SPRT would have a binomial distribution, and it would be necessary to apply the SPRT for the binomial distribution. Applying an SPRT using groups would not be quite as ecient as using the individual observations, but it would usually be more ecient than the standard p-chart and CUSUM chart. An issue related to the administrative cost of the SPRT chart is the possibility that the SPRT chart may require more e€ort to plot the control statistic than is required by other types of charts. If the SPRT chart is applied using computer software then plotting would not be an issue. The e€ort required for hand plotting could be reduced by noting that the control statistic moves down at the constant rate of 1=m as long as nondefectives are observed, and jumps up by an amount …m ÿ 1†=m when a defective is observed. Thus, it is not really necessary to plot every point to determine the decision to be made.

Acknowledgement Zachary G. Stoumbos' work was supported in part by a 1996 Rutgers Faculty of Management Research Fellowship and a 1996 Rutgers Faculty of Management Research Grant.

References [1] Gan, F.F. (1993) An optimal design of CUSUM control charts for binomial counts. Journal of Applied Statistics, 20, 445±460. [2] Reynolds, M.R., Jr., Amin, R.W., Arnold, J.C. and Nachlas, J.A. (1988) X charts with variable sampling intervals. Technometrics, 30, 181±192. [3] Reynolds, M.R., Jr., Amin, R.W. and Arnold, J.C. (1990) CUSUM charts with variable sampling intervals. Technometrics, 32, 371±384. [4] Vaughan, T.S. (1993) Variable sampling interval np process control chart. Communications in Statistics: Theory and Methods, 22, 147±167. [5] Reynolds, M.R., Jr. (1996) Variable sampling interval control charts with sampling at ®xed times. IIE Transactions, 28, 497±510. [6] Stoumbos, Z.G. and Reynolds, M.R., Jr. (1996) Control charts applying a general sequential test at each sampling point. Sequential Analysis, 15, 159±183. [7] Stoumbos, Z.G. and Reynolds, M.R., Jr. (1997) Control charts applying a sequential test at ®xed sampling intervals. Journal of Quality Technology, 29, 21±40. [8] Prabhu, S.S., Runger, G.C. and Keats, J.B. (1993) An adaptive sample size X chart. International Journal of Production Research, 31, 2895±2909. [9] Costa, A.F.B. (1994) X charts with variable sample size. Journal of Quality Technology, 26, 155±163. [10] Park, C. and Reynolds, M.R., Jr. (1994) Economic design of a variable sample size X chart. Communication in Statistics: Simulation and Computation, 23, 467±483. [11] Rendtel, U. (1990) CUSUM schemes with variable sampling intervals and sample sizes. Statistical Papers, 31, 103±118. [12] Prabhu, S.S., Montgomery, D.C. and Runger, G.C. (1994) A combined adaptive sample size and sampling interval X control scheme. Journal of Quality Technology, 26, 164±176.

557 [13] Costa, A.F.B. (1997) X chart with variable sample size and sampling intervals. Journal of Quality Technology, 29, 197±204. [14] Park, C. and Reynolds, M.R., Jr. (1998) Economic design of a variable sampling rate X chart. Journal of Quality Technology, (to appear). [15] Wald, A. (1947) Sequential Analysis, Dover Publications, Inc., New York, NY. [16] Ghosh, B.K. (1970) Sequential Tests of Statistical Hypotheses, Addison±Wesley Publishing Company, Boston, MA. [17] Daudin, J.J. (1992) Double sampling X charts. Journal of Quality Technology, 24, 78±87. [18] Stoumbos, Z.G. and Reynolds, M.R., Jr. (1996) Variable sampling rate control charts with sampling at ®xed intervals, in Proceedings of the International Industrial Engineering Research Conference, Institute of Industrial Engineers, Norcross, GA. pp. 687±692. [19] Stoumbos, Z.G. and Reynolds, M.R., Jr. (1997) Corrected di€usion theory approximations in evaluating properties of SPRT charts for monitoring a process mean. Nonlinear Analysts, 30, 3987±3996. [20] Reynolds, M.R., Jr. and Stoumbos, Z.G. (1998) A CUSUM chart for monitoring a proportion when inspecting continuously. Journal of Quality Technology, (in press). [21] Siegmund, D. (1979) Corrected di€usion approximations in certain random walk problems. Advances in Applied Probability, 11, 701±719. [22] Siegmund, D. (1985) Sequential Analysis, Springer±Verlag, New York, NY. [23] Cheney, W. and Kincaid, D. (1985) Numerical Mathematics and Computing, 2nd edn., Brooks/Cole Publishing Company, Paci®c Grove, CA. [24] Burden, R.L. and Faires, J.D. (1993) Numerical Analysis, 5th edn., PWS Publishing Company, Boston, MA.

Appendix Appendix A: computing exact properties of the SPRT chart For practical applications, the SPRT chart can be designed and evaluated using the CD approximations that were given in the main text. For precise evaluations and comparisons, exact methods based on a Markov chain model can be used. These exact methods will be outlined in this appendix. Consider ®rst the case of p0 < p1 . The possible values of the observation Xi are 0 and 1, and thus the possible values of the SPRT statistic Tj ÿ cj in (7) are determined by the choice of c ˆ r1 =r2 . When c is a rational number, there will be a ®nite number of possible values for Tj ÿ cj, and then these values will correspond to the states of the Markov chain. In most quality control applications, the values of p0 and p1 will be relatively small, and in this case it will be reasonable to make a slight adjustment in p1 so that c ˆ r1 =r2 ˆ 1=m, where m is a positive integer. To make the presentation of the Markov chain as simple as possible, the case in which c ˆ 1=m will be presented here. In this case, g and h can be taken to be integer multiples of 1=m, and then there are t ˆ mh ÿ mg ÿ 1 possible values of Tj ÿ cj which are between g and h. Let the transient state i of the Markov chain correspond to the value g ‡ …i=m†, for i ˆ 1; 2; . . . ; t. The possible values of

558

Reynolds and Stoumbos

…Xi ÿ 1=m† are ÿ1=m and …m ÿ 1†=m, and thus after each observation, the transition will either be down 1 state or up …m ÿ 1† states. Let c ˆ t ÿ m ‡ 1 represent the highest state from which it is not possible to reject H0 . This means that if the current state is i  c, then observing a defective will result in a transition to state i ‡ m ÿ 1  t, but if the current state is i > c, then observing a defective will result in rejecting H0 and a signal. The transition probability matrix for this Markov chain is relatively simple. In particular, if pij is the transition probability from state i to state j, then: p1m ˆ p; pi;iÿ1 ˆ q; pi;iÿ1 ˆ q;

pi;i‡mÿ1 ˆ p;

for i ˆ 2; 3; . . . ; c;

.. . Nc ˆ 1 ‡ qNcÿ1 ‡ pNt ; Nc‡1 ˆ 1 ‡ qNc ; Nc‡2 ˆ 1 ‡ qNc‡1 ; .. . Nt ˆ 1 ‡ qNtÿ1 :

The approach used to solve these equations is to ®nd Nc , and then for i 6ˆ c, obtain Ni from Nc . For i > c, it is relatively easy to show that:

…A1†

Ni ˆ

for i ˆ c ‡ 1; c ‡ 2; . . . ; t;

where q ˆ 1 ÿ p. Let Ni be the ASN when starting in state i, and let N ˆ …N1 ; N2 ; . . . ; Nt †0 be the vector of ASN values for the t transient states. If Q is the transition probability matrix determined by (A1) for these states, then it follows from basic properties of Markov chains that the ASN vector is: N ˆ …I ÿ Q†ÿ1 1;

…A2†

where 1 is a column vector of 1's. The ASN of interest would usually be the component of N corresponding to starting at the value 0. Similarly, if Ai is the OC function when starting in state i, and A is the vector of OC values corresponding to the t transient states, it follows that: A ˆ …I ÿ Q†ÿ1 f;

…A3†

where f is the vector of probabilities of accepting for the t states. It is possible to accept H0 only from state 1, so all components of f are 0 except for the ®rst, which is q. Writing a computer program to ®nd the ASN and OC using (A2) and (A3) is straightforward, but when the values of p0 and p1 are small, the number of states will be quite large, and then it may not be possible to work directly with the transition probability matrix Q. For example, an SPRT chart for p0 =0.001 in Appendix C has Q of dimension 1818  1818. Although the dimension of Q may be very large, the structure of Q is actually very simple. This means that it is possible to solve systems of equations and obtain explicit expressions for the OC and ASN. The resulting expressions are rather complicated, but they avoid the problem of directly manipulating the large matrix Q. The direct solution will be given here for the case of p0 < p1 . Using (A2) or a direct argument shows that N satis®es the equation: N ˆ 1 ‡ QN: Writing out this system of equations gives: N1 ˆ 1 ‡ pNm ; N2 ˆ 1 ‡ qN1 ‡ pNm‡1 ; N3 ˆ 1 ‡ qN2 ‡ pNm‡2 ;

…A5†

1 ÿ qiÿc ‡ qiÿc Nc : p

…A6†

Once Nc and Ni , for i > c, are obtained, Ni , for i < c, can be obtained by ®nding Ncÿ1 ; Ncÿ2 ; . . . ; N2 successively, using the equation: Ni ˆ 1q…Ni‡1 ÿ pNi‡m ÿ 1†:

…A7†

Equation (A7) is obtained from the equation Ni‡1 ˆ 1 ‡ qNi ‡ pNi‡m , which, from (A5), holds for i ˆ 2; 3; . . . ; c. N1 is obtained using the ®rst equation in (A5). The expression for Nc is obtained by working up through the equations in (A5), with each Ni being expressed in terms of Ni‡1 ; Ni‡2 ; . . .. Once Nc is reached in the sequence, (A6) can be used to give an equation with terms involving only Nc . This equation can then be solved explicitly for Nc . The resulting solution can be expressed using a set of constants W …i0 ; i† de®ned by: W …0; i† ˆ 1;

for i ˆ 0; 1; . . . ; m;

and for i0  1, 8 W …i0 ÿ 1; m† > < W …i0 ÿ 1; m† ÿ pqmÿ1 W …i0 ÿ 1; 1† W …i0 ; i† ˆ 0 mÿ1 0 > : W …i ; i ÿ 1† ÿ pq W …i ÿ 1; i†; if i ˆ 2; 3; . . . ; m:

…A8† if i ˆ 0, if i ˆ 1, (A9)

The constants W …i0 ; i† can be evaluated recursively. Express c as c ˆ i0 m ‡ k, for some i0 ˆ 1; 2; . . . and k ˆ 0; 1; . . . ; m ÿ 1. Then, the result of solving for Nc as described above is: "0 i ÿ1 m X X 1 jm‡k q qmÿi W …i0 ÿ 1 ÿ j; i† Nc ˆ W …i0 ‡ 1; k† jˆ0 iˆ1 ‡

m X iˆk‡2

…A4† ‡

k X

W …i0 ÿ 1; i†…qm‡kÿi ÿ qmÿ1 † #

W …i0 ; i†…2qkÿi ÿ qmÿ1 † :

…A10†

iˆ1

This expression will allow the evaluation of the ASN for p0 < p1 for any number of states in the Markov chain. However, (A8), (A9), and (A10) are suciently compli-

The SPRT chart for monitoring a proportion

559

cated that a program would usually be required for evaluation. For the OC function, the system of equations corresponding to (A3) is: A ˆ f ‡ QA:

…A11†

Writing out this system of equations and solving gives: Ai ˆ qiÿc Ac ;

for

i ˆ c ‡ 1; c ‡ 2; . . . ; t;

0

…A12†

0

and for c ˆ i m ‡ k, for some i ˆ 1; 2; . . . and k ˆ 0; 1; . . . ; m ÿ 1, 0

qi m‡k Ac ˆ : W …i0 ‡ 1; k†

…A13†

To ®nd Ai , for i < c, start with Ac and work down using: Ai ˆ

Ai‡1 ÿ pAm‡i ; q

for i ˆ 2; 3; . . . ; c ÿ 1;

…A14†

and A1 ˆ q ‡ pAm . Now consider the case of p1 < p0 in which the critical inequality of the SPRT is (27). If c ˆ r1 =r2 ˆ 1=m, where m is a positive integer, then as before, g and h can be taken to be integer multiples of 1=m. Then, there will be t ˆ mg ÿ mh ÿ 1 transient states in the Markov chain, where transient state i corresponds to the value h ‡ …i=m†, for i ˆ 1; 2; . . . ; t. Let c ˆ t ÿ m ‡ 1 represent the highest state from which it is not possible to accept H0 . Then the transition probability matrix Q is determined from (A1), and the ASN can be obtained from (A2). The OC can be obtained from (A3), except that the ®rst c components of f are now 0 and the last …t ÿ c† components are p. Appendix B: the CD approximations Wald [15] originally developed the SPRT and derived some very simple approximations to its OC and ASN functions. These approximations ignore the excesses (overshoots) of the SPRT's test statistic over the acceptance and rejection limits at the termination of the SPRT and typically are not very accurate. Much later, for a certain exponential family of continuous distributions, Siegmund [21,22] developed corrected di€usion (CD) approximations by deriving the limiting expected excesses of the SPRT's test statistic over the acceptance and rejection limits and correcting for these excesses in Wald's approximations. Siegmund's corrections substantially improve the accuracy of Wald's approximations. The Bernoulli distribution being considered in this paper does not belong to the exponential family of continuous distributions that Siegmund [21,22] considered. Therefore, Siegmund's approximations do not apply to the SPRT for monitoring a Bernoulli proportion p. Here, we extend the work of Siegmund [21,22] for the OC and ASN of an SPRT for p, by standardizing a Bernoulli random variable and approximating this standardized variable by a Brownian motion process. Then, we use a

smoothing argument to derive the limiting expected excess of the SPRT's test statistic Tj ÿ cj over h. When p0 < p1 the issue of expected excess below g does not arise, because, by design, the test statistic drops by steps of size c and will exactly hit g when accepting H0 . The derivation of the CD approximations to the ASN and OC of the SPRT for p is outlined below. The technical, mathematical details of this derivation are beyond the scope of this paper, and therefore will be suppressed and presented in a future theoretical paper. Let X denote a Bernoulli random variable with < p0 < 1, and standardize X to P …X ˆ 1† ˆ p0 , wherep0 obtain Z ˆ …X ÿ p0 †= p0 …1 ÿ p0 †. Then, it can be shown that:     1 ÿ p0 1=2 1 ÿ p0 ÿ1=2 ÿ ; …A15† Ep0 …Z 3 † ˆ p0 p0 and

(   ) 1 ÿ p0 1=2 Ep0 …expfikZg† ˆ exp ik p0 p0 (  )  1 ÿ p0 ÿ1=2 ‡ exp ÿik …1ÿp0 †; …A16† p0 p where i ˆ ÿ1. Further, let: e…p0 † ˆ

1 Ep …Z 3 † 6 0

( " 1 2 Relog 2 1 ÿ Ep0 k2 k 0 ( )#) …ks†2 …expfikZg†exp ÿ dk: 2 1 ÿ lims!0 p

Z1

…A17†

Then, it can be shown that the limiting expected excess of the p SPRT's statistic over h is equal to test  e…p0 † p0 …1 ÿ p0 †. It can also be shown that e…p0 † satis®es the relationship: e…p0 † ˆ ÿ13E1ÿp0 …Z 3 † ‡ e…1 ÿ p0 † ˆ 13Ep0 …Z 3 † ‡ e…1 ÿ p0 †: …A18† For 0 < p0  0:5, a simple approximation to e…p0 † in equation (A17) that provides very good approximations to the OC and ASN of the SPRT for p is: e…p0 †  Ep0 …Z 3 †=3:

…A19†

Then, for 0 < p0  0:5, an approximation to the expected excess over h is: p …A20† Ep0 …Z 3 † p0 …1 ÿ p0 †=3 ˆ …1 ÿ 2p0 †=3: For 0:5 < p0 < 1, a very good approximation to e…p0 † is obtained by evaluating e…1 ÿ p0 † using approximation

560 (A19), and then substituting the result in relation (A18). Because e…p0 † requires a complicated numerical evaluation, in many practical situations approximations (A19) and (A20) will be very useful. The CD approximations in (18) and (19) to the OC and ASN functions of the SPRT for p are obtained by using Wald's original approximations, and replacing h with h given by (17) if p0 < p1 , and replacing g with g given by (28) if p0 > p1 . It should be noted that the target proportion p0 is used throughout Equations (A15)±(A20) in determining the expected excess over h for both p ˆ p0 and p 6ˆ p0 , because the adjustment for di€erent values of p has been accounted for in Wald's original approximations through n…p†.

Appendix C: the accuracy of the CD approximations The CD approximations were used in designing the SPRT charts in the examples in Sections 5 and 6. In Section 5 the exact values of quantities such as the ASN and ANSS which were reported were very close to the values given by the CD approximations. In this appendix, some additional results and comparisons for the CD approximations will be provided. Also here, the use of (18) in (11) to approximate the ANSS of the SPRT chart for p 6ˆ p0 ; p1 , or r1 =r2 will be discussed. To evaluate the accuracy of the CD approximations in the design of SPRT charts for a case in which p0 is larger than the value 0.01 used in the example in Section 5, consider the example of Section 6 which had p0 ˆ 0:05 and p1 ˆ 0:075. The objective was to design an SPRT chart with in-control ASN  100 and in-control ATS  1000. Going through the design procedure gave g ˆ ÿ21=16 and h ˆ 160=16. The exact in-control ASN is 101.21 and the exact in-control ATS is 977.68. For p ˆ p1 , the exact SSATS is 6.82, and the CD approximation also gives 6.82. Thus, in this case, the CD approximations yield very accurate results. For another example, consider a case in which the value of p0 is very small. It might be expected that the design and evaluation of the SPRT chart would be the most dicult when p0 is very small. Suppose that p0 ˆ 0:001 and p1 ˆ 0:003, and the objective is to design an SPRT chart with in-control ASN  200 and in-control ANSS  250. If p1 is adjusted slightly to about 0.003 0019, then this will give c ˆ 1=549. The design procedure gives g ˆ ÿ98=549 and h ˆ 1721=549. The Markov chain for this case has t ˆ 1818 transient states, so to ®nd exact properties, it is necessary to use the expressions given in Appendix A, which do not require the direct manipulation of the transition matrix Q. The exact in-control ASN is 199.63, and the exact in-control ANSS is 248.74. For p ˆ p1 , the exact ANSS is 5.50, and the CD approximation also gives 5.50. Thus, in this case as well, the CD approximations give very accurate results.

Reynolds and Stoumbos In a very extensive number of cases considered, the design procedure that is based on the CD approximations yielded SPRT charts with in-control ASN and in-control ANSS values that were well within 5% of the speci®ed values for the in-control ASN and in-control ANSS. In fact, as demonstrated by the examples presented in this paper, the great majority of these cases yielded SPRT charts with in-control ASN and in-control ANSS values well within 1 or 2% of the speci®ed values for the incontrol ASN and ANSS. The CD approximations provide a highly accurate and simple procedure for the design of SPRT charts for practical applications. To use (18) to approximate the OC function (and in turn the ANSS) for p 6ˆ p0 ; p1 , or r1 =r2 , the value of n…p† that satis®es Equation (20) must be determined numerically. The iterative Newton±Raphson procedure (discussed in Cheney and Kincaid [23] and Burden and Faires [24]) can be used to ®nd n…p†. The Newton±Raphson procedure applied directly to Equation (20) was found to be quite sensitive to the choice of starting value. It was found that applying the Newton±Raphson procedure to the equivalent equation  pÿ 1ÿ

1 ÿ p1 1 ÿ p0

n…p† !, 

p1 p0

n…p†   ! 1 ÿ p1 n…p† ˆ0 ÿ 1 ÿ p0 …A21†

gives a procedure that is insensitive to the choice of starting value. Extensive computational results indicated that the starting values 0:1 would work very well and give rapid convergence in a few iterations. Tables A1 and A2 give exact and approximate ASN and ANSS values for the SPRT charts discussed earlier in this appendix. The values in the columns labeled UD are based on the original uncorrected di€usion (UD) approximations developed by Wald [15]. These UD approximations use h instead of h in (18) and (19). Tables A1 and A2 also give n…p† and the number of iterations in the Newton±Raphson procedure used to ®nd n…p† using (A21). The CD approximations to the ASN and ANSS are all very close to their exact counterparts, and their determination only requires 2±4 iterations of Equation (A21). The CD approximations o€er a substantial improvement in accuracy over their UD counterparts, especially for values of p near p0 . The CD approximations can be used to conveniently obtain accurate ANSS values for shifts in p of interest, which can be used to decide whether a chosen SPRT chart will provide acceptable detection times for these shifts. Note that in Tables A1 and A2, the approximate in-control ASN is not exactly equal to the speci®ed values 100 and 200, respectively, and the approximate in-control ANSS is not exactly equal to the speci®ed value of 250. The reason for this is the use of g and h that have been rounded to the nearest integer multiple of c.

The SPRT chart for monitoring a proportion

561

Table A1. Exact and approximate ASN and ANSS values for an SPRT chart with p0 = 0.050, p1 = 0.077, m = 16, g = )21/16, and h = 161/16 Approximations to ASN p

Approximations to ANSS

Exact ASN

CD

UD

Exact ANSS

CD

UD

n(p)

Iterations

101.3 108.5 116.3 125.0 144.6 166.9 203.1 256.0 281.7

101.3 108.5 116.4 125.0 144.6 166.9 203.2 256.1 281.8

100.9 107.9 115.6 123.9 142.8 164.1 198.3 247.8 272.1

251.5 178.6 128.3 93.2 51.1 29.5 14.5 6.0 3.4

252.6 179.3 128.7 93.5 51.2 29.6 14.5 6.0 3.4

220.0 158.0 114.7 84.2 47.1 27.7 13.9 5.9 3.4

* 0.914 0.830 0.746 0.582 0.422 0.189 )0.184 )0.540

* 3 3 3 3 3 2 3 3

0.050 0.051 0.052 0.053 0.055 0.057 0.060 0.065 0.070

Table A2. Exact and approximate ASN and ANSS values for an SPRT chart with p0 = 0.001, p1 = 0.003, m = 549, g = )98/549, and h = 1721/549 Approximations to ASN p

Approximations to ANSS

Exact ASN

CD

UD

Exact ANSS

CD

UD

n(p)

Iterations

199.6 216.3 233.9 252.1 288.6 322.9 365.1 413.9 395.4

199.6 216.3 233.7 251.8 288.0 322.0 363.5 410.1 390.6

193.9 208.2 222.9 237.8 266.9 293.5 325.6 363.1 347.9

248.7 161.2 109.3 77.0 42.4 26.1 15.0 5.5 3.5

250.6 162.2 109.7 77.2 42.4 26.2 15.0 5.5 3.5

172.3 117.2 82.9 60.8 35.7 23.2 14.0 5.5 3.5

* 0.851 0.713 0.583 0.343 0.124 )0.173 )0.999 )1.685

* 3 3 3 3 2 3 4 4

0.0010 0.0011 0.0012 0.0013 0.0015 0.0017 0.0020 0.0030 0.0040

Finally, the CD approximations can also be used in SPRTs with non-integer r1 =r2 , where the issue of expected excess below g when p0 < p1 may arise when accepting H0 . A general theoretical investigation of CD approximations is beyond the scope of this paper, and therefore will be withheld and presented in a future theoretical paper.

Biographies Marion R. Reynolds, Jr. is a Professor in the Departments of Statistics and Forestry at Virginia Tech. He received a B.S. in Statistics from Virginia Tech, an M.S. in Statistics from Stanford University, and a Ph.D. in Operations Research from Stanford University. He has published papers on statistical quality control, sequential analysis,

validation of simulation models, and applications of statistics to forestry. He is a member of ASA, ASQ, IIE, and INFORMS, and is on the editorial boards of the Journal of Quality Technology and Sequential Analysis. Zachary G. Stoumbos is an Assistant Professor in the Department of Management Science and Information Systems and a Member of the Rutgers Center for Operations Research (RUTCOR) at Rutgers, The State University of New Jersey. He received a B.S. in Mathematics from the University of North Carolina at Charlotte, an M.S. in Statistics from Virginia Tech, and a Ph.D. in Statistics from Virginia Tech. He has published papers on statistical quality control, sequential analysis, and control systems. He is a member of ASA and ASQ, and senior member of IIE. Contributed by On-line Quality Engineering Department.