Mixed effects models have become one of the major approaches to the analysis of longitudinal studies. Random effects in those models play a twofold role. First, they address heterogeneity among individual temporal trajectories, and, second, they induce a correlational structure among temporal observations of the same subject. If both the exposure and outcome vary with time, it is natural to specify mixed effects model for both. If heterogeneity in temporal trajectories is related to unknown subject-level confounders, the corresponding random effects will be correlated, inducing correlation between random effects in the outcome model and the exposure. In this case, there are three different effects of the exposure on outcome, the within-subject or individual level effect, the between-subject effect of mean individual exposure, and the marginal or the population-average effect. If the existing correlation between random effects and exposure is ignored, the estimated exposure effect(s) will be biased. If exposure is measured with error, there will always be a nonzero correlation between random effects in the outcome model and error-prone exposure, even if this correlation was zero in the model with true exposure.

Due to this critical result, the unbiased estimation of the effect of measurement error in the mixed model requires taking the correlation between random effects and error-prone exposure into account. Theoretical developments are exemplified by the analysis of data on physical activity energy expenditure from a large validation study of different physical activity instruments using doubly labeled water as unbiased reference measurements.

Mixed effects models have become one of the major approaches to the analysis of longitudinal studies. Random effects in those models play a twofold role. First, they address heterogeneity among individual temporal trajectories, and, second, they induce a correlational structure among temporal observations of the same subject. If both the exposure and outcome vary with time, it is natural to specify mixed effects model for both. If heterogeneity in temporal trajectories is related to unknown subject-level confounders, the corresponding random effects will be correlated, inducing correlation between random effects in the outcome model and the exposure. In this case, there are three different effects of the exposure on outcome, the within-subject or individual level effect, the between-subject effect of mean individual exposure, and the marginal or the population-average effect. If the existing correlation between random effects and exposure is ignored, the estimated exposure effect(s) will be biased. If exposure is measured with error, there will always be a nonzero correlation between random effects in the outcome model and error-prone exposure, even if this correlation was zero in the model with true exposure.

Due to this critical result, the unbiased estimation of the effect of measurement error in the mixed model requires taking the correlation between random effects and error-prone exposure into account. Theoretical developments are exemplified by the analysis of data on physical activity energy expenditure from a large validation study of different physical activity instruments using doubly labeled water as unbiased reference measurements.