County-Level Correlations, Ecological Fallacy, and a Press Release About Fast Food
A county health department pooled 2019 data from 298 counties. Adult obesity prevalence came from the Behavioral Risk Factor Surveillance System (BRFSS), which uses self-reported height and weight. Median household income came from census records, and fast-food outlet density (outlets per 10,000 residents) came from a commercial business registry. Across counties, median income correlated with obesity at r = -0.42, and outlet density correlated with obesity at r = 0.36. After adjusting for median income, the outlet-density correlation fell to r = 0.11. County obesity prevalence ranged from 19% to 41%. BRFSS response rates were 45% in the highest-income quartile and 31% in the lowest. Validation studies in this population show that self-reported BMI understates measured BMI, and the understatement is larger among respondents with higher measured BMI. The department's press release announced that restricting fast-food outlets could reduce obesity by up to 15%.
1. What can and cannot be inferred from these correlations? Address the 15% claim specifically.
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The design is cross-sectional and ecological, so it cannot establish that any individual's obesity is related to that individual's food environment, and it cannot establish temporal order. The attenuation from r = 0.36 to r = 0.11 after adjusting for income is evidence that the outlet-density association is largely confounded by county income, not that income is the only confounder. Reverse causation is plausible: outlets locate where demand is high, so obesity and outlet density may both follow population characteristics. The 15% figure has no derivation here. r = 0.36 corresponds to r-squared of about 0.13, which is the proportion of variance shared at the county level, not a policy effect size; converting a correlation to a percentage reduction in obesity would require an intervention estimate from a different design. With n = 298, spatial autocorrelation also inflates the effective sample size, so the apparent precision of these correlations is overstated.
2. Reason about the direction of bias from the survey response pattern and the BMI underreporting. Do they strengthen or weaken the case for an income gradient, and do they rescue the outlet-density claim?
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Differential response means low-income counties are underrepresented. If nonrespondents in those counties have higher obesity, obesity is underestimated precisely where income is low, which would weaken the observed income-obesity correlation, so the true county-level gradient may be steeper than -0.42. Underreporting of BMI that is greater at higher BMI is differential misclassification with respect to the outcome: high-obesity counties have their prevalence pulled down more, which typically attenuates associations toward the null, though it can distort in either direction. Both biases suggest the observed correlations are conservative estimates of county-level association. That does not rehabilitate the outlet-density claim, because those biases act on the outcome and income axes, not on the confounding structure that drove r from 0.36 to 0.11. A more accurate outcome measure would sharpen the estimates but would not convert a county-level association into an intervention effect.
3. Design a study or analysis that could support a causal claim about outlet density. What result would be genuinely persuasive?
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Move to individual-level, longitudinal data with measured height and weight and geocoded residential exposure, adjusting for individual income, education, and neighborhood deprivation. Stronger still is a natural experiment: identify a jurisdiction that adopted a zoning change capping new outlets in a defined area, then use difference-in-differences against comparable jurisdictions with parallel pre-trends. Pre-specify obesity and purchasing outcomes, follow long enough for the outlet mix to change, and verify that the policy actually changed exposure. A persuasive result would be a relative decline in measured obesity or in objectively recorded purchases in the restricted area, graded by the magnitude of exposure change, with a plausible mediation pathway and no similar change in controls. A null or a reversal would be equally informative. Because restrictions typically bind only new openings, effects should be expected to accumulate slowly, which is another reason a single cross-section cannot settle the question.