MPH 550 Week 4 Comparing Groups Example

Are Colorado's Smaller Counties Less Healthy? Comparing Two Groups of Counties With t-Tests, a Rank Test and Effect Sizes

[Student Name]

University of Phoenix

MPH/550: Public Health Statistics

Week 4 Assignment

[Instructor Name]

[Date]

The health department, its analyst and the state association request are composites written for a model paper; county estimates come from the CDC PLACES 2025 release, and other findings come from the sources cited.

The state association of local health officials asked the county's analyst to help answer a question for its legislative agenda: do Colorado's smaller, mostly rural counties carry a heavier chronic disease burden than larger ones? If so, the association would argue for funding weighted toward small counties. This paper compares the two groups and reports what the data support.

The Data and Groups

The analyst used CDC PLACES age-adjusted estimates for 2023 for all 64 Colorado counties (Centers for Disease Control and Prevention [CDC], 2025). She divided counties at 50,000 residents: 16 larger counties, which together hold about 90% of the state's population, and 48 smaller ones. The cutoff is simple and commonly used, though any cutoff is somewhat arbitrary.

Outcomes Chosen in Advance

Before running tests, she chose five outcomes: current smoking, diagnosed diabetes, physical inactivity, obesity and lack of health insurance among adults aged 18 to 64. Choosing in advance limits the temptation to search many outcomes and report only the ones that differ.

Looking Before Testing

The analyst first plotted each outcome for the two groups side by side. The plots showed that larger counties spread more widely on obesity and inactivity, running from Boulder's very low values to higher values in Pueblo and Weld, while smaller counties clustered more tightly. Diabetes showed a right tail among small plains counties. These pictures shaped the choice of test and warned that averages might hide variation within groups.

Choosing the Test

Each outcome is a continuous percentage, and there are two independent groups, pointing to a two-sample t-test. The classic Student's t-test assumes equal variances, but the groups differed in spread; for obesity, the standard deviation was 4.82 among larger counties and 3.34 among smaller ones, and the groups differ threefold in size.

Why Unequal Variances

A review of the question recommended the unequal variance, or Welch, t-test as a default whenever two groups are compared on a continuous outcome: it performs about as well as Student's test when variances are equal and far better when they are not, especially with unequal group sizes (Ruxton, 2006). The analyst used it for all five outcomes.

Results: Smoking

Adult smoking averaged 11.24% in larger counties and 13.30% in smaller ones, a difference of 2.05 points, with a 95% interval of 0.72 to 3.39 points; t was 3.13 with about 32 degrees of freedom and p was 0.004. Adults in smaller counties smoke at noticeably higher rates, a gap that survived every check.

Results: Diabetes

Diabetes averaged 7.98% in larger counties and 8.78% in smaller ones, a difference of 0.80 points, with an interval of 0.01 to 1.59 and p of 0.049.

Results: The Other Three

Physical inactivity was 2.07 points lower in larger counties, with an interval from 4.46 lower to 0.32 higher and p of 0.087. Obesity was 1.50 points lower, with an interval from 4.21 lower to 1.21 higher and p of 0.26. Lack of insurance was 1.39 points lower, with an interval from 3.46 lower to 0.68 higher and p of 0.18. All three point in the same direction, but each interval includes zero.

Checking Normality

The t-test assumes that group means are approximately normally distributed. With 48 smaller counties, the central limit theorem makes that reasonable even if individual values are skewed. With only 16 larger counties, skewness matters more, which is one reason the analyst added a rank test as a check rather than relying on the t-test alone.

Effect Sizes

A p-value shows whether a difference is distinguishable from zero, not how large it is. Effect sizes fill that gap, and reporting them helps readers judge practical importance, since large samples can make trivial differences significant and small samples can hide important ones (Sullivan & Feinn, 2012). Using Cohen's d, the smoking difference was 0.81 standard deviations, a large effect by common conventions; diabetes and inactivity were about 0.5, moderate; obesity was 0.40 and insurance 0.33, smaller.

A Robustness Check

Diabetes values were right-skewed, so the analyst ran a rank-based Mann-Whitney test, which compares the ordering of values rather than means. For smoking, the result held: medians of 10.7% and 13.9%, p of 0.007. For diabetes, the rank test gave p of 0.12. The diabetes finding depends on the method chosen, which signals fragility.

Multiple Testing

Five tests at the 0.05 level carry a much higher chance of at least one false positive than a single test. A review of adjustment methods explains that corrections are needed when a claim rests on any one of several tests being significant, while they matter less for a few prespecified primary questions interpreted individually (Bender & Lange, 2001). Because the association planned to cite whichever outcomes differed, the analyst applied a Bonferroni threshold of 0.01. Only smoking passed.

Interpreting the Diabetes Result

The diabetes p-value of 0.049 sits right at the conventional line, the interval nearly touches zero, the rank test does not confirm it and it fails the corrected threshold. The honest summary is that smaller counties may have modestly higher diabetes, but this comparison cannot establish it.

Other Tests for Other Questions

Different questions call for different tests. Comparing three groups, such as metropolitan, small city and rural counties, would use one-way ANOVA, followed by adjusted pairwise comparisons, or the Kruskal-Wallis test for skewed data. Comparing categories, such as the share of counties above a smoking threshold in each group, would call for chi-square, switching to Fisher's exact method if expected cell counts fall below five.

Power and Group Size

With only 16 larger counties, the comparison has limited power to detect moderate differences. The obesity and insurance results, with wide intervals, could reflect real differences the data were too thin to confirm. Adding more years would not help much, because PLACES estimates for adjacent years share the same model; a better route would be individual-level survey data with a rural and urban indicator.

Limits of the Comparison

The unit of analysis is the county, so results describe counties, not individuals. The estimates are modeled from survey data and demographic information, which may exaggerate or blur differences. A cutoff at 50,000 lumps resort counties with high incomes together with poorer farm counties. Age adjustment removes differences due to age, which is appropriate for comparison but hides the actual burden in older rural counties.

What the Association Can Say

The analyst advised the association to lead with smoking, where the difference is large, consistent across methods and robust to adjustment, and to describe the other measures as pointing in the same direction but not clearly different. That framing will hold up if legislators' staff check the numbers.

Conclusion

Comparing 16 larger and 48 smaller Colorado counties showed a clear, large difference in adult smoking and weaker, inconsistent differences in diabetes and other measures. Choosing a test suited to unequal variances, reporting effect sizes, checking results with a rank test and adjusting for multiple outcomes kept the analysis honest and gave the association claims it can defend.

References

Bender, R., & Lange, S. (2001). Adjusting for multiple testing: When and how? Journal of Clinical Epidemiology, 54(4), 343-349. https://doi.org/10.1016/S0895-4356(00)00314-0

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

Ruxton, G. D. (2006). The unequal variance t-test is an underused alternative to Student's t-test and the Mann-Whitney U test. Behavioral Ecology, 17(4), 688-690. https://doi.org/10.1093/beheco/ark016

Sullivan, G. M., & Feinn, R. (2012). Using effect size, or why the P value is not enough. Journal of Graduate Medical Education, 4(3), 279-282. https://doi.org/10.4300/JGME-D-12-00156.1