Quasi-likelihood estimation for parameters in regression model with compositional response

Compositional data are the data reflecting the relative information. For the regression analysis with a compositional response variable, the linear regression models assume that the response variable follows a normal distribution on the simplex. The Dirichlet regression model accounts for the response variable being distributed according to a Dirichlet distribution. However, the distribution of the data may not satisfy this assumption. This paper proposes a quasi-likelihood estimation for parameters in simplex regression model. First, the relationship between the mean of the compositional response variable and the isometric logratio coordinates of the compositional covariates is established, then the likelihood function of the compositional response variable is derived, and parameters in the likelihood function are solved using the Newton-Raphson iterative method. Finally, estimation and inference for the parameters of the simplex regression model are made based on the relationships between different logratio transformations and the asymptotic distribution of quasi-likelihood estimation. The effectiveness of the quasi-likelihood estimation for the simplex regression model is verified through numerical simulation and analysis of metabolomics data. The results show that the proposed method is a useful approach, and it does not require assumptions about the distribution of the compositional response variable.