Confirmatory Factor Analysis: Zero to Hero Tutorial

This comprehensive tutorial takes you from the foundational concepts of Confirmatory Factor Analysis (CFA) all the way through advanced model specification, estimation, evaluation, modification, and practical usage within the DataStatPro application. Whether you are new to structural modelling or building on prior EFA experience, this guide develops your understanding systematically from the ground up.

Table of Contents

1. Prerequisites and Background Concepts
2. What is Confirmatory Factor Analysis?
3. The Mathematics Behind CFA
4. Assumptions of CFA
5. Types of CFA Models
6. Using the CFA Component
7. Model Specification
8. Model Identification
9. Estimation Methods
10. Model Fit and Evaluation
11. Model Modification and Respecification
12. Interpreting CFA Results
13. Measurement Invariance Testing
14. Worked Examples
15. Common Mistakes and How to Avoid Them
16. Troubleshooting
17. Quick Reference Cheat Sheet

1. Prerequisites and Background Concepts

Before diving into Confirmatory Factor Analysis, it is essential to be comfortable with the following foundational concepts. Each is briefly reviewed here. If you are coming directly from the EFA tutorial, many of these will already be familiar.

1.1 The Common Factor Model (Review)

Recall from the EFA tutorial that the common factor model expresses each observed variable $X_j$ as a linear function of one or more latent factors plus a unique (error) term:

$X_j = \lambda_{j1}F_1 + \lambda_{j2}F_2 + \dots + \lambda_{jm}F_m + \epsilon_j$

In matrix form:

$\mathbf{X} = \boldsymbol{\Lambda}\mathbf{F} + \boldsymbol{\epsilon}$

Where:

• $\boldsymbol{\Lambda}$ is the $p \times m$ factor loading matrix.
• $\mathbf{F}$ is the $m \times 1$ vector of latent factors.
• $\boldsymbol{\epsilon}$ is the $p \times 1$ vector of unique factors (errors).

CFA uses exactly this model — but unlike EFA, the structure of $\boldsymbol{\Lambda}$ is pre-specified by the researcher rather than discovered from the data.

1.2 Variance-Covariance Matrix

The variance-covariance matrix $\boldsymbol{\Sigma}$ (or $\mathbf{S}$ when estimated from data) is a square symmetric matrix that contains variances on its diagonal and covariances on its off-diagonal elements:

$\boldsymbol{\Sigma} = \begin{pmatrix} \sigma^2_1 & \sigma_{12} & \cdots & \sigma_{1p} \\ \sigma_{21} & \sigma^2_2 & \cdots & \sigma_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ \sigma_{p1} & \sigma_{p2} & \cdots & \sigma^2_p \end{pmatrix}$

CFA works by comparing the model-implied covariance matrix $\boldsymbol{\Sigma}(\boldsymbol{\theta})$ (derived from the model parameters) with the observed sample covariance matrix $\mathbf{S}$. The closer these two matrices are, the better the model fits the data.

1.3 Model Parameters and Parameter Vectors

In CFA, the unknown quantities to be estimated are collected into a parameter vector $\boldsymbol{\theta}$, which typically contains:

• $\boldsymbol{\Lambda}$: Factor loadings.
• $\boldsymbol{\Phi}$: Factor variances and covariances (the factor covariance matrix).
• $\boldsymbol{\Theta}_\epsilon$: Unique factor variances (and possibly covariances).

The model implies a specific covariance structure:

$\boldsymbol{\Sigma}(\boldsymbol{\theta}) = \boldsymbol{\Lambda}\boldsymbol{\Phi}\boldsymbol{\Lambda}^T + \boldsymbol{\Theta}_\epsilon$

This is the fundamental equation of CFA — the model-implied covariance matrix expressed as a function of the model parameters.

1.4 Degrees of Freedom in Structural Models

The degrees of freedom (df) for a CFA model are:

$df = \frac{p(p+1)}{2} - q$

Where:

• $p$ is the number of observed variables.
• $\frac{p(p+1)}{2}$ is the number of unique elements in the covariance matrix (known information).
• $q$ is the number of freely estimated parameters.

Degrees of freedom indicate how much information is left over after estimating the model parameters. A model with $df > 0$ is testable — the remaining information can be used to assess fit.

1.5 Hypothesis Testing Framework

CFA operates within a formal hypothesis testing framework. The null hypothesis is that the model-implied covariance matrix equals the population covariance matrix:

$H_0: \boldsymbol{\Sigma} = \boldsymbol{\Sigma}(\boldsymbol{\theta})$

$H_1: \boldsymbol{\Sigma} \neq \boldsymbol{\Sigma}(\boldsymbol{\theta})$

A statistically significant test (rejection of $H_0$) suggests the model does not fit the data. Note that in SEM/CFA, we typically want to retain $H_0$ (non-significant test = good fit), which is the opposite of most hypothesis tests.

2. What is Confirmatory Factor Analysis?

2.1 The Core Idea

Confirmatory Factor Analysis (CFA) is a theory-driven statistical technique used to test whether a hypothesised factor structure — specified by the researcher in advance — is consistent with observed data.

Unlike Exploratory Factor Analysis (EFA), which discovers the factor structure from the data with minimal constraints, CFA begins with an explicit, pre-specified model that states:

• How many factors exist.
• Which observed variables load on which factors.
• Which loadings are constrained to zero (variable does not load on a given factor).
• Whether factors are correlated with each other.

CFA then tests whether this pre-specified structure is statistically tenable — that is, whether it can adequately reproduce the patterns of correlations (or covariances) observed in the data.

2.2 The Role of Theory

Theory is not optional in CFA — it is mandatory. You must be able to state, before looking at the data:

"I hypothesise that variable $X_1$, $X_2$, and $X_3$ are indicators of latent Factor 1, and that $X_4$, $X_5$, and $X_6$ are indicators of latent Factor 2."

This prior specification is what makes CFA confirmatory — you are testing a theory, not searching for one.

2.3 Where CFA Fits in the Research Cycle

CFA typically follows EFA in the scale development cycle: Theory → Item Development → EFA (Discover structure) → New Sample → CFA (Confirm structure) → Revise Theory → Collect New Data → CFA again

Important: EFA and CFA should ideally be conducted on different, independent samples. Using the same sample for both EFA (to discover a structure) and CFA (to confirm the same structure) is circular and will produce artificially good fit.

2.4 CFA as a Special Case of SEM

CFA is a special case of Structural Equation Modelling (SEM). In full SEM:

• The measurement model (CFA) specifies how latent factors are measured by observed indicators.
• The structural model specifies directional relationships between the latent factors.

CFA alone contains only the measurement model — there are no directional paths between factors, only correlations (covariances).

2.5 EFA vs. CFA: A Comprehensive Comparison

2.6 Real-World Applications

• Psychology: Validating a self-report measure of depression against its theoretical subscale structure (e.g., cognitive, affective, somatic dimensions).
• Education: Confirming that a standardised test measures the intended abilities (e.g., verbal, quantitative, and spatial reasoning as separate factors).
• Marketing: Testing whether a customer satisfaction scale measures the hypothesised dimensions (e.g., product quality, service, value, loyalty).
• Health Sciences: Validating quality-of-life instruments with pre-specified subscales (physical, psychological, social, environmental).
• Organisational Psychology: Confirming the factor structure of a job performance or employee engagement measure across different departments or organisations.
• Neuroscience: Testing whether specific cognitive tasks load onto theoretically distinct neural processing systems.

3. The Mathematics Behind CFA

3.1 The CFA Model in Full

For $p$ observed variables $\mathbf{X} = (X_1, X_2, \dots, X_p)^T$, $m$ latent factors $\mathbf{F} = (F_1, F_2, \dots, F_m)^T$, and $p$ unique factors $\boldsymbol{\epsilon} = (\epsilon_1, \epsilon_2, \dots, \epsilon_p)^T$, the CFA measurement model is:

$\mathbf{X} = \boldsymbol{\mu} + \boldsymbol{\Lambda}\mathbf{F} + \boldsymbol{\epsilon}$

Where $\boldsymbol{\mu}$ is the $p \times 1$ vector of variable means (intercepts). When working with mean-centred or standardised variables (as is typical), $\boldsymbol{\mu} = \mathbf{0}$ and the model simplifies to:

$\mathbf{X} = \boldsymbol{\Lambda}\mathbf{F} + \boldsymbol{\epsilon}$

3.2 Model Assumptions (Mathematical)

The formal mathematical assumptions are:

1. $E(\mathbf{F}) = \mathbf{0}$ — Factors have zero mean.
2. $E(\boldsymbol{\epsilon}) = \mathbf{0}$ — Unique factors have zero mean.
3. $\text{Cov}(\mathbf{F}, \boldsymbol{\epsilon}) = \mathbf{0}$ — Factors and unique factors are uncorrelated.
4. $\text{Cov}(\boldsymbol{\epsilon}) = \boldsymbol{\Theta}_\epsilon$ — Unique factor covariance matrix (usually diagonal, i.e., errors are uncorrelated).
5. $\text{Cov}(\mathbf{F}) = \boldsymbol{\Phi}$ — Factor covariance matrix (contains factor variances on the diagonal, covariances off-diagonal).

3.3 The Model-Implied Covariance Matrix

Under the above assumptions, the model-implied covariance matrix of the observed variables is:

$\boldsymbol{\Sigma}(\boldsymbol{\theta}) = \boldsymbol{\Lambda}\boldsymbol{\Phi}\boldsymbol{\Lambda}^T + \boldsymbol{\Theta}_\epsilon$

Where:

• $\boldsymbol{\Lambda}$ ($p \times m$): Factor loading matrix. Most entries are fixed to zero (variable does not load on that factor); free entries are estimated from the data.
• $\boldsymbol{\Phi}$ ($m \times m$): Factor covariance matrix. Diagonal elements are factor variances; off-diagonal elements are factor covariances (correlations if standardised).
• $\boldsymbol{\Theta}_\epsilon$ ($p \times p$): Unique variance matrix. Usually diagonal (containing the error variances $\theta_j = \psi_j = 1 - h_j^2$); off-diagonal elements are fixed to zero unless error covariances are specifically freed.

This equation is the cornerstone of CFA. Every parameter in the model contributes to this implied covariance matrix, and model fit is assessed by comparing this implied matrix to the observed sample covariance matrix $\mathbf{S}$.

3.4 The CFA Discrepancy Function

Estimation finds the parameter vector $\hat{\boldsymbol{\theta}}$ that minimises the discrepancy between $\mathbf{S}$ and $\boldsymbol{\Sigma}(\boldsymbol{\theta})$. For Maximum Likelihood estimation (the most common method), the discrepancy function is:

$F_{ML}(\boldsymbol{\theta}) = \ln|\boldsymbol{\Sigma}(\boldsymbol{\theta})| + \text{tr}[\mathbf{S}\boldsymbol{\Sigma}^{-1}(\boldsymbol{\theta})] - \ln|\mathbf{S}| - p$

Where:

• $\ln|\cdot|$ is the natural log of the matrix determinant.
• $\text{tr}(\cdot)$ is the matrix trace.
• $p$ is the number of observed variables.

$F_{ML}$ equals zero when $\boldsymbol{\Sigma}(\boldsymbol{\theta}) = \mathbf{S}$ exactly (perfect fit) and increases as the two matrices diverge.

3.5 The Model Chi-Squared Statistic

The model chi-squared statistic is derived from the minimised discrepancy function:

$\chi^2_{\text{model}} = (n - 1) \cdot \hat{F}_{ML}$

With degrees of freedom:

$df = \frac{p(p+1)}{2} - q$

Where $n$ is the sample size and $q$ is the number of freely estimated parameters. This statistic tests $H_0: \boldsymbol{\Sigma} = \boldsymbol{\Sigma}(\boldsymbol{\theta})$.

3.6 Communality and Reliability in CFA

In CFA, the communality of observed variable $X_j$ (the proportion of its variance explained by the common factors) is:

$h_j^2 = \frac{\lambda_j^2 \phi_{kk}}{\lambda_j^2 \phi_{kk} + \theta_j}$

For a standardised factor (variance = 1), this simplifies to:

$h_j^2 = \frac{\lambda_j^2}{\lambda_j^2 + \theta_j} = \lambda_j^2$

(when the factor variance is fixed to 1 and the loading is from a standardised solution).

The reliability of each indicator (its squared standardised loading) is also called the item reliability or $R^2$ of the indicator:

$R^2_j = \hat{\lambda}_j^2 \quad \text{(standardised loading squared)}$

3.7 Factor Variances and Covariances

The factor covariance matrix $\boldsymbol{\Phi}$ contains:

• Diagonal elements ($\phi_{kk}$): Variance of factor $k$. Fixed to 1 to set the metric of the factor (marker variable approach alternative), or estimated freely if a marker variable constraint is used.
• Off-diagonal elements ($\phi_{kk'}$): Covariance between factors $k$ and $k'$. Converting to correlations:

$r_{kk'} = \frac{\phi_{kk'}}{\sqrt{\phi_{kk} \phi_{k'k'}}}$

In a standardised solution, all factor variances are set to 1, so $\phi_{kk'} = r_{kk'}$ directly.

3.8 The Residual (Error) Variance

The error variance (unique variance) of variable $X_j$ is:

$\theta_j = \text{Var}(X_j) - \lambda_j^2 \phi_{kk}$

Or in the standardised solution:

$\theta_j = 1 - \lambda_j^2$

Errors are assumed to be uncorrelated (i.e., $\boldsymbol{\Theta}_\epsilon$ is diagonal) unless correlated error terms are specifically freed — typically only when there is a theoretical justification (e.g., two items share method variance because they are both reverse-scored).

4. Assumptions of CFA

4.1 Correct Model Specification

The most fundamental assumption of CFA is that the hypothesised model is correctly specified — that the factor structure, the pattern of zero and non-zero loadings, and the factor correlations reflect the true population structure. Misspecification of any aspect of the model will result in poor fit and/or biased parameter estimates.

⚠️ No amount of statistical power or sample size can compensate for a fundamentally wrong model. Theory must guide model specification.

4.2 Multivariate Normality

The most common estimator (Maximum Likelihood) assumes that the observed variables follow a multivariate normal distribution. This requires:

• Each variable is approximately normally distributed (check histograms, Q-Q plots).
• Variables are jointly normally distributed (check multivariate kurtosis).

Mardia's multivariate kurtosis coefficient is frequently used to assess this assumption. A commonly cited threshold is a normalised multivariate kurtosis value $< 3.0$ to proceed comfortably with standard ML.

When normality is violated:

• Use Robust ML (MLR) estimators that provide Satorra-Bentler scaled chi-squared and robust standard errors.
• Use Weighted Least Squares (WLS) variants for ordinal data.
• Use Bootstrapped standard errors.

4.3 Independence of Observations

Each observation (row in the dataset) must be statistically independent of every other. Clustered or nested data (e.g., students within schools, measurements within individuals) violates this assumption. Address with:

• Multilevel CFA (for nested structures).
• Sandwich estimators (for mild clustering).

4.4 Adequate Sample Size

CFA is a large-sample technique. Parameter estimates and fit statistics are asymptotically derived and may be unreliable in small samples. General guidelines:

⚠️ With small samples, chi-squared tests are underpowered (may fail to detect truly poor fit) and standard errors are inflated. Conversely, with very large samples ($n > 1000$), even trivial model misspecifications produce significant chi-squared tests.

4.5 Sufficient Indicator Reliability

Each latent factor should be measured by indicators with meaningful standardised loadings. As a minimum:

• Standardised loadings $|\hat{\lambda}_j| \geq 0.40$.
• Ideally $|\hat{\lambda}_j| \geq 0.50$, with many at $\geq 0.70$.

Low loadings indicate that the indicators are poor measures of the latent factor.

4.6 No Perfect Multicollinearity or Singularity

Variables that are perfectly correlated (or nearly so) will cause the covariance matrix $\mathbf{S}$ to be singular (non-invertible), making estimation impossible. Remove redundant variables before running CFA.

4.7 Scale of Measurement

Observed variables should be continuous (interval or ratio scale). For ordinal variables with 5 or more categories, standard ML can often be used with little consequence. For truly ordinal data (fewer than 5 categories) or binary variables, use a polychoric/tetrachoric correlation matrix with WLS or WLSMV estimation.

5. Types of CFA Models

5.1 The Single-Factor (One-Factor) Model

All $p$ observed variables load on a single latent factor. This is the simplest CFA model:

$\boldsymbol{\Sigma}(\boldsymbol{\theta}) = \boldsymbol{\lambda}\boldsymbol{\lambda}^T + \boldsymbol{\Theta}_\epsilon$

Where $\boldsymbol{\lambda}$ is a $p \times 1$ vector of loadings.

Use case: Testing whether a scale is truly unidimensional (measures a single construct). Requirement: $p \geq 3$ indicators for identification.

5.2 The Correlated Factors (Standard) Model

The most common CFA model: $m$ factors, each measured by a distinct set of indicators, with factors allowed to correlate freely:

$\boldsymbol{\Sigma}(\boldsymbol{\theta}) = \boldsymbol{\Lambda}\boldsymbol{\Phi}\boldsymbol{\Lambda}^T + \boldsymbol{\Theta}_\epsilon$

Where $\boldsymbol{\Phi}$ has free off-diagonal elements (factor covariances).

Use case: Most scale validation studies with multiple subscales.

5.3 The Orthogonal Factors Model

Same as the correlated factors model, but factor covariances are constrained to zero ($\boldsymbol{\Phi} = \mathbf{I}$):

$\boldsymbol{\Sigma}(\boldsymbol{\theta}) = \boldsymbol{\Lambda}\boldsymbol{\Lambda}^T + \boldsymbol{\Theta}_\epsilon$

Use case: Testing whether factors are truly independent. Rarely realistic in social sciences — use only if theoretically justified.

5.4 The Higher-Order (Second-Order) Factor Model

A higher-order CFA posits that the correlations among the first-order factors are themselves explained by one or more second-order factors. For example:

• First-order factors: Verbal Ability, Quantitative Ability, Spatial Ability.
• Second-order factor: General Intelligence (g).

Model structure:

$\mathbf{F}_1 = \boldsymbol{\Gamma}\mathbf{F}_2 + \boldsymbol{\zeta}$

Where:

• $\mathbf{F}_1$ is the vector of first-order factors.
• $\mathbf{F}_2$ is the vector of second-order factors.
• $\boldsymbol{\Gamma}$ is the matrix of second-order loadings.
• $\boldsymbol{\zeta}$ is the vector of first-order factor residuals (disturbances).

5.5 The Bifactor Model

A bifactor model specifies a general factor ($G$) that loads on all observed variables, plus several group-specific factors (each loading on a subset of variables):

$X_j = \lambda_{jG} G + \lambda_{jk} F_k + \epsilon_j$

The general factor and group factors are orthogonal. This model is useful for:

• Testing the relative contributions of a general vs. specific factors.
• Evaluating whether a composite score (total score) is justified despite subscale structure.

Bifactor model fit statistic — Omega ($\omega$):

$\omega_{\text{general}} = \frac{(\sum_{j=1}^p \lambda_{jG})^2}{(\sum_{j=1}^p \lambda_{jG})^2 + \sum_{k=1}^m (\sum_{j \in F_k} \lambda_{jk})^2 + \sum_{j=1}^p \theta_j}$

5.6 Measurement Invariance Models

A series of increasingly constrained CFA models used to test whether the factor structure is equivalent across groups (e.g., genders, countries, time points). See Section 13 for full details.

5.7 Summary of CFA Model Types

6. Using the CFA Component

The CFA component in DataStatPro provides a complete workflow for specifying, estimating, evaluating, and modifying confirmatory factor models.

Step-by-Step Guide

Step 1 — Select Dataset

Choose the dataset from the "Dataset" dropdown. The dataset should contain the observed indicator variables you wish to include in the CFA. Ensure:

• All indicator variables are numeric (continuous or ordinal).
• The dataset has sufficient sample size ($n \geq 200$ recommended).
• Variables have been screened for outliers, missing data, and non-normality.

💡 Tip: Run basic descriptive statistics before CFA. Check means, standard deviations, skewness ($|z| < 2$), and kurtosis ($|z| < 7$) for each indicator before proceeding.

Step 2 — Specify the Factor Structure

Define your CFA model using the model specification panel:

• Number of Factors: Enter the number of latent factors $m$.
• Factor Names: Give each factor a meaningful theoretical label.
• Indicator Assignments: For each factor, select which observed variables are its indicators. This defines the non-zero entries of $\boldsymbol{\Lambda}$.
• Cross-Loadings (Advanced): Specify any variables that are hypothesised to load on more than one factor. By default, all cross-loadings are fixed to zero.
• Factor Covariances: Specify whether factors are allowed to correlate (default: yes).

⚠️ Important: Factor specification must be theory-driven, not data-driven. Do not inspect EFA loadings from the same dataset and then specify the CFA model — this is circular reasoning and inflates apparent model fit.

Step 3 — Set Identification Constraints

Choose how to set the metric (scale) of each latent factor. Two approaches are available:

• Marker Variable (Reference Indicator): Fix one loading per factor to 1.0. The factor takes the scale of that indicator. (Default in most software.)
• Fixed Factor Variance: Fix the variance of each factor to 1.0. All loadings are freely estimated, and the factor is on a standardised scale.

💡 Tip: The marker variable approach is conventional and easier to interpret in unstandardised solutions. Fixed factor variance is useful for comparing loadings across factors.

Step 4 — Select Estimation Method

Choose from the "Estimation Method" dropdown:

• ML (Maximum Likelihood): Default. Appropriate when multivariate normality holds.
• MLR (Robust ML): Provides Satorra-Bentler scaled $\chi^2$ and robust standard errors. Recommended when normality is mildly to moderately violated.
• WLSMV (Weighted Least Squares Mean and Variance Adjusted): For ordinal indicators (Likert items with fewer than 5 categories). Uses polychoric correlations internally.

💡 Recommendation: Use MLR as the default — it is robust to mild non-normality and performs well under most practical conditions.

Step 5 — Specify Error Covariances (If Applicable)

By default, all error covariances are fixed to zero. If theory justifies correlated errors (e.g., two items share similar wording, reverse-scored items), you can free specific error covariances in the advanced panel.

⚠️ Freeing error covariances purely based on modification indices, without theoretical justification, is a form of capitalising on chance and should be avoided.

Step 6 — Select Fit Indices to Report

Choose which fit statistics to display. The DataStatPro app reports:

• ✅ Model $\chi^2$ and $p$-value
• ✅ CFI (Comparative Fit Index)
• ✅ TLI (Tucker-Lewis Index)
• ✅ RMSEA and 90% CI
• ✅ SRMR
• ✅ AIC and BIC
• ✅ Modification Indices
• ✅ Standardised and Unstandardised Loading Tables
• ✅ Factor Correlation Matrix
• ✅ Communalities ($R^2$) Table
• ✅ Reliability Estimates (AVE, CR, $\omega$)

Step 7 — Run the Analysis

Click "Run CFA". The application will:

1. Build the model-implied covariance matrix $\boldsymbol{\Sigma}(\boldsymbol{\theta})$.
2. Estimate model parameters using the chosen estimator.
3. Compute the chi-squared test, degrees of freedom, and $p$-value.
4. Compute all selected fit indices.
5. Display the loading table (standardised and unstandardised).
6. Generate modification indices and expected parameter change (EPC) statistics.
7. Compute factor reliability and validity statistics (AVE, CR).

7. Model Specification

7.1 Three Types of Parameters

In a CFA model, every possible parameter is either:

7.2 Specifying the Loading Matrix

The loading matrix $\boldsymbol{\Lambda}$ is specified by indicating which loadings are free and which are fixed to zero. A typical 2-factor, 6-indicator specification looks like:

$\boldsymbol{\Lambda} = \begin{pmatrix} \lambda_{11} & 0 \\ \lambda_{21} & 0 \\ \lambda_{31} & 0 \\ 0 & \lambda_{42} \\ 0 & \lambda_{52} \\ 0 & \lambda_{62} \end{pmatrix}$

Where:

• $\lambda_{11}, \lambda_{21}, \lambda_{31}$ are freely estimated loadings on Factor 1.
• $\lambda_{42}, \lambda_{52}, \lambda_{62}$ are freely estimated loadings on Factor 2.
• The 0s are fixed — these variables are constrained to have no relationship with the corresponding factor.

7.3 Identification Constraints in the Loading Matrix

One loading per factor must be fixed (either to 1 or to a specific value) to set the factor's scale. Using the marker variable approach, the first indicator of each factor has its loading fixed to 1:

$\boldsymbol{\Lambda} = \begin{pmatrix} \mathbf{1} & 0 \\ \lambda_{21} & 0 \\ \lambda_{31} & 0 \\ 0 & \mathbf{1} \\ 0 & \lambda_{52} \\ 0 & \lambda_{62} \end{pmatrix}$

7.4 Specifying Factor Covariances

The factor covariance matrix $\boldsymbol{\Phi}$ for a 2-factor model:

$\boldsymbol{\Phi} = \begin{pmatrix} \phi_{11} & \phi_{12} \\ \phi_{12} & \phi_{22} \end{pmatrix}$

• $\phi_{11}$, $\phi_{22}$: Factor variances (free or fixed to 1).
• $\phi_{12}$: Factor covariance (free if factors are allowed to correlate; fixed to 0 if orthogonal).

7.5 Specifying Error Variances

Error variances are always freely estimated:

$\boldsymbol{\Theta}_\epsilon = \begin{pmatrix} \theta_1 & 0 & \cdots & 0 \\ 0 & \theta_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \theta_p \end{pmatrix}$

If theoretically justified, specific off-diagonal elements (error covariances) can be freed:

$\boldsymbol{\Theta}_\epsilon = \begin{pmatrix} \theta_1 & 0 & \theta_{13} & 0 & 0 & 0 \\ 0 & \theta_2 & 0 & 0 & 0 & 0 \\ \theta_{13} & 0 & \theta_3 & 0 & 0 & 0 \\ 0 & 0 & 0 & \theta_4 & 0 & 0 \\ 0 & 0 & 0 & 0 & \theta_5 & 0 \\ 0 & 0 & 0 & 0 & 0 & \theta_6 \end{pmatrix}$

Here, $\theta_{13}$ is a freed error covariance between variables 1 and 3.

7.6 Counting Free Parameters

The total number of free parameters $q$ in a standard CFA model is:

$q = (p - m) + \frac{m(m+1)}{2} + p$

Where:

• $(p - m)$: Free loadings (one per factor is fixed for identification).
• $\frac{m(m+1)}{2}$: Free elements of $\boldsymbol{\Phi}$ (variances and covariances of $m$ factors; factor variances may be fixed to 1 instead, in which case only covariances are free).
• $p$: Error variances (one per indicator).

8. Model Identification

Model identification refers to whether there is a unique set of parameter values $\hat{\boldsymbol{\theta}}$ that can be estimated from the data. An unidentified model cannot be estimated.

8.1 The Three Conditions of Identification

💡 All CFA models should be over-identified ($df > 0$) to allow meaningful fit testing.

8.2 Necessary Conditions for Identification

Global condition: The model must have $df \geq 0$:

$\frac{p(p+1)}{2} \geq q$

Factor metric condition: Each factor must have a fixed scale. Either:

• Fix one loading per factor to 1 (marker variable), OR
• Fix each factor variance to 1 (standardised factor).

Local (sufficient) condition: Each factor must have at least 2 free loadings (preferably 3 or more) for reliable estimation.

8.3 The t-Rule (Necessary Condition)

A necessary (but not sufficient) condition for identification is that the number of free parameters must not exceed the number of known values:

$q \leq \frac{p(p+1)}{2}$

This ensures $df \geq 0$.

8.4 The Two-Indicator Rule

A single-factor model with exactly 2 indicators and factor variance fixed to 1 is just-identified ($df = 0$). There is a unique solution but no degrees of freedom for testing. This is generally inadvisable — use at least 3 indicators per factor.

8.5 The Three-Indicator Rule

A single-factor model with 3 or more indicators (and one loading fixed for scale) is over-identified and can be properly tested. This is the minimum recommended number of indicators per factor.

💡 General recommendation: Each latent factor should have at least 3 indicators, ideally 4 or more, to ensure identification, stability, and reliability of the factor estimate.

8.6 Empirical Under-Identification

Even if a model is theoretically identified, it can be empirically under-identified if:

• A factor has only 2 indicators and they are highly correlated with each other (but not uniquely with any factor).
• Factor correlations are near 1 (factors are nearly indistinguishable).
• An error variance estimate is near 0 or negative (Heywood case).

These situations will cause estimation to fail or produce inadmissible solutions. See the Troubleshooting section for solutions.

9. Estimation Methods

9.1 Maximum Likelihood (ML)

ML is the default and most widely used estimation method for CFA. It finds the parameter estimates $\hat{\boldsymbol{\theta}}$ that maximise the likelihood of observing the sample covariance matrix $\mathbf{S}$, given the model.

The ML fit function (discrepancy function to be minimised):

$F_{ML} = \ln|\boldsymbol{\Sigma}(\boldsymbol{\theta})| + \text{tr}[\mathbf{S}\boldsymbol{\Sigma}^{-1}(\boldsymbol{\theta})] - \ln|\mathbf{S}| - p$

Properties of ML estimates:

• Consistent: Converge to the true population values as $n \to \infty$.
• Asymptotically efficient: Have the smallest possible variance among consistent estimators.
• Asymptotically normally distributed: Enabling z-tests and confidence intervals.
• Assumption: Multivariate normality of the observed variables.

The log-likelihood of the model is:

$\ell(\boldsymbol{\theta}) = -\frac{n}{2}\left[p\ln(2\pi) + \ln|\boldsymbol{\Sigma}(\boldsymbol{\theta})| + \text{tr}(\mathbf{S}\boldsymbol{\Sigma}^{-1}(\boldsymbol{\theta}))\right]$

9.2 Robust Maximum Likelihood (MLR)

MLR (also called Robust ML or the Satorra-Bentler method) adjusts the ML chi-squared statistic and standard errors to account for violations of multivariate normality.

The Satorra-Bentler scaled chi-squared statistic:

$\chi^2_{SB} = \frac{\chi^2_{ML}}{c}$

Where $c$ is a scaling correction factor based on the degree of multivariate kurtosis in the data:

$c = \frac{\text{tr}(\hat{\mathbf{U}}\mathbf{W}^{-1})}{df}$

With $\hat{\mathbf{U}}$ being an estimate of the asymptotic covariance matrix of the sample variances and covariances, and $\mathbf{W}$ being the weight matrix.

Robust standard errors are computed using the sandwich estimator:

$\hat{\text{Var}}_{robust}(\hat{\boldsymbol{\theta}}) = \mathbf{A}^{-1}\mathbf{B}\mathbf{A}^{-1}$

Where $\mathbf{A} = -\partial^2 \ell / \partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^T$ is the Hessian and $\mathbf{B}$ is the cross-product of the score functions.

💡 Recommendation: Use MLR as the default estimator in practice. It performs nearly as well as ML when data are normal, and substantially better when they are not.

9.3 Weighted Least Squares (WLS) and WLSMV

For ordinal or binary indicators, standard ML is inappropriate. Instead:

WLS (Weighted Least Squares): Minimises a weighted version of the discrepancy between the sample and model-implied polychoric correlation matrix:

$F_{WLS} = (\mathbf{s} - \boldsymbol{\sigma}(\boldsymbol{\theta}))^T \mathbf{W}^{-1} (\mathbf{s} - \boldsymbol{\sigma}(\boldsymbol{\theta}))$

Where $\mathbf{s}$ and $\boldsymbol{\sigma}(\boldsymbol{\theta})$ are vectorised forms of the sample and model-implied correlation matrices, and $\mathbf{W}$ is a weight matrix.

WLSMV (Weighted Least Squares Mean and Variance Adjusted): A more robust variant of WLS that requires less extreme sample sizes. Uses polychoric/tetrachoric correlations and provides a mean-and-variance adjusted chi-squared statistic. Recommended for ordinal data in most practical applications.

9.4 Generalised Least Squares (GLS)

GLS minimises a discrepancy function that weights deviations by the inverse of the sample covariance matrix:

$F_{GLS} = \frac{1}{2}\text{tr}\{[\mathbf{I} - \mathbf{S}^{-1}\boldsymbol{\Sigma}(\boldsymbol{\theta})]^2\}$

GLS is less commonly used than ML but does not require normality of the variables (only of the parameter estimates, asymptotically).

9.5 Comparison of Estimation Methods

10. Model Fit and Evaluation

Evaluating model fit in CFA requires examining multiple complementary indices — no single statistic tells the full story. A well-fitting model should show good fit across several indices simultaneously.

10.1 The Model Chi-Squared Test ($\chi^2$)

The chi-squared test is the primary (and oldest) test of exact model fit:

$\chi^2_{\text{model}} = (n - 1) \cdot \hat{F}_{ML}$

$df = \frac{p(p+1)}{2} - q$

• Null hypothesis: $H_0: \boldsymbol{\Sigma} = \boldsymbol{\Sigma}(\boldsymbol{\theta})$ (model fits exactly).
• A non-significant $p$-value ($p > 0.05$) suggests the model cannot be rejected — conventionally interpreted as "good fit."

Limitations:

• With large $n$, almost any model is rejected even with trivially small discrepancies.
• With small $n$, even poor-fitting models may not be rejected.
• Almost no real-world model fits the data exactly — the null hypothesis is essentially always false in large samples.

⚠️ The chi-squared test should never be the sole basis for accepting or rejecting a CFA model. Always supplement with approximate fit indices.

10.2 The Normed Chi-Squared ($\chi^2/df$)

A simple correction for sample size sensitivity is the ratio of chi-squared to degrees of freedom:

$\chi^2/df = \frac{\chi^2_{\text{model}}}{df}$

⚠️ This ratio lacks a formal statistical basis and thresholds vary widely across textbooks (some accept up to 5.0, others only up to 2.0). Use it as a rough indicator only.

10.3 Comparative Fit Index (CFI)

The CFI compares the fit of the target model to the null model (independence model — all observed variables are uncorrelated, no factors):

$\text{CFI} = 1 - \frac{\max(\chi^2_{\text{model}} - df_{\text{model}}, 0)}{\max(\chi^2_{\text{null}} - df_{\text{null}}, 0)}$

CFI ranges from 0 to 1, where higher values indicate better fit.

💡 The widely cited cutoff of CFI $\geq 0.95$ for "good fit" (Hu & Bentler, 1999) was derived from simulations with specific conditions. Apply it as a guideline, not an absolute rule.

10.4 Tucker-Lewis Index (TLI) / Non-Normed Fit Index (NNFI)

The TLI is similar to CFI but penalises for model complexity (additional parameters). It can fall outside [0, 1] in practice:

$\text{TLI} = \frac{\chi^2_{\text{null}}/df_{\text{null}} - \chi^2_{\text{model}}/df_{\text{model}}}{\chi^2_{\text{null}}/df_{\text{null}} - 1}$

10.5 RMSEA (Root Mean Square Error of Approximation)

The RMSEA measures the discrepancy between the model and data per degree of freedom, adjusting for model complexity:

$\text{RMSEA} = \sqrt{\max\left(\frac{\chi^2_{\text{model}} - df}{df \cdot (n - 1)}, 0\right)}$

RMSEA estimates the discrepancy in the population (not just the sample), making it less sensitive to sample size than the chi-squared test.

A 90% confidence interval for RMSEA is routinely reported:

• Lower bound: Test $H_0$: RMSEA $\leq 0.05$ (close fit test).
• Upper bound: Test $H_0$: RMSEA $\geq 0.10$ (poor fit test).

💡 RMSEA favours more parsimonious (simpler) models and penalises for additional parameters. This makes it complementary to CFI, which does not penalise for complexity.

10.6 SRMR (Standardised Root Mean Square Residual)

The SRMR is the standardised average of all residual correlations (differences between observed and model-implied correlations):

$\text{SRMR} = \sqrt{\frac{2\sum_{j \leq j'}\left(\frac{s_{jj'} - \hat{\sigma}_{jj'}}{\sqrt{s_{jj}s_{j'j'}}}\right)^2}{p(p+1)}}$

SRMR is particularly sensitive to misspecified factor loadings (rather than misspecified factor covariances).

10.7 Information Criteria: AIC and BIC

AIC and BIC are used for comparing competing models (e.g., different numbers of factors, different indicator assignments), not for assessing absolute fit:

$\text{AIC} = \chi^2_{\text{model}} - 2 \cdot df$

$\text{BIC} = \chi^2_{\text{model}} - df \cdot \ln(n)$

For both: lower values indicate a better-fitting, more parsimonious model.

• AIC favours predictive accuracy (less penalty for complexity).
• BIC penalises complexity more heavily and favours simpler models.

💡 When comparing two models, a $\Delta\text{AIC} > 10$ or $\Delta\text{BIC} > 10$ is generally considered strong evidence in favour of the model with the lower value.

10.8 The Chi-Squared Difference Test ($\Delta\chi^2$)

For nested models (one model is a restricted version of another), the improvement in fit can be formally tested using the chi-squared difference test:

$\Delta\chi^2 = \chi^2_{\text{restricted}} - \chi^2_{\text{free}}$

$\Delta df = df_{\text{restricted}} - df_{\text{free}}$

Under $H_0$ (the restriction is valid), $\Delta\chi^2$ follows a chi-squared distribution with $\Delta df$ degrees of freedom.

• Significant $\Delta\chi^2$ ($p < 0.05$): The restricted model fits significantly worse — the freed parameters are statistically necessary.
• Non-significant $\Delta\chi^2$: The restriction does not significantly worsen fit — the simpler model is preferred (parsimony principle).

For MLR estimation, use the Satorra-Bentler scaled difference test:

$\Delta\chi^2_{SB} = \frac{\chi^2_{SB,\text{restricted}} \cdot df_{\text{restricted}} - \chi^2_{SB,\text{free}} \cdot df_{\text{free}}}{cd}$

Where $cd$ is a scaling correction. This is computed automatically by DataStatPro when MLR is used.

10.9 Composite Reliability (CR) and Average Variance Extracted (AVE)

Beyond model fit, reliability and validity of the measurement model are assessed using:

Composite Reliability (CR) (also called construct reliability or Raykov's $\omega$):

$\text{CR}_k = \frac{(\sum_{j \in F_k} \hat{\lambda}_j)^2}{(\sum_{j \in F_k} \hat{\lambda}_j)^2 + \sum_{j \in F_k} \hat{\theta}_j}$

Where $\hat{\lambda}_j$ are standardised loadings and $\hat{\theta}_j$ are error variances for indicators of factor $k$.

Average Variance Extracted (AVE): Measures the average proportion of variance in the indicators explained by the factor:

$\text{AVE}_k = \frac{\sum_{j \in F_k} \hat{\lambda}_j^2}{\sum_{j \in F_k} \hat{\lambda}_j^2 + \sum_{j \in F_k} \hat{\theta}_j} = \frac{\sum_{j \in F_k} \hat{\lambda}_j^2}{n_k}$

Where $n_k$ is the number of indicators for factor $k$.

Discriminant Validity (Fornell-Larcker Criterion): For each pair of factors $k$ and $k'$:

$\text{AVE}_k > r^2_{kk'}$ and $\text{AVE}_{k'} > r^2_{kk'}$

Where $r_{kk'}$ is the correlation between factors $k$ and $k'$. This criterion checks that each factor shares more variance with its own indicators than with any other factor.

10.10 Comprehensive Fit Evaluation Summary

💡 Best practice: A model should satisfy at least CFI $\geq 0.95$, RMSEA $\leq 0.08$, and SRMR $\leq 0.08$ simultaneously before being considered adequately fitting. Meeting only one criterion is insufficient.

11. Model Modification and Respecification

11.1 When is Modification Justified?

Model modification involves changing the original model specification when fit is inadequate. This is justified only when:

1. The modification is guided by substantive theory, not purely by statistical indices.
2. The modification is theoretically defensible and can be explained.
3. The modified model is cross-validated in an independent sample (or at minimum, the exploratory nature of the modification is clearly acknowledged).

⚠️ Data-driven model modification without theoretical justification is a form of post-hoc model fitting that capitalises on chance. Results of extensively modified models should be treated as exploratory and replicated in new data.

11.2 Modification Indices (MI)

A modification index for a fixed (constrained) parameter represents the expected decrease in the model chi-squared if that parameter were freed (with $\Delta df = 1$):

$\text{MI}_{ij} \approx \Delta\chi^2 \approx z_{ij}^2$

Where $z_{ij}$ is the expected standardised estimate if the parameter were freed.

• Large MI (typically $> 10$ or $> 3.84$ for $p < 0.05$): Suggests freeing this parameter would substantially improve fit.
• Small MI: Parameter is not important to model fit.

The Expected Parameter Change (EPC) accompanies each MI and estimates the magnitude and direction of the change if the parameter were freed:

$\text{EPC}_{ij} \approx \frac{\text{MI}_{ij}}{z_{ij}}$

The Standardised EPC (SEPC) expresses the change in standardised units, facilitating comparison across parameters.

11.3 Types of Modifications

Freeing a fixed loading (cross-loading): Adding a loading that was previously fixed to zero — i.e., allowing a variable to load on an additional factor. Justifiable only if the variable theoretically reflects that additional factor.

Freeing an error covariance: Allowing two indicator errors to correlate. Most commonly justified when:

• Two items share similar wording or are reverse-scored forms of the same item.
• Two items were administered in the same method (e.g., both interviewer-rated; both self-reported).
• Two items share a specific content domain narrower than the factor.

Removing a poorly fitting indicator: If an indicator has:

• A very low standardised loading ($< 0.30$).
• A very large MI involving many other parameters.
• Poor communality ($R^2 < 0.10$).

Then removing it and re-running the model may be justified, especially if the item is theoretically weak or ambiguous.

11.4 The Model Modification Decision Tree

Start: Run specified CFA model | v Is overall fit acceptable? (CFI ≥ 0.95, RMSEA ≤ 0.08, SRMR ≤ 0.08) | Yes | No | | | v | Examine modification indices | | | v | Is there a MI > 10? | | | Yes → Is the modification theoretically justifiable? | | | Yes → Free the parameter → Re-run model → Check fit again | | | No → Do NOT free the parameter | | | v | Is there a poorly fitting indicator (low loading, high MI)? | | | Yes → Is removal theoretically defensible? → Remove → Re-run | | | No → Accept imperfect fit (report as limitation) v Accept model (document all modifications transparently)

11.5 Reporting Model Modifications

All model modifications must be reported transparently, including:

• The original model and its fit.
• Each modification made, the MI that motivated it, and the theoretical justification.
• The final model and its fit.
• An explicit statement that the modified model is exploratory and requires replication.

12. Interpreting CFA Results

12.1 The Unstandardised Solution

The unstandardised (raw) solution reports parameter estimates in the original metric of the observed variables. This solution is:

• Used for hypothesis testing (z-tests, p-values).
• Useful for comparing parameters across groups (e.g., in multi-group CFA).
• Affected by the scale of the variables.

Unstandardised factor loading $\hat{\lambda}_j$: The expected change in $X_j$ for a one-unit increase in the latent factor $F_k$, holding other factors constant (similar to a regression coefficient).

Standard error $SE(\hat{\lambda}_j)$: Estimated standard deviation of $\hat{\lambda}_j$ across hypothetical repeated samples.

z-statistic (Wald test):

$z_j = \frac{\hat{\lambda}_j}{SE(\hat{\lambda}_j)}$

Under $H_0: \lambda_j = 0$, $z_j$ follows $\mathcal{N}(0, 1)$ asymptotically.

p-value:

$p_j = 2 \times (1 - \Phi(|z_j|))$

Confidence interval for $\lambda_j$:

$\hat{\lambda}_j \pm z_{\alpha/2} \cdot SE(\hat{\lambda}_j)$

⚠️ The marker variable (reference indicator) has its loading fixed to 1 — no z-test or confidence interval is reported for this parameter.

12.2 The Standardised Solution

The standardised solution scales all variables (both observed and latent) to have unit variance. This makes loadings interpretable as correlations between the indicator and the factor, and facilitates comparisons across indicators and factors.

Standardised loading formula:

$\hat{\lambda}_j^* = \hat{\lambda}_j \cdot \sqrt{\frac{\hat{\phi}_{kk}}{\hat{\sigma}_{jj}}}$

Where $\hat{\phi}_{kk}$ is the factor variance and $\hat{\sigma}_{jj}$ is the observed variance of $X_j$.

Interpretation of standardised loadings:

| $|\hat{\lambda}_j^*|$ | Interpretation | | :-------------------- | :------------- | | $\geq 0.70$ | Strong indicator — excellent | | $0.55 - 0.69$ | Good indicator | | $0.45 - 0.54$ | Adequate indicator | | $0.30 - 0.44$ | Weak indicator — consider replacing | | $< 0.30$ | Poor indicator — consider removing |

12.3 The $R^2$ (Item Reliability) Values

For each indicator, the $R^2$ is the squared standardised loading — the proportion of the indicator's variance explained by the latent factor:

$R^2_j = (\hat{\lambda}_j^*)^2$

This is the reliability of the item as a measure of the factor:

12.4 Factor Correlations

The factor correlation matrix (standardised $\boldsymbol{\Phi}$) shows how strongly the latent factors are associated. For a two-factor model:

$r_{12} = \frac{\hat{\phi}_{12}}{\sqrt{\hat{\phi}_{11}\hat{\phi}_{22}}}$

• $|r_{12}| < 0.30$: Factors are relatively distinct — discriminant validity is supported.
• $|r_{12}| = 0.30 - 0.60$: Moderate correlation — factors share some construct space but are distinguishable.
• $|r_{12}| > 0.85$: Factors are highly similar — consider whether they truly represent distinct constructs (discriminant validity threat).

12.5 Reading the Full CFA Results Table

A complete CFA results table reports the following for each indicator:

12.6 Residual Correlations (Standardised Residuals)

The standardised residuals are the residual correlations divided by their standard errors:

$z_{jj'} = \frac{r_{jj'} - \hat{r}_{jj'}}{SE(\hat{r}_{jj'})}$

Large standardised residuals ($|z_{jj'}| > 1.96$) indicate that the model under- or over-predicts the relationship between variables $j$ and $j'$. These are diagnostic of model misfit for specific variable pairs and can guide targeted modifications.

13. Measurement Invariance Testing

Measurement invariance (also called measurement equivalence) testing examines whether the CFA model holds equally across different groups (e.g., males vs. females, different countries, different time points). Without invariance, comparing latent factor scores or means across groups is not meaningful.

13.1 Why Invariance Matters

If a scale measuring "Depression" has different factor loadings for men and women, then comparing depression scores between genders is not comparing the same construct — the scale itself functions differently. Invariance testing ensures that comparisons are fair and meaningful.

13.2 The Hierarchy of Invariance Models

Invariance testing proceeds through a sequence of increasingly constrained models:

Level 1 — Configural Invariance (Baseline Model)

The same factor structure (same pattern of free and fixed loadings) holds in all groups, but all parameters are estimated freely within each group:

$H_{\text{config}}: \boldsymbol{\Lambda}_g = \boldsymbol{\Lambda}, \quad \text{same pattern across groups}$

This tests whether the same factors exist in each group. If configural invariance fails, no further testing is meaningful.

Level 2 — Metric Invariance (Weak Invariance)

Factor loadings are constrained to be equal across groups:

$H_{\text{metric}}: \boldsymbol{\Lambda}_1 = \boldsymbol{\Lambda}_2 = \dots = \boldsymbol{\Lambda}_G$

Metric invariance is required for comparing factor covariances and correlations across groups. If loadings differ across groups, the constructs are not measured on the same scale.

Test: $\Delta\chi^2_{\text{metric vs. config}}$ with $\Delta df = (m-1)(G-1)p/m$ (approximately).

Level 3 — Scalar Invariance (Strong Invariance)

Factor loadings and indicator intercepts are constrained to be equal across groups:

$H_{\text{scalar}}: \boldsymbol{\Lambda}_1 = \boldsymbol{\Lambda}_2, \quad \boldsymbol{\tau}_1 = \boldsymbol{\tau}_2$

Where $\boldsymbol{\tau}$ is the vector of item intercepts. Scalar invariance is required for comparing latent factor means across groups.

Test: $\Delta\chi^2_{\text{scalar vs. metric}}$ with $\Delta df = p(G-1)$ (approximately).

Level 4 — Strict Invariance

Factor loadings, intercepts, and error variances are constrained to be equal across groups:

$H_{\text{strict}}: \boldsymbol{\Lambda}_1 = \boldsymbol{\Lambda}_2, \quad \boldsymbol{\tau}_1 = \boldsymbol{\tau}_2, \quad \boldsymbol{\Theta}_{\epsilon,1} = \boldsymbol{\Theta}_{\epsilon,2}$

Strict invariance is rarely required in practice but is necessary for comparing observed (manifest) variable means across groups.

13.3 Evaluating Invariance

For each step, compare fit between the constrained and unconstrained model:

💡 Because $\Delta\chi^2$ is sensitive to sample size, the $\Delta\text{CFI} \leq -0.010$ criterion (Cheung & Rensvold, 2002) is increasingly preferred as a more robust indicator of meaningful invariance violations.

13.4 Partial Invariance

If full metric or scalar invariance fails, it is possible to test for partial invariance — freeing specific loadings or intercepts that are non-invariant while constraining the rest. Partial scalar invariance still allows latent mean comparisons if at least 2 intercepts per factor are invariant, though conclusions should be made with caution and caveats.

14. Worked Examples

Example 1: Single-Factor CFA — Unidimensionality of a Depression Scale

A researcher hypothesises that five items from a depression questionnaire measure a single unidimensional construct. Items are rated 1–5 (Never to Always).

Items:

• D1: I feel sad.
• D2: I feel hopeless about the future.
• D3: I have lost interest in activities I used to enjoy.
• D4: I feel worthless.
• D5: I have difficulty making decisions.

Sample: $n = 350$. Estimator: MLR.

Model Specification:

$\boldsymbol{\Lambda} = \begin{pmatrix} 1 \\ \lambda_{21} \\ \lambda_{31} \\ \lambda_{41} \\ \lambda_{51} \end{pmatrix}$

Free parameters: $q = 4$ (loadings) + $5$ (error variances) + $1$ (factor variance) = $10$.

$df = \frac{5(6)}{2} - 10 = 15 - 10 = 5$

Results:

Fit Statistics:

Parameter Estimates:

Reliability:

• CR = 0.88 → Good reliability
• AVE = 0.59 → AVE $\geq 0.50$: Convergent validity supported

Conclusion: The single-factor model fits well across all indices. All standardised loadings are $\geq 0.63$, and the factor explains between 40% and 76% of each item's variance. The five depression items form a reliable unidimensional scale. D5 (difficulty making decisions) has the weakest loading — it may reflect a broader cognitive dimension but is still acceptable.

Example 2: Two-Factor Correlated CFA — Physical and Psychological Well-Being

A researcher proposes that a 8-item well-being scale measures two correlated latent factors: Physical Well-Being (PWB) and Psychological Well-Being (PSYWB).

Items:

Sample: $n = 480$. Estimator: MLR.

Model Specification:

$\boldsymbol{\Lambda} = \begin{pmatrix} 1 & 0 \\ \lambda_{21} & 0 \\ \lambda_{31} & 0 \\ \lambda_{41} & 0 \\ 0 & 1 \\ 0 & \lambda_{62} \\ 0 & \lambda_{72} \\ 0 & \lambda_{82} \end{pmatrix}$, \quad $\boldsymbol{\Phi} = \begin{pmatrix} \phi_{11} & \phi_{12} \\ \phi_{12} & \phi_{22} \end{pmatrix}$

Free parameters: $q = 6$ (loadings) + $8$ (error variances) + $2$ (factor variances) + $1$ (factor covariance) = $17$.

$df = \frac{8(9)}{2} - 17 = 36 - 17 = 19$

Fit Statistics:

Standardised Parameter Estimates:

Factor Correlation:

$r_{\text{PWB, PSYWB}} = 0.54$ ($p < 0.001$)

The two factors are moderately correlated ($r = 0.54$), confirming they are related but distinct constructs. Using oblique factor specification (correlated factors) was therefore appropriate.

Reliability and Validity:

Discriminant Validity (Fornell-Larcker):

$r^2_{\text{PWB, PSYWB}} = 0.54^2 = 0.29$

• $\text{AVE}_{\text{PWB}} = 0.57 > 0.29$ ✅
• $\text{AVE}_{\text{PSYWB}} = 0.67 > 0.29$ ✅

Discriminant validity is supported — each factor shares more variance with its own indicators than with the other factor.

Conclusion: The two-factor CFA model fits the data well. Both PWB and PSYWB are reliable and valid latent factors with strong indicator loadings. The moderate inter-factor correlation confirms that physical and psychological well-being are related but distinct aspects of overall well-being.

Example 3: Testing an Alternative Model and Chi-Squared Difference Test

Using the same data as Example 2, suppose a reviewer suggests that all 8 items measure a single general Well-Being factor. We can formally test this using a $\Delta\chi^2$ test.

Model A: Two-factor correlated model (from Example 2)

• $\chi^2_A(19) = 28.41$, CFI = 0.983, RMSEA = 0.033

Model B: Single-factor model (all 8 items → 1 factor)

• $\chi^2_B(20) = 98.71$, CFI = 0.881, RMSEA = 0.098

Chi-Squared Difference Test:

$\Delta\chi^2 = 98.71 - 28.41 = 70.30$

$\Delta df = 20 - 19 = 1$

$p < 0.001$

The $\Delta\chi^2(1) = 70.30$ is highly significant. The two-factor model fits significantly better than the single-factor model. The hypothesis of a single general well-being factor is rejected in favour of the two-factor structure.

15. Common Mistakes and How to Avoid Them

Mistake 1: Specifying the CFA Model Based on EFA from the Same Sample

Problem: Running EFA on a dataset to discover the factor structure, then running CFA on the same dataset to "confirm" it is circular — the model will fit well simply because it was built from that data. This inflates apparent model fit and provides no genuine confirmation.
Solution: Use independent samples for EFA (development sample) and CFA (validation sample). If only one sample is available, randomly split it in half and use each half for one analysis.

Mistake 2: Relying Solely on the Chi-Squared Test

Problem: The chi-squared test is excessively sensitive to sample size. With $n > 500$, almost any model is rejected; with $n < 100$, even poorly fitting models are not rejected.
Solution: Always report a battery of fit indices — at minimum: $\chi^2/df$, CFI, RMSEA, and SRMR. Make model acceptance decisions based on convergent evidence across multiple indices.

Mistake 3: Over-Modifying the Model Based on Modification Indices Alone

Problem: Systematically freeing parameters with the largest modification indices without theoretical justification is a form of capitalising on chance. The resulting model is essentially an exploratory model in disguise, and fit is spuriously improved.
Solution: Only free parameters that have both large MIs AND theoretical justification. Every modification should be explainable by a substantive argument, not just a statistical one. Clearly report all modifications and treat the modified model as exploratory.

Mistake 4: Ignoring Non-Convergence or Inadmissible Solutions

Problem: Treating estimation outputs as valid when the algorithm has not converged or has produced Heywood cases (negative error variances or correlations $> 1$).
Solution: Always check for convergence warnings. If a Heywood case occurs, investigate the model specification — the problem is usually too few indicators per factor, near-perfect correlations between factors, or too many factors. Do not report or interpret inadmissible solutions.

Mistake 5: Comparing Non-Nested Models Using $\Delta\chi^2$

Problem: Using the chi-squared difference test to compare models that are not nested (one is not a restricted version of the other). The $\Delta\chi^2$ test is only valid for nested models.
Solution: Use AIC or BIC to compare non-nested models. Use $\Delta\chi^2$ only for nested model comparisons.

Mistake 6: Forgetting to Set the Scale of the Latent Factor

Problem: Failing to impose a metric constraint (either fixing one loading to 1 or fixing the factor variance to 1) results in an under-identified model that cannot be estimated.
Solution: Always apply one of the two scaling approaches for each latent factor. The DataStatPro application applies the marker variable constraint automatically by default.

Mistake 7: Using Only 2 Indicators Per Factor

Problem: A factor with exactly 2 indicators is just-identified when the factor variance is fixed — no degrees of freedom are available to test the model, and the factor score estimate is unreliable.
Solution: Use at least 3 indicators per factor, ideally 4 or more. Three indicators is the minimum for an over-identified single-factor model.

Mistake 8: Ignoring Measurement Invariance When Comparing Groups

Problem: Comparing latent factor means or covariances across groups (e.g., men vs. women) without testing whether the measurement model is invariant across those groups.
Solution: Always conduct a formal measurement invariance analysis before making cross-group comparisons. At minimum, metric invariance must hold for comparing factor covariances; scalar invariance must hold for comparing factor means.

Mistake 9: Treating Standardised and Unstandardised Solutions Interchangeably

Problem: Reporting unstandardised loadings as if they were standardised (or vice versa), leading to incorrect interpretations (e.g., interpreting a raw loading of 2.5 as a "very strong correlation," which is impossible for a standardised loading).
Solution: Always clearly label which solution is being reported. For interpretation of loading strength, use the standardised solution. For formal hypothesis testing, use the unstandardised solution.

Mistake 10: Ignoring Composite Reliability and AVE

Problem: Reporting only factor loadings and fit indices without assessing whether the factors are reliably measured and distinguish from each other (convergent and discriminant validity).
Solution: Always compute and report CR, AVE, and the Fornell-Larcker discriminant validity criterion as part of a comprehensive CFA report.

16. Troubleshooting

17. Quick Reference Cheat Sheet

Core Equations