Confirmatory Factor Analysis (CFA)

Confirmatory factor analysis fits a model in which the researcher states beforehand which item measures which latent construct. Unlike the exploratory approach, the structure is not discovered but tested: the question is how well a model derived from theory or earlier work reproduces the observed covariance pattern. In the panel you do not write model syntax; you declare the factors and pick the items that load on each, and the measurement model is built from that. The result brings together item loadings, fit indices and criteria for construct validity.

• Does the original three-factor structure of an adapted scale hold in this sample?
• Do the items we developed separate into the two dimensions theory predicts?
• Are two closely related constructs genuinely distinct, or are they measured as a single dimension?
• Does the short form of a scale preserve the measurement structure of the long form?

• When the factor structure is specified in advance by theory or previous findings.
• When construct validity is examined in an adaptation or validation study.
• When a structure found through exploratory analysis needs confirming in an independent sample.
• When the soundness of the measurement model must be shown before modelling relationships between constructs.

• Indicators: item columns coded numerically and treated as continuous.
• At least two indicators per factor; an indicator can be assigned to only one factor.
• One row per participant; incomplete rows are removed before the analysis.

A model specified in advance

Which item loads on which factor comes from theory or earlier work, not from inspecting the data. Adjusting the model to the data and retesting it in the same sample turns a confirmatory study into an exploratory one.

Continuously measured indicators

Estimation assumes the indicators are continuous and approximately normal. With ordinal items that have few categories the estimates can be biased.

Adequate sample size

Common guidance suggests ten observations per item or at least two hundred participants; fit indices become unstable in small samples.

An identified model

The number of free parameters must not exceed the observed covariances. Two-indicator factors and complex models can create identification problems.

• A model specified in advance: The panel does not check this assumption automatically; the researcher evaluates it.
• Continuously measured indicators: The panel requires numeric columns and stops with an explanation when labelled columns cannot be converted to numbers.
• Adequate sample size: The engine only requires a minimum number of valid rows. It does not enforce the sample-size guidance from the literature, which remains the researcher's judgement.
• An identified model: At least two indicators per factor is enforced, as is the rule that an indicator belongs to one factor only; when estimation fails the engine reports the error directly.

1. 1Upload the data file containing the item columns.
2. 2On the Variable tab confirm the items are read as numeric and that reverse-coded items have been corrected.
3. 3On the Analysis tab choose Confirmatory Factor Analysis.
4. 4In the parameter form define the factors and select the items that load on each one.
5. 5Run the analysis. Item loadings, fit indices, composite reliability, average variance extracted and discriminant validity open as separate sections.
6. 6Download the results as Excel; the credits card names the library and its version.

Item loadings

Loading estimates for the items of each factor. They are returned unstandardised; standardised values are not requested separately.

Fit indices

Chi-square with its degrees of freedom and p value, plus CFI, TLI, RMSEA, SRMR, AIC and BIC.

Fit interpretation

The indices are labelled against thresholds: 0.95 and above excellent and 0.90 and above acceptable for CFI and TLI; 0.05 and below excellent and 0.08 and below acceptable for RMSEA and SRMR.

Composite reliability

A reliability measure computed from the loadings of each factor, grounded in the measurement model rather than in item covariances alone.

Average variance extracted

How much of the variance in its indicators each factor accounts for, used when judging convergent validity.

Discriminant validity table

The Fornell-Larcker criterion, comparing the squared factor correlation with the average variance extracted of the factors involved.

Composite reliability and average variance extracted

In this model the counterpart of an effect size is the set of construct quality measures. The literature commonly cites 0.70 for composite reliability and 0.50 for average variance extracted; a factor below those levels suggests its items measure the construct weakly.

Item loadings

Higher loadings indicate a stronger relation between item and factor. Because the estimates are unstandardised, the scales of the items have to be taken into account before comparing them against thresholds.

No confidence interval is returned for the loadings, the fit indices or the reliability measures. Modification indices and bootstrapped intervals are not computed either, so any model revision should be driven by theory and preferably tested on a new sample.

The numbers below are a representative example, not data from a real study or a real user.

Research question

Does a twelve-item job satisfaction scale, assumed to have three dimensions, keep that structure in this sample?

Variables

First factor: intrinsic satisfaction (four items) · Second factor: relationship with the supervisor (four items) · Third factor: pay and benefits (four items)

Example result

n = 412. Chi-square(51) = 118.4, p < 0.001; CFI = 0.957; TLI = 0.944; RMSEA = 0.057; SRMR = 0.048. Composite reliability was 0.84, 0.88 and 0.79 and average variance extracted 0.57, 0.64 and 0.51. The Fornell-Larcker criterion held for all three factor pairs.

Interpretation

The significant chi-square is expected in a sample of this size. CFI and SRMR are good while TLI and RMSEA sit in the acceptable range, so the three-factor structure is consistent with the data and the factors are distinct from one another. Because the loadings are unstandardised, no item-level threshold comparison was made.

Research question: Does the assumption that the items belong to two separate constructs fit the data?

Latent factors and their items

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Computation credits: scipy 1.18.0 · statsmodels 0.14.6 · scikit-learn 1.9.0 · numpy 2.5.1 · pandas 3.0.5 · semopy 2.3.11 · 89fc29a · Data seed: 20260915

The three-factor measurement model showed acceptable fit, chi-square(51) = 118.4, p < .001, CFI = .957, TLI = .944, RMSEA = .057, SRMR = .048, with composite reliability between .79 and .88.

An example sentence close to APA style; the numbers are representative.

• Not appropriate when the structure is unknown; exploratory factor analysis suits discovery.
• Revising the model against the data and retesting it in the same sample undermines the confirmatory claim.
• Because the panel produces no modification indices, locating the source of poor fit takes additional work.
• Estimates can be biased with ordinal items that have few categories, and the panel offers no choice of estimator for such cases.
• Fit indices are unstable in small samples, so strong conclusions about the structure should be avoided.
• Causal paths between constructs are not tested in this model; structural equation modelling covers that.

• Exploratory Factor Analysis (EFA)Why: When the factor structure is not yet known and the dimensions behind the items have to be discovered.
• Structural Equation Modeling (SEM)Why: When the measurement model has been confirmed and the paths between constructs need testing.
• McDonald's OmegaWhy: When only reliability is in question and a full measurement model is not needed.

• Exploratory Factor Analysis (EFA)
• Structural Equation Modeling (SEM)
• Cronbach's Alpha
• McDonald's Omega

Chi-square is significant, should the model be rejected?

Chi-square is sensitive to sample size and turns significant on small deviations once there are a few hundred participants. The literature bases the decision not on it alone but on CFI, TLI, RMSEA and SRMR considered together.

The fit indices disagree, what should I do?

They look at different things: CFI and TLI measure comparative fit while RMSEA and SRMR summarise residual error. Report all of them, discuss which part of the model is weak and state the reasoning behind your decision.

Where do I get standardised loadings?

The panel returns unstandardised loadings and does not compute standardised values. If your report requires standardised loadings you have to derive them yourself or use another tool.

How many participants do I need?

The engine only enforces a minimum number of valid rows. Common guidance in the literature is ten observations per item or at least two hundred participants, and the requirement grows with model complexity.

• Brown, T. A. (2015). Confirmatory Factor Analysis for Applied Research
• Hu, L., & Bentler, P. M. (1999). Cutoff Criteria for Fit Indexes in Covariance Structure Analysis
• Fornell, C., & Larcker, D. F. (1981). Evaluating Structural Equation Models with Unobservable Variables and Measurement Error
• semopy documentation