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Estimating Net Effects of Treatments in Treatment Sequence without the Assumption of Strongly Ignorable Treatment Assignment

In sequential causal inference, one estimates the causal net effect of treatment in treatment sequence on an outcome after last treatment in the presence of time-dependent covariates between treatments, improves the estimation by the untestable assumption of strongly ignorable treatment assignment, and obtains consistent but non-genuine likelihood-based estimate. In this article, we introduce the net effect of treatment as parameter for the conditional distribution of outcome given all treatments and time-dependent covariates and show that it is equal to the causal net effect of treatment under the assumption of strongly ignorable treatment assignment. As a result, we can estimate the net effect of treatment and evaluate its causal interpretation in two separate steps. The first step is fucus of this article while the second step can be accomplished by usual sensitivity analyses. We construct point parametrization for the conditional outcome distribution in which the parameters of interest are the point effects of single-point treatments. With point parametrization and without the untestable assumption, we estimate the net effect of treatment by maximum likelihood, improve the estimation by testable pattern of the net effect of treatment, and obtain unbiased consistent maximum-likelihood estimate for the net effect of treatment with finite-dimensional pattern.