Evaluating confounding and missing data in two-arm longitudinal studies - applying the longitudinal E-value

Establishing causal inference in observational studies has always been a challenge. One of the most well-known biases is unmeasured confounding, which occurs when an unmeasured variable influences both the exposure and outcome. One method for quantifying the potential impact of this bias is through the E-value, which provides the strength a confounder needs to have to explain away a proposed causal association. However, the E-value is typically only derived when the exposure, confounder and outcome do not vary over time. Therefore the E-value is not applicable to longitudinal designs, as it does not handle the complexities of multiple measures of the outcome when analysed simultaneously. In this paper we will apply a recent extension of the E-value for interrupted time series to a two-arm longitudinal study design, concerning the willingness to compete hurt and the use of analgesics in youth elite athletes. In the example, we explain how the interpretation of this extension differs from the interrupted time series design. Moreover, we compare the extension with the original E-value in terms of interpretability and assumptions regarding the confounder. In the example, we demonstrate that the original E-value suggest our association is less robust to confounding than the longitudinal extension, and we explain the practical implications of this discrepancy. Overall this paper highlights the importance of considering the temporal ordering of the average treatment effect when evaluating unmeasured confounding.