The assumption of positivity or experimental treatment assignment requires that observed treatment levels vary within confounder strata. This article discusses the positivity assumption in the context of assessing model and parameter-speciﬁc identiﬁability of causal eﬀects. Positivity violations occur when certain subgroups in a sample rarely or never receive some treatments of interest. The resulting sparsity in the data may increase bias with or without an increase in variance and can threaten valid inference. The parametric bootstrap is presented as a tool to assess the severity of such threats and its utility as a diagnostic is explored using simulated data. Several approaches for improving the identiﬁability of parameters in the presence of positivity violations are reviewed. Potential responses to data sparsity include restriction of the covariate adjustment set, use of an alternative pro jection function to deﬁne the target parameter within a non-parametric marginal structural model, restriction of the sample, and modiﬁcation of the target intervention. All of these approaches can be understood as trading oﬀ proximity to the initial target of inference for identiﬁability; we advocate approaching this tradeoﬀ systematically.
Petersen, Maya L.; Porter, Kristin; Gruber, Susan; Wang, Yue; and van der Laan, Mark J., "Diagnosing and Responding to Violations in the Positivity Assumption" (September 2010). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 269.