A transformation‐free linear regression for compositional outcomes and predictors

Abstract

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 logratio 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.

Issues of under-representation in quantitative DNA metabarcoding weaken the inference about diet of the tundra vole Microtus oeconomus

“…Based on our experience with these analyses, we have two main messages for future developments of DNA metabarcoding diet studies. First , comparisons between observed and expected diets will benefit from using the direct regression approach of Fiksel, Zeger & Datta (2021) , where both response and predictor variables are compositional. Although this method does not require transformation and is easier to interpret, it does not allow for further covariates.…”

“…We assessed the quantitative accuracy of dietary metabarcoding by using a multivariate regression model that establishes a linear function between the multiple compositional outcomes (responses) and compositional predictors ( Fiksel, Zeger & Datta, 2021 ). Here we used the composition of relative read abundance of each of the three plant species (RRA from faeces or meal mixtures) as response variables and the expected plant species composition ( i.e., known biomass composition) as predictor variables.…”

“…Here we used the composition of relative read abundance of each of the three plant species (RRA from faeces or meal mixtures) as response variables and the expected plant species composition ( i.e., known biomass composition) as predictor variables. This type of compositional analysis accounts for the fact that an increase in one taxon’s proportion will force a decrease in other taxon(s) proportion within the same sample ( Alenazi, 2019 ; Chen, Zhang & Li, 2017 ; Fiksel, Zeger & Datta, 2021 ). The model allows, without transformation, for direct interpretation of the relationship between expected and observed compositions through a Markov transition matrix “B” based on the estimated regression coefficients.…”

“…We computed the regression coefficient (B matrices) and tested for overall linear independence via permutation tests (using an α-level of 0.05) with the codalm package ( Fiksel & Datta, 2020 ; Fiksel, Zeger & Datta, 2021 ) in the R software (R version 4.0.3). To assess the model’s goodness of fit, we plotted the predicted values versus the observed RRA of the faecal samples, using leave-one-out cross-validation (LOOCV) ( Fiksel, Zeger & Datta, 2021 , see Fig. S2 ).…”

Differences in visual field loss pattern when transitioning from SITA standard to SITA faster

“…We fit a linear regression model between AT composition weights in VF pairs in the Standard-Standard sequences, and AT composition weights in VF pairs in the Standard-Faster sequence (Fig. 1 bottom right) 24 . The 16 by 16 matrix of regression coefficients generated by the model can be interpreted as the association between a first VF exam’s set of 16 AT compositional weights with the second VF exam’s set of 16 AT compositional within each sequence (Standard-Standard or Standard-Faster).…”