Factor Analysis

Factor analysis is a statistical method that reduces a large number of observed variables to a smaller set of underlying factors or latent constructs by identifying patterns of correlation among the variables. It assumes that observed variables that correlate with each other do so because they share a common underlying cause (a factor) that is not directly measured.

Researchers frequently collect data on many variables, but not all of these variables are independent. Factor analysis helps researchers understand the underlying structure of their data by identifying which variables cluster together and what latent constructs might explain those clusters. In psychology, factor analysis has been foundational in identifying personality dimensions (such as the Big Five), intelligence factors, and symptom clusters. In health research, it is used to validate questionnaires by confirming that items intended to measure the same construct actually load on the same factor. Without factor analysis, researchers might treat individual items as independent measures when they are actually indicators of a smaller number of underlying constructs, leading to redundant analyses and many separate statistical tests of what is largely one construct.

Factor analysis comes in two main forms. Exploratory factor analysis (EFA) is used when the researcher does not have a predetermined theory about the factor structure and wants to discover how variables cluster together. The procedure begins with a correlation matrix of all variables. An extraction method (such as principal axis factoring or maximum likelihood) identifies factors that account for as much as possible of the variance the variables share (their correlations), rather than their total variance. A rotation method (orthogonal rotations like varimax, or oblique rotations like promax) is then applied to produce a more interpretable factor solution. The researcher examines the rotated factor loadings (after an orthogonal rotation, the correlations between each variable and each factor; after an oblique rotation, the pattern coefficients, which work like regression weights because the factors are allowed to correlate) to identify which variables belong to which factor and to name the factors.

Confirmatory factor analysis (CFA) is used when the researcher has a specific hypothesis about the factor structure (for example, based on theory or prior EFA results) and wants to test whether the data fit this hypothesized model. CFA is a form of structural equation modeling and provides fit statistics (such as chi square, RMSEA, CFI, and TLI) that indicate how well the proposed model matches the observed data. Commonly cited cutoffs, such as CFI and TLI of about 0.95 or higher and RMSEA of about 0.06 or lower (Hu and Bentler, 1999), are guidelines rather than strict pass or fail rules.

Key decisions in factor analysis include how many factors to retain (informed by eigenvalues, scree plots, parallel analysis, and interpretability), which extraction and rotation methods to use, and what constitutes a meaningful factor loading (typically above 0.30 or 0.40). Sample size recommendations vary, but a common guideline is at least 5 to 10 observations per variable, with a minimum of 200 to 300 participants.

Factor analysis is appropriate when developing or validating questionnaires and measurement instruments, when exploring the underlying structure of a dataset with many variables, when reducing data dimensionality for subsequent analyses, or when testing theoretical models about how constructs relate to observed measures. EFA is used in the early stages of instrument development, while CFA is used to confirm the factor structure in a new sample or test competing theoretical models.

A common error is using EFA and CFA on the same dataset, which capitalizes on chance. Best practice is to conduct EFA on one sample and CFA on a separate sample (split half or independent). Retaining too many or too few factors is another frequent mistake; researchers should use multiple criteria to determine the number of factors, because the rule of keeping every factor with an eigenvalue greater than 1 (the Kaiser criterion), used alone, tends to retain too many, while parallel analysis is generally more accurate. Ignoring cross loadings (variables that load substantially on more than one factor) can lead to poorly defined factors. Using factor analysis with too small a sample produces unstable solutions that will not replicate. Finally, some researchers confuse factor analysis with principal component analysis (PCA); although they are computationally similar, PCA reduces variables to components that maximize total variance, while factor analysis models the shared variance among variables.

A research team develops a 30 item questionnaire measuring attitudes toward telemedicine. An initial EFA with 350 respondents reveals four factors: perceived usefulness (8 items, loadings 0.52 to 0.81), ease of use (7 items, loadings 0.48 to 0.76), trust in technology (6 items, loadings 0.55 to 0.73), and perceived privacy (5 items, loadings 0.44 to 0.69). The other four items are removed due to low loadings or cross loadings, leaving 26 items. A subsequent CFA with a new sample of 400 respondents supports the four factor structure with acceptable but not excellent fit (RMSEA = 0.048, CFI = 0.94, TLI = 0.93): RMSEA is within the commonly cited 0.06 cutoff, while CFI and TLI fall just below 0.95.