APPLICATION OF LAGRANGE MULTIPLIERS IN TWO-WAY CLASSIFICATION ANALYSIS OF VARIANCE WITH INTERACTION

Resumen

The objective of this study was to use Lagrange multipliers to obtain the solution to the normal equations, followed by the sums of squares of the analysis of variance, for a two-way classification model with interaction. The model included the main effects of rations and sires at two and three levels, respectively, as well as their interaction. The least squares method was used in an unbalanced data set with all cells filled. The modified normal equations were formed with the matrix (X’X)_(12,12) in the upper part, together with the matrix P_(12x6) in the lower part, its transpose P’_(6x12) appears along with the zero matrix 0_(6x6), which together constitutes the left-hand side (LHS) of the modified normal equations. Both the parameter vector and the right-hand side (RHS) were augmented with six q’s and six 0’s, respectively. In the inverse of the LHS of the modified equations, within the section corresponding to (X’X), the inverse of the coefficient matrix of the incomplete rank matrix was obtained, which resulted in the same property. It was verified, however, that this inverse is the generalized inverse corresponding to the system of equations when the sum of the constants is assumed to be zero for each effect. The direct procedure was used for the decomposition of the models’ sum of squares, where both the vector of parameter estimators and the inverse elements of the submatrix for each effect were reduced according to their corresponding degrees of freedom. The results were identical to those previously obtained with conventional procedures, with the advantage that all inverse elements were available for calculating standard errors, reducing the possibility of errors.