IHP 340 Module 3 Correlation Short Paper example

Booked Further Out, Missed More Often? Scatterplot and Correlation Analysis of Lead Time and No-Show Rates in 15 Primary Care Clinics

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Southern New Hampshire University

IHP 340: Statistics for Healthcare Professionals

Module Three Short Paper

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[Date]

The organization, setting and figures below are a composite written as a model document. No real employer, client, colleague or patient is described.

Booked Further Out, Missed More Often? Scatterplot and Correlation Analysis of Lead Time and No-Show Rates in 15 Primary Care Clinics

The Question and the Data

A composite network of 15 primary care clinics loses appointment slots every day to patients who do not arrive and do not cancel. The operations director has noticed that some clinics book routine appointments weeks in advance while others can see patients within days, and she asks whether longer booking lead times go along with higher no-show rates. For each clinic, the network calculated the median lead time, the number of days between the day an appointment was booked and the appointment itself, and the no-show rate, the percentage of scheduled appointments missed without cancellation, over the last six months.

Both variables are ratio-level measurements, which makes them suitable for a scatterplot and a Pearson correlation. The data are shown below.

Table 1

Median Lead Time and No-Show Rate by Clinic

Note. Mean lead time 14.5 days (SD 7.8); mean no-show rate 12.4 percent (SD 3.0).

What the Scatterplot Shows

With lead time on the horizontal axis and no-show rate on the vertical axis, the scatterplot shows a positive direction: as lead time increases, no-show rates tend to rise. The form is roughly linear, with no obvious curve. The strength appears moderate to strong, since most points lie near an upward-sloping line but with visible scatter. One point stands apart. Clinic O has the longest lead time, 28 days, but a no-show rate of only 11.2 percent, well below what the pattern of the other clinics would predict. A careful reader notices that point before calculating anything, because one unusual clinic can move a correlation coefficient a long way in a sample this small.

The Correlation and What It Means

The Pearson correlation coefficient for all 15 clinics is r = 0.73, and the relationship is statistically significant (t = 3.82, df = 13, p = .002). By a commonly used guide for medical research, a coefficient between 0.70 and 0.90 represents a high positive correlation (Mukaka, 2012). The coefficient of determination is r squared = 0.53, meaning that about 53 percent of the variation in no-show rates among these clinics is associated with the linear relationship with lead time. The remaining 47 percent is associated with other factors, such as the patients each clinic serves, transportation access and reminder practices.

The least squares line has a slope of about 0.28, which means that, on average, each additional day of median lead time is associated with a no-show rate about 0.28 percentage points higher. Across the range of the data, from 3 to 28 days, that amounts to a difference of roughly 7 percentage points.

Checking the Assumptions and the Uncertainty

A Pearson correlation assumes a linear relationship, quantitative variables and observations that are independent of each other. The first two conditions are met. Independence is reasonable because each clinic has its own patients and schedule, although clinics in the same network share some policies. The larger concern is sample size. With only 15 clinics, the estimate of r is imprecise: a 95 percent confidence interval for the correlation, calculated with the Fisher transformation, runs from about 0.34 to 0.90. The true relationship across clinics like these could therefore be anywhere from modest to very strong. That width is another reason to treat the finding as a signal worth testing rather than a precise measure.

The Outlier

When Clinic O is removed, the correlation rises to r = 0.89 and r squared to 0.79. A single clinic therefore lowers the apparent strength of the relationship considerably. Before deciding whether to exclude it, the analyst should ask why it differs. The clinic manager reports that Clinic O introduced a two-way text reminder system a year ago that asks patients to confirm or cancel three days before the visit. That explanation suggests the clinic is not a data error but a meaningful exception: it may show that the effect of long lead times can be reduced. The outlier should be reported, not deleted, and both coefficients should be shown. Anscombe (1973) demonstrated with four very different datasets that share the same correlation coefficient that summary statistics can hide important features that only a graph reveals, which is why the scatterplot comes first.

Correlation Is Not Causation

The strong correlation does not show that long lead times cause no-shows. Several other explanations are possible. Clinics with long lead times may be busier because they serve larger populations with more social barriers, and those barriers could cause both long waits and missed visits. Clinics with short lead times may be newer or in areas with better public transportation. The direction could even run partly the other way: clinics with many no-shows might overbook to compensate, which lengthens lead times. Because the data are observational and measured at the clinic level, they also cannot show that individual patients who book further ahead are more likely to miss appointments. Drawing that conclusion from group data would be an ecological fallacy.

Recommendation

The analysis supports a cautious recommendation. The network should treat lead time as a warning sign and test interventions rather than assuming a cause. Two tests are reasonable: expanding the two-way reminder system used at Clinic O to two clinics with long lead times, and reserving a share of appointment slots at one clinic for booking within 48 hours. Comparing no-show rates in these clinics before and after the changes, against similar clinics without them, would provide stronger evidence about cause than the correlation alone.

Conclusion

Across 15 clinics, median booking lead time and no-show rate show a high positive linear correlation, r = 0.73, with lead time associated with about half of the variation in no-show rates. One clinic with a reminder system weakens the pattern and points to a possible solution. Because correlation cannot establish causation, the finding should guide testing rather than immediate changes to scheduling policy (Daniel & Cross, 2018).

References

Anscombe, F. J. (1973). Graphs in statistical analysis. The American Statistician, 27(1), 17-21. https://doi.org/10.1080/00031305.1973.10478966

Daniel, W. W., & Cross, C. L. (2018). Biostatistics: A foundation for analysis in the health sciences (11th ed.). Wiley.

Mukaka, M. M. (2012). A guide to appropriate use of correlation coefficient in medical research. Malawi Medical Journal, 24(3), 69-71.