Concept Architecture
Concept
Theoretically, Factor Analysis is a multivariate statistical method that explains the relationships among a set of observed variables through a smaller number of unobserved latent factors. It is based on common factor theory, which assumes that correlations among measured variables arise because they are influenced by shared underlying constructs. The method exists to identify latent dimensions, reduce data complexity and improve the interpretation of multivariate data.
Mathematically, factor analysis models each observed variable as a linear combination of common factors and a unique error component. The covariance structure of the observed variables is decomposed into variance explained by common factors and variance unique to each variable. Model parameters, including factor loadings, communalities and unique variances, are estimated using recognised estimation methods such as maximum likelihood or principal axis factoring.
In practice, factor analysis is applied by constructing a covariance or correlation matrix, estimating latent factors, determining the appropriate number of factors, evaluating model fit and interpreting the resulting factor structure. In health economics, it is widely used in the development and validation of patient-reported outcome measures, health-related quality-of-life instruments and preference measurement questionnaires.
Purpose
Used to identify latent constructs, explain correlations among observed variables, reduce dimensionality, validate measurement instruments and support multivariate statistical analysis in health economics.
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�? = ????�
u�? = 1 ? h�?
where:
- � = covariance matrix
- � = covariance matrix of common factors
- ? = diagonal matrix of unique variances
- h�? = communality
- u�? = uniqueness
- ??? = factor loading
Related Mathematical Methods
- Exploratory Factor Analysis
- Confirmatory Factor Analysis
- Maximum Likelihood Estimation
- Principal Axis Factoring
- Eigenvalue Decomposition
- Varimax Rotation
- Promax Rotation
- Parallel Analysis
Example
A health economist analyses responses from a 30-item health-related quality-of-life questionnaire completed by 1,200 patients. Factor analysis identifies four latent constructs representing physical health, emotional wellbeing, social functioning and symptom burden. Variables with strong loadings on the same factor are interpreted as measuring the same underlying health domain, supporting questionnaire validation.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| CORREL | =CORREL(B2:B201,C2:C201) | Calculate correlations between questionnaire items |
| MMULT | =MMULT(A1:D4,E1:H4) | Matrix multiplication for factor calculations |
| TRANSPOSE | =TRANSPOSE(A1:D4) | Construct transpose of loading matrices |
| MINVERSE | =MINVERSE(A1:D4) | Matrix inversion during estimation |
| MDETERM | =MDETERM(A1:D4) | Evaluate covariance matrix properties |
VBA (Optional)
Automate factor analysis workflows by importing survey data, generating correlation matrices, estimating factor solutions and producing formatted loading and communality reports.
Sources
- Harman HH. Modern Factor Analysis.
- Bartholomew DJ, Knott M, Moustaki I. Latent Variable Models and Factor Analysis. 3rd ed.
- Hair JF, Black WC, Babin BJ, Anderson RE. Multivariate Data Analysis.
- Johnson RA, Wichern DW. Applied Multivariate Statistical Analysis.
- ISPOR Task Force reports on patient-reported outcome instrument development.
Related Concepts (5)
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 factor analysis?
A technique identifying a smaller number of underlying, unobserved factors accounting for the pattern of correlations among a larger set of measured variables.
Source: Spearman 1904
What does factor analysis reduce a set of variables to?
Factor analysis reduces a large set of correlated measured variables to a smaller number of underlying, unobserved factors that account for the pattern among them. It works on the idea that variables which move together may all reflect a common hidden dimension, so many items can be summarised by a few factors. This helps make sense of complex data and underpins the construction of scales, where several questions are taken to measure one latent trait. Distilling many variables into a few factors is its purpose. Kline (2015) describes this technique.
Source: Kline 2015
How does factor analysis work?
Factor analysis works by analysing the correlations among the observed variables and extracting factors that account for their shared variance, estimating how strongly each variable loads on each factor. Decisions are made about how many factors to retain, often using eigenvalues, and the factors may be rotated to aid interpretation. So factor analysis works by modelling the observed variables as combinations of a few latent factors plus unique variance, extracting and interpreting those factors from the correlation structure, which allows a large set of variables to be summarised by a smaller number of underlying dimensions that explain much of their common variation.
Source: Spearman 1904
What is factor analysis used for?
Factor analysis is used for data reduction, summarising many correlated variables into a few underlying factors; for uncovering latent constructs, such as the dimensions underlying a set of questionnaire items; and for developing and validating measurement scales. So factor analysis is used to simplify complex data and to reveal the structure of latent variables behind observed measures, which is valuable in psychometrics, scale development, and any setting where many measured variables are thought to reflect a smaller number of underlying constructs, allowing those constructs to be identified and the variables organised around them.
Source: Spearman 1904
What are the types of factor analysis?
The types of factor analysis are exploratory factor analysis, which discovers the factor structure from the data without a prior hypothesis, letting the number of factors and loadings emerge, and confirmatory factor analysis, which tests a prespecified structure hypothesised in advance. Exploratory analysis is used when the structure is unknown, confirmatory when it is theorised. So factor analysis comprises exploratory and confirmatory approaches, differing in whether the structure is uncovered from the data or tested against a hypothesis, and the two are often used in sequence, exploratory work suggesting a structure that confirmatory analysis then tests on independent data.
Source: Spearman 1904
What are the limitations of factor analysis?
The limitations of factor analysis include that it involves subjective decisions, such as the number of factors to retain and the rotation, which affect the results; that the factors are interpreted by the analyst and may not correspond to real constructs; that it assumes linear relationships and requires adequate sample sizes; and that different methods can give different solutions. So factor analysis is applied and interpreted with care, since its results depend on methodological choices and judgement, and the factors it produces are statistical constructs whose meaning must be reasoned rather than assumed, which is why replication and attention to interpretability are important in relying on a factor solution.
Source: Spearman 1904
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Verified by Dr Darrin Baines
British health economist
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Verification date: 15 Dec 2025
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