Design aspects of calibration studies in nutrition, with analysis of missing data in linear measurement error models

Motivated by an example in nutritional epidemiology, we investigate some design and analysis aspects of linear measurement error models with missing surrogate data. The specific problem investigated consists of an initial large sample in which the response (a food frequency questionnaire, FFQ) is observed and then a smaller calibration study in which replicates of the error prone predictor are observed (food records or recalls, FR). The difference between our analysis and most of the measurement error model literature is that, in our study, the selection into the calibration study can depend on the value of the response. Rationale for this type of design is given. Two major problems are investigated. In the design of a calibration study, one has the option of larger sample sizes and fewer replicates or smaller sample sizes and more replicates. Somewhat surprisingly, neither strategy is uniformly preferable in cases of practical interest. The answers depend on the instrument used (recalls or records) and the parameters of interest. The second problem investigated is one of analysis. In the usual linear model with no missing data, method of moments estimates and normal-theory maximum likelihood estimates are approximately equivalent, with the former method in most use because it can be calculated easily and explicitly. Both estimates are valid without any distributional assumptions. In contrast, in the missing data problem under consideration, only the moments estimate is distribution-free, but the maximum likelihood estimate has at least 50% greater precision in practical situations when normality obtains. Implications for the design of nutritional calibration studies are discussed.