Numerous studies have shown that microbial metabolites, which represent the products of bacteria in the human gut, play a key role in shaping cancer risk and response to treatment. However, metabolite data typically contain a large proportion of missing values, which may result from either low abundance or technical challenges in data processing. Moreover, given the compositionality of microbiome data, where the observed abundances can only be interpreted on a relative scale, standard variable selection methods are not applicable. In this project, we propose a novel Bayesian regression method to address these challenges in the integration of metabolite and microbiome data. Key features of our proposed model include modeling the two different mechanisms of missingness for the metabolite data and adopting a Bayesian prior designed to address the compositional characteristics of microbiome data. We demonstrate on simulated data that our proposed model can accurately impute the unobserved true metabolite values and correctly select the relevant microbiome predictors. We further illustrate our method using real data from a study focused on understanding the interplay between the microbiome and metabolome in colorectal cancer.
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