  In experimental design and causal inference, it may happen that the treatment
is not defined on individual experimental units, but rather on pairs or, more
generally, on groups of units. For example, teachers may choose pairs of
students who do not know each other to teach a new curriculum; regulators might
allow or disallow merging of firms, and biologists may introduce or inhibit
interactions between genes or proteins. In this paper, we formalize this
experimental setting, and we refer to the individual treatments in such setting
as entangled treatments. We then consider the special case where individual
treatments depend on a common population quantity, and develop theory and
methodology to deal with this case. In our target applications, the common
population quantity is a network, and the individual treatments are defined as
functions of the change in the network between two specific time points. Our
focus is on estimating the causal effect of entangled treatments in
observational studies where entangled treatments are endogenous and cannot be
directly manipulated. When treatment cannot be manipulated, be it entangled or
not, it is necessary to account for the treatment assignment mechanism to avoid
selection bias, commonly through a propensity score methodology. In this paper,
we quantify the extent to which classical propensity score methodology ignores
treatment entanglement, and characterize the bias in the estimated causal
effects. To characterize such bias we introduce a novel similarity function
between propensity score models, and a practical approximation of it, which we
use to quantify model misspecification of propensity scores due to
entanglement. One solution to avoid the bias in the presence of entangled
treatments is to model the change in the network, directly, and calculate an
individual unit's propensity score by averaging treatment assignments over this
change.
