The influences of the between- and within-run components of variation on the mean rule

Most assays performed in chemistry and endocrinology laboratories have values ofsb which are non-zero with the ratio Sb/S, varying from 0.3 to 3 [2 and 3]. Westgard, Falk and Groth have recently used computer simulations to show how Sb and sw affected the performance characteristics ofvarious control rules [4 and 5]. Their simulations were based on a model that incorporated Sb, Sw and errors of varying sizes, either systematic or random. The simulations were used to obtain the probability of false rejection (Pfr), the probability of a rejection signal when no analytical error was present; and the probability of error detection (Peel), the probability of a rejection signal when an analytical error was present. They found that with significant Sb, there was deterioration in the performance characteristics of rules sensitive to systematic error, i.e. there was a tendency for Pfr tO increase and Pe, to decrease. They concluded that the optimal detection of systematic errors was difficult in the presence of significant Sb.


Introduction
Quality control materials are usually described by their means and 'total' standard deviation (st). Analysis of variance can be used to separate the total standard deviation into its within-run (Sw) and between-run (Sb) components [1] show how Sb and sw affected the performance characteristics of various control rules [4 and 5]. Their simulations were based on a model that incorporated Sb, Sw and errors of varying sizes, either systematic or random. The simulations were used to obtain the probability of false rejection (Pfr), the probability of a rejection signal when no analytical error was present; and the probability of error detection (Peel), the probability of a rejection signal when an analytical error was present. They found that with significant Sb, there was deterioration in the performance characteristics of rules sensitive to systematic error, i.e. there was a tendency for Pfr tO increase and Pe, to decrease. They concluded that the optimal detection of systematic errors was difficult in the presence of significant Sb.
An alternative model, which more realistically describes the interpretation of control observations by the laboratorian, is presented. Use of this model yields performance characteristics which differ significantly from the Westgard approach and demonstrates how a specific control procedure, the mean rule, can be optimized for analytical methods. The mean rule is a simple, powerful procedure for the detection of systematic errors and has been recommended in various quality control schemes [6]. Its simplicity permits direct calculation of the probability of rejection without computer simulations. * Pierre Douville can be contacted at L'Hotel-Dieu de Quebec City, Quebec, Canada. ]" Author to whom correspondence should be sent.

Models and methods
Mean rule A mean rule, 0.01, refers to the control rule for which the mean of a set of N measurements exceeds the control limits which give a 1% frequency of false rejections (Pfr 0.01) [7]. When a method is in control, the mean of the controls for each run is distributed about a grand mean with a standard deviation of (sw2/N + Sb)/ [5]  The probability that the control mean is outside either the upper or lower control limits is equal to the sum of the probabilities that correspond to the upper and lower normalized distances (found in any Z-distribution table).

Alternative Model (Model 2)
To determine the presence of systematic error in an analytical run, one or more control rules are used to compare the control results to the stable mean. In the application of the mean rule, the total shift (shiftt) or the difference between the stable mean and the current batch mean is monitored. The total shift is equal to the sum of any systematic error plus the shift due to the between run component. Model  The probabilities that the new mean is outside either the upper or lower .control limit can be calculated as for Model 1.

Methods
For Model 1, the systematic error was varied from 0 to 5 st in multiples of 0.5 st for the following values ofsb/sw: O, .5, 1, 2, and infinity (s 0). The probability of the mean of one and four observations exceeding its control limits was calculated with equations (4) and (5). The probability was then plotted against the size of the systematic error, expressed in multiples of st.
For Model 2, the total shift was varied from 0 to 5 st in multiples of 0.5 st for the following values ofsb/Sw: O, .5, 1, 2, and infinity (Sw 0). The probability of the mean of one and four observations exceeding its control limits was calculated with equations (6) and (7). The probability was then plotted against the size ofthe total error expressed in multiples of st.
To illustrate the differences between Model and Model 2, power functions were generated as described above for Sb/So and N 1,2, 4 and 8.

Discussion
In Model 1, for each run generated, the total systematic deviation from the true mean is given by ASE plus the contribution given by Sb. Since the contribution of Sb can be either positive or negative, the total deviation can be greater or smaller than ASE. For all the runs generated, the mean deviation is thus ASE. ASE may be thought to represent a long-term error that occurs over many runs, while Sb represents the short-term shifts occurring from run to run. In Model 1, an effort is made to separate the contribution ofsb from that of ASE; Sb becomes a source of noise that impairs the detection of the long-term systematic error ASE. The distinction between ASE and Sb explains the general degradation in performance for rules used to detect systematic error (ASE) with increasing Sb/Sw. The laboratorian does not distinguish, however, between the long-term and short-term shifts at the end of an analytical run. He is interested in their total effect. These considerations prompted the adoption of Model 2 in which only two components are required to describe a run, the systematic comporrent, expressed by shiftt and the random component by sw.
The power function plots of the mean rule derived from Model 2 for nonzero Sb differ significantly from the Model plots, both for Pjs and Ped. The Pfr of the mean rule, 0.01, is fixed at 0"01 for both models. In Model 1, PYr is represented by they intercept. For Model 2, Sb represents the standard deviation of shiftt for a series of acceptable runs in which no systematic error is present. Approximately 95% of thse acceptable runs will have a shiftt below two times Sb. By transforming this two times Sb to a corresponding multiple of st, a zone of low shiftt can be defined which includes 95% ofthese acceptable runs. The probability of rejection should be low in this zone. The runs with a very low shiftt will have a probability of rejection far below 0"01 for the x0.01 rule. Acceptable runs with a shift close to twice Sb will have a probability of rejection higher than 0"01. Overall, for the 0.01 rule, the average pj for the acceptable runs will be 0"01.
Two observations emerge regarding the detection of large shifts. First, for high Sb/S, the probability of rejection changes abruptly, from a very low level to a high level as shifq increases, This reflects a high certainty about the size of the shift when the noise produced by Sw is low. The power function curves do not change significantly with replicate analysis. Second, with low Sb/S, the increased Peal with increasing shifh is more gradual and reflects the noise produced by Sw. Replicates improve the efficiency markedly and allow detection of even small shifh.
Use of the mean rule with actual laboratory data yields a low pfi for 'in control' runs and a high Ped with significant total shifts. The creatinine method, for example, on the Hitachi 705 has been shown to have ratios OfSb/Sw ranging from 0.5 to depending on the concentration of the control [2]. For a ratio Sb/Sw of 1, Sb s]V'2--equation (1). A shifh of 2Sb is equivalent to 1"4 st and implies that approximately 95% of the 'in control' runs have a shiftt below 1.4 st. As expected, figure 4(b) indicates a low probability of rejection for shifh below 1"4 st, because the mean rule with a low Pfr was used. The probability of detecting a significant shift increases greatly with increased numbers of observations. Figure 4(b) shows that for a total shift of2"5 st, the probability increases from 0"45 (X 1) to 0"99 (X 8).
Model 2 indicates that the performance of the mean rule is not degraded with Sb. On the contrary, only one or two controls are required to detect shifts larger than usual when the ratio Sb/Sw is high. A high Sb/Sw must not be perceived as ideal as it only reflects the fact that large shifts are accepted as part of the normal variation. The ideal situation, in which no significant shift (Sb/Sw is low) occurs between runs, necessitates meticulous attention, to reduce the between-run sources of errors, for example calibration errors. In addition, as shown in figure 3, many control measurements are necessary to obtain the desired probability of error detection in order to maintain the performance. In practice, there is often a trade-off between the effort for control and the size of the tolerated shifts.
While the mean rule can be a powerful tool for the detection of systematic deviation, its use is not always practical. For most instruments, two different levels of control material are analysed per batch. Averaging the normalized values of each level may not be acceptable because the systematic shifts might be different at each level. In this situation, a multi-rule approach is probably more appropriate [9]. However, long-term averages of successive values of control material can easily be computed. Cembrowski et al. described an application of the mean of controls that permits an accuracy trend analysis by the continuous monitoring of the mean [10]. We think that this accuracy trend analysis will increase the usefulness of the information provided by reference samples. Moreover, because of its excellent performance characteristics for the detection of systematic shifts, the mean should be used to analyse patient data [8].

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