Concept Architecture
Concept
Theoretically, Exploratory Factor Analysis (EFA) is a multivariate statistical method used to identify latent constructs that explain the correlation structure among a set of observed variables. It is founded on common factor theory, which assumes that observed variables share underlying factors responsible for their covariance. EFA exists to reduce dimensionality, identify underlying domains and support the development and validation of measurement instruments.
Mathematically, EFA represents each observed variable as a linear combination of one or more common factors together with a unique error component. The factor loading matrix quantifies the relationship between observed variables and latent factors, while communalities represent the proportion of variance explained by the common factors. Parameters are estimated using methods such as maximum likelihood or principal axis factoring, followed by factor rotation to improve interpretability.
In practice, EFA is implemented by constructing a correlation matrix, selecting an extraction method, determining the appropriate number of factors using statistical criteria, estimating factor loadings and applying orthogonal or oblique rotation. It is widely used in health economics to develop patient-reported outcome measures, quality-of-life instruments and preference elicitation questionnaires.
Purpose
Used to identify latent constructs, reduce the dimensionality of correlated variables, evaluate questionnaire structure and support the development and validation of health economic measurement instruments.
Mathematical Formulae
Primary Formula
x = ?f + �
where:
- x = vector of observed variables
- ? = factor loading matrix
- f = vector of common factors
- � = vector of unique errors
Supporting Formulae
� = ?�?? + ?
h�? = ????�
where:
- � = covariance matrix
- � = covariance matrix of common factors
- ? = diagonal matrix of unique variances
- h�? = communality of variable i
- ??? = loading of variable i on factor j
Related Mathematical Methods
- Maximum Likelihood Estimation
- Principal Axis Factoring
- Eigenvalue Decomposition
- Varimax Rotation
- Promax Rotation
- Oblimin Rotation
- Parallel Analysis
- Scree Plot Analysis
Example
A health economist develops a 20-item quality-of-life questionnaire for patients with chronic heart failure. Exploratory factor analysis identifies three latent factors representing physical functioning, emotional wellbeing and social participation. One questionnaire item has loadings of 0.82, 0.14 and 0.09 across the three factors, indicating that it primarily measures physical functioning. Items with low communalities or substantial cross-loadings are revised or removed before subsequent confirmatory analysis.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| CORREL | =CORREL(B2:B101,C2:C101) | Construct correlation matrix prior to factor analysis |
| MMULT | =MMULT(A1:D4,E1:H4) | Matrix multiplication during factor calculations |
| TRANSPOSE | =TRANSPOSE(A1:D4) | Create transpose of loading matrix |
| MINVERSE | =MINVERSE(A1:D4) | Matrix inversion for estimation procedures |
| MDETERM | =MDETERM(A1:D4) | Matrix determinant in covariance calculations |
VBA (Optional)
Automate exploratory factor analysis workflows by importing questionnaire data, generating correlation matrices, estimating factor solutions and exporting rotated loading tables.
Sources
- Bartholomew DJ, Knott M, Moustaki I. Latent Variable Models and Factor Analysis. 3rd ed.
- Harman HH. Modern Factor Analysis.
- Fabrigar LR, Wegener DT. Exploratory Factor Analysis.
- Hair JF, Black WC, Babin BJ, Anderson RE. Multivariate Data Analysis.
- ISPOR Task Force reports on patient-reported outcome instrument development.
Related Concepts (6)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is exploratory factor analysis?
A technique identifying an underlying set of factors explaining correlations among observed variables, without a predetermined hypothesis, unlike confirmatory factor analysis.
Source: Spearman 1904
What does exploratory factor analysis seek without a prior hypothesis?
Exploratory factor analysis seeks the underlying factors that explain the pattern of correlations among a set of measured variables, without any prior hypothesis about how many there are or which variables belong to which. It lets the structure emerge from the data, revealing how items group together into a smaller number of latent dimensions. This makes it a tool for the early stage of developing a measure, before a specific structure can be proposed and tested. Letting the factor structure surface from the data is its purpose. Kline (2015) describes this technique.
Source: Kline 2015
How does exploratory factor analysis work?
Exploratory factor analysis works by analysing the correlations among the observed variables to extract factors that account for the shared variance, then determining how many factors to retain, often using eigenvalues, and rotating the factors to make them more interpretable, so that each factor is associated with a subset of variables through their loadings. So exploratory factor analysis works by extracting and rotating factors from the correlation structure of the variables, identifying groups of variables that load together on common factors, which are then interpreted as latent dimensions, allowing the underlying structure of the data to be uncovered and described without a prior hypothesis.
Source: Spearman 1904
How does exploratory factor analysis differ from confirmatory factor analysis?
Exploratory factor analysis discovers the factor structure from the data without a predetermined hypothesis, letting the number of factors and their loadings emerge, while confirmatory factor analysis tests a prespecified structure, with the factors and which variables load on them fixed in advance. Exploratory analysis is used when the structure is unknown, confirmatory when it is hypothesised. So the two differ in whether the structure is uncovered or tested, with exploratory factor analysis being hypothesis-generating and data-driven and confirmatory factor analysis hypothesis-testing, and exploratory work often precedes confirmatory analysis, the former suggesting a structure that the latter then formally tests on new data.
Source: Spearman 1904
When is exploratory factor analysis used?
Exploratory factor analysis is used when the underlying structure of a set of variables is unknown and the aim is to discover the latent factors that explain their correlations, for example in developing a questionnaire or exploring the dimensions underlying a set of measures. So exploratory factor analysis is used in the early stages of understanding a set of variables, to reveal how many factors underlie them and how the variables relate to those factors, which is valuable for scale development and for data reduction, and its findings often inform a later confirmatory factor analysis that tests the suggested structure on independent data.
Source: Spearman 1904
What are the key decisions in exploratory factor analysis?
The key decisions in exploratory factor analysis include how many factors to retain, guided by eigenvalues, scree plots, and interpretability; which extraction method to use; and which rotation to apply to aid interpretation, whether one that keeps factors uncorrelated or one that allows them to correlate. These choices affect the results. So exploratory factor analysis involves several judgement-laden decisions about the number of factors, extraction, and rotation, which shape the factor solution, and because different choices can yield different structures, these decisions are made carefully and with attention to interpretability, since the aim is a meaningful and replicable set of factors rather than an arbitrary one.
Source: Spearman 1904
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 15 Dec 2025
Content version: 1.0.0
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- HE-ES-SA-060
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