## FACTOR ANALYSIS

Factor analysis is a method used to reduce many variables to a few dimensions. For example, it is possible to reduce 128 adjectives in the Interpersonal Adjective Scales to only two dimensions, named dominance and friendliness. The naming of the resulting dimensions is somewhat subjective--different researchers may prefer to call the first dimension power or control, and the second dimension love or affiliation. The factors are named by looking at the items that are most highly correlated with the underlying factor (i.e., those items with the highest **factor loading**) and trying to intuit the general trait that those items represent. By using factor analysis, researchers lose a little accuracy but gain a great deal of parsimony.
Factor analysis, technically, is an attempt to recover the correlation matrix of each variable with all other variables. One starts with a **data matrix** with a score for each participant on each variable. One then finds the **correlation matrix**, or correlation of each variable with every other variable. By submitting the correlation matrix to a standard computer program, one finds the **factor matrix**, which is made up of a correlation of each variable with each factor. By turning this factor matrix on its side one derives the factor matrix transpose, and by multiplying the factor matrix by the factor matrix transpose, one arrives at an **approximation matrix** comprised of the correlation of each variable with every other variable. The approximation matrix is approximately equal to the correlation matrix. The object of factor analysis is to recover the correlation matrix using a few factors. While some information is lost in the process, hopefully not very much information is lost.

Once one obtains a factor matrix (comprised of correlations of items with factors), it is possible to **rotate** the factor matrix. Although there are many methods of rotation--actually, infinitely many--the method most favored by psychologists is rotation to simple structure. **Simple structure** means that each item has a high loading on (or correlation with) one and only one factor. Simple structure is intended to improve the interpretability of factors by including in each factor only items that are good measures of that factor.

Factor analysis is the basis for such **trait approaches** as the five-factor model and **general intelligence**. Indeed, factor analysis is probably the most widely used **method** in personality psychology.