<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Regression analysis with compositional response, observations carrying relative information, is an appropriate tool for statistical modelling in many scientific areas (e.g. medicine, geochemistry, geology, economics). Even though this technique has been recently intensively studied, there are still some practical aspects that deserve to be further analysed. Here we discuss the issue related to the coordinate representation of compositional data. It is shown that linear relation between particular orthonormal coordinates and centred log-ratio coordinates can be utilized to simplify the computation concerning regression parameters estimation and hypothesis testing. To enhance interpretation of regression parameters, the orthogonal coordinates and their relation with orthonormal and centred log-ratio coordinates are presented. Further we discuss the quality of prediction in different coordinate system. It is shown that the mean squared error (MSE) for orthonormal coordinates is less or equal to the MSE for log-transformed data. Finally, an illustrative real-world example from geology is presented.</p></abstract>

Regression analysis with compositional response, observations carrying relative information, is an appropriate tool for statistical modelling in many scientific areas (e.g. medicine, geochemistry, geology, economics). Even though this technique has been recently intensively studied, there are still some practical aspects that deserve to be further analysed. Here we discuss the issue related to the coordinate representation of compositional data. It is shown that linear relation between particular orthonormal coordinates and centred log-ratio coordinates can be utilized to simplify the computation concerning regression parameters estimation and hypothesis testing. To enhance interpretation of regression parameters, the orthogonal coordinates and their relation with orthonormal and centred log-ratio coordinates are presented. Further we discuss the quality of prediction in different coordinate system. It is shown that the mean squared error (MSE) for orthonormal coordinates is less or equal to the MSE for log-transformed data. Finally, an illustrative real-world example from geology is presented.

Practical Aspects of Log-ratio Coordinate Representations in Regression with Compositional Response

Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc,

Department of Geology, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc,

Measurement Science Review

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Regression analysis with compositional response, observations carrying relative information, is an appropriate tool for statistical modelling in many...

Practical Aspects of Log-ratio Coordinate Representations in Regression with Compositional Response

Published Online: Oct 26, 2016

Page range: 235 - 243

Received: Jun 01, 2016

Accepted: Oct 17, 2016

DOI: https://doi.org/10.1515/msr-2016-0029

Keywords
Orthonormal coordinates, centred -ratio coordinates, orthogonal coordinates, log-transformation, multivariate regression model, mean squared error

© 2016 Eva Fišerová et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Practical Aspects of Log-ratio Coordinate Representations in Regression with Compositional Response

Published Online: Oct 26, 2016

Page range: 235 - 243

Received: Jun 01, 2016

Accepted: Oct 17, 2016

DOI: https://doi.org/10.1515/msr-2016-0029

KeywordsOrthonormal coordinates, centred -ratio coordinates, orthogonal coordinates, log-transformation, multivariate regression model, mean squared error

© 2016 Eva Fišerová et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Orthonormal coordinates, centred -ratio coordinates, orthogonal coordinates, log-transformation, multivariate regression model, mean squared error