Rare Events & Conventional SPC
Rare events in healthcare pose a problem in deciding how to monitor them with SPC charts. There are many examples of rare event data in healthcare including medication errors, patient falls, nosocomial infections, surgical complications, needlestick accidents and VAP events all representing important measures. Because these events occur so infrequently, conventional charts exhibit problems.
These problems are related to data collection and to theory behind the conventional chart. In the u chart and the p chart, we need some knowledge of the denominator which may not be easy to come by. With the u chart, we often use a surrogate number such as patient days to represent the area of opportunity simply because the opportunities are very difficult, if not impossible, to count. This helps us over the issue with collecting denominator data. With the p chart, using a binomial distribution, we would need to count each opportunity for an event to occur. It may be difficult to collect this data as well.
In order for these control charts to be effective, there are some 'rules of thumb' for the amount of data related to the theory behind them. For the p chart, the rule of thumb is that np ≥ 5 where n is the subgroup size and p is the probability that the event will occur.
This means that for a process with a p of 0.01 we need an n of 500. If we have a medication error probability of 1 in 100 doses, our subgroups would need to be at least 500 doses to properly use a p chart.
In other words, our period length would need to be long enough to capture this many doses for our p chart to be effective, in this case 500 doses in the period. This also translates into a longer period before we can detect that something has changed affecting the med error rate. Ideally, we would like to detect this process shift as soon as possible. If our probability is smaller, we would need even larger subgroups. Similar caveats exist for the u chart.
Ideally, we would like to detect this process shift as soon as possible. If our probability is smaller, we would need even larger subgroups. Similar caveats exist for the u chart.
Control Chart for Intervals Between Events
Where we have very few of these events, charts drawn using a reasonable time period subgroup would probably show a lot of zeros with little valuable information.
With rare events, it may take several years to get 24 points (for a conventional control chart) and expected counts below one are the rule.
Hence, for time-ordered analyses, inter-arrival intervals must be used. The inter-arrival interval is the date of an event minus the date of the preceding event. Interarrival intervals are expected to have an exponential distribution, which is quite well met with real data sets.
The simplest implementation of the g chart (Benneyan) is measuring time between events such as days between patient falls. This data is relatively easy to collect. You only need to record the dates that events occurred.
A change in thinking is required to evaluate a g chart.
A higher value on the chart means that the rate of the event occurring has actually decreased because the time between events is longer. For adverse events this is a good thing.
A smaller value plotted on the chart means that the rate of the event occurring has increased.
Alternatively, a fourth root transformation is said to work well with such data sets (Hart & Hart).
Benneyan, James C.
Number-between g-type statistical control charts for monitoring adverse events.
Benneyan, James C.
Performance of number-between g-type statistical control charts for monitoring adverse events.
- Hart MK, Hart RF.
Statistical process control for health care. Wadsworth Group, 2002.
Risk-Adjusted Cumulative Sum (CUSUM) Control Charts
The surveillance of hospital-acquired infection (HAI) can be frustrated by the infrequent nature of many infections. Traditional methods of analysis may fail to detect unacceptable increases quickly enough, resulting in preventable risk to patient.
Statistical process control methods such as the Cumulative Sum (CUSUM) and Exponentially Weighted Moving Average (EWMA) are particularly suitable for the detection of small sustained increases in uncommon events.
When employed with individual instances of infection, they can be based on the Poisson distribution and used without denominator data, which are often difficult to identify and frequently time-consuming to collect.
The CUSUM chart directly incorporates all the information in the sequence of sample values by plotting the cumulative sums of the deviations of the sample values from a target value.
Because they combine information from several samples, CUSUM charts are more effective than Shewhart charts for detecting small process shifts.
They are particularly effective with samples of size n=1. If an upward trend develops in the plotted points, this is evidence that the process mean has shifted, and a search for some assignable cause should be performed.
There are two ways to represent CUSUMS, the "tabular" and the "V-mask" forms. Since the leader in this field  says the "tabular form is preferable", this is the only one used here.
Risk-adjustment can be incorporated easily, especially if monitoring common surgical procedures. We use the NHSN  reports and the NNIS risk index for surgical operations.
By pre-selecting a specfied risk level before plotting the CUSUM chart, further risk adjustment is unnecessary.
Alteration in infection rates may occur as a sudden large increase in infections numbers, or as a sustained rise in the mean infection rate.
As the Shewhart chart is better at detecting the former pattern (a shift of 1.5σ or larger), and CUSUM and EWMA charts are better at detecting the latter (samples of size n≥1), they are best employed together.
- Montgomery DC. Introduction to statistical quality control. Wiley & Sons, USA. 1997 3rd edn.
Dudeck MA, Weiner LM, Allen-Bridson K et al.
National Healthcare Safety Network (NHSN) report, data summary for 2012, Device-associated module.