Estimating ratios of means of multicategory data observed with sample and category perturbations

Summay We consider the problem of estimating ratios of means of a multivariate outcome across covariates when the data are observed with unknown sample-specific and category-specific perturbations. Our model admits a partially identifiable estimand, and we establish full identifiability by imposing interpretable parameter constraints. To reduce bias and guarantee the existence of estimators in the presence of sparse observations, we apply an asymptotically negligible and constraint-invariant penalty to the loss 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 under violated distributional assumptions. The flexibility of the approach, and comparisons with related methods, are illustrated through a simulation study and a meta-analysis of microbial associations with colorectal cancer.