|A statistical procedure for transforming (observations by variables) a data matrix so that the variables in the new matrix are uncorrelated. Unlike principal components analysis, which has a similar goal, factor analysis does not identify as many new variables (termed factors) as there are in the original matrix because it ignores that portion of the variance in each of the original variables which is unique to it â€” i.e. is uncorrelated with any other variable.
The first stage in a factor analysis involves creating a matrix of similarities between the variables in the original data matrix, usually employing correlation coefficients. It then identifies and eliminates the unique variance (that part of each variable which is uncorrelated with any other), and subsequently follows the same general sequence of procedures as in principal components analysis, with the successive extraction of factors that maximize the common variance accounted for. The results â€” the eigenvalues and the matrices of factor loadings and factor scores â€” are interpreted in the same way as the comparable matrices of component loadings and scores.
Factor analysis concentrates on identifying the commonalities in the interrelationships among variables. It can be used either inductively (as in exploratory data analysis), to separate groups of variables with common relative distributions across the observations, or deductively, to test hypotheses regarding the existence of such groups. Few geographical applications have rigorously followed the second route, because the available theory gives only very general expectations concerning the loadings.
To facilitate the inductive search for groups of related variables (as in applications which fall under the general term factorial ecology), the factor loading matrix may be rotated mathematically to maximize the relationship of each of the original variables to just one factor. Of the many rotation procedures available in computer statistical packages, most fall into one of two types: (1) orthogonal rotations (of which the most popular is Varimax), which maintain the uncorrelated nature of the factors; and (2) oblique rotations, which allow correlations among the factors.Â (RJJ)
Suggested Reading Johnston, R.J. 1978: Multivariate statistical analysis in geography: a primer on the general linear model. London and New York: Longman.