Access to healthy food is key to maintaining a healthy lifestyle and can be quantified by the distance to the nearest grocery store. However, calculating this distance forces a trade-off between cost and correctness. Accurate route-based distances following passable roads are cost-prohibitive, while simple straight-line distances ignoring infrastructure and natural barriers are accessible yet error-prone. Categorizing low-access neighborhoods based on these straight-line distances induces misclassification and introduces bias into standard regression models estimating the relationship between disease prevalence and access. Yet, fully observing the more accurate, route-based food access measure is often impossible, which induces a missing data problem. We combat bias and address missingness with a new maximum likelihood estimator for Poisson regression with a binary, misclassified exposure (access to healthy food within some threshold), where the misclassification may depend on additional error-free covariates. In simulations, we show the consequence of ignoring the misclassification (bias) and how the proposed estimator corrects for bias while preserving more statistical efficiency than the complete case analysis (i.e., deleting observations with missing data). Finally, we apply our estimator to model the relationship between census tract diabetes prevalence and access to healthy food in northwestern North Carolina.

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