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In experimental design and causal inference, it may happen that the treatmentis not defined on individual experimental units, but rather on pairs or, moregenerally, on groups of units. For example, teachers may choose pairs ofstudents who do not know each other to teach a new curriculum; regulators mightallow or disallow merging of firms, and biologists may introduce or inhibitinteractions between genes or proteins. In this paper, we formalize thisexperimental setting, and we refer to the individual treatments in such settingas entangled treatments. We then consider the special case where individualtreatments depend on a common population quantity, and develop theory andmethodology to deal with this case. In our target applications, the commonpopulation quantity is a network, and the individual treatments are defined asfunctions of the change in the network between two specific time points. Ourfocus is on estimating the causal effect of entangled treatments inobservational studies where entangled treatments are endogenous and cannot bedirectly manipulated. When treatment cannot be manipulated, be it entangled ornot, it is necessary to account for the treatment assignment mechanism to avoidselection bias, commonly through a propensity score methodology. In this paper,we quantify the extent to which classical propensity score methodology ignorestreatment entanglement, and characterize the bias in the estimated causaleffects. To characterize such bias we introduce a novel similarity functionbetween propensity score models, and a practical approximation of it, which weuse to quantify model misspecification of propensity scores due toentanglement. One solution to avoid the bias in the presence of entangledtreatments is to model the change in the network, directly, and calculate anindividual unit's propensity score by averaging treatment assignments over thischange.

Propensity score methodology in the presence of network entanglement between treatments