13.1 Data reduction techniques

Both Principal Components Analysis (PCA) and Factor Analysis (FA) are data reduction techniques. There is an on-going debate on how to use each method and in which situations (Fabrigar et al., 1999; Velicer & Jackson, 1990). However, there are important differences between these two analytical techniques that should be noted.

13.1.1 Principal Components Analysis (PCA)

Principal Components Analysis (PCA) is a method to reduce quantitative and correlated data. It is mainly used to transform a group of correlated variables into a new group of independent variables. It is also used to find linear combinations of the observed variables to generate components that will explain the maximum variability of these original variables. Principal Components Analysis does not need to assume normality.

PCA's key feature is that the correlated variables form the components (Figure 13.1). For this reason, these components explain all the variance of the original variables (the sum of the principal components' variances is equal to the sum of the original variables' variances). Consequently, PCA does not estimate the measurement error of the original variables.

Figure 13.1: Reflective and formative indicator models.

13.1.2 Factor Analysis (FA)

Factor Analysis (FA) is a very popular multivariate technique in the fields of personality, intelligence, social psychology, individual differences, behavioral economics, or even marketing. Although FA is a method that reduces data as PCA does, its popularity lies in its ability to reveal the internal structure (i.e., dimensionality) of tests. In sharp contrast to PCA, FA explains the common or shared variance of the items (i.e., the commonality) (Figure 13.1). Moreover, in PCA we do not estimate the error of the original variables, whereas in FA the variance is decomposed into shared variance (i.e., the variance that the items share due to a common factor) and the unique variance (i.e., the item's error).

FA allows us to reduce a set of observed variables or measures (e.g., items in a psychological well-being inventory) to a small set of latent variables (e.g., autonomy, personal growth, positive relationships with others) that are responsible of the behaviors or observed responses provided to those items.

In Figure 13.2, the items of the psychological well-being inventory reflect (i.e., are caused by) the six latent variables (first order factors) found after computing an Exploratory Factor Analysis (EFA) or when imposing some theoretical or empirical model to validate it (Confirmatory Factor Analysis or CFA).

Figure 13.2: Factorial structure of psychological well-being (Ryff & Keyes, 1995).