In causal inference for longitudinal data, standard methods usually assume that the underlying processes are discrete time processes, and that the observational time points are the time points when the processes change values. The identification of these standard models often relies on the sequential randomization assumption, which assumes that the treatment assignment at each time point only depends on current covariates and the covariates and treatment that are observed in the past. However, in many real world data sets, it is more reasonable to assume that the underlying processes are continuous time processes, and that they are only observed at discrete time points. When this happens, the sequential randomization assumption may not be true even if it is still a reasonable abstraction of the treatment decision mechanism at the continuous time level. For example, in a multi-round survey study, the decision of treatment can be made by the subject and the subject's physician in continuous time, while the treatment level and covariates are only collected in discrete times by a third party survey organization. The mismatch in the treatment decision time and the observational time makes the sequential randomization assumption false in the observed data. In this dissertation, we show that the standard methods could produce severely biased estimates, and we would explore what further assumptions need to be made to warrant the use of standard methods. If these assumptions are false, we advocate the use of controlling-the-future method of Joffe and Robins (2009) when we are able to reconstruct the potential outcomes from the discretely observed data. We propose a full modeling approach and demonstrate it by an example of estimating the effect of vitamin A deficiency on children's respiratory infection, when we are not able to do so. We also provide a semi-parametric analysis of the controlling-the-future method, giving the semi-parametric efficient estimator.
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