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Jay Arthur—The KnowWare Man

Health Care

The Ambiguous Control Chart Trend Rule

Trend rules are helpful in service industries but you need to know which one to use

Published: Tuesday, September 21, 2021 - 12:03

There are many control chart rules to detect special causes (i.e., out-of-control conditions). Although most of these rules are clear, the one that seems to befuddle most people is the rule about trends. Is it six points (including the first point), six points (excluding the first point), or seven points including the first point? Confusing, isn’t it? The goal of this article is to identify the usable answer to this question. But first, it might be interesting to take a look at the various rules proposed over the years.

Western Electric

The original Western Electric rules1 did not include a trend rule. These four rules compare a series of points in the data set to zones created by the 1, 2,  and 3 sigma lines.


Figure 1: Nelson’s original drawing illustrating the trend rule

Lloyd Nelson

In 1984, Lloyd Nelson added control chart rules, including the trend rule: six consecutive points increasing or decreasing.2 I suspect this rule was particularly hard to describe because Nelson included a visual representation of the rule (figure 1). Note that his written description lists “six points in a row steadily increasing or decreasing,” and each of the data sets shown in the illustration contain exactly six points. So, we can infer that Nelson’s definition of the trend rule is six points in a row steadily increasing or decreasing including the first point.

Joseph Juran

Joseph Juran’s Quality Control Handbook, 4th Edition3 uses Nelson’s rules and diagram.


Figure 2: HCDG’s illustration of the trend rule

The Health Care Data Guide

In 2011, Provost and Murray published The Health Care Data Guide (HCDG).4 This book defines statistical rules and practices that the healthcare field adopted as standards for data analysis. In this book, Provost and Murry define a trend in a Shewhart (i.e., control) chart as “Six consecutive points increasing (trend up) or decreasing (trend down).” Figure 2 shows six points decreasing (all circled) and eight points increasing (all circled). So again, this rule includes the first point.

Douglas Montgomery

In Douglas Montgomery’s Introduction to Statistical Quality Control,5 the rule is: “six points in a row steadily increasing or decreasing,” but without a visual, leaving the rule open to ambiguity.

AIAG

In the Automotive Industry Action Group’s (AIAG) Statistical Process Control, 2nd Edition,6 the trend rule is defined as six points in a row, all increasing or all decreasing (Table II.1, p. 75). Again, ambiguous. The first edition used seven points in a row as a trend, but the rule was changed in the second edition.


Figure 3: Westgard’s trend rule

James Westgard

In Basic QC Practices,7 James Westgard introduces “Westgard Rules,” which are a variation on the Nelson rules for Levey Jennings charts used in laboratories. Levey Jennings charts use standard deviation instead of sigma estimator to calculate limits. His 7T rule (figure 3) adds one additional point to the trends in figure 1. Seven points trending in the same direction, including the initial point, violates the rule. So, the rule is seven points including the first point.

On the other hand: Donald Wheeler

Incidentally, it must be noted that not everyone in the statistical community agrees that the “trend” rule adds value, regardless of the number of points involved. In Understanding Statistical Process Control, 2nd Edition8 Donald Wheeler says that a “runs up” or “runs-down” (i.e., a trend) does not increase the sensitivity of the control chart and will result in more false alarms. For a more detailed understanding of his point, see his article here.9

Some examples

People sometimes forget that control chart rules can detect something positive—solutions, not just problems. Working extensively with healthcare, I have found that the trend rule helps confirm the effects of process changes. Because services involve people and processes, not just machines, the improvement often takes place over time.

If we use Western Electric rules on healthcare patient falls data (figure 4), it shows an out-of-control condition in May and June but misses the process improvement that occurred later in October through March (figure 5). An improvement team discovered that falls were concentrated in an orthopedic-recovery nursing unit and involved men between the ages of 20 to 40.


Figure 4: Healthcare falls per 1,000 days X chart using Western Electric rules. (Created with QI Macros.)


Figure 5: Healthcare falls per 1,000 days X chart (using healthcare rules) showing a trend and process change. (Created with QI Macros.)

The trend rule confirms improvement before the process stabilizes and shows a run. Healthcare is chasing a goal of zero harm, popularized by the Joint Commission. Once a hospital approaches zero, it becomes more difficult to detect changes.

As another example, increases in hospital-acquired infections happen slowly but steadily. The trend rule will detect these kinds of problems before they become a hospitalwide epidemic.

Where to use the trend rules? It depends.

As Wheeler and other who work in manufacturing environments have pointed out, trend rules may not be as useful in those situations. However, based on my experience, they are useful in service industries. And if we are going to use them, we need to be clear about the rule.

After reviewing these references, it is clear that the trend rule is either six or seven points in a row increasing or decreasing, including the initial point as shown in figures 1 and 3. Let’s stop arguing about whether the trend rule includes the first point. According to Nelson, it does, and the correct number of points is six. The vast majority of cited sources agree. So, the choice of trend rule is seven points for labs using Levey-Jennings charts and six points for everyone else.

Similarly, there’s disagreement about how many points constitute a run above or below the center line. Most sources say seven, eight, or nine points in a row. The most common number of points is eight, so consider that to be a good starting point. I have found Montgomery’s rules to be the best hybrid of Nelson’s and AIAG’s rules.

Concerning the question of whether to use the trend rule or not, I would rather be alerted to a potential unstable condition so that I can investigate rather than be blindsided after the fact. If you are analyzing these runs manually or mechanically, use your customer’s desired rule set. (I use healthcare rules when working with hospitals and AIAG rules when working with automotive.)

When using SPC software, you will need the flexibility to use any of these rules to meet your customer’s requirements. Figure 6 is a summary of the commonly used rules.


Figure 6: Control chart rules by source

References
1. Western Electric Co. Statistical Quality Control Handbook, 2nd Edition. ‎ AT&T Technologies/Western Electric, 1982.
2. Nelson, Lloyd S. “The Shewhart Control Chart—Tests for Special Causes.” Journal of Quality Technology, Feb. 22, 2018.
3. Juran, Joseph M. Juran’s Quality Control Handbook, 4th Edition. McGraw-Hill, 1988.
4. Provost, Lloyd P. and Murray, Sandra K. The Health Care Data Guide. Josey-Bass, 2011.
5. Montgomery, Douglas C. Introduction to Statistical Quality Control, 7th Edition. Wiley, 2013.
6. Automotive Industry Action Group (AIAG). Statistical Process Control—SPC, 2nd edition. AIAG, 2005.
7. Westgard, James O. Basic QC Practices 3rd Edition. Westgard QC, 2010.
8. Wheeler, Donald J. and Chambers, David S. Understanding Statistical Process Control, 2nd Edition. SPC Press, 1992.
9. Wheeler, Donald J. and Stauffer, Rip. “When Should We Use Extra Detection Rules?Quality Digest, Oct. 9, 2017.

Discuss

About The Author

Jay Arthur—The KnowWare Man’s picture

Jay Arthur—The KnowWare Man

Jay Arthur, speaker, trainer, founder of KnowWare International Inc., and developer of QI Macros for Excel, understands how to pinpoint areas for improvement in processes, people, and technology. He uses data to pinpoint broken processes and helps teams understand their communication styles and restore broken connections. Arthur is the author of Lean Six Sigma for Hospitals (McGraw-Hill, 2011), and Lean Six Sigma Demystified (McGraw-Hill, 2010), and QI Macros SPC Software for Excel. He has 30 years experience developing software. Located in Denver, KnowWare International helps service and manufacturing businesses use lean Six Sigma tools to drive dramatic performance improvements.

Comments

The Western Electric Zone

The Western Electric Zone Tests includes 8 successive points on the same side of the central line. In the example there are 7. But these are man-made rules that the underlying process does not care about. Maybe your conclusion about a real change would have been reasonable after only 4 successive points! 

Most important rule is to understand the data in its context. In the example given, the conclusion would likely have been the same with or without the extra trend rule. It then becomes unclear why the extra rule would do anything else than add complexity and increase the number of false signals.

KEEP IT SIMPLE

Dr Wheeler explains why you should only use Rule #1https://www.qualitydigest.com/inside/statistics-column/when-should-we-us...

Montgomery makes a mess of Process Behavior Charts.  They are NOT probability charts. 

https://www.linkedin.com/pulse/control-charts-keep-simple-dr-tony-burns/

It's not about what rules to use

This article is about what is the correct way to interpret the trend rule, not about which rules you choose to use. That's up to you and your customer.

KEEP IT SIMPLE

Dr Wheeler explains why you only need [Western Electric] Rule #1 here:https://www.qualitydigest.com/inside/statistics-column/when-should-we-us...

Counting points - the fencepost problem

Fun with integer math. Do we count the fenceposts, or do we count the fence sections built on those posts? A 1-D fence needs one more fencepost than fence sections.

So is the trend rule that we need 6 fenceposts or 6 fence segments?

"six consecutive points increasing or decreasing" implies that we need 6 fence segments since it's the fence segments (line segments connecting points) that can have increasing or decreasing slopes. But common usage is to use 6 fenceposts (points).

Fence Posts

Nelson counted posts, not segments.

Rule 4 or Rule 5

In the example for Figure 4 and 5 you cite the Trend rule (5) as the reason to review the limits of the process and adjust them. Isn't the real indicator a long-term shift below the calculated mean line of the process? Rule 4 is the one that indicates a major process shift in this case.

Also in all the examples that you cite as references you seem to have over looked the ISO series on control charting (ISO 7870-1-9). This series provides excellent guidance on the usage and development of control charts for many applications.

Trends Signal Process Shifts Too

First, I doubt that a 21-page, $120 ISO standard will shed any startlingly new clarity on control charts that is not contained in the references cited.

Second, I will argue that a trend signifies a process shift just as much as a run of 8 points below the center line. If patient falls started increasing, any hospital CEO worth his or her salt wouldn't wait for a run of 8 points above the centerline to take corrective action.

The goal of any control chart is to tell a story—one of concern (instability), stability or one of improvement. This example shows the improvement in a way that can be easily grasped by anyone.

It is possible that if the

It is possible that if the 21-page, $120 ISO standard was used the UCL and LCL wouldn’t have changed between Figures 4 and 5. No out of control point for 2019/06/30 in Figure 5. Also how can the process be stabilized after only 6 or 7 points? Don't you need to have more data to show the process is now in control at the ’new’ limits?

Trends signal process shifts

Every reference cited says the same thing as the ISO standard. That's great when you are working on a manufacturing line and sampling data every hour. You get results in a day or less. I can understand that point of view.

In service industries, some indicators, such as Falls per 1000 Patient Days are tracked weekly or monthly. Service industries can't wait eight months to take corrective action when an indicator goes wrong, nor should they wait eight months to declare improvement.

While you are arguing the letter of the law developed for manufacturing over the last century, I believe that we can use the intent of the law for services. Since most people work in service industries, we need rules that work for them as well.

The limits still changed

The limits still changed bewtween Fig 4 and 5 and an out of control point suddenly was ok.

Why not follow a standard so that all service industries are using the same terms and defintions? You wouldn't have this issue about fence posts vs. fence sections.

Still I guess a trend of 5 points in a downward direction is a start in testing for assignable cause. Your example took 6 months to see if the the new process change worked. Would another way of measuring improvement have been faster? What if the new process failed?

Work with the data you have

Yes, it would have been nice to know the results faster. In this case I was just working with the data available.

A g Chart of time between patient falls (instead of falls per 1000 patient days) might have demonstrated the change faster, but that data was not available.

Again, I'm not overly concerned with whether people use the trend rule or not. If they do, I'd like them to apply it correctly.