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Brief Communication |
Address correspondence to Wallace E. Carroll, M.D., 1556 San Leandro Lane, Montecito, CA 93108, USA; tel 805 969 2758; fax 805 969 3796; e-mail wallml{at}cox.net.
Abstract
Shewhart introduced industry to the concept of three standard deviation (3S) limits for quality control and Levey, Jennings, Henry, and Westgard adapted this idea to clinical laboratory medicine. Westgard ultimately formulated a system of rules to enable clinical laboratory scientists to decide whether the tests they were doing were "in control" and reportable, or "out of control." The present communication explains mathematically how Shewhart’s "beyond 3S" (ie, >1 chance in 370) is an indication for corrective laboratory action.
(received 8 June 2002; accepted 19 September 2002)
Keywords: quality assurance, Westgard’s rules, probability, clinical laboratory testing
This brief report applies probability to "Westgard’s Rules of Quality Control" in laboratory medicine, in order to explain and clarify these rules [1].
In 1931, Shewhart [2] proposed the use of control charts for monitoring quality in manufacturing operations. With them, he designated a measurement that is beyond 3S from the mean "as an indication of a significant variation from standard quality or as an indication of trouble." In 1950, Levey and Jennings [3] recommended that Shewhart’s control charts be used in clinical laboratories to provide "a constant check on the reliability of the numerous determinations run each day" and to make it "possible to determine at a glance whether errors of analysis [were] beyond the statistical variation of the procedures employed." The control limit was, as for Shewhart, 3S.
In 1959, Henry [4] described the use of 1S, 2S, or 3S control limits. Routinely, he recommended employing 95% confidence limits. For Gaussian distributions, this would be ± 2S, but Shewhart’s ± 3S was adopted when ± 3S data were considered within the "desired precision." Hence, a decision had to be made whether 2S or 3S should be used.
Westgard et al [5] added the following rule limits, which provide definitive guidelines for the interpretation of a control result that is 1S, 2S, or 3S from the mean:
| Clinical reference controls are considered "out of control" in the following circumstances: (Rule a: 1–3s). If 1 control observation is more than 3S beyond either side of the mean. (Rule b: 2–2s). If 2 consecutive control observations exceed 2S from the mean and both observations are on the same side of the mean. (Rule c: 4–1s). If 4 consecutive control observations exceed 1S and all are on the same side of the mean. (Rule d: R–4s). If the difference between the largest and the smallest observations exceeds 4S. (These control values would lie on opposite sides of the mean.) (Rule e: 10X). If 10 consecutive control observations fall on the same side of the mean. (Rule f: 1–2s). If 1 control value exceeds 2S on either side of the mean. (This is not considered "out of control," but a "warning sign.")
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Westgard’s rules are readily adaptable to processing by a computer. Westgard explained his rules by control rules graphs [5] and decision limit control charts [1], but the reasons that clinical laboratory reference controls are "out of control" seem more easily explained in terms of probability.
For 1–3s, 99.7% of values would be included within 3S, so 3/1000 or 1/333 would be outside; but, if 99.7% is taken to one additional place, one gets 99.73%, so: 2.70/1000 or 1/370.37, in which the denominator can be rounded to give 1/370. Simply stated: If the reference control is > 3S beyond the mean, the odds are only 1 in 370 that this is due to chance. The odds are 369 out of 370 that something is wrong with the analytical system. For this reason and, as proposed by Shewhart [2], 3S (1 chance in 370) has been arbitrarily accepted as indicating "out of control."
For 2–2s, 95.45% of values would be included, so 4.55% or 1/22 would be excluded on either side of the mean. There is then a 1/44 chance that the second result will be on the same side as the first result, giving a 1/22 x 1/44 = 1/968 chance that there is a 2–2s rule violation, ie, beyond 1/370.
For 4–1s, 68.27% are included, so 31.73% are excluded on each side of the mean. Thus, 1/2 x 0.3173 accounts for one side. This gives 0.164 = 0.000634 or 1/1577 for the one side. Since the rule violation can occur on both sides, multiplication by 2 gives the probability of one in 789 that the reference is "out of control" because of chance. This is also beyond 1/370.
For R–4s, each control is 2S on the opposite side of the mean, so the odds would be 1/22 x 1/22 or 1 in 484 that the control is out of range because of chance, still beyond 1/370.
For 10X, the odds that the controls are on the same side of the mean 10 consecutive times are 1/2n-1 or 1 chance in 512 that the control is "out of control" because of chance.
For 1–2s, a single event beyond 2s has odds of about 1 in 44. To account for the other side, multiplication by 2 is performed to give 1 in 22. This 1/22 is greater than the 1/370 cutoff, so 1 occurrence in 22 is probably due to chance; 1/22 is, therefore, a "warning sign" and not evidence of an "out of control" condition.
It is thus evident that one can account for a reference control result being "out of control" or "in-range" based on simple odds. To indicate a real laboratory problem, one need only accept Shewhart’s 3S (1 chance in 370) as the appropriate limit.
Acknowledgment
We thank Karen Eugene for typing the manuscript.
References
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