MPH 550 Week 5 Correlation and Regression Analysis Example

An R-Squared of 0.95 That Proves Less Than It Seems: Regressing County Diabetes on Physical Inactivity in Colorado

[Student Name]

University of Phoenix

MPH/550: Public Health Statistics

Week 5 Assignment

[Instructor Name]

[Date]

The health department, its analyst and the program proposal are composites written for a model paper; county estimates come from the CDC PLACES 2025 release, regression results were calculated from those estimates and other findings come from the sources cited.

A program manager at the county health department brought a chart to a planning meeting: Colorado counties with more physically inactive adults had more diabetes, and the points lay almost on a straight line. She proposed a county walking program and projected how much diabetes it would prevent using the line's slope. The analyst was asked to check the chart. This paper presents her correlation and regression analysis and what it can and cannot support.

The Data

Her source was the PLACES county file from CDC, age-adjusted and covering 2023, for all 64 Colorado counties: diagnosed diabetes as the outcome and physical inactivity, meaning adults who reported no exercise or other activity outside work during the previous 30 days, as the predictor (Centers for Disease Control and Prevention [CDC], 2025). The unit of analysis is the county.

The Scatterplot

The scatterplot showed a tight, upward, linear pattern. Boulder County sat at the lower left, with 12.2% inactivity, and Bent County at the upper right, with 28.9% inactivity and 13.1% diabetes. No curve or separate clusters appeared.

The Projection on the Table

The program manager's projection used the line directly. Pueblo County's inactivity estimate was 25.7%. If a walking program cut that to 20.7%, the slope of 0.365 implied diabetes would fall by about 1.8 points, which she translated into roughly 2,400 fewer adults with diabetes. The arithmetic was correct. The question was whether the slope could bear that interpretation.

Correlation

The Pearson correlation was 0.977, an extraordinarily strong positive linear association. For county-level health data, correlations this high are unusual and deserve suspicion before celebration.

Simple Linear Regression

The fitted line was: predicted diabetes equals 1.65 plus 0.365 times inactivity. The slope means that counties with one percentage point more inactive adults had, on average, 0.365 points more diabetes; its 95% confidence interval ran from 0.345 to 0.385. The intercept, 1.65, is the predicted diabetes at zero inactivity, far outside the data and not meaningful by itself.

R-Squared and Fit

The R-squared was 0.955: inactivity accounted for about 95% of the variation in diabetes across counties. The typical prediction error, the residual standard deviation, was 0.34 percentage points. The p-value for the slope was far below 0.001. By every standard summary, the model fits almost perfectly, which is exactly why the analyst kept checking. A fit this close in observational county data usually means the variables share a common source, not that one fully explains the other.

Residuals

Residuals showed no curve or widening pattern, so linearity and constant variance were reasonable. The largest positive residuals, counties with more diabetes than predicted, were Bent, Jefferson and Costilla, each about 0.9 points above the line. Pueblo County had the largest negative residual: predicted 11.0%, estimated 10.4%. The county the program targeted had less diabetes than its inactivity alone would predict.

Influential Points

The analyst checked whether a few counties drove the result. Removing Bent and Boulder, the two extremes, barely changed the slope, from 0.365 to 0.360, and no single county pulled the line noticeably on its own. The relationship is not an artifact of one or two unusual places.

Other Correlates

Diabetes correlated strongly with nearly everything related to disadvantage: 0.96 with lack of insurance, 0.90 with high blood pressure, 0.87 with obesity and 0.85 with smoking. It correlated weakly with depression, at minus 0.03. When one measure correlates above 0.85 with many others, a single-cause story becomes hard to defend.

Multiple Regression

The analyst added current smoking and lack of insurance as predictors. The inactivity coefficient fell from 0.365 to 0.213, with a standard error of 0.043; the insurance coefficient was 0.142, with a standard error of 0.032; smoking added little, 0.034 with a standard error of 0.033. R-squared rose only to 0.967.

Interpreting the Change

The inactivity slope fell because inactivity and lack of insurance travel together; their correlation was 0.95. Part of what looked like an inactivity effect reflected the broader disadvantage that both measure. With predictors this correlated, collinearity makes each coefficient unstable, and small changes in data or model could shift them considerably.

Why the Fit Is So Tight

Part of the answer lies in how the data were made. PLACES estimates come from multilevel models that combine survey responses with each county's demographic makeup, using the same census characteristics for every measure. CDC researchers showed that such model-based estimates track direct survey estimates closely at the county level, with correlations of 0.88 to 0.95 for the condition they studied (Zhang et al., 2014). Because each county's estimates share inputs, measures built from the same demographics will correlate more tightly than direct measurements would.

The Ecological Fallacy

Even if the correlation were fully real, it describes counties. In a classic demonstration using 1930 census data, the correlation between two characteristics across nine geographic divisions was 0.946, while the correlation for the same characteristics among individuals was only 0.203 (Robinson, 1950). A county-level slope cannot say how much a particular person's risk changes when they become active.

What Individual Studies Show

Individual-level research answers the program manager's real question more directly. A meta-analysis of 10 prospective cohort studies with 301,221 participants found a relative risk of type 2 diabetes of 0.69 for regular moderate physical activity compared with being sedentary, and 0.70 for regular brisk walking, with associations persisting after adjustment for body mass index (Jeon et al., 2007).

Advice to the Program Manager

The analyst supported the walking program but not the projection. The county regression cannot estimate cases prevented, because the slope reflects modeled data, shared disadvantage and county-level grouping. The cohort evidence supports physical activity as protective for individuals, and a program evaluation could measure participants' activity and later outcomes directly.

What a Better Analysis Would Use

The question of how much diabetes a walking program prevents needs individual data. The department's household survey, which records activity, diabetes, age, income and insurance for each respondent, supports a logistic regression at the person level. Better still, a program evaluation could compare participants with similar non-participants over several years. Either approach avoids the ecological fallacy and the shared modeling that inflate the county results.

Communicating the Finding

The analyst presented the chart with a caption the program manager agreed to: counties with more inactivity have more diabetes, but county patterns cannot estimate how many cases a program will prevent. She kept the scatterplot, because the pattern is real and persuasive, while removing the projection from the funding request.

Assumptions and Limits

The regression assumed independent observations, but neighboring counties share conditions, which could make standard errors too small. The county data are one year. Small counties' estimates rely heavily on the model. Age adjustment allows fair comparison but changes the values from the actual burden.

Conclusion

A correlation of 0.977 and an R-squared of 0.955 looked like strong evidence that inactivity drives diabetes across Colorado counties. Checking residuals, adding confounders and understanding how the estimates were built showed a more modest picture: a real association entangled with broader disadvantage and amplified by modeling. The ecological fallacy and individual cohort evidence placed the finding in context, supporting the program while retiring the projection.

References

Centers for Disease Control and Prevention. (2025). PLACES: Local data for better health, county data, 2025 release [Data set]. https://data.cdc.gov/d/swc5-untb

Jeon, C. Y., Lokken, R. P., Hu, F. B., & van Dam, R. M. (2007). Physical activity of moderate intensity and risk of type 2 diabetes: A systematic review. Diabetes Care, 30(3), 744-752. https://doi.org/10.2337/dc06-1842

Robinson, W. S. (1950). Ecological correlations and the behavior of individuals. American Sociological Review, 15(3), 351-357. https://doi.org/10.2307/2087176

Zhang, X., Holt, J. B., Lu, H., Wheaton, A. G., Ford, E. S., Greenlund, K. J., & Croft, J. B. (2014). Multilevel regression and poststratification for small-area estimation of population health outcomes: A case study of chronic obstructive pulmonary disease prevalence using the Behavioral Risk Factor Surveillance System. American Journal of Epidemiology, 179(8), 1025-1033. https://doi.org/10.1093/aje/kwu018