Factor Analysis

Exploratory Factor Analysis

EFA is a method used to detect the optimal number of factors that accounts for the correlation of [Page 882]items. Each factor is interpreted and named based on the items that have high loadings on this factor. Its character is inductive as the number of factors, the amount of correlations between factors, and the assignments between items and factors are performed empirically without a precise deductive theoretical model. There is no unique solution for the relation between observed items and latent factors. There are two different approaches to the factor model—(a) common-factor model and (b) principal component analysis—and each implies different substantive assumptions. In the common-factor model, it is assumed that every item is measured with some error. Therefore, the common-factor model is most appropriate in survey research, where items are always measured with some error. However, in the principal component model, the researcher assumes that there is no measurement error in the items. All analyses are based on correlation matrices as input.

EFA is performed in five steps:

• 1.
  Factor extraction: The method that is most often used is the method of principal axis factoring and is based on the computation of the eigenvalues and vectors of the correlation matrix between all measured variables. Its goal is the maximization of the variance of each successively extracted factor.
• 2.
  Number of factors: Different procedures determine the number of factors: (a) According to the Kaiser-Guttman criterion, the number of factors should correspond to the number of eigenvalues of the full input correlation matrix that exceed one. (b) The scree plot is a graph of each eigenvalue plotted in descending order. The number of factors is determined by visual inspection by observing whether, suddenly, the plot shows no real difference between the last two eigenvalues in the scree plot. (c) In parallel analysis, a second factor analysis is calculated with a random data set with the same numbers of variables and cases as in the original analysis. Only the factors with higher eigenvalues than the factors of the random data are accepted. (d) In maximum likelihood estimation, one studies the amount of residuals left over after introducing more factors and evaluates them by goodness-of-fit tests of the whole model.
• 3.
  Communality estimation as a measure for the common variance of every item: The most often used method for an initial estimate is the squared multiple correlation of one item with all other items. Depending on the number of factors extracted, the final solution contains the explained variance of every item.
• 4.
  Factor rotation: The goal of rotation is to achieve “a simple structure” that allows a good interpretation of the resulting solution. Rotations can be orthogonal, assuming that there are no correlations between factors (e.g., varimax, quartimax, and equimax), or oblique (e.g., promax, oblimin), assuming that there exist correlations between factors. In the case of correlated factors, the factor loadings of items on factors contained in the pattern factor matrix represent standardized partialized regression coefficients. In the case of noncorrelated factors, these coefficients are simply correlation coefficients between items and factors. The factor structure matrix contains the simple bivariate correlations between items and constructs. The values differ from the factor pattern matrix only if factors are correlated.
• 5.
  Factor score estimation: As a final step, one may be interested in estimating the factor scores. The factor scores are often used as weights for raw scores in the computation of indices that are used in subsequent analyses.

Confirmatory Factor Analysis

In contrast to EFA, in CFA, a theoretical model is needed that contains a series of a priori hypotheses. These hypotheses should be drawn from the substantive literature. For example, in the theory of values of Shalom Schwartz, 10 values are postulated and theoretically derived. Therefore, the 10 theoretical constructs can be formalized and tested as latent variables in a CFA model. However, it is not definitely determined in the literature which and how many dimensions and items are adequate to measure the concepts of nationalism and patriotism and whether one can differentiate between them. The hypotheses in a CFA model refer to the following aspects of a measurement model: (a) determination of the number of factors, (b) assumption regarding whether the factors are correlated or not, (c) determination of which items load on which factor and where to set loadings a priori to zero, and (d) assumptions about the correlations of the errors. CFA allows the specification and testing of these assumptions. It is possible to fix parameters to a [Page 883]certain value or to constrain parameters that are set equal to other parameters. One assumes that the expected value of random measurement error is zero (εs = 0) and the correlation of the random measurement error with the latent variable (factor) is also zero. Furthermore, it is necessary to fix one of the loadings to one or to standardize the variance of the latent variable to reach identification of the model, that is, to achieve a unique solution.

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