Sums of Squares and Cross Products Matrix

This lesson introduces the sums of squares and cross products matrix (aka, SSCP matrix). We show how to use matrix methods to compute the SSCP matrix, using both raw scores and deviation scores.

Sum of Squares: Vectors

In statistics, many formulas require the calculation of sums of squares; i.e., squaring all of the elements in a set and then taking the sum of those squares.

Using matrix algebra, the sum of squares for all the elements of a vector is calculated according to the following formula:

Σ x_i² = x'x

where

x is an n x 1 column vector of scores: x₁, x₂, . . . , x_n
Σ x_i² is the sum of the squared values from vector x

To illustrate, let's find the sum of squares for the elements of vector x, where x' = [ 1 2 3 ].

Σ x_i² = ( 1 * 1 ) + ( 2 * 2 ) + ( 3 * 3 )

Σ x_i² = 1 + 4 + 9 = 14

Thus, the sum of the squared elements from vector x is 14.

Sums of Squares and Cross Products: Matrices

With matrices, we can compute not only sums of squares but also sums of cross products. For an r x c matrix, an individual cross product is represented by X_rjX_rk. The sum of cross products between all the elements of columns j and k is represented by Σ X_rjX_rk, summed over r. A matrix of sums of squares and sums of cross products is represented by X' X, as shown below.

where

X is an r x c matrix of raw scores: X₁₁, X₁₂, . . . , X_rc
X' X is a c x c matrix of sums of squares and sums of cross products
Σ X_i² is the sum of the square of all elements in column i of matrix X
Σ X_i X_j is the sum of cross products produced by multiplying each element in column i of matrix X with the corresponding element from column j and summing the result

Thus, the diagonal elements of X' X are sums of squares, and the off-diagonal elements are cross products. Note that the cross product matrix X' X is a symmetric matrix.

See problem 1 for an example showing how to create a cross product matrix.

Sums of Squares and Deviation Scores

In the previous lesson, we showed how to transform a matrix of raw scores into a matrix of deviation scores. There are advantages to working with deviation scores.

• Computations can be easier.
• Equations are often more comprehensible.

As a result, researchers often transform their raw data into deviation scores before they calculate sums of squares and cross products. This is such a common practice that the term "sums of squares" has two meanings. It can refer to raw score sums of squares or to deviation score sums of squares.

See problem 2 for an example showing how to create a cross product matrix, using deviation scores.