Seemingly unrelated measurement error models, with application to nutritional epidemiology

Motivated by an important biomarker study in nutritional epidemiology, we consider the combination of the linear mixed measurement error model and the linear seemingly unrelated regression model, hence Seemingly Unrelated Measurement Error Models. In our context, we have data on protein intake and energy (caloric) intake from both a food frequency questionnaire (FFQ) and a biomarker, and wish to understand the measurement error properties of the FFQ for each nutrient. Our idea is to develop separate marginal mixed measurement error models for each nutrient, and then combine them into a larger multivariate measurement error model: the two measurement error models are seemingly unrelated because they concern different nutrients, but aspects of each model are highly correlated. As in any seemingly unrelated regression context, the hope is to achieve gains in statistical efficiency compared to fitting each model separately. We show that if we employ a "full" model (fully parameterized), the combination of the two measurement error models leads to no gain over considering each model separately. However, there is also a scientifically motivated "reduced" model that sets certain parameters in the "full" model equal to zero, and for which the combination of the two measurement error models leads to considerable gain over considering each model separately, e.g., 40% decrease in standard errors. We use the Akaike information criterion to distinguish between the two possibilities, and show that the resulting estimates achieve major gains in efficiency. We also describe theoretical and serious practical problems with the Bayes information criterion in this context.