By Spencer Bennett, David Bowers

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**Extra resources for An Introduction to Multivariate Techniques for Social and Behavioural Sciences**

**Example text**

E. Otf1 = Q/1 /Q/1 (max). e. 3) until successive sets of values for et11 agree to the desired degree (in our case, to two places of decimals). In our example, the procedure only has to be carried out once. 8 gives details of the calculations. 8, F11 , gives the values for the first factor loadings. jt which equals unity. 57. 1) (re-stated here for convenience), r;i = L F;kFik k relates the original correlations between variables, r;j, to the correlations between the variables and factors. In a similar way, we may in the principal factor analysis proceed to test the hypothesis that only one factor is needed to account for the original r;j's.

If there are n factors, there should ben non-significant loadings in each factor. For every pair of factors there should be several variables with non-significant loadings in one, but significant loadings in the other. Where there are four or more variables for every pair of factors, a large proportion of loadings should have non-significant values in both. For every pair of factors there should be only a small proportion of significant loadings in both. 1 suggests that a clockwise rotation through 45° might improve the situation.

If A is a linear combination of B and C, and the latter two variables tend to load highly on different factors, A is quite likely to be a hybrid variable, making interpretation difficult, and attempts to make it conform to the 'simple structure' solution can distort the interpretation. Moreover, since A will very probably correlate with B or Cor both, its inclusion may lead to the extraction of spurious factors. The purpose of orthogonal rotation is to locate clusters of related variables which are relatively independent of(or orthogonal to) other clusters; the factor 'body bulk' is independent of factors concerned with body length.