  When drawing causal inferences about the effects of multiple treatments on
clustered survival outcomes using observational data, we need to address
implications of the multilevel data structure, multiple treatments, censoring
and unmeasured confounding for causal analyses. Few off-the-shelf causal
inference tools are available to simultaneously tackle these issues. We develop
a flexible random-intercept accelerated failure time model, in which we use
Bayesian additive regression trees to capture arbitrarily complex relationships
between censored survival times and pre-treatment covariates and use the random
intercepts to capture cluster-specific main effects. We develop an efficient
Markov chain Monte Carlo algorithm to draw posterior inferences about the
population survival effects of multiple treatments and examine the variability
in cluster-level effects. We further propose an interpretable sensitivity
analysis approach to evaluate the sensitivity of drawn causal inferences about
treatment effect to the potential magnitude of departure from the causal
assumption of no unmeasured confounding. Expansive simulations empirically
validate and demonstrate good practical operating characteristics of our
proposed methods. Applying the proposed methods to a dataset on older high-risk
localized prostate cancer patients drawn from the National Cancer Database, we
evaluate the comparative effects of three treatment approaches on patient
survival, and assess the ramifications of potential unmeasured confounding. The
methods developed in this work are readily available in the $\textsf{R}$
package $\textsf{riAFTBART}$.
