Choosing the Right Type of Rotation in PCA and EFA

What Is Rotation?

In the PCA/EFA literature, definitions of rotation abound. For example, McDonald (1985, p. 40) 

defines rotation as “performing arithmetic to obtain a new set of factor loadings (v-ƒ regression 

weights) from a given set,” and Bryant and Yarnold (1995, p. 132) define it as “a procedure in which 

the eigenvectors (factors) are rotated in an attempt to achieve simple structure.” Perhaps a bit more 

helpful is the definition supplied in Vogt (1993, p. 91): “Any of several methods in factor analysis by 

which the researcher attempts to relate the calculated factors to theoretical entities. This is done 

differently depending upon whether the factors are believed to be correlated (oblique) or uncorrelated 

(orthogonal).” And even more helpful is Yaremko, Harari, Harrison, and Lynn (1986), who define 

factor rotation as follows: “In factor or principal-components analysis, rotation of the factor axes 

(dimensions) identified in the initial extraction of factors, in order to obtain simple and interpretable 

factors.” They then go on to explain and list some of the types of orthogonal and oblique procedures. 

 How can a concept with a goal of simplification be so complicated? Let me try defining rotation

from the perspective of a language researcher, while trying to keep it simple. I think of rotation as any 

of a variety of methods (explained below) used to further analyze initial PCA or EFA results with the 

goal of making the pattern of loadings clearer, or more pronounced. This process is designed to reveal 

the simple structure. 

The choices that researchers make among the orthogonal and oblique varieties of these rotation 

methods and the notion of simple structure will be the main topics in the rest of this column. 21

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What Are the Different Types of Rotation?

As mentioned earlier, rotation methods are either orthogonal or oblique. Simply put, orthogonal 

rotation methods assume that the factors in the analysis are uncorrelated. Gorsuch (1983, pp. 203-204) 

lists four different orthogonal methods: equamax, orthomax, quartimax, and varimax. In contrast, 

oblique rotation methods assume that the factors are correlated. Gorsuch (1983, pp. 203-204) lists 15 

different oblique methods.1

Version 16 of SPSS offers five rotation methods: varimax, direct oblimin, quartimax, equamax, and 

promax, in that order. Three of those are orthogonal (varimax, quartimax, & equimax), and two are 

oblique (direct oblimin & promax). Factor analysis is not the focus of my life, nor am I eager to learn 

how to use a new statistical program or calculate rotations by hand (though I’m sure I could do it if I 

had a couple of spare weeks), so those five SPSS options serve as boundaries for the choices I make. 

But how should I choose which one to use? 

Tabachnick and Fiddell (2007, p. 646) argue that “Perhaps the best way to decide between 

orthogonal and oblique rotation is to request oblique rotation [e.g., direct oblimin or promax from 

SPSS] with the desired number of factors [see Brown, 2009b] and look at the correlations among 

factors…if factor correlations are not driven by the data, the solution remains nearly orthogonal. Look 

at the factor correlation matrix for correlations around .32 and above. If correlations exceed .32, then 

there is 10% (or more) overlap in variance among factors, enough variance to warrant oblique rotation 

unless there are compelling reasons for orthogonal rotation.” 

 For example, using the same Brazilian data I used for examples in Brown 2009a and b (based on 

the 12 subtests of the Y/G Personality Inventory from Guilford & Yatabe, 1957), I ran a three- factor 

EFA followed by a direct oblimin rotation. The resulting correlation matrix for the factors that the 

analysis produced is shown in Table 1. Notice that the highest correlation is .084. Since none of the 

correlations exceeds the Tabachnick and Fiddell threshold of .32 described in the previous paragraph, 

“the solution remains nearly orthogonal.” Thus, I could just as well run an orthogonal rotation. 

Table 1. Correlation Matrix for the Three Factors in an EFA with Direct Oblimin Rotation for the Brazilian Y/GPI Data

Factor 1 2 3

1 1.000 -0.082 0.084

2 -0.082 1.000 -0.001

3 0.084 -0.001 1.000

Moreover, as Kim and Mueller (1978, p. 50) put it, “Even the issue of whether factors are 

correlated or not may not make much difference in the exploratory stages of analysis. It even can be 

argued that employing a method of orthogonal rotation (or maintaining the arbitrary imposition that the 

factors remain orthogonal) may be preferred over oblique rotation, if for no other reason than that the 

former is much simpler to understand and interpret.” 

How Do Researchers Decide Which Particular Type of Rotation to Use?

We can think of the goal of rotation and of choosing a particular type of rotation as seeking 

something called simple structure, or put another way, one way we know if we have selected an 

adequate rotation method is if the results achieve simple structure. But what is simple structure? 

Bryant and Yarnold (1995, p. 132-133) define simple structure as:

A condition in which variables load at near 1 (in absolute value) or at near 0 on an eigenvector (factor). Variables 

that load near 1 are clearly important in the interpretation of the factor, and variables that load near 0 are clearly 

unimportant. Simple structure thus simplifies the task of interpreting the factors.

Using logic like that in the preceding quote, Thurstone (1947) first proposed and argued for five 

criteria that needed to be met for simple structure to be achieved: 

1. Each variable should produce at least one zero loading on some factor. 

2. Each factor should have at least as many zero loadings as there are factors.

3. Each pair of factors should have variables with significant loadings on one 

and zero loadings on the other. 

4. Each pair of factors should have a large proportion of zero loadings on both factors 

(if there are say four or more factors total).

5. Each pair of factors should have only a few complex variables