DNP/701 Week 6: Correlation and Regression, sample paper

Fourteen Units, One Correlation of Minus 0.48 and a Regression the Data Cannot Support: Relating Nurse Staffing to Falls, and What Two Landmark Staffing Studies Did Differently

[Student Name]

University of Phoenix

DNP/701: Biostatistics and Epidemiology

Week 6 Assignment

[Instructor Name]

[Date]

The hospital and its figures are a composite written for a model paper.

I am a DNP student and clinical nurse specialist at a 300-bed community hospital, where I also teach part time in our affiliated rural clinic network. Our chief nursing officer asked whether units with more nursing hours have fewer falls. I have data from 14 inpatient units for the past year: nursing hours per patient day and falls per 1,000 patient days. This paper presents the correlation, its interpretation and what regression could and could not add.

The Data

Nursing hours per patient day ranged from 7.2 to 11.8 across units, with a median of 8.9. Falls ranged from 1.4 to 5.1 per 1,000 patient days, with a median of 3.0. A scatterplot shows a general downward pattern: units with more hours tend to have fewer falls, with two units well off the trend.

The Correlation

The Pearson correlation between hours and falls is minus 0.48. Squaring it gives 0.23, meaning about 23% of the variation in fall rates across units is linearly associated with variation in nursing hours. The 95% confidence interval is wide, roughly minus 0.80 to plus 0.07, and the p value is about .08.

Interpreting a Correlation

Schober et al. (2018) explain that correlation coefficients describe the strength and direction of a relationship, that Pearson's coefficient assumes a linear relationship and is sensitive to outliers, that Spearman's rank correlation is preferable for skewed data or outliers and that correlation does not imply causation. They caution that labels such as weak or moderate are arbitrary and that the coefficient should be interpreted in context with its confidence interval.

Checking Robustness

Because two units are outliers, I calculated Spearman's rank correlation, which was minus 0.41. The direction and approximate size are similar. One outlier is a rehabilitation unit with high fall risk due to its mobility-focused care, and the other is a short-stay surgical unit with low risk. Unit type likely confounds the relationship.

With 14 units, the relationship could be anything from strong to nonexistent; the interval, not the coefficient, is the honest answer.

Why Regression on 14 Units Would Mislead

A simple regression of falls on hours would give a slope, perhaps showing that each additional hour per patient day is associated with 0.4 fewer falls per 1,000 patient days. But with only 14 units, adding adjustments for unit type, patient age or acuity would quickly overfit the model. Each additional predictor needs many more observations to be estimated reliably.

Nonlinear Possibilities

The benefit of added hours may level off: moving from 7 to 8 hours might matter more than moving from 11 to 12. A Pearson correlation assumes a straight line and would understate a curved relationship. With more data, a scatterplot with a smoothed line would show whether such a threshold exists.

Ecological Fallacy

This is a unit-level analysis. Even if units with more hours have fewer falls, it does not show that individual patients cared for by nurses with lighter assignments fall less. Conclusions about individuals from group-level data risk the ecological fallacy.

How a Landmark Study Did It

Aiken et al. (2002) linked data from 168 Pennsylvania hospitals, surveys of more than 10,000 nurses and outcomes for more than 232,000 surgical patients. Using logistic regression adjusted for patient and hospital characteristics, they reported an odds ratio of 1.07 for 30-day mortality with each additional patient in a nurse's assignment, an identical odds ratio for death after a complication, and an odds ratio of 1.23 for nurse burnout. The patient-level data, large sample and adjustment for confounders made the estimates far more credible than a correlation across a few units.

A Different Design

Needleman et al. (2011) studied inpatient mortality in one academic medical center, linking each patient's exposure to individual shifts where staffing was eight hours or more below target. Using survival models, they reported a hazard ratio of 1.02 for each understaffed shift a patient experienced and 1.04 for each high-turnover shift. By measuring exposure at the shift level for each patient, they avoided the ecological problem.

What Our Hospital Could Do

Borrowing the shift-level approach of Needleman et al. (2011), I could link each patient's falls to the staffing on the shifts they experienced, using our staffing and incident systems. A logistic or survival model at the patient level, adjusted for age, fall risk score and unit type, would give a more credible estimate than a correlation across units.

Measurement Issues

Nursing hours per patient day combine registered nurse, licensed practical nurse and aide hours, which may have different effects. Fall rates depend on how consistently falls are reported, which varies by unit culture. A unit with strong safety reporting may appear to have more falls than one where falls go unreported. Both measurement problems add noise and possible bias to the correlation.

Direction of the Relationship

The correlation does not show which way the relationship runs. Units with more falls might receive additional staff as a response, which would weaken or even reverse the observed correlation. Time-ordered data, with staffing measured before falls, would help separate cause from response.

What a Slope Would Mean

If a regression gave a slope of minus 0.4 falls per 1,000 patient days per added hour, that would suggest about one fewer fall per month on a 30-bed unit for each added hour per patient day, but only if the relationship were causal and unconfounded, which our data cannot establish.

Assumptions of Regression

Linear regression assumes a linear relationship, independent observations, constant variance and normally distributed residuals. Logistic regression, used for yes-or-no outcomes like falling, assumes a linear relationship between predictors and the log odds of the outcome. Patients within the same unit are not independent, so a multilevel model would account for clustering.

Sample Size for a Patient-Level Model

Our hospital has about 18,000 admissions a year and roughly 300 falls. A common rule of thumb for logistic regression calls for about 10 outcome events per predictor, so 300 falls could support a model with up to about 30 predictors, far more than a unit-level analysis of 14 observations.

Reporting to the Chief Nursing Officer

My message: "Units with more nursing hours tended to have fewer falls, but with 14 units the estimate is imprecise and partly reflects differences in unit type. Published patient-level studies show that staffing affects outcomes; a patient-level analysis of our own data would tell us more."

Conclusion

The correlation of minus 0.48 between nursing hours and falls across 14 units is consistent with a protective effect but is imprecise, sensitive to outliers and confounded by unit type. Guidance on interpreting correlation emphasizes confidence intervals and context. Landmark staffing studies used patient-level regression with adjustment, large samples or shift-level exposure to produce more credible estimates, a model for our hospital's next analysis.

References

Aiken, L. H., Clarke, S. P., Sloane, D. M., Sochalski, J., & Silber, J. H. (2002). Hospital nurse staffing and patient mortality, nurse burnout, and job dissatisfaction. JAMA, 288(16), 1987-1993. https://doi.org/10.1001/jama.288.16.1987

Needleman, J., Buerhaus, P., Pankratz, V. S., Leibson, C. L., Stevens, S. R., & Harris, M. (2011). Nurse staffing and inpatient hospital mortality. New England Journal of Medicine, 364(11), 1037-1045. https://doi.org/10.1056/NEJMsa1001025

Schober, P., Boer, C., & Schwarte, L. A. (2018). Correlation coefficients: Appropriate use and interpretation. Anesthesia and Analgesia, 126(5), 1763-1768. https://doi.org/10.1213/ANE.0000000000002864