Rejoinder: ‘Estimating time-varying causal excursion effects in mobile health with binary outcomes’

Abstract

1. Introduction We would like to thank the editor for making this a discussion paper. We are honoured that these three groups of researchers agreed to be discussants. Reading their discussions has been an enjoyable and thought-provoking journey for us. We organize our rejoinder by topic, as some topics have been discussed by multiple discussants. 2. Dependence of the causal excursion effect on treatment protocol In general, the causal excursion effect in (2) of the main manuscript depends on the randomization probability of past treatment assignments ( ⁠ A ¯ t − 1 ⁠), which we refer to as the treatment protocol. This feature distinguishes the excursion effect from most other causal effects in time-varying treatment settings, such as causal effects in marginal structural models or structural nested mean models. Guo et al. (2021) and Zhang & Laber (2021) provided numerical examples to illustrate this feature of the excursion effect. Guo et al. (2021, Equation 6) considered as one example the following data-generating process: ... where the prior treatment, A 1 ⁠ , interacts with the present treatment, A 2 ⁠ , to affect Y 2 ⁠ . They showed that the magnitude and sign of the causal excursion effect of A 2 on Y 2 ⁠ , which marginalizes over A 1 ⁠ , depends on the randomization probability of A 1 ⁠ . In the second example they considered, there is a three-way interaction between A 1 ⁠ , X 2 and A 2 on Y 2 ⁠ , and they showed that the effect modification by X 2 of the causal excursion effect of A 2 can vary depending on the randomization probability of A 1 ⁠ . Zhang & Laber (2021) provided two examples where the current treatment interacts with a prior treatment as well as a time-varying covariate, and showed the dependence of the excursion effect on the randomization probability of prior treatments. Zhang & Laber (2021) wrote ‘a large negative causal excursion effect does not mean a treatment is harmful; rather it may mean that the protocol drove patients into states for which treatment appeared harmful.’ We very much agree. Indeed, the comparison between two excursions starting at time t is likely to greatly differ by the treatment protocol up to time t ⁠ ; an example is to consider (i) a treatment protocol that is feasible to implement in practice, and (ii) a treatment protocol that is infeasible or excessively onerous. Below we argue that this consequence makes sense in a continual learning setting such as in mobile health.