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Davis Balestracci

Statistics

Proper Use of a U-Chart

The simple power of plotting data in their naturally occurring time order wins the day

Published: Wednesday, March 21, 2018 - 12:03

Because of a growing movement in the health insurance industry toward not reimbursing hospitals for any expenses caused by a system-acquired infection, one health system made efforts to improve its infection rate starting in the last quarter of 2016. In June 2017, a year-over-year graph was presented to show progress to date.

Despite the impressive progress, there was obviously more work to do to eradicate these “should never happen events”:

The system consisted of five hospitals, and one analyst discovered monthly data for each all the way back to January 2015. He created the following graph (that I have dubbed a “copulating earthworm graph”), which for some strange reason executives (and many analysts) seem to love:

But he also knew the bottom line is the individual rates. He aggregated each of the five hospitals’ individual data into its overall rate, feeling this would result in the most accurate assessment, and came up with the following bar graph comparison (also showing the overall average):

A Black Belt told him it might be more insightful to present these data as an SPC chart. She showed him the table of “which chart to use,” and the decision tree suggested a u-chart. In addition to the default three-sigma limits, the Black Belt suggested adding the one- and two-sigma limits as well to better detect any outliers.

Do these three additional graphs have any value in suggesting how to proceed? Might the best strategy at this point just be “staying the course” because the initial year-over-year bar graph comparison showed so much progress?

Let’s apply some data sanity

When someone comes to me with a problem and asks, “Which chart should I use?” I always ask the following questions:
1. Could you please show me the data, or describe an actual situation, that are making you ask me this question? (Checked off, in this case.)
2. Please tell me why this situation is important. (Check.)
3. Please show me a run chart of this indicator plotted over time. (Copulating earthworms ≠ run chart; need individual run charts.)
4. What ultimate actions would you like to take with these data? (“Get better.” Sorry: Too vague!)

As I hope will be obvious to you from the charts below, a run chart would have easily identified the recent special cause “needle bumps” for hospitals 1 (last 8 points) and 4 (last 9 points). They also show that hospitals 2, 3, and 5 are common cause (systems 1 and 2 are from my last column). All of this was factored into the simultaneous Individuals control chart display below; note that all of these are on the same scale.

Note how putting the individual charts on the same scale is so much more useful than putting them all on one graph. Sometimes, this is all one needs to “tell the story” and know how to proceed. And believe it or not, I’ve had people frequently say to me, “This is too confusing. Couldn’t you just put them all on one graph?” Sigh....

Isn’t it also interesting to note how this stratification shows that the difference touted in the first bar graph’s year-over-year comparison was due to the improvement in hospitals 1 and 4? Hospitals 2, 3, and 5 have shown no improvement, even with the increased and vague, “We need to do better” emphasis at the end of 2016.

With the information provided from these comparative Individuals charts, one could now proceed to do a u-chart analysis of means (ANOM). This type of analysis is a statistical stratification that takes the overall system rate and stratifies it by the five individual hospital performances that aggregate into this overall average. This allows the five rates to be compared to see whether they are statistically different from the “system” average of which they are a part.

Please note: This is an appropriate use of a u-chart. It is when the u-chart is used for plotting data over time that one gets into trouble. Also, many times, the varying denominators due to differing sizes of hospitals being compared can significantly affect individual limits of the Individuals chart (although that’s not so true with these data). This can make direct visual comparison difficult if the individual averages don’t look differentiated enough by their same-scale placement. So, the only way to compare the actual rates is via a u-chart ANOM.

The denominator issue also invalidates a naïve use of traditional statistical calculations, including analysis of variance (ANOVA), on the rates themselves.

Aggregating the most recent, stable history for each hospital:

Uavg = 259 / 38.607 = 6.71

Like the p-chart ANOM, the common cause limits depend on the “window of opportunity” over which the count occurred, which in this case is the number of patient days for each hospital. More days imply more accuracy in their calculated average, which results in narrower limits.

U-chart common-cause limits:

For these data:

Applying this to the previous data:

The resulting u-chart:

Hospital 2 is indeed “above average,” and based on these data, the rates of the other four hospitals are neither significantly different from each other nor from the overall average.

The u-chart earlier in this article had two serious flaws:
• It used the entire histories of hospitals 1 and 4 in its analysis.
• It included one- and two-sigma limits.

Even now, with the correct u-chart ANOM, only when it’s used in conjunction with the display of the Individuals charts on the same scale does it become useful to determine subsequent action. If possible:
• Aggregate the 19 infections of hospital 1, the 109 infections of hospital 2, the 60 infections of hospital 3, the three infections of hospital 4, and the 68 infections of hospital 5.
• Do a Pareto analysis of each (difficult in the case of hospital 4, but see next steps)
• In the case of hospitals 1 and 4, compare these Pareto analyses to a Pareto analysis of “pre-needle bump” data: 152 infections for hospital 1 and 38 for hospital 4.
• Of course, investigate what happened at months 22–23 of hospital 1, and months 21–22 of hospital 4.

Compare these actions to those usually resulting from a meeting where the first four graphs of this article might be presented:



Once again, the simple power of plotting data in their naturally occurring time order wins the day. Until there is an app for critical thinking (that could take awhile), you would do well to start there in most situations and throw away all those flowcharts for “Which chart should I use for which situation?”

Discuss

About The Author

Davis Balestracci’s picture

Davis Balestracci

Davis Balestracci is a past chair of ASQ’s statistics division. He has synthesized W. Edwards Deming’s philosophy as Deming intended—as an approach to leadership—in the second edition of Data Sanity (Medical Group Management Association, 2015), with a foreword by Donald Berwick, M.D. Shipped free or as an ebook, Data Sanity offers a new way of thinking using a common organizational language based in process and understanding variation (data sanity), applied to everyday data and management. It also integrates Balestracci’s 20 years of studying organizational psychology into an “improvement as built in” approach as opposed to most current “quality as bolt-on” programs. Balestracci would love to wake up your conferences with his dynamic style and entertaining insights into the places where process, statistics, organizational culture, and quality meet.

Comments

XmR as a replacement

"Thus, if you do not have advanced degrees in statistics, or if you simply have a hard time
determining if your counts can be characterized by a Binomial or a Poisson distribution, you can
still verify your choice of specialty chart for your count-based data by comparing the theoretical
limits with the empirical limits of an XmR"  - Dr Wheeler

chart.