Worksheet - Variance-Covariance Matrix and Correlation Matrix

Raw data (i.e., Raw score matrix)

Calculate the mean vector

Obtain a listing of deviations from the mean vector (i.e., Deviation matrix)

Matrix multiplication - this procedure calculates the crossproduct of the original matrix of differences (A) with a transposed copy (A^') of itself (i.e., one in which rows and columns have been inverted). This results in a new matrix with dimensions of 3 (i.e., rows of A) x 3 (i.e., (columns of A^')

Deviation SSCP (Sums of Squares and Cross Products) matrix. This procedure calculates the sums of squares along the diagonal and the sums of Cross product in the lower triangle off the diagonal. All products are calculated as length vector 1 * length of vector 2 * sine of angle between vector 1 and 2. As the different variables are orthogonal to each other (i.e., the angle between them is 90^o), the angle term of the product formula therefore is always 1 (i.e., sine(90^o) = 1) and can thus be dropped from these calculations. Note there are three different, mathematically equivalent techniques to calculate the deviation SSCP. The symbol X in all tables below refers to the differences from the mean and not the raw variable measures.

Variance-Covariance Matrix. This procedure calculates the variance (i.e., MS) for each variable along the diagonal and the covariances the lower triangle off the diagonal. Recall that variance estimates refer to a sum of squares divided by its degrees of freedom

Standard Deviation Matrix. This procedure calculates the standard deviations (s) as the quare root of the variances along the diagonal and the covariances in the lower triangle off the diagonal.

Correlation matrix. i.e., Standardized variance/covariance matrix - Divide each entry by two standard deviations (s), one for each relevant row and column. Understand why this is identical to the Variance/covariance matrix calculated from the deviation matrix in which deviations have been standardized (i.e. replaced with their standard normal deviates). Terms in the variance/covariance matrix are standardized to a s² of 1 by dividing all variance terms by its variance and all covariance terms by its covariance.

Matrix of Coefficients of Determination (r²) - Note: Do not use in Correlation analysis. Square each entry from above.

If you have difficulties in performing these steps, review some of the basic concepts in matrix algebra.