Research Methodology - Dr. Krishan K. Pandey
Research Methodology - Dr. Krishan K. Pandey Research Methodology - Dr. Krishan K. Pandey
Multivariate Analysis Techniques 325 correlation, R, and the result is the first matrix of residual coefficients, R 1 . * After obtaining R 1 , one must reflect some of the variables in it, meaning thereby that some of the variables are given negative signs in the sum [This is usually done by inspection. The aim in doing this should be to obtain a reflected matrix, R' 1 , which will have the highest possible sum of coefficients (T)]. For any 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 loadings are obtained in the usual way (already explained in the context of first 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 R' 1 . (v) For subsequent factors (C, D, etc.) the same process outlined above is repeated. After the second centroid factor is obtained, cross products are computed forming, matrix, Q 2 . This is then subtracted from R 1 (and not from R' 1 ) resulting in R 2 . To obtain a third factor (C), one should operate on R 2 in the same way as on R 1 . First, some of the variables would have to be reflected to maximize the sum of loadings, which would produce R' 2 . Loadings would be computed from R' 2 as they were from R' 1 . Again, it would be necessary to give negative signs to the loadings of variables which were reflected which would result in third centroid factor (C). We may now illustrate this method by an example. Illustration 1 Given is the following correlation matrix, R, relating to eight variables with unities in the diagonal spaces: Variables 1 2 3 4 5 6 7 8 1 1.000 .709 .204 .081 .626 .113 .155 .774 2 .709 1.000 .051 .089 .581 .098 .083 .652 3 .204 .051 1.000 .671 .123 .689 .582 .072 Variables 4 .081 .089 .671 1.000 .022 .798 .613 .111 5 .626 .581 .123 .022 1.000 .047 .201 .724 6 .113 .098 .689 .798 .047 1.000 .801 .120 7 .155 .083 .582 .613 .201 .801 1.000 .152 8 .774 .652 .072 .111 .724 .120 .152 1.000 Using the centroid method of factor analysis, work out the first and second centroid factors from the above information. * One should understand the nature of the elements in R1 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 r12 r1A r2A r12⋅ A 2 2 1 r1A 1 r2A = − ⋅ − − (The numerator in the above formula is what is found in R corresponding to the entry for variables 1 and 2. In the 1 denominator, the square of the term on the left is exactly what is found in the diagonal element for variable 1 in R . Likewise 1 the partial variance for 2 is found in the diagonal space for that variable in the residual matrix.) contd.
326 Research Methodology Solution: Given correlation matrix, R, is a positive manifold and as such the weights for all variables be +1.0. Accordingly, we calculate the first centroid factor (A) as under: Table 13.1(a) Variables 1 2 3 4 5 6 7 8 1 1.000 .709 .204 .081 .626 .113 .155 .774 2 .709 1.000 .051 .089 .581 .098 .083 .652 3 .204 .051 1.000 .671 .123 .689 .582 .072 Variables 4 .081 .089 .671 1.000 .022 .798 .613 .111 5 .626 .581 .123 .022 1.000 .047 .201 .724 6 .113 .098 .689 .798 .047 1.000 .801 .120 7 .155 .083 .582 .613 .201 .801 1.000 .152 8 .774 .652 .072 .111 .724 .120 .152 1.000 Column sums 3.662 3.263 3.392 3.385 3.324 3.666 3.587 3.605 Sum of the column sums (T) = 27.884 ∴ T = 5281 . First centroid factor A = 3662 . 3263 . 3392 . 3385 . 3324 . 3666 . 3587 . 3605 . , , , , , , , 5281 . 5281 . 5281 . 5281 . 5. 281 5281 . 5281 . 5281 . = .693, .618, .642, .641, .629, .694, .679, .683 We can also state this information as under: Table 13.1 (b) Variables Factor loadings concerning first Centroid factor A 1 .693 2 .618 3 .642 4 .641 5 .629 6 .694 7 .679 8 .683 To obtain the second centroid factor B, we first of all develop (as shown on the next page) the first matrix of factor cross product, Q 1 : Since in R 1 the diagonal terms are partial variances and the off-diagonal terms are partial covariances, 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 squareroot of the diagonal element for that row and then dividing the elements in each column by the square-root of the diagonal element for that column.
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326 <strong>Research</strong> <strong>Methodology</strong><br />
Solution: Given correlation matrix, R, is a positive manifold and as such the weights for all variables<br />
be +1.0. Accordingly, we calculate the first centroid factor (A) as under:<br />
Table 13.1(a)<br />
Variables<br />
1 2 3 4 5 6 7 8<br />
1 1.000 .709 .204 .081 .626 .113 .155 .774<br />
2 .709 1.000 .051 .089 .581 .098 .083 .652<br />
3 .204 .051 1.000 .671 .123 .689 .582 .072<br />
Variables 4 .081 .089 .671 1.000 .022 .798 .613 .111<br />
5 .626 .581 .123 .022 1.000 .047 .201 .724<br />
6 .113 .098 .689 .798 .047 1.000 .801 .120<br />
7 .155 .083 .582 .613 .201 .801 1.000 .152<br />
8 .774 .652 .072 .111 .724 .120 .152 1.000<br />
Column sums 3.662 3.263 3.392 3.385 3.324 3.666 3.587 3.605<br />
Sum of the column sums (T) = 27.884 ∴ T = 5281 .<br />
First centroid factor A = 3662 . 3263 . 3392 . 3385 . 3324 . 3666 . 3587 . 3605 .<br />
, , , , , , ,<br />
5281 . 5281 . 5281 . 5281 . 5. 281 5281 . 5281 . 5281 .<br />
= .693, .618, .642, .641, .629, .694, .679, .683<br />
We can also state this information as under:<br />
Table 13.1 (b)<br />
Variables Factor loadings concerning<br />
first Centroid factor A<br />
1 .693<br />
2 .618<br />
3 .642<br />
4 .641<br />
5 .629<br />
6 .694<br />
7 .679<br />
8 .683<br />
To obtain the second centroid factor B, we first of all develop (as shown on the next page) the<br />
first matrix of factor cross product, Q 1 :<br />
Since in R 1 the diagonal terms are partial variances and the off-diagonal terms are partial covariances, it is easy to convert<br />
the entire table to a matrix of partial correlations. For this purpose one has to divide the elements in each row by the squareroot<br />
of the diagonal element for that row and then dividing the elements in each column by the square-root of the diagonal<br />
element for that column.
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