Confirmatory Factor Analysis (CFA): A Complete Guide

Confirmatory factor analysis (CFA) is a statistical technique used to test whether a set of observed variables (survey items, test scores, indicators) fit a measurement model that was specified in advance, based on theory or prior research. That single feature — the model is fixed before the data are analyzed — is what separates CFA from exploratory factor analysis (EFA), and it is the source of most confusion between the two. In EFA, the number of factors and which items load on which factor are discovered from the data. In CFA, the researcher specifies which items belong to which factor, how many factors exist, and (often) which factors correlate with one another, and then tests whether the data are consistent with that structure.

CFA is a special case of structural equation modeling (SEM) restricted to the measurement part of a model — the relationships between latent variables and their observed indicators — without the structural (regression) paths between latent variables that a full SEM would add. It is the standard tool for validating a scale or questionnaire’s factor structure in a new sample, confirming that a previously developed instrument still behaves the way it did in the original development study, and establishing that a set of items can legitimately be treated as a single composite score before that score is used in further analysis.

CFA vs. EFA: Why the Distinction Matters

The core difference is confirmatory versus exploratory logic, not a difference in the underlying factor-analytic machinery:

• EFA is used when there is no strong prior theory about how many factors underlie a set of items or which items belong together. It lets the data suggest a structure — typically during initial scale development, when a researcher has generated a pool of candidate items and needs to see how they group empirically.
• CFA is used when a structure already exists — from a prior EFA, from established theory, or from a previously published and validated instrument — and the question is whether that specific structure holds in a new sample, population, language, or context.

A well-established methodological rule follows directly from this: EFA and CFA should not be run on the same sample. Using EFA to find a structure and then immediately running CFA on the same data to “confirm” it is circular — the CFA is almost guaranteed to fit well because the factor structure was extracted from that exact data, capitalizing on sample-specific noise rather than testing anything independent. The standard practice is to split the sample (commonly via random split-half) and run EFA on one half and CFA on the other, or to run the EFA on one sample and reserve CFA for a genuinely separate sample or a later replication study. If only one sample is available and no prior theory exists, that is a signal to run EFA and stop there, being explicit that the structure is exploratory and needs confirmation elsewhere.

Specifying a CFA Model

A CFA model consists of several building blocks:

• Latent variables (factors) — the unobserved constructs the researcher is trying to measure (e.g., “anxiety,” “job satisfaction,” “instrument usability”). They cannot be measured directly, which is the entire reason a measurement model is needed.
• Observed indicators — the actual items, test scores, or ratings collected from participants, each hypothesized to reflect one (or occasionally more than one) latent factor.
• Factor loadings — the estimated strength of the relationship between each indicator and the latent factor it belongs to, analogous to a regression coefficient predicting the observed item from the unobserved factor.
• Error (residual) terms — the portion of each indicator’s variance not explained by the latent factor, capturing measurement error and anything specific to that one item.
• Correlated factors — in most applied CFA models, latent factors are allowed to correlate with one another rather than being forced orthogonal (uncorrelated), since psychological and organizational constructs are rarely independent in reality.
• Correlated errors — occasionally two indicators’ error terms are allowed to correlate, usually because they share wording, method, or some other source of overlap not accounted for by the latent factor. This should be theoretically justified, not added purely to improve fit (see the section on modification indices below).

Identification

Before a CFA model can be estimated, it must be identified — there must be enough known information (the observed variances and covariances) to solve for the unknown parameters (loadings, factor variances, error variances) uniquely. Two mechanical requirements govern this:

• The t-rule: the number of freely estimated parameters cannot exceed the number of unique elements in the observed variance-covariance matrix (p(p+1)/2 for p indicators). This is a necessary but not sufficient condition for identification.
• The three-indicator rule: a standard, sensible practice is that every latent factor should have at least three observed indicators (two-indicator factors can be identified but are fragile and more sensitive to specification error; single-indicator factors require fixing measurement error externally).

A further, separate requirement is scaling the latent variable, which has no inherent metric of its own. This is done one of two equivalent ways: fixing one indicator’s loading to 1 (the “marker variable” or “reference indicator” method, which gives the factor the scale of that one item), or fixing the latent factor’s variance to 1 (the “standardized” method, which puts all loadings on a correlation-like metric). Most CFA software defaults to the marker-variable approach; both give mathematically equivalent model fit.

Estimation: ML, MLR, and WLSMV

The default estimator in most CFA software is maximum likelihood (ML), which assumes the observed indicators are continuous and multivariate normal. In practice, this assumption is routinely violated — and two robust alternatives are standard practice depending on why it fails:

• MLR (robust/Huber-White maximum likelihood) — used when indicators are continuous but non-normally distributed (skewed or kurtotic). MLR produces the same point estimates as ML but corrects the standard errors and the chi-square test statistic (the Satorra-Bentler correction) for non-normality, without requiring a different estimation approach.
• WLSMV (weighted least squares with mean- and variance-adjustment) — used when indicators are ordinal or categorical, which is the typical case for Likert-type survey items. WLSMV treats each item as a discretized version of an underlying continuous variable and estimates polychoric (rather than Pearson) correlations among items before fitting the model.

A common and consequential mistake is running ordinary ML on Likert-scale data treated as if it were continuous. With five or more roughly symmetric response categories the practical distortion is often modest, but with fewer categories, skewed distributions, or ordinal data with only two or three levels, ML on Likert data understates factor loadings and distorts fit indices, typically making a genuinely acceptable model look worse than it is. WLSMV (or a robust ML variant applied to polychoric correlations) is the methodologically preferred choice for ordinal indicators, and is what most published psychometric validation studies now use for Likert-based instruments.

Evaluating Model Fit

No single statistic determines whether a CFA model “fits.” The convention is to report a small set of complementary indices together:

• Chi-square (χ²) test — tests the null hypothesis that the model-implied covariance matrix equals the population covariance matrix exactly. A non-significant χ² technically indicates good fit, but the test is well known to over-reject at large sample sizes: with a large enough N, even trivial, substantively unimportant discrepancies between the model and the data become statistically significant. For this reason χ² is reported (often alongside its degrees of freedom, and sometimes the χ²/df ratio) but is rarely treated as the deciding criterion on its own in applied work.
• CFI (Comparative Fit Index) and TLI (Tucker-Lewis Index) — incremental fit indices that compare the fit of the specified model to a baseline (null) model in which all variables are uncorrelated. Conventional guidance treats values at or above .95 as indicating good fit.
• RMSEA (Root Mean Square Error of Approximation) — a “badness of fit” index that estimates how far the model is from the population, per degree of freedom, with a built-in penalty for model complexity. Conventional guidance treats values at or below .06 as indicating good fit, and RMSEA is typically reported with a 90% confidence interval.
• SRMR (Standardized Root Mean Square Residual) — the average standardized discrepancy between the observed and model-implied correlations. Conventional guidance treats values at or below .08 as indicating good fit.

These specific cutoffs (CFI ≥ .95, RMSEA ≤ .06, SRMR ≤ .08) trace back to Hu and Bentler’s widely cited 1999 simulation study in Structural Equation Modeling (Hu, L. T., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1), 1–55), which recommended a “two-index strategy” — reporting CFI together with either RMSEA or SRMR — based on which combinations best distinguished correctly specified from misspecified models across their simulated conditions.

It is important to be explicit about what these numbers are and are not. Hu and Bentler’s simulations used continuous, normally distributed indicators, specific sample sizes, and small-to-moderate models; the cutoffs do not automatically transfer to every combination of estimator, indicator type, model size, and sample size a researcher might encounter. They are widely used conventions born from one influential simulation study, not universal laws of model fit, and methodologists have repeatedly cautioned against treating them as a rigid pass/fail checklist. Chasing a specific cutoff by mechanically adding parameters until CFI crosses .95 is a well-documented way to produce a model that fits the sample but does not reflect a defensible theoretical structure — the same problem addressed directly in the next section.

Modification Indices and the Risk of Data-Driven Respecification

Most CFA software reports modification indices (MIs) — an estimate of how much the model χ² would drop if a currently fixed parameter (most often a correlated error, or a cross-loading of an item onto a factor it was not specified to belong to) were freed and estimated instead. Because a large MI tells you precisely which single change would most improve fit, it is tempting to add that parameter, re-run the model, and repeat until the fit indices clear their conventional thresholds.

This practice carries a real risk that CASRAI treats as a genuine research-integrity concern, not just a technical caveat: respecifying a model purely because a modification index suggests it, with no theoretical justification for why that specific error correlation or cross-loading should exist, is a form of overfitting. The revised model is now partly fit to sampling-specific noise in that one dataset rather than to a structure that will replicate. A model modified this way and then presented as if it had been confirmatory from the start misrepresents what was actually tested — the same underlying problem as HARKing (hypothesizing after results are known) in other areas of quantitative research.

The defensible practice is: (1) only add a modification that has a substantive, a priori justification (for example, two items that share near-identical wording, which plausibly correlate for reasons beyond the latent factor); (2) disclose every modification made, and why, in the reporting; and (3) treat a model that required substantial post hoc modification as exploratory rather than confirmatory, ideally cross-validated in an independent sample before being relied upon.

Measurement Invariance

Before comparing latent factor means, variances, or structural relationships across groups — different countries, languages, time points, demographic groups, or experimental conditions — a researcher first has to establish that the instrument is measuring the construct the same way in each group. This is measurement invariance (also called measurement equivalence) testing, and it proceeds through a standard, increasingly restrictive sequence of nested models, each compared to the one before it:

1. Configural invariance — the same factor structure (same number of factors, same pattern of which items load on which factor) holds across groups, with all parameters estimated freely in each group. This is the baseline model against which every later step is compared.
2. Metric (weak) invariance — factor loadings are constrained to be equal across groups. Establishing metric invariance justifies comparing relationships involving the latent variable (e.g., correlations, regression paths) across groups.
3. Scalar (strong) invariance — item intercepts are additionally constrained to be equal across groups, on top of the equal loadings from metric invariance. Scalar invariance is the level required to justify comparing latent factor means across groups — without it, an apparent group difference in the observed scores could reflect a difference in how the items function rather than a genuine difference in the underlying construct.
4. Strict (residual) invariance — item residual (error) variances are additionally constrained to be equal. This is the most restrictive level and is not required for most substantive group comparisons; it is more relevant when the researcher wants to treat observed and latent scores as fully interchangeable across groups.

Each step is typically evaluated by comparing model fit to the previous, less restrictive step, using a χ² difference test and/or a change in CFI (a commonly used rough guideline is a drop in CFI of no more than .01 between nested models, alongside corresponding changes in RMSEA), rather than judging each model’s absolute fit in isolation. This nested-model logic for testing invariance is most closely associated with methodological syntheses such as Vandenberg and Lance’s (2000) review of the practice in organizational research, building on earlier multi-group CFA work. Comparing latent means across groups without first establishing at least scalar invariance is a common and consequential error — it risks attributing a measurement artifact to a substantive group difference.

Reliability and Validity from a CFA

A CFA model provides more precise reliability and validity estimates than simpler approaches like Cronbach’s alpha, because it accounts for each item’s actual estimated relationship to the latent factor rather than treating all items as equally good indicators. See CASRAI’s guide to Cronbach’s alpha for how the two approaches differ and why alpha assumes essential tau-equivalence that a CFA does not require.

• Composite reliability (CR), sometimes called construct reliability, and McDonald’s omega are both computed from the standardized factor loadings and error variances of a CFA model, and are generally preferred over Cronbach’s alpha when a scale’s items are not assumed to be equally strong indicators of the underlying factor.
• Average variance extracted (AVE) — the average amount of variance a latent factor explains in its own indicators, relative to the variance due to measurement error. AVE is used as a convergent-validity check (a common informal benchmark is AVE ≥ .50, meaning the factor explains more variance in its indicators than error does) and feeds into discriminant-validity tests.
• Discriminant validity asks whether factors that are supposed to be distinct constructs are, in fact, statistically distinguishable from one another. The traditional test is the Fornell-Larcker criterion (Fornell & Larcker, 1981): a factor’s AVE should exceed the squared correlation between that factor and any other factor in the model. A more recently developed and increasingly used alternative is the heterotrait-monotrait ratio of correlations (HTMT) (Henseler, Ringle, & Sarstedt, 2015), which several methodological studies have found detects discriminant validity problems the Fornell-Larcker criterion misses, particularly with highly correlated factors.

See CASRAI’s guide to types of validity in research for how construct validity, established in part through CFA, relates to the broader family of validity concepts a study needs to address.

Sample Size

CFA, like other SEM techniques, is a large-sample method, and there is no single sample size that is correct for every model. Commonly cited heuristics include a minimum of roughly 200 observations for models of typical size and complexity, or ratios such as 10 (or, more conservatively, 20) observations per estimated parameter. These figures are useful as a first-pass sanity check, but they are rough heuristics, not formal power calculations — actual power to detect model misspecification, and the stability of parameter estimates, depends on the number of indicators per factor, the size of the factor loadings, the number of factors, the estimator used, and the amount of missing data, not on sample size alone. Where feasible, a model-specific Monte Carlo power analysis (simulating data under the hypothesized model and checking how often the fitting procedure recovers stable, well-identified estimates at a given N) is a more defensible basis for a sample-size justification than citing a generic N:parameter ratio.

Reporting a CFA

A complete, reproducible CFA report typically includes:

• The full model specification — which items load on which factors, which factors are allowed to correlate, and any correlated error terms — ideally as a path diagram or an explicit list, not just prose.
• The estimator used (ML, MLR, WLSMV, etc.) and the software and version.
• Sample size and how missing data were handled.
• All conventional fit indices together (χ² and df, CFI, TLI, RMSEA with its confidence interval, SRMR), not a single favorable statistic selected after the fact.
• Standardized factor loadings for every indicator, and composite reliability/AVE if reported.
• Any modification made to the originally specified model, with the justification for it, reported transparently as noted above — not silently incorporated into a “final” model presented as if it had been the a priori specification.

Software

• R (lavaan) — a free, widely used, and well-documented package for CFA and SEM in R, with syntax that closely mirrors standard SEM notation and native support for ML, MLR, and WLSMV estimation.
• Mplus — a commercial, specialist SEM program with broad support for categorical, count, and mixture modeling extensions beyond standard CFA, widely used in psychology and education research.
• SPSS AMOS — a commercial add-on to SPSS with a graphical path-diagram interface, commonly used where a point-and-click SEM tool is preferred over syntax-based specification.

Frequently Asked Questions

What is the difference between confirmatory factor analysis and exploratory factor analysis?

EFA discovers a factor structure from the data with no fixed prior specification; CFA tests whether a specific, previously specified factor structure fits new data. EFA is typically used earlier, during scale development; CFA is used to confirm a structure in a new sample or context, and the two should not be run on the same sample.

What are good CFA fit indices?

The most commonly cited conventions, from Hu and Bentler (1999), are CFI and TLI at or above .95, RMSEA at or below .06, and SRMR at or below .08 — reported together rather than relying on any single index, and treated as guidelines rather than fixed pass/fail rules.

Why shouldn’t I chase modification indices until my model fits?

Adding parameters purely because a modification index flags them, with no theoretical justification, fits the model to noise specific to that one sample rather than to a structure likely to replicate. Any such change should be theoretically justified, disclosed, and treated as exploratory rather than confirmatory.

What is measurement invariance and why does it matter?

Measurement invariance testing establishes whether an instrument measures a construct the same way across groups, through a sequence of increasingly restrictive models (configural, metric, scalar, strict). At minimum, scalar invariance is required before comparing latent factor means across groups, since without it an apparent group difference could reflect how the items function rather than a real difference in the underlying construct.

Can I use Cronbach’s alpha instead of a CFA-based reliability estimate?

Alpha is a simpler, widely reported statistic, but it assumes all items are equally strong indicators of the same construct (essential tau-equivalence) and does not itself confirm unidimensionality. Composite reliability and omega, both computed from a CFA’s factor loadings, do not require that assumption and are generally preferred once a CFA has been run. See CASRAI’s Cronbach’s alpha guide for the full comparison.

What sample size do I need for a CFA?

Common heuristics suggest a minimum of roughly 200 observations, or 10-20 observations per estimated parameter, but these are rough starting points, not formal power calculations. Actual requirements depend on the number of factors and indicators, loading strength, and estimator used.

Related CASRAI Guides

Confirmatory factor analysis sits within CASRAI’s broader research methods coverage. See also Cronbach’s alpha and types of validity in research for the reliability and validity concepts a CFA is often used to establish; correlation coefficient and descriptive statistics for the foundational statistics a CFA builds on; regression analysis, which shares much of its underlying logic with the factor model; and questionnaire design for how the items that go into a CFA are constructed in the first place.