Factor Analysis - Hierarchical Factor Analysis

Oblique Factors. Some authors (e.g., Cattell & Khanna; Harman, 1976; Jennrich & Sampson, 1966; Clarkson & Jennrich, 1988) have discussed in some detail the concept of oblique (non-orthogonal) factors in order to achieve more interpretable simple structure (refer to the Introductory Overview). Specifically, computer (algorithmic) strategies have been developed to rotate factors so as to best represent "clusters" of variables, without the constraint of orthogonality of factors. However, the oblique factors produced by such rotations are often not easily interpreted. For example, suppose we analyzed the responses to a questionnaire asking about people's satisfaction with various aspects of their lives. Let us assume that the questionnaire contains 3 items designed to measure satisfaction with one's work, 3 items designed to measure satisfaction with one's home life, and 4 items designed to measure overall (miscellaneous) satisfaction. Let us also assume that people's responses to those last 4 items are affected about equally by their satisfaction at home (Factor 1) and at work (Factor 2). In this case, an oblique rotation will likely produce two correlated factors with less-than-obvious meaning, that is, with generally many cross-loadings.

Hierarchical Factor Analysis. Instead of computing loadings for often difficult to interpret oblique factors, the Factor Analysis module in STATISTICA uses a strategy first proposed by Thompson (1951) and Schmid and Leiman (1957), which has been elaborated and popularized in the detailed discussions by Wherry (1959, 1975, 1984). In this strategy, STATISTICA first identifies clusters of items and rotates axes through those clusters; next the correlations between those (oblique) factors are computed, and that correlation matrix of oblique factors is further factor-analyzed to yield a set of orthogonal factors that divide the variability in the items into that due to shared or common variance (secondary factors), and unique variance due to the clusters of similar variables (items) in the analysis (primary factors). To return to the example above, such a hierarchical analysis might yield the following factor loadings:

STATISTICA

FACTOR

ANALYSIS

Secondary & Primary Factor Loadings

 

 

Factor

Second. 1

Primary 1

Primary 2

WORK_1

.483178

.649499

.187074

WORK_2

.570953

.687056

.140627

WORK_3

.565624

.656790

.115461

HOME_1

.535812

.117278

.630076

HOME_2

.615403

.079910

.668880

HOME_3

.586405

.065512

.626730

MISCEL_1

.780488

.466823

.280141

MISCEL_2

.734854

.464779

.238512

MISCEL_3

.776013

.439010

.303672

MISCEL_4

.714183

.455157

.228351

Careful examination of these loadings lead to the following conclusions:

  1. There is a general (secondary) satisfaction factor that likely affects all types of satisfaction measured by the 10 items;

  2. There appear to be two primary unique areas of satisfaction that can best be described as satisfaction with work and satisfaction with home life.

Wherry (1984) discusses examples of such hierarchical analyses in great detail and how meaningful and interpretable secondary factors can be derived.