Analysis of Variance Using Matrix Algebra

Expressing variance as standard variance components is most meaningful. Variance expressed as extended variance components (sums of squares) facilitate the use of a spreadsheet for the computation of analysis of variance components. Variance in its expanded form, discussed in detail in this chapter, facilitates computation of variance within the context of matrix algebra.

Single Classification Analysis of Variance

Consider comparing the data vector of control or reference measurements Y₀ = [1 2 3] with another data vector of experimental measurements Y₁ = [2 4 3]. A parent vector X = [0 0 0 1 1 1] can be constructed to indicate the membership in either the control group (0) or experimental group (1), and vectors Y₀ and Y₁ can be catenated into vector Y [1 2 3 2 4 3]. The regression solution is

 

 

The above solution can be also presented in the tabular form as

 

 

In terms of 'sums of squares,' called here the extended variance components, the variance of the above data can be partitioned by using the Microsoft Excel as

 

 

And summarized in the tabular form as

 

 

Note that you can standardize the extended variance components by dividing them by the total, which for the example equals 5.5.

          Using the matrix algebra notation, you can obtain the column variance component as

 

 

which for the example equals

 

 

Subtracting matrices within the parentheses

 

 

 

and by triangulating the skew-symmetric matrix into the skew-positive matrix, squaring at the same time the skew-positive matrix elements

 

 

 

the expanded column variance component can be obtained as equal to 9.0. The total expanded variance component can be calculated either as n times the extended variance component, for the example as 6(5.5) which equals 33.0, or, by using matrix algebra, as

 

 

 

where T is the decomposed data matrix X. For the example,

 

 

 

Subtracting data vectors within the parentheses

 

 

 

and by triangulating the skew-symmetric matrix into the skew-positive matrix

 

 

 

 

the expanded total variance component can be computed by summing the squared elements of the skew-symmetric above matrix as equal to 33.0. In tabular representation,

 

 

Source of Variance	Variance Components	Standard Variance Components	Extended Variance Components	Expanded Variance Components
Columns	.25	.27	1.5	9
Residual	.67	.73	4.0	24
Total	.92	1.00	5.5	33

 

As in the case of the other non-standard variance components, you can standardize the expanded variance components by dividing each expanded variance component by the total, for the example, by 33.0.

Double Classification Analysis of Variance

Using the extended variance components, the variance of the data for our example can be partitioned by using the Microsoft Excel spreadsheet as

 

 

In tabular representation

 

Source of Variance	Extended Variance Components	Standard Variance Components	Variance Components
Columns	1.5	.27	.25
Rows	3.0	.55	.50
Residual	1.0	.18	.17
Total	5.5	1.00	.92

 

In the above table, division by the total (5.5) standardized the extended variance components. The variance components were obtained by decomposing the data matrix into vector T [1 2 3 3 2 4] and computing its variance as equal to .92. The standard variance components then were multiplied by .92.

          In matrix algebra notation, the variance component for the rows of the data matrix can be conceptualized as

 

 

For the example,

 

 

 

Subtracting matrices within the parentheses

 

 

 

and by triangulating the skew-symmetric matrix into the skew-positive matrix

 

 

 

the expanded row variance component can be computed by summing the squared elements of the skew-positive above matrix as equal to 18.0. In tabular representation,

 

Source of Variance	Standard Variance Components	Variance Components	Extended Variance Components	Expanded Variance Components
Columns	.27	.25	1.5	9
Rows	.55	.50	3.0	18
Residual	.18	.17	1.0	9
Total	1.00	.92	5.5	33

Summary

The discussion in this chapter, outlining computation of single and double-classification analysis of variance in matrix algebra notation can be summarized as