by John Kelley
Variables
----------------
| Extro Intro |
| ----- ----- |
1 | 5 1 |
2 | 1 3 |
Subjects 3 | 2 5 |
4 | 4 0 |
5 | 3 1 |
----------------
To compute the Sums of Squares/Cross Products matrix (SSQ/CP),pre-multiply
X by its transpose X':
X' * X = X'X
5 1 2 4 3 5 1 55 21
1 3 5 0 1 1 3 21 36
2 5
4 0 SSQext CP
3 1 CP SSQint
2x5 5x2 2x2
To sum variables (columns) across subjects (rows), pre-multiply X by the
transpose of the unit vector (1'):
1' * X = 1'X
1 1 1 1 1 5 1 15 10
1 3
2 5
4 0
3 1
1x5 5x2 1x2
To find the means of the variables, multiply the vector
representing the sums of the variables by the scalar n-1 to
yield a vector containing the variables' means:
n-1 * 1'X = n-11'X
1/5 15 10 3 2
You can expand the vector containing the means into a matrix of the same
dimensions as the original data matrix by pre-multiplying by the unit vector:
1 * (n-11'X) = 1(n-11'X)
1 3 2 3 2
1 3 2
1 3 2
1 3 2
1 3 2
5x1 1x2 5x2
_
Thus, 1(n-11'X) is the matrix formula for deriving X, a matrix
that is the same size as the original data matrix and that contains the mean
for each variable in every cell of the appropriate column.
In addition, one can rearrange the formula as follows:
_
X = 1(n-11'X)
_
X = n-1[(1)(1')]X = n-1(E)X,
where E is a square matrix of dimensions nxn and contains all ones. The
example below illustrates this formula:
n-1 * E * X = n-1(E)X = X
(1/5) 1 1 1 1 1 5 1 3 2
1 1 1 1 1 1 3 3 2
1 1 1 1 1 2 5 3 2
1 1 1 1 1 4 0 3 2
1 1 1 1 1 3 1 3 2
5x5 5x2 5x2
A matrix of mean deviations (Y) can be derived next:
_
X - X = Y
X - n-1(E)X = Y
5 1 3 2 2 -1
1 3 3 2 -2 1
2 5 - 3 2 = -1 3
4 0 3 2 1 -2
3 1 3 2 0 -1
Now we can generate a matrix of the Sums of Squares and CrossProducts of
the Mean Deviations (note the similarity of this operation to the very first
operation we performed):
Y' * Y = Y'Y
2 -2 -1 1 0 2 -1 10 -9
-1 1 3 -2 -1 -2 1 -9 16
-1 3
1 -2 SSQext CP
0 -1 CP SSQint
2x5 5x2 2x2
One can then simply multiply by the scalar (n-1)-1to produce the
Variance/Covariance Matrix (S):
(n-1)-1 * Y'Y = V = S (some denote it with V, some with S)
= (1/4) * 10 -9 = 2.50 -2.25 = s2ext Cov
-9 16 -2.25 4.00 Cov s2int
To produce the Correlation Matrix, R, from the Variance/Covariance
Matrix, S, we need to backtrack a bit. First, we need to produce a data matrix
of standardized scores (Z). To do so, we simply take the mean deviation data
matrix (Y) that we computed earlier and post-multiply it by a diagonal matrix
consisting of the reciprocals of the standard deviations of the appropriate
variables(Ds-1).
Note: s-1ext =[sqrt(s2ext)]-1 = [sqrt(2.50)]-1= .63,
s-1int =[sqrt(s2int)]-1 = [sqrt(4.00)]-1= .50.
Y * Ds-1 = Z
2 -1 0.63 0.00 1.26 -0.50
-2 1 0.00 0.50 -1.26 0.50
-1 3 -0.63 1.50
1 -2 0.63 -1.00
0 -1 0.00 -.50
5x2 2x2 5x2
Now for some algebraic manipulation. The formula for producing a
Correlation Matrix is R = Z'Z(n-1)-1.
Therefore:
R = Z'Z(n-1)-1
= Z' * Z * (n-1)-1 then, by substitution:
= (YDs-1)' * (YDs-1) * (n-1)-1 then, b/c the transpose of
a product is the product
of the transposes in the
reverse order:
= (Ds-1)'Y'* (YDs-1) * (n-1)-1 then, we can group the
terms b/c matrix
multiplication is
associative:
= (Ds-1)'[Y'Y (n-1)-1] * Ds-1 Now, we substitute:
= (Ds-1)'*S*(Ds-1) Finally, we substitue
(Ds-1) for (Ds-1)' b/c
transposing a diagonal
matrix doesn't change it:
R = (Ds-1)*S*(Ds-1)
Therefore, the correlation matrix (R) can be generated from the
Variance/Covariance Matrix (S) by simply pre- and post-multiplying by a
diagonal matrix composed of the reciprocals of the standard deviations of the
appropriate variables as illustrated below for our data set:
(Ds-1) * S *(Ds-1) =
R
0.63 0.00 2.50 -2.25 0.63 0.00
0.00 0.50 -2.25 4.00 0.00 0.50
= 1.575 -1.4175 * 0.63 0.00 = 1.00 -.71
-1.125 2.0000 0.00 0.50 -.71 1.00
Begin with a data matrix X. The variables are Extroversion and
Introversion, and there are five subjects.