Alternative attribute control charts based on improved square root transformation.

Abstract

In the real world applications nowadays, the situation of low defect level in process often e[x.sub.i]sts, and it results that the performances of classical attribute control charts become bad. Historically, classical attribute control charts have been developed by using the normal appro[x.sub.i]mation. However, the normal appro[x.sub.i]mations are far from adequate for the situation of low defect level and the sample size is not large enough, mainly due to skewness in the exact distribution. In this paper, an improved square root transformation, named ISRT, is used to construct the ISRT p-chart, np-chart and c-chart for charting the binomial data and Poisson data. Comparing the ISRT p-chart with several known p-charts, the minimum sample sizes required for obtaining positive lower control limits for the ISRT p-chart are small. Numerical results also indicate that the ISRT control charts can match any specific percentile point of run length distribution of the true limits when the parameter is unknown.

Keywords and Phrases: Attribute control charts, Control limits, Improved square root transformation, Normal appro[x.sub.i]mation, Q-chart.

1. Introduction

Many quality characteristic cannot be conveniently measured numerically. In such case, the attribute control charts, for example p-chart, np-chart and c-chart, are often used to chart the parameters of process. Attribute control charts have been historically developed using the normal appro[x.sub.i]mation. Suppose the process observations are taken from a binomial distribution with parameters n and p, where n denotes the number of items produced and inspected and p denotes the defect level of process. The classical p-chart and np-chart have been developed for charting the binomial data. When p is known, the classical p-chart and np-chart can be constructed with the center lines (CLs) p and np and the control limits are p [+ or -] 3 [square root of p (1 - p)]/n and np [+ or -] 3[square root of np (1 - p)], respectively.

The normal appro[x.sub.i]mation is often used to deal with binomial data when the conditions np [greater than or equal to] 5 and n(1 - p) [greater than or equal to] 5 are fulfilled. However, Schader-Schmid-1989 showed that the normal appro[x.sub.i]mation of binomial distribution performs poorly even when the often-used rules of thumb such as np [greater than or equal to] 5 are met and furthermore, the accuracy of the appro[x.sub.i]mation depends heavily upon the value of p. Ryan-Schwertman-1997 showed that the upper tail probability in the classical np-chart is usually too large and the lower tail probability is usually too small, especially when p is small. This inadequacy in appro[x.sub.i]mation is mainly due to the skewness of the exact distribution. For improving performances of the classical p-chart and np-chart, some alternative appro[x.sub.i]mation rules were provided so that tail areas of the charts can be close to those of the exact distribution.

In Section ??, we describe four di_erent charting approaches proposed in the literature. In Section ??, we introduce a new method called the improved square root transformation (ISRT). In Section ??, some numerical results are presented to assess the performances of these methods. Some conclusions are made in Section ??.

2. Recent Developments

In this section, we briefly introduce four known methods which improve the performances of tail probabilities in a p-chart or an np-chart. Ryan-Schwertman-1997 proposed the Arcsine p-chart. For known p, define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [x.sub.i] is the number of nonconforming items in the ith sample. It can be shown that Yi is approximately standard normal. Hence, we can plot [Y.sub.1], [Y.sub.2],..., on a chart with the center line CL = 0 and upper control limit at UCL = 3 and lower control limit at LCL = -3. When p is unknown, we can use [[??].sub.i] = [x.sub.1] + [x.sub.2] +...+ [x.sub.i]/[n.sub.1] + [n.sub.2] +...+ [n.sub.i] to estimate p, and thus define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for i [greater than or equal to] 2. The Arcsine p-chart has two beneficial e_ects: (1) the tail areas will be close to the tail areas under normality, and (2) for a given p, the minimum sample size necessary to obtain a positive LCL is much smaller than that using the classical 3-sigma p-chart.

The second method is the Q-chart proposed by Quesenberry-1991a Quesenberry-1991a,Quesenberry-1991b,Quesenberry-1991c. For known p, let [u.sub.i] = B([x.sub.i]; [n.sub.i], p) denote the probability that [x.sub.i] [less than or equal to] [x.sub.i] for a binomial random variable [x.sub.i]. Quesenberry-1991a suggested that the statistics [Q.sub.i] = = [[PHI].sup.-1] ([u.sub.i]), i = 1, 2,..., are plotted on a chart with CL = 0, UCL = 3, and LCL = -3, where [PHI](x) denotes the cumulative distribution function of the standard normal distribution. In practice, p is usually unknown. We can define [s.sub.i] = [t.sub.i] = [i.summation over (j-1]) [x.sub.j] and let [u.sub.i] = H([x.sub.i]; [t.sub.i], [n.sub.i], [s.sub.i-1]) be the probability that [X.sub.i] _[less than or equal to] [x.sub.i] for a hypergeometric random variable [X.sub.i], and [Q.sub.i] = [[PHI].sup.-1] ([u.sub.i]). Then, we can plot [Q.sub.1],[Q.sub.2],..., on a chart with center line CL = 0, and control limits UCL = 3, and LCL = -3.

Quesenberry-1991b indicated that, in general, the Q-chart has tail probability less than 0.00135 for the lower tail and more than 0.00135 for the upper tail. Moreover, Quesenberry-1991b concluded that the Arcsine p-chart gives a better approximation to the nominal lower tail area, and the Q-chart provides a better approximation to the nominal upper tail area. Basically, both Arcsine p-chart and Q-chart plot a transformed statistic rather than the statistic of interest, and computations for both transformed statistics are not simple.

Winterbottom-1993 and Chen-1998 used the Cornish-Fisher expansion of quantiles to construct a modified p-chart. For a given p, we can plot [w.sub.i] = [x.sub.i]/[n.sub.i], for i = 1, 2,..., on a chart with the center line CL = p and the control limits

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If p is unknown, we can plot wi's on a chart with the center line CL = [[??].sub.i-1] and the control limits

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[??].sub.i] = [x.sub.1] + [x.sub.2] + ... + [x.sub.i] [n.sub.1] + [n.sub.2] + ... + [n.sub.i] Moreover, Winterbottom-1993 used the Cornish and Fisher expansion of quantiles to develop a modified c-chart for charting the Poisson data. The center line of the modified c-chart is CL = c and its control limits are

UCL = c + 3 [square root of (c) + 4/3, LCL = c - 3 [square root of (c) + 4/3,

where c is the mean of the Poisson distribution and it can be either specified or replaced by an estimate c computed using an in-control baseline data.

In order to avoid the cumbersome expressions found in some confidence limits approximations, Ryan-Schwertman-1997 used regression equations to produce a regression-based np-chart which has the center line CL = np and control limits

UCL = 0.6195 + 1.0052np + 2.983pnp,

LCL = 2.9529 + 1.01956np - 3.2729pnp. (1)

The value of p can be either specified or replaced by the estimate p computed from an in-control baseline data. In practice, the control limits in equation (??) are suggested to be rounded to the nearest integer. The corresponding center line and control limits of the regression-based p-chart can be obtained by dividing the center line and the control limits in equation (??) by n. Replacing np by c, a regression-based c-chart can be obtained.

3. ISRT Attribute Control Charts

Let X be a binomial random variable with parameters n and p, where p denotes the defect level of process, and let [??] = X/n be the sample defect level. If p is small, the normal approximation to the binomial distribution is inadequate, mainly due to skewness in the exact distribution. To overcome this defect, an improved square root transformation, named the ISRT, is used to construct an ISRT p-chart.

Suppose that g([??]) = [square root of ([?/?])]. Using the Taylor series expansion for g to the second order, we have

g([??] [congruent to] g(p) + g' (p)([??] - p]) + g"(p)/2[([??] - p).sup.2].

Equivalently,

[square root of (n)] [g ([??]) - g(p) - g"(p)/2 [([??] - (p).sup.2]] [congruent to] g'(p) [square root of (n)] ([??] - p).

Hence, both [square root of (n)] [g ([??]) - g(p) - g"(p)/2 [([??] - (p).sup.2]] and g'(p)[square root of (n)] ([??] - (p) have the same limiting distribution, that is, [square root of (n)] [g ([??]) - g(p) - g"(p)/2 [([??] - (p).sup.2]] is asymptotically normally distributed with mean 0 and variance [[g'(p)].sup.2] be the absolute estimating error in process, and let [[sigma].sub.[??] = [square root of (p(1-p)/n] be the standard deviation of sample proportion. It can be shown that

Z = g([??] - g (p)/|g'(p)|g'(p)[[sigma].sub.[??] - g"(p) [e.sup.2]/2 |g'(p)| [[sigma].sup.[??]]

is asymptotically standard normal. If the 3-sigma control limits are taken, we can show that

0.0027 = P(Z < -3 or Z > 3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, the ISRT p-chart can be constructed with the center line C[L.sub.p] = g(p) = [square root of (p) and the control limits

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the binomial distribution is positively skewed when the defect level p is small, it is not adequate to choose the control limits with equal-tails. After some numerical investigation on some cases of the low defect level of p [less than or equal to] 0.1, the absolute estimating error e is suggested to be 3 [square root of (p(1-p)/n)] n for the LC[L.sub.p], and to be [square root of (p(1-p)/n]) for the UC[L.sub.p]. Accordingly, the control limits can be rewritten as

UC[L.sub.p] = [square root of (p)] + 3/2 [square root of (1 - p)/n] - 1/2 (1 - p/n[square root of (p)]),

LC[L.sub.p] = [square root of (p)] - 3/2 [square root of (1 - p)/n] - 9/8 (1- p/n [square root of 9p)]). (2)

By multiplying [square root of (n)] to the center line C[L.sub.p] = [square root of (p)] and the control limits in equation (??), the ISRT np-chart has the center line C[L.sub.np] = [square root of (np)] and the control limits

UC[Ln.sub.p] = [square root of (np)] + [square root of (np)] + 3/2 [square root of (1 - p)] - 1/2 (1 - p)/ [square root of (np)],

LC[L.sub.np] = [square root of (np)] + 3/2 [square root of (1 - p])] - 1/2 (1 - p)/ [square root of (np)].

The value of p can be either specified or replaced by the estimate [??] computed from an in-control baseline data. That is, when the defect level p is unknown, we can select m pre-samples each of size n. As a general rule, m should be 20 or 25. Then if there are [D.sub.i] defective items in the sample i, we compute the

[??] = [[SIMGA].sup.m.sub.i = 1][D.sub.i]/mn = [[SIGMA].sup.m.sub.i=1] [[??].sub.i]/m, (3)

where [[??].sub.i] = [D.sub.i]/n, I = 1,2,...,m.

If the sample size n is large and p is small enough so that np has a moderate size c, the ISRT c-chart can be constructed with the center line C[L.sub.c] = pc and the control limits

UC[L.sub.c] = [square root of (c)] + 3/2 - 1/2 (1/[square root of (c)]).

LC[L.sub.c] = [square root of (c)] - 3/2 - 9/8 (1/[square root of (c)],

by using the Poisson approximation to binomial. The value of c can be either specified or replaced by the estimate c computed from an in-control baseline data with

[??] = Total nonconformities/m. (4)

4. Numerical Study

Three criteria are provided for investigating the performance of the ISRT p-chart here. They are: (1) the minimum value of the sample size n_ required for the LCL to be positive, (2) the closeness of the false alarm probabilities to the nominal values for both over UCL and under LCL, and (3) how close the chart can match to a specified percentile point of run length (RL) distribution when the parameter is unknown.

First, we compare the regression-based p-chart, the Arcsine p-chart, the modified p-chart, the Q-chart, the classical 3-sigma p-chart with the ISRT p-chart based on the value of [n.sup.*]. For the Arcsine p-chart and the Q-chart, the positive lower control limit means that P([Y.sub.i] < LCL) > 0 and P([Q.sub.i] < LCL) > 0, respectively, and it means that LCL > 0 for others. This criterion is also a good reference for determining the sample size of establishing p-chart in practical applications. Table ?? shows the values of [n.sup.*] for these charts when the value of p is smaller than 0.1. The practitioners can use the sample sizes suggested in Table ?? to construct the selected attribute control charts. We can find that values of [n.sup.*] for the ISRT p-chart, the regression-based p-chart and the Arcsine p-chart are smaller than others. Likewise, for the classical c-chart, the value of LCL is positive if c > 9; for the regression-based c-chart, the value of LCL is positive if c > 4.07; for the modified c-chart, the value of LCL is positive if c > 6.04 and for the ISRT c-chart, the value of LCL is positive if c > 4.20.

Because of the good performance of the modified c-chart and the regression-based c-chart based on the first criterion, they are selected to be compared the closeness of the false alarm probabilities to the nominal values for both over UCL and under LCL with the ISRT c-chart. Let [Z.sub.i] denote any one of the three plotting statistics on these charts and the false alarm probabilities P([Z.sub.i] > UCL) and P([Z.sub.i] < LCL) are computed for c = 4 to 25. Since the Poisson distribution is discrete, it is hard to require that both exact tail areas equal to 0.00135. For a given value of c, we define the exact c-chart as the one whose _L (the lower tail area under LCL) and [[alpha].sub.U] (the upper tail area over the UCL) are as close to 0.00135 as possible, but are not [sigma]% larger than 0.00135, where 0 [less than or equal to] [sigma] [less than or equal to] 100. In this paper, we took [sigma] = 50, that is, both [[alpha].sub.L] and [[alpha].sub.U] are restricted to be less than or equal to 0.00315(1+0.5) = 0.002025. All numerical values are calculated by Poisson probability and shown in Table ??. In Table ??, we see that the di_erences among three c-charts are very small and all of them agree with the exact c-chart. Likewise, the simulation results provided by Lin-2002 for the p-chart with parameters p = 0.1 and n = 10(10)200, and p = 0.01 and n = 100(100)2000, also showed that all the modified p-chart, regression-based p-chart and the ISRT p-chart agree with the exact p-chart.

The above discussion are made under the assumption that the parameter p or c is known, but this assumption is usually inadequate in practice. For evaluating the performance of the charts when the parameters are unknown, how close the control limits of the three charts can match to a specific percentile point of RL distribution for the true limits is considered. The criterion was proposed by Nedumaran-Pignatiello-2001. Assume that the process is in-control and m initial subgroups of the binomial data with parameters n and p are observed, and then k future subgroup observations are obtained immediately after m initial subgroup observations. Let [gamma] be the predetermined probability of a signal within observations m + 1,m + 2,...,m + k. Then

P(LCL < [Z.sub.i] < UCL, i = m + 1,m + 2,...,m + k) = 1 - [gamma].

For the estimated limits which perform similarly to the true limits, is set to be equal to the corresponding RL distribution percentile with the true limits. That is,

[gamma] = P(RL _ k) = 1 - [(1 - [alpha]).sup.k],

where [alpha] is the false alarm probability for one subgroup. A benchmark for the probability is [alpha] = 0.0027 or [gamma] = 1 - [(1 - 0.0027).sup.k]. For example, if k = 10, the benchmark probability is 0.0267. However, when p is unknown, [Z.sub.m]+1,[Z.sub.m+2],...,[Z.sub.m+k] are dependent and the probability depends upon the initial subgroup size m. The correlation among [Z.sub.i]'s vanishes if m is large enough. Nevertheless, we can adopt the value of [gamma] as measures of the false alarm probability of the three charts and estimate them.

In our numerical study, the future k subgroup observations were generated one at a time, and the sample proportions of each future subgroup were plotted on each chart until a false alarm was issued or until all k subgroups were plotted. This procedure was replicated 50000 times and the probabilities of a signal within k subgroups were estimated. The estimated probabilities for the ISRT np-chart, the regression-based np-chart and the modified np-chart are listed in Table ??. It can be seen that the control limits of the ISRT np-chart perform similarly to the true limits for almost all combinations of n, p and k, and the control limits of the regression-based np-chart produce a larger number of false alarms than the true limits when the value of p is unknown. The false alarm probability of the modified np-chart becomes large when the value of k is large or the value of p is small. The numerical results recommend that the ISRT np-chart is adequate for charting the binomial data when the defect level p is low and unknown.

5. Conclusions

This paper provides a new method based on the improved square root transformation which can be applied to three attribute control charts. They are the ISRT p-chart, ISRT np-chart and ISRT c-chart. The false alarm probabilities of these charts are close to the nominal values whenever the parameter is known or unknown. Numerical results indicate that the ISRT p-chart and the ISRT c-chart almost coincide with the exact p-chart and exact c-chart, respectively, under reasonable restrictions. Moreover, the ISRT np-chart can match any specific percentile point of run length distribution of the true limits when the parameter is unknown. Though the ISRT attribute control chart plots a square root transformed statistic rather than the statistic of interest, the link between the plotted statistic and the statistic of interest is simple to be interpreted.

Acknowledgments 

The authors would like to thank the Managing Editor and referee for providing helpful comments.

Received January 28, 2005, Accepted July 29, 2005.

References

[1] G. Chen, An improved p chart through simple adjustments, Journal of Quality Technology 30 (1998), 142-151.

[2] C. C. Lin, Improved Square Root Transformation for Attribute Control Charts (2002), Master thesis, Tamkang University, Department of Statistics (in Chinese).

[3] G. Nedumaran, and J. J. Pignatiello, On estimating X control chart limits, Journal of Quality Technology 33 (2001), 206-212.

[4] C. P. Quesenberry, SPC Q charts for start-up process and short or long runs, Journal of Quality Technology 23 (1991a), 213-224.

[5] C. P. Quesenberry, SPC Q charts for a binomial parameter: Short or long runs, Journal of Quality Technology 23 (1991b), 239-246.

[6] C. P. Quesenberry, SPC Q charts for a Poisson parameter _: short or long runs, Journal of Quality Technology 23 (1991c), 296-303.

[7] T. P. Ryan, and N. C. Schwertman, Optimal limits for attributes control charts, Journal of Quality Technology 29 (1997), 86-98.

[8] M. Schader, and F. Schmid, Two rules of thumb for the approximation of the binomial distribution by the normal distribution, The American Statistician 43 (1989), 23-24.

[9] A. Winterbottom, Simple adjustments to improve control limits on attribute charts, Quality and Reliability Engineering International 9 (1993), 105-109.

Tzong-Ru Tsai *, Chien-Chih Lin, and Shuo-Jye Wu

Department of Statistics, Tamkang University, Tamsui, Taipei, Taiwan

* E-mail: [e-mail]

Table 1: Minimum value of sample size required for the LCL is positive

                                    p

chart             0.1   0.05   0.02   0.01   0.005   0.001

ISRT               38     80    206    416     836    4195
regression-based   41     82    204    408     815    4072
Arcsine            43     88    222    445     891    4461
modi ed            58    119    300    602    1206    6037
Q-chart            63    129    328    658    1319    6605
classical          81    171    441    891    1791    8991

Table 2: False alarm probabilities for four different c-charts

                      under LCL

                 regression
c     modified       based      ISRT     exact

4          NA           NA        NA        NA
5          NA       0.00674   0.00674       NA
6          NA       0.00248   0.00248       NA
7      0.00091      0.00091   0.00091   0.00091
8      0.00034      0.00302   0.00034   0.00034
9      0.00123      0.00123   0.00123   0.00123
10     0.00050      0.00277   0.00050   0.00050
11     0.00121      0.00121   0.00121   0.00121
12     0.00052      0.00229   0.00052   0.00052
13     0.00105      0.00105   0.00105   0.00105
14     0.00181      0.00181   0.00047   0.00181
15     0.00086      0.00279   0.00086   0.00086
16     0.00138      0.00138   0.00040   0.00138
17     0.00067      0.00206   0.00067   0.00067
18     0.00104      0.00104   0.00104   0.00104
19     0.00151      0.00151   0.00052   0.00151
20     0.00078      0.00209   0.00078   0.00078
21     0.00111      0.00111   0.00111   0.00111
22     0.00150      0.00150   0.00058   0.00150
23     0.00081      0.00198   0.00081   0.00198
24     0.00108      0.00108   0.00108   0.00108
25     0.00142      0.00142   0.00059   0.00142

                       over UCL

                 regression
c     modified       based      ISRT     exact

4      0.00092      0.00284   0.00284   0.00092
5      0.00070      0.00545   0.00202   0.00202
6      0.00140      0.00363   0.00140   0.00140
7      0.00096      0.00241   0.00241   0.00096
8      0.00159      0.00372   0.00159   0.00159
9      0.00106      0.00243   0.00243   0.00106
10     0.00159      0.00345   0.00159   0.00159
11     0.00104      0.00225   0.00225   0.00104
12     0.00147      0.00305   0.00147   0.00147
13     0.00097      0.00397   0.00199   0.00199
14     0.00131      0.00261   0.00131   0.00131
15     0.00172      0.00331   0.00172   0.00172
16     0.00113      0.00219   0.00219   0.00113
17     0.00145      0.00273   0.00145   0.00145
18     0.00096      0.00333   0.00181   0.00181
19     0.00121      0.00223   0.00223   0.00121
20     0.00149      0.00269   0.00149   0.00149
21     0.00100      0.00320   0.00181   0.00181
22     0.00121      0.00216   0.00121   0.00121
23     0.00146      0.00255   0.00146   0.00146
24     0.00099      0.00298   0.00173   0.00173
25     0.00118      0.00204   0.00204   0.00118

Note: NA denotes 'not available'.

Table 3: Performance assessment for [alpha] = 0.0027

                              regression
n      p      k    modified        based      ISRT   [gamma]

150    0.10   10    0.02876      0.06008   0.02966   0.02667
300    0.10   10    0.02922      0.03864   0.02664   0.02667
150    0.10   20    0.05492      0.11662   0.05620   0.05263
300    0.10   20    0.05704      0.07488   0.05286   0.05263
150    0.10   30    0.08102      0.16910   0.08276   0.07790
300    0.10   30    0.08328      0.11190   0.07810   0.07790
450    0.05   10    0.03988      0.07312   0.03250   0.02667
600    0.05   10    0.03938      0.06320   0.03292   0.02667
450    0.05   20    0.07518      0.14040   0.06390   0.05263
600    0.05   20    0.07206      0.11586   0.06132   0.05263
450    0.05   30    0.11482      0.20650   0.09730   0.07790
600    0.05   30    0.10512      0.16882   0.08932   0.07790
2000   0.01   10    0.04308      0.09312   0.03502   0.02667
2000   0.01   20    0.08174      0.17378   0.06734   0.05263
2000   0.01   30    0.12082      0.25072   0.10222   0.07790