Latent class inverse probability weighting to estimate causal effects of sequential treatments under unobserved confounding

In many fields, it is of interest to analyze longitudinal studies designed to assess the causal effect of a sequential treatment on an outcome measured at the end of the period. We consider the common case of a binary treatment assigned repeatedly over time, where the treatment assignment at a given occasion depends on the sequence of previous assignments, as well as on time-varying confounders. A popular modeling strategy is represented by Marginal Structural Models; within this approach, the causal effect of the treatment is estimated by the Inverse Probability Weighted (IPW) estimator. This estimator is consistent provided that all the confounders are observed (sequential ignobility). To alleviate this serious limitation, we propose an extension of the IPW estimator to account for unobserved pre-treatment confounders. The proposed approach is based on the assumption that the unobserved confounders are summarized by a discrete latent variable, thus we estimate the probabilities of treatment using a latent class model. The new estimator, called Latent Class Inverse Probability Weighted (LC-IPW), is based on two steps. Its properties are assessed by a simulation study: the LC-IPW estimator outperforms the IPW estimator for all combinations of sample size and number of occasions considered in the simulation study, even when there is no unobserved confounding. The proposed approach is applied to the estimation of causal effects of wage subsidies on employment, using a dataset of Finnish firms observed for eight years.