On the problem of factor significance in the models of variance analysis

In analysis of variance models the hypotheses about factor significance form comparisons of the levels of the same factor. If there are no reasons according to the criterion for testing such hypotheses to reject the hypothesis about the equality of some levels of factors, then these levels are considered significantly different. The goal of the article is to find out whether the levels of factors have a significant effect on the response variable. Linear models of analysis of variance (AV) under conditions of complete and incomplete factor design are considered. As a computational scheme for processing the AV models the procedure for transforming incomplete rank models into a full rank models and representing the space of linear estimated forms as a direct sum of orthogonal subspaces corresponding to each of the qualitative facts of the model has been developed. Choosing different groups of linearly independent columns in the initial observation matrix and orthogonalizing this system of vectors (as columns of the matrix) we can obtain various orthogonal bases for the space of linear forms under estimation. Projections of the response vector to the vectors of orthogonal basis corresponding to the same basis determine the contribution of this factor to the total sum of squares obtained as a result of the response projection to the entire space of linear forms being estimated. With different orthogonal bases of the space of linear estimated forms, these contributions of factors change. Under conditions of incomplete factor design, one can distinguish an orthogonal basis of the space of linear estimated forms which provides the greatest significance of any comparison of one of the factors, and, consequently, of the factor itself. This allows determination of the best (in a certain sense) experimental design which ensures the greatest significance of the factors selected. To prove the results obtained the method of ranking factors LASSO was used.

analysis of variance, qualitative factors, space of linear estimated forms, comparisons, full and incomplete rank models, LASSO

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