Epidemiological studies provide evidence that environmental exposures may affect health
through complex mixtures. Formal investigation of the effect of exposure mixtures is usually
achieved by modelling interactions, which relies on strong assumptions relating to the identity
and the number of the exposures involved in such interactions, and on the order and
parametric form of these interactions. These hypotheses become difficult to formulate and
justify in an exposome context, where influential exposures are numerous and heterogeneous.
To capture both the complexity of the exposome, and its possibly pleiotropic effects, models
handling multivariate predictors and responses, such as partial least squares (PLS) algorithms,
can prove useful. As an illustrative example, we applied PLS models to data from a study
investigating the inflammatory response (blood concentration of 13 immune markers) to the
exposure to four disinfection by-products (one brominated and three chlorinated compounds),
while swimming in a pool. To accommodate the multiple observations per participant (N=60;
before and after the swim), we adopted a multi-level extension of the PLS algorithms, including
sparse PLS models shrinking loadings coefficients of unimportant predictors (exposures), and/or
responses (protein levels). Despite the strong correlation among co-occurring exposures, our
approach identified a subset of exposures (N=3/4) affecting the exhaled levels of 8 (out of 13)
immune markers. PLS algorithms can easily scale to high-dimensional exposures and responses,
and prove useful for exposome research to identify sparse sets of exposures jointly affecting a
set of (selected) biological markers. Our descriptive work may guide these extensions for higher
dimensional data.