DUGi

Orthogonal Regression for Three-part Compositional Data Via Linear Model with Type-II Constraints

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Orthogonal regression is a proper tool for fitting two-dimensional data points when errors occurin both the variables. This type of modelling technique is also called the total least squares (TLS)in the statistical literature. In its simplest form it attempts to fit a line that explains the set ofn two-dimensional data points in such a way that the sum of squared distances from data pointsto the estimated line is minimal. Orthogonal regression is invariant under the orthogonal rotation ofcoordinates and thus it is convenient for regression analysis of three-part compositional data, performedafter isometric logratio transformation.The difficulty or even impossibility of deeper statistical analysis (confidence regions, hypothesestesting) using the standard solution for orthogonal regression based on maximum-likelihood methodcan be overcome by calibration line technique based on linear statistical models, namely linear modelswith type-II constraints (constraints involve in addition to the unknown model’s parameters the otherunobservable ones). The main advantage of the linear model approach is its validity for finite samplesin contrast to the standard techniques. It means we can determine exact variances and covariancesof estimated line’s coefficients (for the standard technique we have only asymptotic variances andcovariances). Further, under assumption of normality, we can make any standard statistical inference,e.g., construct confidence regions and bounds and test hypotheses that is then also easy to interpreton the simplex sample space. Consequently, we can apply various standard approaches to checkingthe model and its assumptions for adequacy and validity, e.g. coefficient of determination, residualsanalysis or normality tests.The only restrictive condition that must be fulfilled in order to ensure a meaningful analysis ofcompositional data, concretely invariance of the regression line’s parameters in the sample space ofcompositions, the simplex, with respect to the choice of the orthonormal basis for the ilr transformation,concerns the covariance structure of variables that needs to be very simple (homoscedastic) in thiscase. Moreover, from the theory of linear statistical models it follows that estimation by linear models(least squares method) and the orthogonal regression give the same results under this condition.The aim of the contribution is to present an iterative algorithm for estimating the regression line vialinear models with type-II constraints and some statistical inference, together with the correspondinginterpretation for compositional data. The theoretical results will be applied to real-world exampl

Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada