Nonlinear models with measurement error: Application to vitamin D
Adequate vitamin D status is essential to maintain healthy bones and to reduce risk of fracture. It is difficult, however, to determine recommended intake levels due to the complexity of the metabolism of the vitamin D we consume. Vitamin D status depends on factors other than consumption of vitamin D from food and supplements; for example, status depends on sun exposure, skin pigmentation, adiposity and several other environmental and physiological factors. A biomarker for vitamin D status is a person's 25-hydroxyvitamin D (or 25(OH)D) serum level. From a practical viewpoint, we cannot make public health recommendations using 25(OH)D levels. Ideally we want to be able to make recommendations for intakes of vitamin D. In our work, we model the association between intake of vitamin D from all sources and the level of 25(OH)D in the serum. Since we can only obtain noisy measurements of vitamin D intake, we propose a nonlinear measurement error model to describe the dependency of 25(OH)D serum levels on vitamin D intake which accounts for the nuisance day-to-day variance when estimating long-term average intake.
Initially, we assume that the unobservable usual vitamin D intakes are normally distributed, and discuss maximum likelihood estimation in that context. We then propose an extended model where we no longer assume that the distribution for the unobservable predictor is normal, but instead is a finite mixture of discrete distributions. We show an application of the nonlinear measurement error model using data from the 2005-2006 National Health and Nutrition Examination Survey (NHANES). In the application, we implement the extended nonlinear measurement error model and also assume that the measurement errors follow a truncated normal distribution with mean and variance depending on usual intake.