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Types of Factoring



There are different methods of extracting the factors from a set of data. The method chosen will matter more to the extent that the sample is small, the variables are few, and/or the communality estimates of the variables differ.





• Principal components analysis (PCA): By far the most common form of factor analysis, PCA seeks a linear combination of variables such that the maximum variance is extracted from the variables. It then removes this variance and seeks a second linear combination which explains the maximum proportion of the remaining variance, and so on. This is called the principal axis method and results in orthogonal (uncorrelated) factors. PCA analyzes total (common and unique) variance.
  

  
  

  • SPSS procedure: Select Analyze - Data Reduction - Factor - Variables (input variables) - Descriptives - Under Correlation Matrix, check KMO and Anti-image to get overall and individual KMO statistics - Extraction - Method (principal components) and Analyze (correlation matrix) and Display (Scree Plot) and Extract (eigenvalues over 1.0) - Continue - Rotation - under Method, choose Varimax - Continue - Scores - Save as variables - Continue - OK.
    

    
    
    

  • Canonical factor analysis , also called Rao's canonical factoring, is a different method of computing the same model as PCA, which uses the principal axis method. CFA seeks factors which have the highest canonical correlation with the observed variables. CFA is unaffected by arbitrary rescaling of the data.
    

    
    

  • Common factor analysis, also called principal factor analysis (PFA) or principal axis factoring (PAF), Common factor analysis is a form of factor analysis which seeks the least number of factors which can account for the common variance (correlation) of a set of variables, whereas the more common principal components analysis (PCA) in its full form seeks the set of factors which can account for all the common and unique (specific plus error) variance in a set of variables. PFA uses a PCA strategy but applies it to a correlation matrix in which the diagonal elements are not 1's, as in PCA, but iteratively-derived estimates of the communalities (R² of a variable using all factors as predictors; see below).
    

    
    

    • Common factor analysis and SEM: Common factor analysis is preferred for purposes of modeling, as in structural equation modeling (SEM). Common factor analysius accounts for the covariation among variables, whereas PCA accounts for the total variance of variables. Because of this difference, in theory it is possible under common factor analysis but not under PCA to add variables to a model without affecting the factor loadings of the original variables in the model. Widaman (1993) notes, "principal component analysis should not be used if a researcher wishes to obtain parameters reflecting latent constructs or factors." However, when commonalities are similar under common factor analysis and PCA, then similar results will follow.
      

      
      

    • PCA vs. common factor analysis. For most datasets, PCA and common factor analysis will lead to similar substantive conclusions (Wilkinson, Blank, and Gruber, 1996). PCA is generally preferred for purposes of data reduction (translating variable space into optimal factor space), while common factor analysis is generally preferred when the research purpose is detecting data structure or causal modeling.

The selection of one technique over the other is based upon several criteria. First of
all, what is the objective of the analysis? Common factor analysis and principal component
analysis are similar in the sense that the purpose of both is to reduce the original
variables into fewer composite variables, called factors or principal components.
However, they are distinct in the sense that the obtained composite variables serve
different purposes. In common factor analysis, a small number of factors are extracted to
account for the intercorrelations among the observed variables--to identify the latent
dimensions that explain why the variables are correlated with each other. In principal
component analysis, the objective is to account for the maximum portion of the variance
present in the original set of variables with a minimum number of composite variables
called principal components.

Secondly, what are the assumptions about the variance in the original variables? If the
observed variables are measured relatively error free, (for example, age, years of
education, or number of family members), or if it is assumed that the error and specific
variance represent a small portion of the total variance in the original set of the
variables, then principal component analysis is appropriate. But if the observed variables
are only indicators of the latent constructs to be measured (such as test scores or
responses to attitude scales), or if the error (unique) variance represents a significant
portion of the total variance, then the appropriate technique to select is common factor
analysis. Since the two methods often yield similar results, only CFA will be illustrated
here.