Target parameters and bias in non-causal change-score analyses with measurement errors

In studies where the outcome is a change-score, it is often debated whether or not the analysis should adjust for the baseline score. When the aim is to make causal inference, it has been argued that the two analyses (adjusted vs. unadjusted) target different causal parameters, which may both be relevant. However, these arguments are not applicable when the aim is to make predictions rather than to estimate causal effects. When the scores are measured with error, there have been attempts to quantify the bias resulting from adjustment for the (mis-)measured baseline score or lack thereof. However, these bias results have been derived under an unrealistically simple model, and assuming that the target parameter is the unadjusted (for the true baseline score) association, thus dismissing the adjusted association as a possibly relevant target parameter. In this paper we address these limitations. We argue that, even if the aim is to make predictions, there are two possibly relevant target parameters; one adjusted for the baseline score and one unadjusted. We consider both the simple case when there are no measurement errors, and the more complex case when the scores are measured with error. For the latter case, we consider a more realistic model than previous authors. Under this model we derive analytic expressions for the biases that arise when adjusting or not adjusting for the (mis-)measured baseline score, with respect to the two possible target parameters. Finally, we use these expressions to discuss when adjustment is warranted in change-score analyses.