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Adapted from Krus, D. J., & Wilkinson, S. M. (1986) Matrix differencing as a concise expression of test variance: a computer implementation. Educational and Psychological Measurement, 46 179-183. 

MATRIX DIFFERENCING AS A CONCISE EXPRESSION OF TEST VARIANCE
David J. Krus and Susan M. Wilkinson
Arizona State University

 

The operation of matrix differencing is introduced as a concise method for the computation of test variance. This procedure is viewed in contrast to computational methods utilized with the traditional analysis of variance framework.

 

Intuitively, the minor and major products of vectors should have as their counterpart the minor and major sums and differences of their corresponding summands, minuends, and subtrahends. These operations are easy to imagine. However, they are not often contemplated. Perhaps they seem too primitive, probably because of a lack of perceived utility. The purpose of this communication was to describe the operation of matrix differencing. its utility for concise computation of test variance, and its applicability for solving some analysis of variance designs.

Differencing Data Matrices

The formation of the minor and major differences of a data vector is analogous to the formation of its minor and major products; however, the elements of a data vector are subtracted prior to their cumulation, instead of being multiplied. This suggested matrix operation can be explicated by comparing the code for matrix multiplication, C = AB’, with the code for matrix differencing (Table 1)

 

(1)

 

where the minus sign symbolizes matrix subtraction and NOT the subtraction of its elements,. The dimension of the A and B matrices is n by k, the order of the C matrix is n,

 

Table 1. Code for matrix multiplication C = AB’ and matrix differencing C = A – B’

Note that the above code is identical, except for the last line in which the multiplication sign for the AB product was
replaced by the minus sign The same code can be used for matrix addition by changing this sign from minus to plus.

As in the course of the multiplicative matrix operations, minor and major difference of matrices can be computed of which only the major differences are of interest. For instance, consider a data matrix X

(2)

the expanded (exp) variance for the rows of A can be written as

 

(3)

 

and the expanded variance for the columns as

 

(4)

 

The total expanded variance can be computed as

 

(5)

 

where  symbolizes elements of the decomposed matrix X.

 

 

For the discussed instance of the data matrix X, the X - X’ expression on the right-hand side of Eq. 3 equals

 

(6)

 

and the expanded variance component for the rows of this instance of the data matrix equals 283. The X’ – X expression on the right-hand side of Eq. 4 equals

 

(7)

 

and the expanded variance component for the columns of this instance of the data matrix X is 81. The variance of the elements of the decomposed matrix X equals 6.234 and the expanded variance component T can be computed (Eq. 5) as (4x2)² 6.234 = 399. These variance components were arranged in Table 2.

 

Table 2. Analysis of variance solved by the matrix differencing.

The variance components obtained by the sum of squares –means squares method are shown in Table 3.

Table 3. Analysis of variance solved by the sum of squares – mean squares method.

The division of either the sums of squares values in Table 3 or the values in the expanded variance components in Table 2 by their respective total components of variance will result in the standardized solution of the analysis of variance, as shown in Table 4.

Table 4. Standardized analysis of variance.

Discussion

If one considers the multitude of steps necessary to explain and to compute the traditional sums of squares – mean squares analysis of variance, matrix differencing, within the computerized environment, offers substantial conceptual and computational advantages over the traditional approach.

 

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