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# Factor Analysis - General Thoughts about Orthogonal and Oblique Factor Rotation

## Factor Rotation

Factor Analysis is a multivariate statistical analytical technique with a primary purpose to define the underlying structure among variables in the analysis of a certain set of data. Factors represent the underlying dimensions that summarize or account for the original set of observed variables. One technique employed to find which variables line up with each other is called factor rotation. Factor Rotation refers to the rotation of the axes on an xy-graph in order to align related components or dimensions of a research study.

**Factor Rotation**

Thurstone argued that in factor analysis it is necessary to rotate factor matrices if wanted to interpret them adequately (Kerlinger & Lee, 2004). He showed that original factor matrices were arbitrary with an infinite number of frames of reference. Moreover, Kerlinger and Lee explained that "principal factor matrices do not in general provide scentifically meaningful structures and it is the configurations of tests or variables in factor space that are of fundamental concern" (p.839). In order to make better sense of the factors in factor space, Kerlinger and Lee further noted that Thurstone posited guidelines for more meaningful factor solutions through the ideas of simple structure and factor axes rotation. According to Hair et al. factor rotation became the most important tool in interpreting factors in the factor analysis method of data analysis.

**Orthogonal and Oblique Oblimin Factor Rotation**

Since Thurstone's invention, two general types of factor rotation have emerged including orthogonal and oblique factor rotation (or as in SPSS software, oblimin rotation). Orthogonal factor rotation is the simplest and most commonly used form of rotation in which the axes are rotated from the point of origin maintaining a 90^{o} angle to each other. The second factor rotation procedure is called oblique and is not constrained to the 90^{o} . Hair et al. explain, "oblique rotations are similar to orthogonal rotations except oblique rotations allow correlated factors instead of maintaining independence between the rotated factors" (p. 127). Furthermore Hair et al. note that "oblique rotation methods are best suited to the goal of obtaining several theoretically meaningful factors or constructs because, realistically, few constructs in the real world are uncorrelated." One caution is that oblique or oblimin rotated factored need extra care in validation because they have a tendency to be specifc to the sample and not generalizable.

Rummel explains the following about oblique factor analysis:

"Oblique rotation takes place in one of two coordinate systems: either a system of *primary* axes or a system of *reference* axes. The reference axes give a slightly better definition of the clusters of interrelated variables than do the primary ones. For each set of axes there are two possible matrices: factor *structure* and factor *pattern* matrices."

**Oblique Rotation: Factor Pattern Matrix and Factor Structured Matrix**

The *primary factor pattern* matrix and the *reference factor structure* matrix delineate the oblique patterns or clusters of interrelationship among the variables. Their loadings define the separate patterns and degree of involvement in the patterns for each variable. Unlike the unrotated or the orthogonally rotated factors, however, their loadings cannot be strictly interpreted as the correlation of a variable with a pattern, and the squared loadings do not precisely give the percent of variation of a variable involved in a pattern.The *primary factor structure* matrix and the *reference factor pattern* matrix give the correlation of each variable with each pattern. The loadings are strictly interpretable as correlations.

Hair et al. (2006). *Multivariate Data Analysis, 6e.* Upper Saddle, NJ: Pearson-Preston Hill.

Kerlinger & Lee (2004). *Foundations of Behavioral Research, 4e*. Belmont, CA: Cengage Learning.

Rummel, R. J. (2002). Understanding Factor Analysis.http://www.hawaii.edu/powerkills/UFA.HTM

Thurstone, L. L. & Thurstone, T. (1941). *Factor Studies of Intelligence*. Chicago, Ill: University of Chicago Press.

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