A Diagnostic to Find and Help Combat Stochastic Positivity Issues -- with a Focus on Continuous Treatments

The positivity assumption is central in the identification of a causal effect, and especially the stochastic variant is an issue many applied researchers face, yet is rarely discussed, especially in conjunction with continuous treatments or Modified Treatment Policies. One common recommendation for dealing with a violation is to change the estimand. However, an applied researcher is faced with two problems: First, how can she tell whether there is a stochastic positivity violation given her estimand of interest, preferably without having to estimate a model first? Second, if she finds a problem with stochastic positivity, how should she change her estimand in order to arrive at an estimand which does not face the same issues? We suggest a novel diagnostic which allows the researcher to answer both questions by providing insights into how well an estimation for a certain estimand can be made for each observation using the data at hand. We provide a simulation study on the general behaviour of different Modified Treatment Policies (MTPs) at different levels of stochastic positivity violations and show how the diagnostic helps understand where bias is to be expected. We illustrate the application of our proposed diagnostic in a pharmacoepidemiological study based on data from CHAPAS-3, a trial comparing different treatment regimens for children living with HIV.