  In the analysis of cluster randomized trials, two typical features are that
individuals within a cluster are correlated and that the total number of
clusters can sometimes be limited. While model-robust treatment effect
estimators have been recently developed, their asymptotic theory requires the
number of clusters to approach infinity, and one often has to empirically
assess the applicability of those methods in finite samples. To address this
challenge, we propose a conformal causal inference framework that achieves the
target coverage probability of treatment effects in finite samples without the
need for asymptotic approximations. Meanwhile, we prove that this framework is
compatible with arbitrary working models, including machine learning algorithms
leveraging baseline covariates, possesses robustness against arbitrary
misspecification of working models, and accommodates a variety of
within-cluster correlations. Under this framework, we offer efficient
algorithms to make inferences on treatment effects at both the cluster and
individual levels, applicable to user-specified covariate subgroups and two
types of test data. Finally, we demonstrate our methods via simulations and a
real data application based on a cluster randomized trial for treating chronic
pain.
