  We systematically investigate issues due to mis-specification that arise in
estimating causal effects when (treatment) interference is informed by a
network available pre-intervention, i.e., in situations where the outcome of a
unit may depend on the treatment assigned to other units. We develop theory for
several forms of interference through the concept of exposure neighborhood, and
develop the corresponding semi-parametric representation for potential outcomes
as a function of the exposure neighborhood. Using this representation, we
extend the definition of two popular classes of causal estimands, marginal and
average causal effects, to the case of network interference. We characterize
the bias and variance one incurs when combining classical randomization
strategies (namely, Bernoulli, Completely Randomized, and Cluster Randomized
designs) and estimators (namely, difference-in-means and Horvitz-Thompson) used
to estimate average treatment effect and on the total treatment effect, under
misspecification due to interference. We illustrate how difference-in-means
estimators can have arbitrarily large bias when estimating average causal
effects, depending on the form and strength of interference, which is unknown
at design stage. Horvitz-Thompson (HT) estimators are unbiased when the correct
weights are specified. Here, we derive the HT weights for unbiased estimation
of different estimands, and illustrate how they depend on the design, the form
of interference, which is unknown at design stage, and the estimand. More
importantly, we show that HT estimators are in-admissible for a large class of
randomization strategies, in the presence of interference. We develop new
model-assisted and model-dependent strategies to improve HT estimators, and we
develop new randomization strategies for estimating the average treatment
effect and total treatment effect.
