Cusum charts

When data of a single type accumulate, new forms of statistical analysis become possible. In the following, conventional control and Cusum charts will be presented. In the authors opinion, newer developments in the form of tight (multiple) specifications and the proliferation of PCs have increased the value of control charts especially in the case of on-line in-process controlling, monitors depicting several stacked charts allow floor supervi-... 

A disadvantage of the conventional control charts is that a small or gradual shift in the observed process parameter is only confirmed long after it has occurred, because the shift is swamped in statistical (analytical) noise. A simple way out is the Cusum chart (cumulated sum of residuals, see program CUSUM.exe), because changes in a parameter s average quickly show up, see Fig. 1.32. The... 

The quality control unit in a cosmetics company supervised the processing of the weekly batch of shampoo by determining, among other parameters, the viscosity and the dry residue. Control charts showed nothing spectacular. (See Fig. 4.10, top.) The cusum charts were just as uneventful, except for that displaying the dry residue (Fig. 4.10, middle and bottom) The change in trend in the middle of the chart was unmistakable. Since the analytical method was very simple and well-proven, no change in laboratory personnel had taken place in the period, and the calibration of the balances was done on a weekly basis, suspicions turned elsewhere. A first hypothesis,... 

Figure 4.10. At the top the raw data for dry residue for 63 successive batches is shown in a standard control chart format. The fact that as of batch 34 (arrow ) a different composition was manufactured can barely be discerned, see the horizontals that indicate the means DRi 33 resp. DR34 g3- A hypothesis that a change occurred as of batch 37 would find support, though. Cusum charts for base period 1. .. 63 resp. base period 1. .. 37 make the change fairly obvious, but the causative event cannot be pinpointed without further information. Starting with batch 55 (second arrow ), production switched back to the old composition.

The validation process begun in Phase I is extended during Phase II. In this phase, selectivity is investigated using various batches of drugs, available impurities, excipients, and samples from stability studies. Accuracy should be determined using at least three levels of concentration, and the intermediate precision and the quantitation limit should be tested. For quality assurance evaluation of the analysis results, control charts can be used, such as the Shewart-charts, the R-charts, or the Cusum-charts. In this phase, the analytical method is refined for routine use. 

Because it uses all of the data, the CUSUM chart is the best way of detecting small changes in the mean. Consider a process for which there is a known target value, T. For each new measurement, the difference between the measurement and T is calculated and added to a running total. This running total is plotted against successive measurements (CUSUM is short for cumulative sum). 

Figure 6.8 CUSUM charts showing (a) a step-change and (b) drift.

Figure 6.9 (a) The V-mask - for interpretation of CUSUM charts and (b) a CUSUM chart illustrating the use of a V-mask. 

Guide to Data Analysis and Quality Control Using CUSUM Techniques. Uses and Value of CUSUM Charts in Business, Industry, Commerce and Public Service , BS 5703-1 2003, British Standards Institute (BSI), London, UK, 2003. 

Figure SAQ 6.2 (a) Shewhart chart with warning and action limits at 2 and 3 standard deviations, respectively, (b) Moving average chart (n = 5) with warning and action limits at 2 Jn and 3 Jn standard deviations, respectively, (c) CUSUM chart, (d) CUSUM chart with V-mask.

One method of data presentation which is in widespread use is the control chart. A number of types of chart are used but where chemical data are concerned the most common types used are Shewhart charts and cusum charts. Only these types are discussed here. The charts can also be used to monitor the performance of analytical methods in analytical laboratories. 

Probability plot Q-Q plot P-P Plot Hanging histogram Rootagram Poissonness plot Average versus standard deviation Component-plus-residual plot Partial-residual plot Residual plots Control chart Cusum chart Half-normal plot Ridge trace Youden plot... 

This shde gives an example of the use of a Cusum Chart. The upper chart is a conventional X-chart, the lower one a Cusum Chart. Starting from the 11th value, all values are below the target value originating from slow between-batch-drift in the analyses. This can be seen in the Cusum Chart by a descending cumulative sum. 

For the evaluation of a Cusum Chart a V-mask is used. This mask is laid on the chart so that the last marked value has a distance of d from the peak. 

The Cusum Chart shows very clearly the point from which the process ran ont of control. The average mn length, i.e. the time needed to detect an ont-of-control sitnation is shorter than for other control charts. Fnrthermore, the size of a change in the process can be detected from the slope of the chart. 

Shewhart charts are adept at detecting mean value shifts on the order of 3(7 or higher. To detect more subtle shifts in the mean value, the CUSUM chart has been developed [10,11]. The cumulative sum is defined as ... 

CUSUM Control Chart A CUSUM chart provides an efficient way of detecting small shifts in the mean of a process (l/2 a), the chart is usually used.The CUSUM chart incorporates information contained in a sequence of sample points. It keeps track of the cumulative sum of the deviations between each sample point (a sample mean) and a target value. Unlike the x chart, which often bases its out-of-control decision on just the most recently collected sample, the CUSUM calculated for a sample point carries the history prior to that sample. For example, a sequence of sample points above the centerline can trigger an out-of-control signal although all of them stayed well below the UCLs of the x chart. 

There are two forms of the CUSUM chart, the tabular form and the V-mask form. Due to its practicality, the tabular form is more preferred in industrial settings. The tabular CUSUM accumulates deviations from a target value (or a known process mean p0). Deviations above that target value are cumulated as a one-sided upper CUSUM (C+) and deviations below the target value are cumulated as one-sided lower CUSUM (C ) ... 

EWMA Control Chart An EWMA control chart plots weighted moving average values for variables data. A weighting factor is chosen by the user to determine how older data points affect the mean value compared to more recent ones. Because the EWMA chart uses information from all samples, it is a good alternative to the CUSUM chart in detecting smaller process shifts. 

FIGURE 12 CUSUM chart for Pet Tabs manufacturing example. 

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