13.2 Exploratory Factor Analysis (EFA)
Exploratory Factor Analysis (EFA) is a type of factor analysis that determines how many latent variables underlie a group of items to explain S (i.e., the covariance matrix) or P (i.e., the correlation matrix). The key feature of EFA is that it explores the underlying structure of a set of measures. For this reason, EFA is considered a key technique when we need to find evidence of internal validity (e.g., the internal structure of a psychological test) (Figure 13.3).
Figure 13.3: Exploratory and Confirmatory Factor Analysis models.
Some psychological tests and scales are classified as ability or performance tests because their goal is to reliably measure one domain of knowledge to assess the ability of respondents (e.g., an exam to evaluate students' understanding of psychometric principles and analytical techniques). Ability tests assume unidimensionality because we are only interested in measuring one underlying factor (e.g., the student's knowledge of psychometrics). However, if we run an EFA and we discover that our ability test has more than one factor (i.e., it is not unidimensional), we are probably measuring other things (e.g., language comprehension) that will have a negative impact on the ability scores that we intend to measure reliably. Although a key feature of EFA is that it reveals the internal structure of a test, it also identifies which items perform better or worse. We can exclude items that represent a poor fit to the internal structure of the test or that simply are not properly related to the underlying latent variables.
Confirmatory Factor Analysis (CFA)
EFA explores the latent variables of observed measures, whereas CFA confirms or tests the fit of the observed measures to a particular model or theory (Figure 13.3). The distinctive features of CFA are:
- CFA assumes the existence of a certain number of common factors and which indicators measure each of these factors, either guided by a theory or by previous "agnostic" factor analyses (i.e., EFA)
- In contrast to EFA, the relationship or the absence of relationship between factors is specified a priori in CFA
- In CFA, we can correlate unique variances (i.e., items' measurement error)
- In EFA, we rotate the factors (i.e., orthogonal or oblique rotations) to find the simplest structure, whereas in CFA there is no reason to find the simplest structure as we already know (assume) the optimal factorial structure of the scale
13.2.1 The basic EFA model
The basic EFA model can be written as differential scores or using matrix notation. In the first case, we must model a group of p observed variables related to a set of q factors (i.e., latent variables) that explain them plus the error term.
\[\begin{aligned} Y_{1} = \lambda_{11}f_{1} \ + \lambda_{12}f_{2} \ + \ ...\ + \lambda_{1q}f_{q} \ + \epsilon_{1}\\ Y_{2} = \lambda_{21}f_{1} \ + \lambda_{22}f_{2} \ + \ ...\ + \lambda_{2q}f_{q} \ + \epsilon_{2}\\ \vdots\\ Y_{p} = \lambda_{p1}f_{1} \ + \lambda_{p2}f_{2} \ + \ ...\ + \lambda_{pq}f_{q} \ + \epsilon_{p}\\ \end{aligned}\]
When using matrix notation, we find four components. Y is the vector with the p observed variables. \(\Lambda\) is the \(p\times q\) matrix that includes the factor loadings (\(\lambda_{pq}\)). f is the vector of factors. Last, \(\epsilon\) is the vector including the error term for each observed variable.
\[Y = \Lambda \ f \ + e\]
The matrix including the factor loadings (\(\Lambda\)) is key to understand the outputs produced by all statistical packages computing factor analysis. Factor loadings (\(\lambda_{pq}\)) are the regression coefficients that show the relative contribution that an observed variable makes to a latent variable or factor. Put differently, factor loadings show us the degree of relationship between each observed variable and the latent variables.
\[\boldsymbol{\Lambda} = \begin{pmatrix} \lambda_{11} & \lambda_{12} & \cdots & \lambda_{1q} \\ \lambda_{21} & \lambda_{22} & \cdots & \lambda_{2q} \\ \vdots & \vdots & \ddots & \vdots \\ \lambda_{p1} & \lambda_{p2} & \cdots & \lambda_{pq} \end{pmatrix}\]
13.2.2 Steps to conduct an Exploratory Factor Analysis (EFA)
In the following sections, we will cover the steps that we need to follow to conduct an Exploratory Factor Analysis on a 12-item scale measuring job burnout (Table 13.2). The file containing the data set is named burnout.sav.
| Item | Statement |
|---|---|
| 1 | I feel frustrated by my job |
| 2 | I feel emotionally exhausted with my job |
| 3 | At the end of the day, I feel rewarded by my employer |
| 4 | I do not see prospects of promotion no matter how hard I work |
| 5 | I do not feel valued by my boss |
| 6 | I feel burned out from my job |
| 7 | I do not feel respected by my peers |
| 8 | I do not feel fatigued on Monday mornings |
| 9 | I never get rewarded when I accomplish my goals |
| 10 | Going to work makes me feel stressed |
| 11 | Working hard does not pay off |
| 12 | I have become more callous toward my peers since I got this job |
| Note. Items 3 and 8 are reverse worded. |
There are five steps that we need to follow to conduct an EFA:
- To tidy the data.
- To generate data visualizations.
- To extract the appropriate number of latent variables or factors.
- To rotate the factors to find the simplest factorial structure.
- To interpret the factorial structure.