  We consider the problem of estimating fold-changes in the expected value of a
multivariate outcome that is observed subject to unknown sample-specific and
category-specific perturbations. We are motivated by high-throughput sequencing
studies of the abundance of microbial taxa, in which microbes are
systematically over- and under-detected relative to their true abundances. Our
log-linear model admits a partially identifiable estimand, and we establish
full identifiability by imposing interpretable parameter constraints. To reduce
bias and guarantee the existence of parameter estimates in the presence of
sparse observations, we apply an asymptotically negligible and
constraint-invariant penalty to our estimating function. We develop a fast
coordinate descent algorithm for estimation, and an augmented Lagrangian
algorithm for estimation under null hypotheses. We construct a model-robust
score test, and demonstrate valid inference even for small sample sizes and
violated distributional assumptions. The flexibility of the approach and
comparisons to related methods are illustrated via a meta-analysis of microbial
associations with colorectal cancer.
