Exploratory Factor Analysis (EFA) vs Confirmatory Factor Analysis (CFA)
Exploratory Factor Analysis (EFA) vs Confirmatory Factor Analysis (CFA)
Factor analysis is a commonly used method of modelling the variability of observed variables as a function of a set of unobserved entities. Such entities, usually called constructs or latent factors, are not directly observable, and factor analysis helps us identify them, understand their nature, and assess their relationship with observed variables, or indicators. There are two prominent classes of factor analysis methods: exploratory factor analysis (EFA) and confirmatory factor analysis (CFA). While both of these methods model the observed covariance among variables through latent predictors, there are significant differences between the two approaches. This article reviews the key ideas behind EFA and CFA, explores their strengths and weaknesses, and investigates when researchers should use one over the other.
Common Factor Model
Both EFA and CFA are rooted in the same framework, namely the common factor model, and as such it is valuable to explore this model first. The common factor model is the dominant paradigm for factor analysis, and the main idea behind it is expressing observed variables as a simple linear combination of common factors: y = Λη + ε.
Observed variables, or indicators, may come from scale items, subscale scores, or direct measures of certain characteristics. Every indicator y is assumed to linearly depend on the common factors η, and our goal is to estimate the coefficients Λ. Each common factor is allowed to influence more than a single observed variable. The regression coefficient corresponding to a factor-indicator pair is called a factor loading. Unique factors ε influence only one observed variable, and it is assumed that the residuals are independent of the explanatory variables. While unique factors are uncorrelated with the common factors, it is possible for them to be correlated with each other. Estimating the model refers to estimating the elements of the factor loading matrix Λ. Once we have the values of factor loadings, we are able to express each indicator y as a linear combination of the factors.
Path diagram for a common factor model with one factor F1. From https://cran.r-project.org/web/packages/OpenMx/vignettes/factor_analysis.html.
EFA and CFA share the goal of expressing the relationships between a set of indicators using a small number of common factors. In other words, EFA and CFA are not disparate approaches as both arise from the common factor model. The key difference between the two models is the treatment of the factor loading matrix Λ. EFA does not put any constrains on factor loadings, and thus all elements of the matrix are freely estimated. Meanwhile, CFA restricts certain factor loadings to zero to ensure that there are no cross-loadings as per the researcher’s hypothesised model. Why does this distinction arise, and when would we want to use one approach over the other? To answer these questions and more, the methods of EFA and CFA are examined in greater detail below to provide more insight on the choice between EFA and CFA.
Exploratory Factor Analysis (EFA)
EFA is a data-driven method for determining the common factors influencing observed variables. As discussed above, EFA is based on the common factor model. The key assumption of the model is that the total variance of each item can be expressed as a sum of common variance, specific variance, and error variance. Common variance refers to the variance of a variable that is shared with other variables, while specific variance corresponds to the variance that does not correlate with other items. Since there is no underlying theory behind the model, all variables are free to load on any factor.
Decomposition of total variance in a common factor model with one factor ‘SPSS Anxiety’. From https://stats.oarc.ucla.edu/spss/seminars/introduction-to-factor-analysis/a-practical-introduction-to-factor-analysis/.
One of the key steps of EFA is determining the number of factors. This choice is data-driven and is dictated by the observed variables as well as the complexity of constructs. How do researchers decide on how many factors to retain? A variety of methods have been proposed which can be roughly categorised as statistical and heuristic. Major statistical methods are Bartlett’s test, comparison data method, and parallel analysis, while examples of heuristic methods include the scree plot as well as retaining factors with eigenvalues greater than one. Overall, there is no consensus on the best method, and researchers continue to compare performance of different approaches.
Since EFA is typically meant to inform theories, it is crucial to express the results in a way that is easy to interpret. One approach for facilitating interpretaion of extracted factors is using rotations. Essentially, a rotation can be understood as a reallocation of factor loadings. If the first factor was constructed to maximise the amount of variance explained, then the corresponding rotated factor may no longer account for the maximum amount of variance. Rotations can be oblique or orthogonal, depending on whether they lead to correlated or uncorrelated factors. Two major examples of rotations are quartimax and varimax. A quartimax rotation is performed so that each variable loads mostly on a single factor, whereas varimax results in each factor loading high on a small number of variables. Overall, the reliance on rotations for interpretation reflects the descriptive nature of EFA, and suggests that there is also some arbitrariness to the choice of the rotation.
A typical output from EFA: retained and rotated factors with highlighted primary loadings. From Flora, D. B., LaBrish, C., & Chalmers, R. P. (2012). Old and new ideas for data screening and assumption testing for exploratory and confirmatory factor analysis. Frontiers in Psychology, 3, 55. https://doi.org/10.3389/fpsyg.2012.00055.
Thus, EFA is mainly used in a descriptive manner which sets it apart from inferential statistical methods. EFA is performed to identify latent constructs and generate hypotheses about structures between constructs, making EFA an exploratory, theory-generating method.
Strengths and weaknesses
If there is insufficient research on the structure of the construct or measures of interest, then it may be ill-advised to restrict the relationships between measures and constructs. This is where EFA can be especially useful as it is not restrictive. Assuming a certain covariance structure without sufficient theoretical basis could result in poor fit of the model, and it might not be obvious how to fix it due to the large number of alternative specifications. Meanwhile, EFA is much easier to adjust since it has fewer parameters with more limited range of values, such as the number of factors or the type of rotation. This is a key strength of the EFA as it offers flexibility and freedom that can be valuable if the goal is to generate, rather than test, theories.
One of the key shortcomings of EFA is that determining the number of retained factors and the rotation method is, to some degree, subjective. For example, blindly using heuristic criteria for retaining components, such as the scree plot, can result in omitting factors that are practically significant but don’t account for a large amount of variance. Similarly, applying an orthogonal rotation to a structure that is actually oblique would cause variables to load on multiple factors. As such, the obtained factor structure might not be the only possible representation of the relations in the data. Variables with similar levels of skewness or kurtosis could lead to artifactual, or ‘difficulty’, factors that may mislead the interpretation of EFA. Another weakness of EFA is that it is not inferential. EFA cannot be used to test measurement invariance or specific hypotheses.
Confirmatory Factor Analysis (CFA)
The purpose of CFA is to evaluate structures of latent constructs that are hypothesised based on theories. CFA can be viewed as a form of structural equation modelling (SEM) as CFA directly corresponds to the measurement model in SEM. Similar to EFA, CFA is rooted in the common factor model and assumes that the relationships between indicators arise from effects of common latent constructs. In contrast to EFA, CFA requires researchers to specify the hypothesised characteristics of factor loadings and relationships between factors. This specification is typically linked to the tested theory and allows researchers to test hypotheses about the relationships between observed data and constructs, as well as the relationships between constructs. Besides assessing how well a specific model fits the data, CFA can also be employed to compare the fit of different models, or to determine which model provides the best fit to the observed data.
CFA models are typically presented in the form of path diagrams.
Path diagram of one-factor CFA with estimated coefficients. From https://stats.oarc.ucla.edu/r/seminars/rcfa/.
Path diagrams summarise the output of CFA by showing the structure of the model, estimated factor loadings, and relationships between indicators and constructs. Since it is possible that an existing theory would not converge on a single model, it can be valuable to specify alternative path diagrams that correspond to other theories. This reflects the main advantage of CFA which is the ability to evaluate competing models against each other using statistical methods. At the same time, CFA may not be not appropriate to use if there are no hypothesised theory-based models.
Strengths and weaknesses
As noted above, the main strength of CFA is that researchers can use it to compare a priori specified models. CFA also makes it possible to carry out systematic tests of measurement invariance which is a major assumption in popular research designs such as randomised control trials. Furthermore, latent CFA factors can be more easily incorporated into subsequent analyses such as SEM. At the same time, CFA requires a more detailed specification of the model compared to EFA which makes it especially important to understand the underlying theoretical frameworks guiding the model. As CFA is much more restrictive than EFA, it is not recommended to perform CFA if only minimal research has been conducted on the structure of latent factors.
Another issue with CFA is that it can be challenging to assess the model’s goodness-of-fit. CFA methodologies agree that model fit should be assessed using several indices and procedures including the chi-square test, incremental measures such as Tucker-Lewis Index (TLI) or Comparative Fit Index (CFI), and absolute indicators such as root mean square error of approximation (RMSEA). A minor technical shortcoming of CFA is that it requires specialised software such as AMOS, EQS, or LISREL, whereas EFA can be performed using conventional statistics software including SAS and SPSS.
EFA or CFA?
The main goal of both EFA and CFA methods is to model relationships among observed variables using a smaller number of unobserved variables. In EFA, the researcher would not have a strong prior theory about how indicators relate to factors or about the number of factors. In CFA, both the number of factors and the nature of relationships between factors and indicators are is hypothesized a priori. This implies that the main criterion of deciding between EFA and CFA is how strong is the theoretical basis for a hypothesised model and relationships. However, there are other considerations that may affect the choice of the procedure including research goals, goodness-of-fit of the CFA model, need for testing specific hypotheses and assumptions, and sensitivity of EFA to parameters. The fact that EFA freely estimates cross-loadings that would be constrained to zero in CFA could be an advantage in cases where CFA fails to support instruments that are well established in EFA research. Strategies that are employed to address this, such as parcelling or ad hoc correlated uniqueness, tend to be counterproductive or misleading.
EFA and CFA can also be used together, but it is crucial to use different samples for the two procedures. It is not appropriate to conduct CFA in order to ‘confirm’ an EFA model using the same data. One way this can be circumvented is by considering disjoint subsamples for each procedure. Another use of EFA in conjunction with CFA is for exploring the underlying structure when the a priori specified model used in CFA does not fit the data.
To sum up, none of the two procedures is strictly better than the other. EFA leans towards theory generation as it is heuristic and allows for a weak literature base. EFA does not restrict factor loadings, with variables being free to load on any factor. However, it requires the researcher to determine the number of factors, and to decide whether the factors should be correlated or uncorrelated. In contrast, CFA is theory-testing in nature. It requires a strong theory or empirical base. The researcher a priori fixes both the number of factors and whether they are correlated or uncorrelated, with variables constrained to load only on specific factors. Ultimately, the choice between CFA and EFA depends on the strength of the theoretical basis of examined constructs and hypothesised relationships.
