  Statistical methods for identifying harmful chemicals in a correlated mixture
often assume linearity in exposure-response relationships. Non-monotonic
relationships are increasingly recognised (e.g., for endocrine-disrupting
chemicals); however, the impact of non-monotonicity on exposure selection has
not been evaluated. In a simulation study, we assessed the performance of
Bayesian kernel machine regression (BKMR), Bayesian additive regression trees
(BART), Bayesian structured additive regression with spike-slab priors
(BSTARSS), and lasso penalised regression. We used data on exposure to 12
phthalates and phenols in pregnant women from the U.S. National Health and
Nutrition Examination Survey to simulate realistic exposure data using a
multivariate copula. We simulated datasets of size N = 250 and compared methods
across 32 scenarios, varying by model size and sparsity, signal-to-noise ratio,
correlation structure, and exposure-response relationship shapes. We compared
methods in terms of their sensitivity, specificity, and estimation accuracy. In
most scenarios, BKMR and BSTARSS achieved moderate to high specificity
(0.56--0.91 and 0.57--0.96, respectively) and sensitivity (0.49--0.98 and
0.25--0.97, respectively). BART achieved high specificity ($\geq$ 0.96), but
low to moderate sensitivity (0.13--0.66). Lasso was highly sensitive
(0.75--0.99), except for symmetric inverse-U-shaped relationships ($\leq$ 0.2).
Performance was affected by the signal-to-noise ratio, but not substantially by
the correlation structure. Penalised regression methods that assume linearity,
such as lasso, may not be suitable for studies of environmental chemicals
hypothesised to have non-monotonic relationships with outcomes. Instead, BKMR
and BSTARSS are attractive methods for flexibly estimating the shapes of
exposure-response relationships and selecting among correlated exposures.
