Positivity

Abstract

The identifiability of causal effects requires sufficient variability in treatment or exposure assignment within strata of confounders. The causal inference literature refers to the assumption of adequate exposure variability within confounder strata as the assumption of positivity or experimental treatment assignment. Positivity violations can arise for two reasons. First, it may be theoretically impossible for individuals with certain covariate values to receive a given exposure of interest. For example, certain patient characteristics may constitute an absolute contraindication to receipt of a particular treatment. The threat to causal inference posed by such structural or theoretical violations of positivity does not improve with increasing sample size. Second, violations or near violations of positivity can arise in finite samples due to chance. This is a particular problem in small samples but also occurs frequently in moderate to large samples when the treatment is continuous or can take multiple levels, or when the covariate adjustment set is large or contains continuous or multilevel covariates. Regardless of the cause, causal effects may be poorly or nonidentified when certain subgroups in a finite sample do not receive some of the treatment levels of interest. In this chapter we will use the term “sparsity” to refer to positivity violations and near-violations arising from either of these causes, recognizing that other types of sparsity can also threaten valid inference.