  In this article, we investigate group differences in phthalate exposure
profiles using NHANES data. Phthalates are a family of industrial chemicals
used in plastics and as solvents. There is increasing evidence of adverse
health effects of exposure to phthalates on reproduction and neuro-development,
and concern about racial disparities in exposure. We would like to identify a
single set of low-dimensional factors summarizing exposure to different
chemicals, while allowing differences across groups. Improving on current
multi-group additive factor models, we propose a class of Perturbed Factor
Analysis (PFA) models that assume a common factor structure after perturbing
the data via multiplication by a group-specific matrix. Bayesian inference
algorithms are defined using a matrix normal hierarchical model for the
perturbation matrices. The resulting model is just as flexible as current
approaches in allowing arbitrarily large differences across groups but has
substantial advantages that we illustrate in simulation studies. Applying PFA
to NHANES data, we learn common factors summarizing exposures to phthalates,
while showing clear differences across groups.
