Regression Analysis for Blip Effects of Treatments in Sequence

ABSTRACT The blip effects are the net effects of treatments in sequence on an outcome of interest, which is estimable from the marginal distribution of all the treatments, covariates and outcome. However, regression requires that the blip effects are estimated from the conditional distribution of the outcome given all the treatments and covariates. In this article, we introduce a regression method for the blip effects that can deal with the non‐ancillarity of treatments and covariates. First, we estimate the blip effects by regression based on the conditional likelihood of the outcome given all the treatments and covariates. Here, the regression model expresses the point effects of treatments in sequence (response variables) as sums of blip effects with respect to some probabilities of treatments and covariates (explanatory variables). The point effect is the one of single‐point treatment, estimated by available methods in causal inference for single‐point treatments. The probabilities are estimated by the corresponding proportions. Second, we estimate the covariance matrix of the estimated blip effects by bootstrap method based on the complete likelihood of all the treatments, covariates and outcome. With the estimate from the first step and the covariance matrix from the second step, we test the blip effects. Besides the non‐ancillarity, this method allows for shared blip effects across different treatments in sequence while avoiding complex models for time‐varying baseline potential outcomes. It may pave the way for leveraging these effects to estimate and test optimal dynamic treatment regimes.