The Authors Respond

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We sincerely thank Schumacher and Sheppard1 for their letter; they uncovered a mistake in the Stata code provided in the eAppendix; ( of our article. The initial code outputs wrong estimations in some particular cases of the simulated samples; the corrected code they provided gives the correct estimations and should be used to run the simulations.
Schumacher and Sheppard1 also advocate the use of mixed models in longitudinal observational studies with repeated measures of the outcome (change from baseline) and time-varying exposures or covariates. They cite two studies analyzing the association between several factors and the coronary artery calcium (hereafter referred to as “calcification”) progression. Mixed models are appropriate for the analysis of repeated measures of the outcome and can be used to adjust for time-varying exposures and covariates. However, the causal interpretation of the estimations obtained can be uneasy if every exposure/covariate is considered as an “exposure of interest.” The relevance of adjusting for the baseline calcification value will depend on the causal effect of interest. For example, focusing on the causal effect of sex on calcification progression, the causal structure corresponds to an exposure which starts before the baseline calcification value. Focusing on the causal effect of intermediate statin use, the exposure of interest could be influenced by the baseline calcification value. Following the graphical approach presented in our article, it seems appropriate to adjust for the baseline calcification value in the second example, but not in the first example: a single model cannot answer every causal question.
Specific statistical methods have been described to study rigorously the causal effect of time-varying exposures or intermediate variables, allowing a proper control for baseline and intermediate confounding.2,3 These methods require to focus on a particular exposure (or repeated exposure) and to define a target parameter corresponding to the scientific objective. For example, to study the causal effect of repeated exposure to particulate matter (PM), one could compare the average potential outcome that would be observed had every individual been repeatedly exposed to PM above a given cut-off, to the average potential outcome had they been exposed below the cut-off.4
Back to our simulated data, the parameter of interest can be expressed using counterfactual language as the difference between the mean outcome ΔBP (a difference in blood pressure) that would have been observed had every individual been exposed to E = 1 versus the mean outcome had every individual been exposed to E = 0. Along with the initial linear models, we applied targeted maximum likelihood estimation5 to estimate the causal effect of interest, adjusted or unadjusted for the baseline value of the outcome (cf. eAppendix; Targeted maximum likelihood estimation provides an efficient double-robust estimator that can be combined with data adaptive procedures (statistical learning). The results show unbiased and efficient estimations consistent with the graphical analysis in our article. They highlight how critical are the underlying causal structural assumptions to interpret the statistical estimations, whether we apply a simple linear regression or more sophisticated approaches like targeted maximum likelihood estimation.
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