The central statistics of the general linear model are the algebraic mean, variance, followed by the coefficients of covariance and correlation. The concepts of correlational analysis can be extended to include the coefficients of statistical significance as t and F, obtained in the course of computations of t-test and analysis of variance. These univariate and bivariate statistical methods are usually expressed by using the summation notation, but can be expressed as well by using the matrix algebra notation.
Using summation notation, the algebraic mean of a variable X can be written as
where n signifies the number of cases. In the notation of matrix algebra, the mean of the vector X can be written as
where 0 is a null vector and n is the length of both the X and 0 vectors.
Consider vector X [1 2 3 4 5]. Its length equals 5 and its arithmetic mean is computed as
Alternatively,
where 1 is a unit vector. Thus,
Using the summation notation in obtained scores, the true variance of a variable X is
For the example of the variable X = [1 2 3 4 5] its variance can be computed, using the obtained scores and their squares, as
For the example, variance of the variable X can be computed as (5(55) - (15)2) / 25 which equals (275 - 225) / 25 which, in turn, equals 2.
In the notation of matrix algebra, the same expression can be written as
where 1 is a unit vector, and the Greek letter delta signifies triangulation of the skew matrix into a skew-positive matrix. For the vector X [1 2 3 4 5] the above expression is written as
Subtracting the X - X' expression within the parentheses,
and triangulating the skew-symmetric matrix
Squaring the matrix elements
the true variance of the variable X can
be computed as 50/25 that is 2.
Using summation notation, the covariance of variables X and Y can be written as
Consider the following example
Using the deviation scores, covariance can be computed as 16/4, which equals 4.0.
In the notation of matrix algebra, the covariance of the matrix X can be written as
where D signifies the matrix X,
linearly transformed into deviation scores, and n is the
number of rows in the matrix D. For the above example,
and its corresponding matrix of deviation scores D is
The covariance of the matrix is computed as
The matrix C is also called the variance-covariance matrix, since the variance of each variable is in its principal diagonal and the covariance among its variables is in the off-diagonal elements.
Using summation notation, the correlation of variables X and Y can be written as
where
and
are standard scores, obtained from the
deviation scores corresponding to variables X and Y by
linear transformations
and
. The
n signifies the number of cases. In the notation of matrix
algebra, the correlation matrix R corresponding to the data
matrix X can be written as
where Z signifies the matrix X, linearly transformed into standard scores, and n is the number of rows in the matrix Z. Consider matrix X
and its corresponding matrix of standard scores Z
The correlation matrix R corresponding to the matrix X is computed as
Multiplying matrices in the numerator
and dividing by the scalar number in the denominator
It is important to realize that all discussed operations, done with respect to columns (attributes) of the data matrix, can be also done with respect to its rows (entities).
The statistical formulae in both the summation and the matrix notation are summarized
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Summation Notation |
Matrix Algebra
Notation |
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Mean |
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Variance |
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for means and variances in the preceding table. For covariance and correlation, the key formulae are summarized as
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Summation Notation |
Matrix Algebra Notation |
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Covariance
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Correlation |
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