Exposure assessment is often subject to measurement errors. We consider here the analysis of studies aimed at reducing exposure to potential health hazards, in which exposure is the outcome variable. In these studies, the intervention effect may be estimated using either biomarkers or self-report data, but it is not common to combine these measures of exposure. Bias in the self-reported measures of exposure is a well-known fact; however, only few studies attempt to correct it. Recently, Keogh et al addressed this problem, presenting a model for measurement error in this setting and investigating how self-report and biomarker data can be combined. Keogh et al find the maximum likelihood estimate for the intervention effect in their model via direct numerical maximization of the likelihood. Here, we exploit an alternative presentation of the model that leads us to a closed formula for the MLE and also for its variance, when the number of biomarker replicates is the same for all subjects in the substudy. The variance formula enables efficient design of such intervention studies. When the number of biomarker replicates is not constant, our approach can be used along with the EM-algorithm to quickly compute the MLE. We compare the MLE to Buonaccorsi's method (Buonaccorsi, 1996) and find that they have similar efficiency when most subjects have biomarker data, but that the MLE has clear advantages when only a small fraction of subjects has biomarker data. This conclusion extends the findings of Keogh et al (2016) and has practical importance for efficiently designing studies.
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