Background Within lifecourse epidemiology, there is substantial interest in relationships between exposures (X) measured longitudinally (e.g. at time 0, 1, 2; hence X₀, X₁, X₂) and outcomes (Y) measured cross-sectionally once (e.g. Y₂ or Y₂ + t). Within a causal framework, modelling presents many challenges, such as how to account for time-invariant confounding. It has been demonstrated that the information contained in separate models for each exposure can be combined into an overall model using unexplained residuals; it is suggested this also reduces standard errors of estimated effect sizes. This study explored how confounders are incorporated into this framework and whether standard errors are indeed reduced.

Methods Directed acyclic graphs (DAGs) depict a range of potential causal relationships between a longitudinal exposure (BMI: body mass index, aged 2, 15 and 64 years; BMI₂, BMI₁₅ and BMI₆₄), and a later-life outcome (CRP: C-reactive protein, a marker for cardiovascular disease) measured once around the time of the last exposure assessment (CRP₆₄), in the presence of a time-invariant confounder (S: sex) measured only at baseline. The DAG guides plausible covariance structures in the data simulation and steers thinking about at which stage(s) in the residual increase modelling process one should incorporate confounders. Residual increase models were then contrasted to standard regression models.

Results With standard regression we model the impact of BMI₂ using CRP₆₄~BMI₂+S, the impact of BMI₁₅ using CRP₆₄~BMI₁₅+BMI₂+S, and the impact of BMI₆₄ using CRP₆₄~BMI₆₄+BMI₁₅+BMI₂+S. In residual increase models, BMI₁₅~BMI₂+S and BMI₆₄~BMI₁₅+BMI₂+S yield residuals E₁₅ and E₆₄, representing differences between expected and actual values of BMI at each time point (or the ‘residual increase’ in BMI) while controlling for the confounding effect, S. We also control for S in the overall model: CRP₆₄~BMI₂+E₁₅+E₆₄+S. It is shown that the residual increase model has coefficients equivalent to those produced by the separate standard regression models and this model offers a modest reduction in the standard errors of estimated coefficients.

Conclusion A time-invariant confounder must be accommodated when generating ‘residual increases’ and in the overall residual increase model. Incorporating confounders this way allows for the interpretation of multiple effects in a single model, though this offers no new information about variable relationships in and of themselves. It is important to use DAGs to determine how to incorporate confounders into the residual increase model framework, which offers modest reductions in the standard errors of estimated coefficients, though model interpretation is perhaps not as straightforward as with standard regression models.