Variance Components of the Analysis of Variance
Results of the Analysis of Variance are often expressed as the Sums of
Squares. However, more meaningful is to present these results using the
Standard Variance Components. Variance in
its expanded form, discussed in detail in this chapter,
facilitates computation of variance within the context of
matrix algebra.
Select ( Designs, Designs of Experiments )
and under the Classic Designs heading click on he Related Measures command.
On the Designs of Experiments menu, under the Shells heading, click on the Double Classification Analysis of Variance command and Unfold its display.
On the scalar module click on the Right Justify
command and change the variable name to Total.
On the Shell (Prototype Components of the Double Classification Analysis of Variance ) menu, click on the Standard Variance Component (1), (2), (3), and (4) to transfer these components to the scalar module. Multiply the C (1) component by the Total Variance
and replace the values of the C (1) component with the calculated value. Repeat this procedure for the remaining components and drag these components to the Shell Prototype of the Double Classification Analysis of Variance.
Click on the Sum of Squares n * k command and multiply the Total variance as ( 6*.917 = 5.5 ). Store this value as a multiplier for the Sums of Squares values. Also, restore the Standard Variance Components to the scalar display.
Multiply the Standard Variance Components by the SSQ multiplier and replace these components with the new calculated values. Drag these components to the Shell Prototype of the Double Classification Analysis of Variance as
Analysis of Variance Using Matrix Algebra
Replace the values of the prototype for the Classic Related Measures Design on the vector display. Select ( Transfers, Launch Matrix Module ).
Select ( Transfers, Vectors to Matrix ) and store the transferred values in the Matrix Cell 1. Transpose the Matrix 1 and save it in the Matrix 2. Click on the Unfold command.
Conceptually, the analysis of variance within the matrix algebra framework begins by computing the X-X' and X'-X matrix differences. Results of these operations are skew symmetric matrices that are converted into the skew asymmetric matrices by deleting the redundant elements, symmetric along the principal diagonal, that have the same magnitude, but opposing signs. However, we can compute the asymmetric matrices of differences directly, by clicking the Delta X command. This operation is not commutative, returning the asymmetric matrices for attributes (columns) when performed in the matrix transpose - matrix order and the asymmetric matrices for entities (rows) when performed in the matrix - matrix transpose order, as
Matrix 4 has to be converted into the Canonical form, where all asymmetric elements have a zero as their counterpart. Click the Canonical command, selecting the Matrix 4 as input, and store the result in the Matrix 5.
Expanded Variance Components are computed as the sum of squared elements of the canonical matrices. Click on the (Sums, Grand Sums of Squared Elements) for the Matrix 3 and the Matrix 5. The resulting values (9 and 18) are the Expanded Variance Components for the Columns and Rows of the data in the Matrix 1. Click the ( Decompose, Matrix 1 ) commands, Transpose the Decomposed matrix, and click the ( Delta X, Sums, Grand Sum of Squared Matrix Elements ) commands to obtain the Total Expanded Variance Component ( 33 ). Now you have all the information to complete the Double Classification of Analysis of Variance Shell template, as
Note that all the components can be converted into the Standard Variance Components by dividing each column by its Total value.
Dendrograms
The Canonical matrix in the Matrix Cell 3 is adjacent to an ordered graph
which reminds us that the main purpose of the analysis of variance is to find (how much and how reliably) means are different from each other.
