Causal Inference
Assessing Stochastic Positivity with Convex Hulls Mark Klose* Mark Klose Tim Feeney
Positivity is a fundamental causal identification condition that ensures inferences are drawn only among individuals with a counterfactually treated counterpart. Positivity consists of deterministic positivity (structural characteristics preventing treatment receipt) and stochastic positivity (random chance preventing a comparable person from receiving the opposite treatment). Methods to assess stochastic positivity violations include comparing distributions stratified by treatment in descriptive tables, and non-overlap of model-estimated propensity scores. Each of these methods has its faults: descriptive tables are unable to assess violations in high dimensional scenarios and propensity score models rely on strong parametric assumptions. To avoid these limitations, we approach positivity from a geometric point of view. From this perspective, each covariate combination can be plotted in k-dimensional space as a point. These points can be stratified by treatment and summarized by a convex hull, the smallest k-dimension enclosure that contains all points and any convex combinations between points. The intersection between both hulls represents all plausible covariate combinations without stochastic positivity violations.
We demonstrate the convex hull using data from a trial comparing estrogen versus placebo for treatment of prostate cancer. When comparing by age and weight index only (n=505), we find 41/505 (8%) individuals were outside the hull intersection. Some characteristics of these individuals are they were over the age of 85 or their weight index is less than 74 or greater than 140.
By assessing positivity with convex hulls, we only require that any valid future observation is a convex combination of observed data. This framework generalizes to any dimension and works with continuous and discrete variables. This allows epidemiologists to assess high-dimension positivity violations without relying on parametric models.
