u Charts, p Charts and i Charts


In Quality Indicators in Healthcare, much of the data is attribute data; that is, counts of things, events or outcomes. We know, for example, how many patient falls we had, how many AMI patients were given beta blockers at arrival or how many medication errors we have recorded. Often, the data can be represented on an i chart. The i chart is almost always available but it is seldom the best choice. Let’s take a look at the charts that may be better and why. For many examples of such data, view the indicators at http://pimd.statit.com, particularly the CHF indicators under the Quality Group.

While the i chart was designed to analyze measurements in a slow-moving process, one can plot either a count of events or a rate or proportion. However, this application of the i chart is likely to hide important information. For example, if the i chart is a plot of the number of medication errors in a period, it would in no way indicate the number of doses administered in the period. This is critical information in deciding whether a spike in the number is statistically important. The counts of one error vs ten errors may take on a different meaning if we know that in the former, there were only two doses given while in the latter, there were 1000 doses given.

In the same way, plotting the rates of medication errors (errors / total doses) on an i chart does not take into account the size of the subgroup in determining the control limits. Some critical information is hidden from the decision maker. Statistically, in estimating the confidence in a rate of .25, it makes a difference if that rate is 250 events in 1000 trials or 1 in 4. While there is some information of the subgroup size in the rate, the i chart does not take different sizes into account in calculating the control limits.

A c or u chart or a p chart are usually better choices for representing these data. The c and u charts work with what is known as an “Area of Opportunity” in which any number of events can occur. The Area of Opportunity is some unit that presents an opportunity for the event to occur and is also commonly referred to as "Subgroup Size". Patients being in a hospital present opportunities for patient falls. Administering medication presents opportunities for medication errors. Often the choice of a c or u chart over a p chart is determined by the question, "can an event happen more than once in the area of opportunity?" Can a patient fall more than once? Can there be more than one medication error in a dose of medication? If the answer is yes, then your data indicate a c or u chart. These charts allow for those situations where it is possible to have more events than areas of opportunity. If your area of opportunity is patient days, it is possible to have more than one fall per patient day, since it is possible for each patient to fall more than once on a particular day.

Once you have determined that a c or u chart is applicable, the decision of which one to use is dependent on whether the area of opportunity data are available. A c chart simply plots counts and assumes a constant size for the area of opportunity or subgroup. A u chart needs the size of the area of opportunity for its calculations because it plots counts per area of opportunity. If you choose the number of patient days as the area of opportunity for patient falls, then the c chart could be applicable only if the number of patient days was fairly constant across all periods charted. For some indicators, you may not have the actual size of the area of opportunity but can make an educated assumption that it is fairly constant. This may be dangerous, however, without data to back up your assumption.

If the area of opportunity varies and if you have the size of the area of opportunity for each count, the u chart is the better choice. The u chart will actually contain more information than the c chart (or the i chart). As Hart and Hart (2002) says, “Note that the u chart may always be used and that the c chart is never better.”

In u chart data, there are a number of areas of opportunities in each time period. The u chart calculates the plotted point by dividing the number of events by this number of the areas of opportunities in that time period. Be sure to note that the number of areas of opportunities can be fractional.

The area of opportunity for a u chart can be manipulated to a unit that may be more acceptable or reasonable to the user community, such as Patient Falls per 100 Patient Days or Medication Errors per 1000 doses. This is reasonable since these charts are based on the Poisson distribution which is applicable to infrequent events. The u chart divides the number of events by the number of units of areas of opportunities. A chart of the transformed data is what Grant and Leavenworth (1996) calls the ku chart, simply because scaling factor, k, is applied to the count of areas of opportunity, which has the effect of dividing the number of Areas of Opportunities by k. Statistically this manipulation has no affect on the analytical results but may scale the numbers to make more sense to the user. Thus, 6 falls in 635 patient days becomes 6 falls in 6.35 100 Patient Days or .94 falls per 100 patient days. This is a simple transformation of dividing the number of Patient Days for each period by 100. In this example, the Area of Opportunity is Patient Day and the Unit of Area of Opportunity is 100 Patient Days.

The p chart is used when there are only two possible outcomes to some event and the sum of the counts of the two outcomes equal the total area of opportunity (the total number of events). The former indicates a binomial distribution. The latter says that the count of outcomes is a subset of the area of opportunity. Several data in healthcare fall into this group. The medication administered was in error or it was not. Beta Blocker was administered at arrival or it was not.

The area of opportunity for this chart may actually be a recorded event such as number of medication doses administered or number of AMI patients admitted. In this case, the count of an outcome (numerator) is a subset of the count of events (denominator). The count of events is the Area of Opportunity. For all AMI patients admitted, how many were administered beta blocker at arrival? The event is an AMI patient was admitted and the two possible outcomes are that the beta blocker was administered or that it was not. The number of AMI patients who received the beta blocker is a subset of the total AMI patients. With this type of data, the number of a particular outcome divided by the number of events can never exceed 1.

In contrast, Patient Falls is not a subset of Patient Days. A number of Patients fell, but no Patient Days fell.

Another concept to help decide on a u chart or p chart is whether the actual number of opportunities for an event can be counted. If counting the opportunities is very difficult or impossible, it is necessary to use a u chart instead of a p chart. Obviously, the number of opportunities for a patient falling would be very hard to count. This leads one to attempt to find an area of opportunity that would be more manageable, but which will have an indicative relationship to the possibility of an occurrence of the event. Patient days for patient falls works because Patient Days contains an indication of the opportunity for patient falls. How do you think such things as “Nurses on duty” or “Wheelchairs available” or “Miles driven by the Ambulances” would work as Areas of Opportunity for Patient Falls?

Let’s look at some actual data to compare these charts.

Patient Falls

Here are the data for falls in a particular department in a hospital. The Rate is calculated as NumFall/NumDays*100. NumDays_100 is NumDays/100.

Date NumFalls NumDays Rate Numdays_100
Oct-2004 1 1048 0.09542 10.48
Nov-2004 4 896 0.446429 8.96
Dec-2004 3 918 0.326797 9.18
Jan-2005 4 995 0.40201 9.95
Feb-2005 2 866 0.230947 8.66
Mar-2005 3 896 0.334821 8.96
Apr-2005 5 864 0.578704 8.64
May-2005 2 930 0.215054 9.3
Jun-2005 0 932 0 7.32
Jul-2005 2 630 0.31746 6.3
Aug-2005 6 492 1.219512 4.92
Sep-2005 2 622 0.321543 6.22
Oct-2005 5 612 0.816993 6.12

An i chart on NumFalls:

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Figure 1. i chart - Number of Falls (mouse over data point to understand underlying data)

Notice that this chart does not give us any out of control points. We would decide that there are no assignable causes in this process.

An i chart on Rate:

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Figure 2. i chart - Rate (mouse over data point to understand underlying data)

Similarly, this chart shows that we have no assignable causes in the process although there is some information from the size of the subgroup or area of opportunity.

A c chart on NumFalls

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Figure 3. c chart - Number of Falls (mouse over data point to understand underlying data)

If we could assume that the number of patient days was fairly constant, we could get by with this chart. However our number of patient days range from 492 to 1048. We are on very thin ice if we make the assumption of equal area of opportunity.

A u chart on numfalls by numdays:

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Figure 4. u chart - Number of Falls per Patient Day (mouse over data point to understand underlying data)

The u chart gives us an alert for an assignable cause. That is the function of control charts, to find possible areas of improvement. In this case, it appears that August of 2005 may have had something change and we need to investigate to see if we can find a cause.

u chart on numfall per 100 patientDays:

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Figure 5. u chart - Number of Falls per 100 Patient Days (mouse over data point to understand underlying data)

This chart gives the same information as the chart in Figure 4. The transformation of data makes it easier to use. It displays a mean of .371 instead of .00371 rounded to .004. Notice also that this chart has a calculated lower control limit of 0 while the i charts in Figures 1 and 2 have negative control limits. Counts cannot go negative, so Attribute Charts (u, c, p, etc.) take this into account. Such an assumption cannot be made for measurement data. The i chart was designed for measurement data.

AMI 6 - Beta Blockers on Arrival

A similar situation is true for i charts and p charts. Consider the following data on AMI 6 Beta Blockers at Arrival. Rate is Num/Den and RateI is Rate*100.

Qtrstr Num Den Rate RateI
Q1 2004 50 55 0.91 90.91
Q2 2004 58 60 0.97 96.67
Q3 2004 49 56 0.88 87.50
Q4 2004 35 36 0.97 97.22
Q1 2005 28 40 1.00 100.00
Q1 2005 30 33 0.90 90.00
Q1 2005 20 23 0.91 90.97
Q2 2005 29 41 0.79 78.57
Q2 2005 34 38 0.88 87.50
Q2 2005 35 35 1.00 100.00

First an i chart:

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Figure 6. i chart - AMI 6 Rate (mouse over data point to understand underlying data)

And the p chart

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Figure 7. p chart - AMI 6 (mouse over data point to understand underlying data)

Again we see that the i chart does not alert us to possible assignable causes that the p chart does. We are in the business of improving our processes. If we were simply working to show that our processes are in control, the i chart might suffice. If we wish to find areas for improvement, we might choose a different chart more in line with the data.

In conclusion, if you can avoid an i chart, do so. We have several i charts on our Statit piMD demo site. For some, such as LOS charts, the i chart works well. For others, the rate is the only data we had. If you get into such a situation, perhaps you can ask why this is the only data available. After all, the rate needed to be calculated somehow. To summarize:

Look to a u chart if:

  • You are counting events (Patient Falls)
  • The event occurs over some area of opportunity (Patient Days)
  • The event can occur more than once in the Area of Opportunity (One patient falls twice on one day)
  • The number of opportunities for the event to occur is difficult to count (Need an indicative Area of Opportunity)
  • Number of Areas of Opportunity may be fractional (6.35 100 Patient Days)

Use the c chart in place of the u chart

  • Only if you do not have the Area of Opportunity data and
  • Only if you can make an educated assumption that the Area of Opportunity is roughly equal across all periods.

Look to the p chart if:

  • You are counting outcomes of an event class (Beta blockers on arrival for AMI patients admitted)
  • An event has exactly two possible outcomes (The sum of the counts of the outcomes equals the count of events)
  • Each event is countable (Number of AMI patients admitted)
  • Each event is an area of opportunity for the outcome under study (Was the beta blocker given at arrival)

References:

Grant, E.L., & Leavenworth, R.S. (1996). Statistical Quality Control (5th ed.). Boston: McGraw-Hill.
Hart, M.K., & Hart, R.F. (2002). Statistical Process Control for Health Care. University of Wisconsin, Oshkosh: Duxbury.

If you would like additional information, please send email to statit.support@acs-inc.com.