  In large-scale disease etiology studies, epidemiologists often need to use
multiple binary measures of unobserved causes of disease that are not perfectly
sensitive or specific to estimate cause-specific case fractions, referred to as
"population etiologic fractions" (PEFs). Despite recent methodological
advances, the scientific need of incorporating control data to estimate the
effect of explanatory variables upon the PEFs, however, remains unmet. In this
paper, we build on and extend nested partially-latent class model (npLCMs, Wu
et al., 2017) to a general framework for etiology regression analysis in
case-control studies. Data from controls provide requisite information about
measurement specificities and covariations, which is used to correctly assign
cause-specific probabilities for each case given her measurements. We estimate
the distribution of the controls' diagnostic measures given the covariates via
a separate regression model and a priori encourage simpler conditional
dependence structures. We use Markov chain Monte Carlo for posterior inference
of the PEF functions, cases' latent classes and the overall PEFs of policy
interest. We illustrate the regression analysis with simulations and show less
biased estimation and more valid inference of the overall PEFs than an npLCM
analysis omitting covariates. A regression analysis of data from a childhood
pneumonia study site reveals the dependence of pneumonia etiology upon season,
age, disease severity and HIV status.
