By Spencer Bennett, David Bowers
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Additional info for An Introduction to Multivariate Techniques for Social and Behavioural Sciences
62 48 9 (57) In this example, orthogonal rotation does not lead to simple structure, whilst oblique rotation does. 5. The relevant equations for deriving revised factor loadings are: Ft = F 1 cos 8 + F2 sin 8 F; = F 2 cosa- F 1 sina, where 8 and a are the angles through which F 1 and F 2 are rotated, respectively. Following this oblique rotation, the sums of squares of factor loadings for each variable no longer give us the communalities, and the sum of squares in each column can no longer be used to determine the amount of variance extracted or attributable to each factor.
As was noted in Chapter 1, one of the difficulties encountered by behavioural scientists is that the variables which they can measure directly are often impure ones, and factor analysis can help with this difficulty by producing purer, though indirectly measurable, variables for use in research. The test of whether this has been achieved by factor analysis can only be gauged by using factors in research and seeing whether stable generalisations can be inferred. The true test of the optimum position for a rotated axis is whether or not it leads to factors that enable advances in subsequent research to be made, not necessarily fulfilling the criteria for simple structure.
The smaller the angle between axes, the larger the cosine and, hence, the correlation. 31. How this affects the interpretation of the analysis had best be left until the next chapter, and all that need be said here is that oblique rotation has helped locate clusters of similar variables, and thus aided interpretation of the factors. We will also leave the oblique rotation of the three factors involved in our body- size example until Chapter 4. 34 Introduction to Multivariate Techniques One problem with oblique factors is that although they are useful for locating clusters of similar variables, they are themselves correlated, and if this correlation is substantial, then it may be necessary to introduce further underlying variables to explain the inter-correlations between factors, as described briefly in the next section; thus, if this is the case, the result is a loss of parsimony of description.
An Introduction to Multivariate Techniques for Social and Behavioural Sciences by Spencer Bennett, David Bowers