Iii the sum of each column obtained as per a above is

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iii. The sum of each column obtained as per ( ) a above is divided by the square root of T obtained in ( ) b above, resulting in what are called centroid loadings. This way each centroid loading (one loading for one variable) is computed. The full set of loadings so obtained constitutes the first centroid factor ( ) say A . 4) To obtain second centroid factor ( ) say B , one must first obtain a matrix of residual coefficients. For this purpose, the loadings for the two variables on the first centroid factor are multiplied. This is done for all possible pairs of variables (in each diagonal space is the square of the particular factor loading). The resulting matrix of factor cross products may be named as 1 Q then 1 Q is subtracted element by element from the original matrix of correlation, R , and the result is the first matrix of residual coefficients, 1 R One should understand the nature of the elements in 1 R matrix. Each diagonal element is a partial variance i.e., the variance that remains after the influence of the first factor is partialed. Each off-diagonal element is a partial co-variance i.e., the covariance between two variables after the influence of the first factor is removed. This can be verified by looking at the partial correlation coefficient between any two variables say 1 and 2 when factor A is held constant 12 1 2 12. 2 2 2 1 1 A A A A A r r r r r r (The numerator in the above formula is what is found in 1 R corresponding to the entry for variables 1 and 2. In the denominator, the square of the term on the left is exactly what is found in the diagonal element for variable 1 in 1 R . Likewise the partial variance for 2 is found in the diagonal space for that variable in the residual matrix.) Since in 1 R the diagonal terms are partial variances and the off-diagonal terms are partial co- variances, it is easy to convert the entire table to a matrix of partial correlations. For this purpose one has to divide the elements in each row by the square-root of the diagonal element for that row and then dividing the elements in each column by the square-root of the diagonal element for the column. After obtaining 1 R one must reflect some of the variables in it, meaning thereby that some of the variables are give signs in the sum [this is usually done by inspection. The aim in doing this should by to obtain a reflected matrix 1 R which will have the highest possible sum of coefficients ( ) T ]. For any
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220 variable which is so reflected, the signs of all coefficients in that column and row of the residual matrix are changed. When this is done, the matrix is named as ‘reflected matrix’ form which the loading are obtained in the usual way (already explained in the context of firs centroid factor), but the loadings of the variables which were reflected must be given negative signs. The full set of loadings so obtained constitutes the second centroid factor (say B). Thus loadings on the second centroid factor are obtained from, 5) For subsequent factors (C,D, etc.) the same process outlined above is repeated.
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