NUR 8050 Module 5 Data Analysis Paper: Control Charts for a Rare Hospital Infection Example

When the Lower Limit Is Zero: Run Charts, a u-Chart, a g-Chart and Segmented Regression for 24 Months of Non-Ventilator Pneumonia Data

Student Name

Doctor of Nursing Practice, Thomas Edison State University

NUR-8050: Scholarly Inquiry: The Basis for Evidence-Based Practice

Mentor Name

Month Day, Year

When the Lower Limit Is Zero: Run Charts, a u-Chart, a g-Chart and Segmented Regression for 24 Months of Non-Ventilator Pneumonia Data

Introduction

The project introduced a twice-daily toothbrushing protocol on a 34-bed medical-surgical unit to reduce pneumonia in patients who are never ventilated. Module 4 set out an interrupted time series with twelve monthly measurements on each side of the change. This paper analyzes the results. The figures are those of an invented unit, built to illustrate the methods, but the methods are the ones a DNP project would use on real data. The central challenge is that the outcome is rare: about one case a month at baseline. Rare events defeat some common tools, and the analysis has to be chosen with that in mind.

The Process Measure First

Before asking whether pneumonia fell, the analysis asks whether the intervention happened. In the baseline month, 44 percent of patient-days had two documented brushings. Over the first three months of the protocol this rose to 71, 82 and 85 percent, and it stayed between 84 and 91 percent for the rest of the year, averaging 88 percent. Weekly bedside audits agreed with the documentation in 90 percent of checks. The intervention was delivered reliably, so any change in the outcome can reasonably be linked to it.

Run Chart Rules

On a run chart the monthly rates sit in date order around a line drawn at their median. Perla et al. (2011) set out four probability-based tests, named shift, trend, runs and astronomical point, each calibrated so that ordinary month-to-month noise seldom sets one off. Their shift rule asks for at least six consecutive values on the same side of the median; their trend rule asks for five consecutive values moving in one direction. Points that fall on the median are skipped when counting.

The rules were designed so that each has a false alarm rate of about 5 percent or less when the process has not changed, which is why they can be applied to small monthly data sets without formal statistics. They are also easy to explain to staff, who can watch the chart on the unit board and see a shift for themselves.

Across the 24 months, the median monthly rate was 0.85 per 1,000 patient-days. The first eight post-intervention months all fell below the median, which meets the rule for a shift. No trend rule was met, and no single month was astronomical. The run chart therefore signals a nonrandom change that began after the protocol started.

The u-Chart and Its Limit

A u-chart is the standard Shewhart control chart for counts per unit of exposure when the exposure varies, as patient-days do from month to month. The baseline center line is the total cases divided by total thousands of patient-days: 14 divided by 11.7, or 1.20 per 1,000. Each month gets its own limits: take the baseline rate, divide it by that month's patient-days in thousands, square-root the result, triple it and add or subtract it from the baseline rate. For a typical month of 975 patient-days, the square root of 1.20 divided by 0.975 is about 1.11, so the upper limit is about 4.5 per 1,000 and the lower limit, 1.20 minus 3.33, is below zero and is therefore set at zero.

That lower limit is the problem. A month cannot fall below zero, so no single month can signal improvement on this chart, however good it is. The u-chart can only detect improvement through rules about runs of points below the center line, which are the same patterns the run chart already found.

The g-Chart for Rare Events

Benneyan et al. (2003) recommend charts based on the number of days or opportunities between events when events are infrequent, because they detect improvement faster than rate charts. A g-chart plots the interval between consecutive cases. At baseline the intervals between the 14 cases had a median of 21 days. After the protocol, intervals lengthened, and two exceeded 70 days, crossing the upper control limit calculated from the baseline intervals. On a g-chart, points above the upper limit signal improvement, since longer gaps mean fewer cases. This chart shows the change more clearly than the u-chart and allows the team to see improvement as soon as a long gap occurs rather than waiting for a run of months.

Segmented Regression

Penfold and Zhang (2013) describe segmented regression as the statistical test for an interrupted time series. The model fits the starting height and direction of the line, then asks whether the line jumped when brushing began and whether its angle changed afterward. For this project, the model estimated no significant baseline trend, a drop in level of 0.48 per 1,000 patient-days at the start of the protocol and no significant change in slope afterward. The confidence interval for the level change was wide and included values close to zero, which is expected with so few events.

What the Data Show and Do Not Show

The pneumonia rate fell from 1.20 to 0.67 per 1,000 patient-days, a relative reduction of about 44 percent, after a protocol that was delivered reliably. Run chart rules, the g-chart and the segmented regression point in the same direction. The absence of a baseline trend makes it less likely that the fall merely continued an existing decline, which was the main weakness of the published study this project built on. Yet with eight cases after the change, chance cannot be excluded, and other changes in the hospital during the year, such as a mobility program on another unit, would need to be checked. The appropriate conclusion is that the evidence supports continuing the protocol and spreading it with continued measurement, not that it proves the protocol prevents pneumonia.

Conclusion

For a rare outcome, the choice of analytic tool decides what can be seen. Run chart rules and a g-chart detected a change that a u-chart could show only through runs, and segmented regression estimated its size with honest uncertainty. The process data confirm the intervention was delivered. The next module evaluates the project as a whole and plans how to sustain it.

References

Benneyan, J. C., Lloyd, R. C., & Plsek, P. E. (2003). Statistical process control as a tool for research and healthcare improvement. Quality and Safety in Health Care, 12(6), 458-464. https://doi.org/10.1136/qhc.12.6.458

Penfold, R. B., & Zhang, F. (2013). Use of interrupted time series analysis in evaluating health care quality improvements. Academic Pediatrics, 13(6 Suppl.), S38-S44. https://doi.org/10.1016/j.acap.2013.08.002

Perla, R. J., Provost, L. P., & Murray, S. K. (2011). The run chart: A simple analytical tool for learning from variation in healthcare processes. BMJ Quality and Safety, 20(1), 46-51. https://doi.org/10.1136/bmjqs.2009.037895