Shewhart charts for mean values

In Chapter 2 we showed how the mean, x, of a sample of measurements could be used to provide an estimate of the popuiation mean, pi, and how the sample standard deviation, s, provided an estimate of the population standard deviation, a. For a small sample size, n, the confidence iimits of the mean are normally given by equation (2.9), with the t value chosen according to the number of degrees of freedom (k - 1) and the confidence ievei required. The same principles can be applied to quality control work, but with one important difference. Over a long period, the population standard deviation, T, of the pesticide ievei in the fruit (or, in the second example, of the tablet weights) wiii become known from experience. In quality control work, ais given the title process capability. Equation (2.9) can be replaced by equation (2.8) with the estimate s replaced by the known a. In practice z = 1.96 is often rounded to 2 for 95% confidence limits and z = 2.97 is rounded to 3 for 99.7% confidence limits. 

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Figure 4.3 [a] Shewhart chart for mean values [example), (b) Shewhart chart for range... 

The usage of quality control charts in the field of quality assurance is based on the assumption that the determined results are distributed normally. Typical control charts used in a LIMS for routine analysis are, for example, the Shewhart charts for mean and blank value control, the retrieval frequency control chart, and the range and single-value control chart [19]. Quality regulation charts can be displayed graphically in the system or exported to spreadsheet programs. 

Two aspects are important for IQC (1) the analysis of control materials such as reference materials or spiked samples to monitor trueness and (2) replication of analysis to monitor precision. Of high value in IQC are also blank samples and blind samples. Both IQC aspects form a part of statistical control, a tool for monitoring the accuracy of an analytical system. In a control chart, such as a Shewhart control chart, measured values of repeated analyses of a reference material are plotted against the run number. Based on the data in a control chart, a method is defined either as an analytical system under control or as an analytical system out of control. This interpretation is possible by drawing horizontal lines on the chart x(mean value), x + s (SD) and x - s, x + 2s (upper warning limit) and x-2s (lower warning limit), and x + 3s (upper action or control limit) and x- 3s (lower action or control limit). An analytical system is under control if no more than 5% of the measured values exceed the warning limits [2,6, 85]. 

Minitab can be used to produce Shewhart charts for the mean and the range. The program calculates a value for R directly from the data. Figure 4.5 shows Minitab charts for the data in Table 4.2. Minitab (like some texts) calculates the warning and action lines for the range by approximating the (asymmetrical) distribution of by a normal distribution. This is why the positions of these lines differ from those calculated above using equations (4.9) and (4.10). 

The target value for a particular analysis is 120. If preliminary trials show that samples of size 5 give an R value of 7, set up Shewhart charts for the mean and range for samples of the same size. 

We do this by using the statistical ideas outlined above. First of all, the QC sample is measured a number of times (under a variety of conditions which represent normal day-to-day variation). The data produced are used to calculate an average or mean value for the QC sample, and the associated standard deviation. The mean value is frequently used as a target value on the Shewhart chart, i.e. the value to aim for . The standard deviation is used to set action and warning limits on the chart. 

The Shewhart chart graphically tests the hypothesis that a process measurement is not different from the desired target value of that measurement. If one assumes that the measurement, yk, follow a normal distribution about the target value, y assuming that the target value y, represents the true mean ym for the given measurement, and assuming that the standard deviation a for the process is known, then... 

If the initial calculation is shown to contain a value more than 3 standard deviations from the mean, the value should be rejected and the mean and standard deviation recalculated. The standard deviation obtained may be used as the target standard deviation (s ) in preparation of Shewhart charts or a desired standard deviation can be used which might be related to medical need or to a previous period of satisfactory performance. For a practical guide to statistical techniques used in quality control of analytical methods and the preparation of cusum... 

Figure 1b. Shewhart (control chart) of control samples for the determination of lead in blood. Concentration in //g/L, blood samples made in own laboratory. Limits set on 10 % of mean value of control samples determined so far.

The zone control chart (also known as the J-chart) is a control chart for the mean that combines features of the Shewhart chart and the cusum chart. It is simple to use, but effective. First it is necessary to establish a value for a, as was done in Example 4.7.1. Then the chart is set up with horizontal lines at the target value, /r,... 

What remain to be added to the plots in Fig. 4.10 are appropriate control limits. In order to indicate the kind of thinking that stands behind control limits for Shewhart charts, let us concentrate on the issue of limits for the plot of means. The fact is that mathematical theory suggests how the behavior of means y ought to be related to the distribution of individual melt indices y, provided the data-generating process is stable, that is, subject only to random causes. If individual responses y can be described as normal with some mean and standard deviation a, mathematical theory suggests that averages of n such values will behave as if a different normal distribution were generating them, one with a mean fiy that is numerically equal to and with a standard deviation Gy that is numerically equal to... 

Both the EC50 values and the 3-pM point of the 2,3,7,8TCDD ealibration curve serve as quality criteria. For each participant, the results for both data points from all 96-well plates analyzed during the presented study were collected and reeorded in Shewhart control charts. The Shewhart control chart is used to identify variations on performanee of the DR CALUX bioassay brought about by unexpected or unassigned causes. The Shewhart eontrol chart shows the mean of the EC50 and 3-pM control point and the upper and lower eontrol limits. In Figure 2, a typical Shewhart control chart is shown. Over the analysis period, none of the participants exceeded the aetion levels (AVG 3 S). 

The relevance of Fig. 5.11 to the problem of setting control chart limits on means is that if one is furnished with a description of the typical pattern of variation in y, sensible expectations for variation in y follow from simple normal distribution calculations. So Shewhart reasoned that since about 99.7 percent (most) of a Gaussian distribution is within three standard deviations of the center of the distribution, means found to be farther than three theoretical standard deviations (of y) from the theoretical mean (of y) could be safely attributed to other than chance causes. Hence, furnished with standard values for /x and a (describing individual observations), sensible control limits for y become... 

It is clear that the manual preparation and continual updating of the charts shown in Fig. 2 for a multilevel, multi-analyte quality control system involves a great deal of work. However, it is possible in a multilevel control system to represent all individual values at different levels on one chart which is a variant of the Shewhart mean plot. The difference of an individual value (e.g. from the target mean (x ) is divided by the target standard deviation (sQ and thus the position of the individual value is represented relative to the target mean in standard deviation intervals 1), see Fig. 1. The bias of each value, irrespective of its analyte concentration, is therefore represented on the same standard deviation scale. This is very convenient for manual and computer plotting as complex scaling is avoided. Fig. 4 shows an example of this... 

Autocorrelation in data affects the accuracy of the charts developed based on the iid assumption. One way to reduce the impact of autocorrelation is to estimate the value of the observation from a model and compute the error between the measured and estimated values. The errors, also called residuals, are assumed to have a Normal distribution with zero mean. Consequently regular SPM charts such as Shewhart or CUSUM charts could be used on the residuals to monitor process behavior. This method relies on the existence of a process model that can predict the observations at each sampling time. Various techniques for empirical model development are presented in Chapter 4. The most popular modeling technique for SPM has been time series models [1, 202] outlined in Section 4.4, because they have been used extensively in the statistics community, but in reality any dynamic model could be used to estimate the observations. If a good process model is available, the prediction errors (residual) e k) = y k)—y k) can be used to monitor the process status. If the model provides accurate predictions, the residuals have a Normal distribution and are independently distributed with mean zero and constant variance (equal to the prediction error variance). 

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