A nonlinear measurement error model and its application to describing the dependency of health outcomes on dietary intake

Many nutritional studies focus on the relationship between individuals' diets and resulting health outcomes. When examining these relationships, researchers are generally interested in individuals' long-term, average intake of nutrients; however, typically only 1–2 days of data are collected. If analyses are performed without accounting for the error in estimating usual intake, estimates will be biased. In this work, we focus on situations where the association between intake and health outcomes is nonlinear. Since we can only obtain noisy measurements of intake, we propose implementing a nonlinear measurement error model which accounts for the nuisance day-to-day variance when estimating long-term average intake. Estimation of the model is performed using maximum likelihood. Properties of the estimators are explored for a model where we assume that the unobservable usual intake is normally distributed. We then propose an extended model where we no longer assume that the distribution for the unobservable predictor is normal, but is instead a finite mixture of discrete distributions. We finish with an application using data from the 2015–2016 National Health and Nutrition Examination Survey (NHANES) where we examine the association between potassium intake and systolic blood pressure.