Many statistical professionals consider the exponential weighted moving average forecasting (EWMA) the ideal solution for controlling processes that generate autocorrelated data. The EWMA chart is predicted to be better than other control charts at detecting small data shifts. Some clinical vital sign data are at times highly autocorrelated and at other times statistical independent. Our experience has shown that the degree of autocorrelation changes as the patient’s condition changes. In addition, to variations in the levels of autocorrelation, often the vital sign measurements are rounded off, which is likely to increase the degree of autocorrelation. This study compares the performance of EWMA charts to standard Shewhart process control charts with limits adjusted for autocorrelation as the amount of data autocorrelation changes and measurements are rounded off.
Many statistical professionals believe the exponential weighted moving average (EWMA) forecasting chart is a solution for controlling processes with autocorrelated data.1 Autocorrelation is often present in clinical data, although the degree of autocorrelation varies as the condition of the patient changes. In addition to inherent autocorrelation present in clinical measurements of patient vital signs, autocorrelation in clinical data increases when clinical measurements are rounded. Therefore, experts consider the EWMA charts a possible solution for detecting small shifts in autocorrelated data and reducing the number of type I errors for caregivers applying statistical process control in a clinical setting.
EWMA charts are created by weighting individual observations or sample averages. Sometimes the latest observations are assigned the greatest weights and the earlier observations receive the smallest weights, although the weighting scheme could be reversed, with the earlier observations receiving the greatest weights. Therefore the points that are charted are based on weighted averages or sums.
This study compares EWMA charts to standard Shewart charts. The comparison is based on real clinical data sets and simulated data. The simulated data were created with autocorrelation at a low level and then a high level.
Among the clinical data sets, the set one with the slow changes in skin temperature was most interesting. When the slow change in skin temperature occurred, the EWMA chart failed to indicate a change; however the Shewhart chart did detect the change. This finding is opposite of what would expect since one of the proclaimed advantages of EWMA charts is to detect small shifts in correlated data.
The target audience for vital sign process control charts is the physician and other caregivers. Physicians and other caregivers sometimes have little mathematical and statistical education with the exception of some recent graduates and physicians with master degrees and Ph.D.’s in other fields. Therefore, researchers who are applying process control in a healthcare setting must be sensitive to use of terminology. The researchers have found that using the word sigma rather then the standard deviation annoyed some caregivers. Caregivers also did like the term exponential weighted moving average, so we refer to the EWMA chart as a moving average type chart.
Another aspect of the EWMA chart that some physicians did not like was the EWMA chart’s visual complexity (Figure 1). It was difficult for some individuals to understand a control chart with upper and lower control limit lines, an average line, and individual subgroups. Some caregivers did not like to use control charts that contained moving upper and lower control limits and averages. Figure 1 illustrates an EWMA forecasting chart with autocorrelation changing from 0.2 to 0.85. Alpha was set at 0.5.
Figure 1 EWMA forecasting chart
Two options are available to reduce the number of complaints about the EWMA’s chart visual complexity:
No matter how the visual design is simplified, the EWMA chart will always be more complex than the standard Shewhart chart. Using a standard normal distribution type scale transformation (centered and reduced) one can graph all control charts with a fixed centerline and control limits. A chart that is centered and reduced loses information (Figure 2); however, the output does focus on the out-of-limits points. This graph is for the same data set as shown in Figure 1, where autocorrelation was varied from 0.20 to 0.85.
Figure 2 Centered Reduced EWMA forecasting chart
EWMA is based on a time series model; it is not a hypothesis testing procedure. The EWMA chart is considered better at detecting smaller shifts than Shewhart control charts. In patient monitoring, the objective is to detect sustained significant clinical changes, not small brief self-correcting changes, which are common with vital signs. This study demonstrates that with some types of patient monitoring, the EWMA chart does not accomplish the goal of detecting small shifts. Notice that near the right side of the chart the skin temperature has increased to approximately 98 degrees and the control limits should be recalculated; however, the EWMA program did not detect the change. When a large change occurs in a series of small increments, the EWMA chart may fail to identify the change (Figure 3). Slow changing vital signs such as skin temperature produce this type of pattern in control charts.
Figure 3 Small Changes (Oxygen saturation data set)
Figure 4 shows the results using a Shewhart type control chart for the same data set shown in the EWMA chart in Figure 3. The Shewhart patient control chart identified the change in the data while the EWMA chart did not.
Figure 4 Shewhart Small Changes (Oxygen saturation data set)
The EWMA may be used to identify small changes that occur at a rate greater than the forecasted rate. Another solution might be to adjust the EWMA control limits to other than three standard deviations and/or use a run beyond the limits as a decision rule for recalculating the limits.2
EWMA charts require an estimate of the constant alpha for calculations. Alpha controls the age of the data included in the EWMA calculation. When alpha is large (near 1), the more recent observations are the most important. When alpha is low (near 0), the older observations are the most important. Alpha and autocorrelation are inversely related. Autocorrelation is a measure of the importance of history. When autocorrelation is high (near 1), the older observations are the most important. When autocorrelation is low (near 0), the more recent observations are the most important.
Other types of manufacturing processes can calculate and set a constant alpha based on a constant level of autocorrelation. In patient vital sign data, autocorrelation is not constant, but varies depending on the patient’s condition.Therefore, when EWMA charts are used for patient vital sign monitoring, alpha should be continually estimated for the degree of autocorrelation among the observations. With a variable autocorrelation there are two approaches available for estimating alpha:
In addition to variable autocorrelation, some vital sign measurements are rounded to the closest whole number, tenth or hundredth. The round off makes calculation of autocorrelation difficult because the level of autocorrelation will approach .9 to 1, which places the value of alpha near zero. For this study we set alpha at 0.05.
Shewhart charts work with highly autocorrelation data with additional mathematical procedures. The procedure we used to create our Shewhart process control charts included estimating the autocorrelation during the base period and then adjusting the control limits based on the estimated autocorrelation (Figure 4).
When rounded off vital data approaches 100% the value of sigma approached zero. Both the EWMA and Shewhart process control charts have calculation problems when sigma is zero; therefore, the researchers used a minimum default value for sigma.
Changes in autocorrelation also have a direct effect on the standard deviation (sigma). The standard deviation is inversely proportional to the value of autocorrelation (autocorrleation = 1 – sigma). As autocorrelation increases sigma decreases. The EWMA forecasting chart and the Shewhart patient control chart for averages do not work well without the addition of a standard deviation or range process control chart. Figures 3 and 4 included a standard deviation chart.
Figure 5 illustrates the results of a simulation analysis on 1) a control limit adjusted Shewhart chart, 2) a centered reduced EWMA chart, 3) a standard EWMA chart, and 4) a sigma chart. All charts show that the sigma process control chart seems to provide us with the best information of when the autocorrelation level changed.
Figure 5 Comparisons of Shewhart and EWMA Charts
EWMA charts do a satisfactory job of identifying changes in patient vital sign data, with the exception of identifying large changes that occur in small increments. The Shewhart chart with limits adjusted for autocorrelation had no limitations. The researchers found that caregivers in the study did not like the complexity of either control chart, and in particular did not like the additional complexity of the EWMA chart. More experimentation needs to be done with the centered reduced control chart to see if these charts can detect small changes without increasing the type I error rate. An important question remains: “Is the loss of information a fair tradeoff for the simplicity of the chart?” In addition, the centered reduced chart lends itself to output on a small printer strip that does not have room to graph the vital sign data swings. The centered reduced graphing may be used for both EWMA and Shewhart charts. Caregivers do not like statistical terms such as exponential weighted moving average and EWMA. This problem can be eliminated by not using the technical terms. Because Shewhart charts have fewer application problems, the researchers prefer to use Shewhart patient control charts with adjusted limits over EWMA charts.
1) Chandrasekaran, S., J. R. English, and R L. Disney, (1995), “Modeling and Analysis of EWMA control schemes with variance-adjusted control limits,” IIE Transactions, June 1995, vol. 27(3), pp. 282-291.
2) Klein, Morton, (1996), “Composite Shewhart-EWMA Statistical Control Schemes,” IIE Transactions, June 1996, Vol. 28(6), pp. 475-481.
3) Zimmerman, Steven M., Robert N. Zimmerman, Lonnie D. Brown, and Shannon S. Brown, (1992) "Using Moving Average Process Control Charts in Biomedical Applications," Proceedings- Ninth International Conference of the Israel Society of Quality Assurance, 1992, November 1992, p.761-764.
