In the year 1984 Thompson demonstrated how the un-rotated pattern/structure actually misrepresents the true nature of the factors & how factor rotation resolves this misrepresentation. Interpretation of the factor analytical results is therefore almost always aided by factor rotation, as it’s possible to redistribute the common variance across the factors to achieve a more parsimonious solution. After the factor solution is rotated, the first un-rotated factor may not account for the largest portion of the variance & thus may not have the largest variance accounted for the value. Since the variance has been redistributed through-out the factors, any of the factors could account for the largest portion of the total variance.
Five principles of Factor Rotation explained by Gorsuch in the year 1983:
a>
Each variable should have at-least one 0 loadings
b> Each variable should have a set of linearity independent variables whose factor loadings are 0.
c> For every pair of factors, there should be several variables whose loadings are 0 for one factor but not for other
d> For every pair of factors, a large portion of variables should have loading 0 on both factors whenever more than about four factors are extracted
e> For every pair of factors, there should only be a small no. of variables with non-zero loadings on both.
Thus, factor rotation is devised to shift the factors in their factor space so that each variable in the analysis has a large factor pattern coefficient on only one factor & has very small or 0 factor pattern coefficients on the other extracted latent constructs.
Types of Factor Rotation
Orthogonal Factor Rotation: It shifts the factor in the factor space maintaining 90degree angles of the factors to one another to achieve the best simple structure. Since the cosine of the angles between vectors of unit length equals r, & the cosine of a 90 degree angle is 0, this rotation strategy maintains the perfectly uncorrelated nature of the factors after the solution is rotated. Less sampling error occurs at this place due to less capitalization on chance that would occur if more parameters were estimated, as is the case in oblique rotation.
Varimax Rotation: Of the Orthogonal Rotation Technique, it is one of the most popular rotation techniques. In this technique the factors are cleaned up so that every observed variable has a large factor pattern/structure coefficients for a small no. of variables & near-zero or very low pattern coefficients with the other group of variables.
Quartimax Rotation: Another popular orthogonal Rotation Technique, in this technique the factor pattern of a variable is simplified by forcing the variable to correlate highly with one main factor the so called G-factor & very little with other factors. The variables are much easy to interpret in this case, but factors are more difficult to interpret since all variables are primarily associated with one factor.
Oblique Factor Rotation: The second type of factor rotation is Oblique Rotation. This method of rotation provides for correlations among the latent constructs. This is termed as oblique because the angles between the factors become more than 90 degree.
Direct Oblimin: One of the popular Oblique Rotation techniques. This is moderated by a delta value, in which higher value of delta represents higher correlations between factors & negative value represents lesser correlations between the factors. This technique more closely honors the nature of reality & demands careful consideration by the researcher, as the correlation between factors must be set prior to analysis.
Promax : Another most popular techniques of Oblique Rotation. In this technique researchers attempt to achieve the most parsimonious simple structure given that the factors are allowed to be correlated with one another.
It has three distinct steps :
It has three distinct steps :
a> - Rotate the factor orthogonally
b> -Target matrix is contrived by raising the factor pattern coefficients to an exponent greater than 2(Typically exponent 3 or 4 are used). The coefficients in the target matrix become smaller, but the absolute distance between them actually increases.
c> -The final step is Promax Rotation involves the “Procrustean” rotation of the original matrix to a best fit position with the target matrix. Promax is often the oblique rotation strategy of choice, as it’s relatively easy to use, typically provides good solutions & tends to produce more replicable results than the direct oblimin rotations
Reference:
“Orthogonal versus Oblique Rotation: A Review of the literature regarding the pros and cons”
by Kieffer, Kevin M published on 11/04/98
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