Concept Architecture
Concept
Theoretically, Confirmatory Factor Analysis (CFA) is a multivariate statistical modelling technique used to test whether observed variables measure one or more hypothesised latent constructs according to a pre-specified measurement model. It is founded on latent variable theory and structural equation modelling, allowing researchers to evaluate construct validity by comparing an a priori theoretical model with observed data. In health economics, confirmatory factor analysis is widely used to validate health-related quality of life instruments, patient-reported outcome measures, preference-based questionnaires and psychometric scales.
Mathematically, confirmatory factor analysis represents observed variables as linear functions of latent factors and measurement error. The covariance matrix implied by the hypothesised factor model is estimated and compared with the observed covariance matrix using maximum likelihood or related estimation methods. Model adequacy is evaluated using goodness-of-fit statistics, parameter estimates and modification indices.
In practice, confirmatory factor analysis is performed using covariance matrices derived from questionnaire or survey data. Factor loadings, error variances and latent factor correlations are estimated simultaneously, while model fit is assessed using recognised fit indices. CFA is routinely applied during the development, validation and cross-cultural adaptation of health outcome instruments used in economic evaluation and health technology assessment.
Purpose
Used to test hypothesised measurement models, evaluate construct validity, confirm latent factor structures, validate health outcome instruments and support psychometric assessment in health economics.
Mathematical Formulae
Primary Formula
x = ?? + �
where:
x = vector of observed variables
? = factor loading matrix
? = vector of latent factors
� = vector of measurement errors
Supporting Formulae
� = ?�?? + �
where:
� = model-implied covariance matrix
� = covariance matrix of latent factors
� = covariance matrix of measurement errors
FML = ln|�| + tr(S�??) ? ln|S| ? p
where:
FML = maximum likelihood fitting function
S = observed covariance matrix
p = number of observed variables
Related Mathematical Methods
Structural Equation Modelling
Maximum Likelihood Estimation
Exploratory Factor Analysis
Covariance Matrix Analysis
Latent Variable Modelling
Measurement Invariance Testing
Example
A health economist evaluates whether a new health-related quality of life questionnaire measures three hypothesised latent domains: physical functioning, emotional wellbeing and social functioning.
Responses from 850 patients are analysed using confirmatory factor analysis. Standardised factor loadings range from 0.71 to 0.90. Model fit statistics indicate good agreement between the hypothesised measurement model and the observed data (CFI = 0.97, TLI = 0.96, RMSEA = 0.045, SRMR = 0.038), supporting the proposed factor structure for use in economic evaluation.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| CORREL | =CORREL(B2:B501,C2:C501) | Calculate correlations between questionnaire items |
| COVARIANCE.S | =COVARIANCE.S(B2:B501,C2:C501) | Construct covariance matrix for CFA |
| MMULT | =MMULT(A2:D5,E2:H5) | Matrix multiplication for covariance calculations |
| TRANSPOSE | =TRANSPOSE(A2:D5) | Matrix transposition during model calculations |
| MINVERSE | =MINVERSE(A2:D5) | Matrix inversion for covariance estimation |
VBA (Optional)
Automate preparation of covariance matrices, matrix operations and export of psychometric datasets for confirmatory factor analysis software.
Sources
Brown TA. Confirmatory Factor Analysis for Applied Research.
Kline RB. Principles and Practice of Structural Equation Modelling.
Byrne BM. Structural Equation Modelling with AMOS.
Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes.
Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (4)
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 confirmatory factor analysis?
A technique testing whether a hypothesised factor structure adequately explains observed correlations among measured variables, unlike exploratory factor analysis.
Source: Jöreskog 1969
What hypothesis does confirmatory factor analysis put to the test?
Confirmatory factor analysis puts a prespecified idea about structure to the test, asking whether a hypothesised set of underlying factors adequately explains the correlations among measured variables. The researcher states in advance which items are meant to reflect which factor, then checks how well that structure fits the observed data. This differs from exploratory factor analysis, which lets the structure emerge without prior specification, and it is used to confirm a proposed measurement model, such as the domains of a questionnaire. Testing a stated factor structure is its purpose. Kline (2015) describes this technique.
Source: Kline 2015
How does confirmatory factor analysis work?
Confirmatory factor analysis works by specifying a model that states the number of factors and which measured variables are indicators of each, then estimating the model's parameters and assessing how well the implied correlations match the observed ones, using fit indices and tests. Loadings, factor correlations, and error variances are estimated under the specified structure. So confirmatory factor analysis works by fitting a prespecified factor model to the data and judging its adequacy through measures of fit, which indicate whether the hypothesised structure reproduces the observed relationships well, allowing the model to be accepted, rejected, or modified in light of the evidence.
Source: Jöreskog 1969
How does confirmatory factor analysis differ from exploratory factor analysis?
Confirmatory factor analysis tests a prespecified factor structure, with the number of factors and the pattern of loadings fixed in advance by hypothesis, while exploratory factor analysis discovers the structure from the data without such constraints, letting the factors and loadings emerge. Confirmatory analysis is hypothesis-testing, and exploratory analysis is hypothesis-generating. So the two differ in whether the factor structure is imposed and tested or derived from the data, with confirmatory factor analysis used to confirm or refute a theorised structure, often after exploratory work or theory has suggested it, and exploratory factor analysis used when the structure is unknown.
Source: Spearman 1904
When is confirmatory factor analysis used?
Confirmatory factor analysis is used when there is a hypothesised or theory-based factor structure to be tested, for example to validate a questionnaire's proposed subscales or to confirm a measurement model before further analysis. It requires a clear prior specification of the factors and their indicators. So confirmatory factor analysis is used to test whether data support a predefined structure, commonly in validating measurement instruments and in structural equation modelling, where establishing that the measured variables reflect the intended underlying constructs is needed before relationships among those constructs are examined, making it a key tool in psychometrics and measurement validation.
Source: Jöreskog 1969
How is model fit assessed in confirmatory factor analysis?
Model fit in confirmatory factor analysis is assessed using a range of indices and tests that compare the correlations implied by the model with those observed, including a chi-square test of overall fit and approximate fit indices that gauge how closely the model reproduces the data while considering complexity. No single measure is definitive. So model fit in confirmatory factor analysis is judged by multiple fit measures together, since each captures a different aspect of how well the specified structure matches the data, and adequate fit across them supports the hypothesised model, while poor fit indicates the structure should be reconsidered or revised.
Source: Jöreskog 1969
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 12 Dec 2025
Content version: 1.0.0
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