A transformation-free linear regression for compositional outcomes and predictors

Jacob Fiksel, Scott Zeger, Abhirup Datta

Research output: Contribution to journalArticlepeer-review


Compositional data are common in many fields, both as outcomes and predictor variables. The inventory of models for the case when both the outcome and predictor variables are compositional is limited, and the existing models are often difficult to interpret in the compositional space, due to their use of complex log-ratio transformations. We develop a transformation-free linear regression model where the expected value of the compositional outcome is expressed as a single Markov transition from the compositional predictor. Our approach is based on estimating equations thereby not requiring complete specification of data likelihood and is robust to different data-generating mechanisms. Our model is simple to interpret, allows for 0s and 1s in both the compositional outcome and covariates, and subsumes several interesting subcases of interest. We also develop permutation tests for linear independence and equality of effect sizes of two components of the predictor. Finally, we show that despite its simplicity, our model accurately captures the relationship between compositional data using two datasets from education and medical research.

Original languageEnglish (US)
Pages (from-to)974-987
Number of pages14
Issue number3
StatePublished - Sep 2022


  • Kullback–Leibler distance loss function
  • compositional data
  • estimating equation
  • expectation-maximization algorithm
  • transformation-free

ASJC Scopus subject areas

  • Agricultural and Biological Sciences(all)
  • Applied Mathematics
  • Biochemistry, Genetics and Molecular Biology(all)
  • Immunology and Microbiology(all)
  • Statistics and Probability


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