  In environmental health research there is often interest in the effect of an
exposure on a health outcome assessed on the same day and several subsequent
days or lags. Distributed lag nonlinear models (DLNM) are a well-established
statistical framework for estimating an exposure-lag-response function. We
propose methods to allow for prior information to be incorporated into DLNMs.
First, we impose a monotonicity constraint in the exposure-response at lagged
time periods which matches with knowledge on how biological mechanisms respond
to increased levels of exposures. Second, we introduce variable selection into
the DLNM to identify lagged periods of susceptibility with respect to the
outcome of interest. The variable selection approach allows for direct
application of informative priors on which lags have nonzero association with
the outcome. We propose a tree-of-trees model that uses two layers of trees:
one for splitting the exposure time frame and one for fitting exposure-response
functions over different time periods. We introduce a zero-inflated alternative
to the tree splitting prior in Bayesian additive regression trees to allow for
lag selection and the addition of informative priors. We develop a
computational approach for efficient posterior sampling and perform a
comprehensive simulation study to compare our method to existing DLNM
approaches. We apply our method to estimate time-lagged extreme temperature
relationships with mortality during summer or winter in Chicago, IL.
