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Correlation / Covariance Matrix help example

  • Correlation matrices
  • The correlation matrix of n random variables X1, ..., Xn is the n x n matrix whose i,j entry is corr(Xi, Xj). If the measures of correlation used are product-moment coefficients, as in our case, the correlation matrix is the same as the covariance matrix of the standardized random variables divided by σ (Xi) for i = 1, ..., n. This applies to both the matrix of population correlations (in which case "σ" is the population standard deviation), and to the matrix of sample correlations (in which case "σ" denotes the sample standard deviation). The correlation matrix is symmetric because the correlation between Xi and Xj is the same as the correlation between Xj and Xi.
    As the correlation of a variable with itself is of course 1.0, the diagonal elements of a correlation matrix are 1.0.
    our example:

    Correlation Matrix

    	
    	0    1.00000    0.32872    0.16767    0.07686    0.05245   -0.73081
    	1    0.32872    1.00000   -0.14550    0.20932    0.17023   -0.21204
    	2    0.16767   -0.14550    1.00000    0.23016    0.23778   -0.05541
    	3    0.07686    0.20932    0.23016    1.00000    0.98891    0.24929
    	4    0.05245    0.17023    0.23778    0.98891    1.00000    0.30965
    	5   -0.73081   -0.21204   -0.05541    0.24929    0.30965    1.00000			
    
  • Covariance matrices
  • Variance and covariance are also often displayed together in a matrix: A covariance matrix (also known as dispersion matrix or variance-covariance matrix) is a matrix whose element in the i,j position is the covariance between the i th and j th elements of a vector. The diagonal terms are the variances. Our example:

    Covariance Matrix

    		
    	0   14.75568    1.87585    0.25227    0.35653    0.23580   -1.90574
    	1    1.87585    2.20695   -0.08466    0.37551    0.29599   -0.21384
    	2    0.25227   -0.08466    0.15341    0.10886    0.10901   -0.01473
    	3    0.35653    0.37551    0.10886    1.45830    1.39773    0.20437
    	4    0.23580    0.29599    0.10901    1.39773    1.36989    0.24603
    	5   -1.90574   -0.21384   -0.01473    0.20437    0.24603    0.46085				
    

    test if different from zero

                   1     2     3     4     5     6  
    	1      -                           
    	2      n.s   -                      
    	3      n.s   n.s   -                 
    	4      n.s   n.s   n.s   -            
    	5      n.s   n.s   n.s   ***   -       
    	6      n.s   n.s   n.s   n.s   n.s   -  
    
    The tables above are translated into a picture, right/upper part: the correlation coefficents, the whiter the higher. Lower/left indicates the significance, red is high(0.5%), magenta (1%), yellow (5%) The red square at 5,4 corresponds to the correlation coefficient 0.98891 and the three stars in the 'test if different from zero'.
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