  Variance reduction for causal inference in the presence of network
interference is often achieved through either outcome modeling, which is
typically analyzed under unit-randomized Bernoulli designs, or clustered
experimental designs, which are typically analyzed without strong parametric
assumptions. In this work, we study the intersection of these two approaches
and consider the problem of estimation in low-order outcome models using data
from a general experimental design. Our contributions are threefold. First, we
present an estimator of the total treatment effect (also called the global
average treatment effect) in a low-degree outcome model when the data are
collected under general experimental designs, generalizing previous results for
Bernoulli designs. We refer to this estimator as the pseudoinverse estimator
and give bounds on its bias and variance in terms of properties of the
experimental design. Second, we evaluate these bounds for the case of cluster
randomized designs with both Bernoulli and complete randomization. For
clustered Bernoulli randomization, we find that our estimator is always
unbiased and that its variance scales like the smaller of the variance obtained
from a low-order assumption and the variance obtained from cluster
randomization, showing that combining these variance reduction strategies is
preferable to using either individually. For clustered complete randomization,
we find a notable bias-variance trade-off mediated by specific features of the
clustering. Third, when choosing a clustered experimental design, our bounds
can be used to select a clustering from a set of candidate clusterings. Across
a range of graphs and clustering algorithms, we show that our method
consistently selects clusterings that perform well on a range of response
models, suggesting that our bounds are useful to practitioners.
