📋 Example — Cognitive Test with 8 Subtests
300 people complete a cognitive test with 8 subtests: V1–V4 assess verbal ability, S1–S4 spatial reasoning. It's unclear in advance how many latent factors underlie this and which items belong to which factor — that's exactly what exploratory factor analysis (EFA) is meant to uncover. The sliders on the left set the true measurement model (unknown in practice): loading λ, cross-loading λx, factor correlation φ — so you can directly observe how changing the model affects the scree plot, loadings, and the reliability of the total score.
Approach — From Items to Factors
①Items & measurement model. 8 items (V1–V4, S1–S4), N = 300. What is set here via sliders would, in practice, have to be estimated from real item responses.
②Correlation matrix R. From the item responses (or here: from the measurement model) follows how strongly each pair of items is related. → view reduced matrix Rr
③Factor extraction — eigenvalues & scree plot. How many factors explain the shared variance? The Kaiser criterion (eigenvalue > 1) is common but rather permissive — parallel analysis (comparison with eigenvalues of random, uncorrelated data of the same size) is more robust. → view scree plot
④Rotation & loading matrix. The unrotated solution is rotated (varimax/oblique) to obtain an interpretable simple structure — which item loads on which factor? The table and biplot show the same information twice, once as numbers and once spatially. → loading table · → biplot
⑤Final communality h² & reliability. The loading table already shows how much variance of each item is jointly explained by the factors. What follows from this for the reliability (α, ω) of the total score? → view reliability
Correlation Matrix — Reduced Rr (Final h² on the Diagonal)
Each cell is the (measurement-model-implied) correlation between two items — teal = positive, orange = negative. The gray-shaded diagonal is not a correlation value but h². In practice, h² is not known in advance: PAF starts with an initial estimate — usually the squared multiple correlation SMC, i.e. the R² from the joint regression of each item on all p−1 remaining items (not on a single other item) — and improves it iteratively until h² converges. This tool skips the estimation entirely: since the population model (Λ, Φ) is known via the sliders, the model-implied ("true") communality is placed directly on the diagonal.
Scree Plot — Eigenvalues (from R)
Kaiser criterion (dashed line at eigenvalue = 1): keep factors with eigenvalue > 1. Common, but notoriously too permissive — it barely depends on the number of items or sample size and often overestimates the number of factors. Parallel analysis (Horn, 1965) is more robust: many datasets of uncorrelated random variables of the same size (n people, p items) are simulated, and only factors whose eigenvalue exceeds the 95th percentile of the simulated eigenvalues are kept. This removes factors that, in a real scree plot, sit just above 1 and are actually sampling noise.
Factor Loading Table & Final Communalities (PAF)
Factor Loading Biplot (F1 vs. F2)
How to Read the Table & Biplot
Both show the same information twice — once as numbers, once spatially. The table lists, for each of the 8 items, the loadings on F1 and F2 as well as h² (proportion of variance jointly explained by both factors) and u² = 1−h² (remainder). SSL (sum of squared loadings) and % var. in the last row show how much total variance each factor explains across all 8 items.
In the biplot, every point is an item, and its coordinates are exactly the same loadings (F1 = x, F2 = y) — the same numbers as in the table, just shown as position instead of number. Near the F1 axis = loads almost purely on F1; near the F2 axis = almost purely on F2; near the origin = low loadings on both factors (poorly explained item, low communality). Distance from the origin grows with h². After varimax rotation, well-separated items cluster near the axes (simple structure); items between the axes have notable cross-loadings on both factors. Under oblique rotation, the factor axes sit at an angle of arccos(φ) to each other instead of at a right angle — the drawn axes themselves then show the factor correlation.
Only V1 and S1 are shown as representatives of their group — the remaining items sit (in this simplified model) at exactly the same positions.
In the biplot, every point is an item, and its coordinates are exactly the same loadings (F1 = x, F2 = y) — the same numbers as in the table, just shown as position instead of number. Near the F1 axis = loads almost purely on F1; near the F2 axis = almost purely on F2; near the origin = low loadings on both factors (poorly explained item, low communality). Distance from the origin grows with h². After varimax rotation, well-separated items cluster near the axes (simple structure); items between the axes have notable cross-loadings on both factors. Under oblique rotation, the factor axes sit at an angle of arccos(φ) to each other instead of at a right angle — the drawn axes themselves then show the factor correlation.
Only V1 and S1 are shown as representatives of their group — the remaining items sit (in this simplified model) at exactly the same positions.
Reliability of the Total Score — Derived from the EFA Structure
ωh
Omega hierarchical (McDonald)
—
Here, reliability is a byproduct of the factor structure (ωh grows with the
factor correlation φ). The α-vs-ω question itself — when α underestimates or feigns
unidimensionality — is explored in depth by its own tool:
→ Reliability: α vs. ω
Flashcards — Exploratory Factor Analysis
① EFA vs. PCA
Exploratory factor analysis (EFA) only explains the shared variance of the items — it uses a reduced correlation matrix with communalities h² on the diagonal. Principal component analysis (PCA) explains all variance (including error) and uses ones on the diagonal. This is a conceptually important difference.
The communality h² is the proportion of an item's variance explained by the common factors. There are two time points: the initial communality (starting estimate before extraction, usually the squared multiple correlation SMC — the R² from the joint regression of the item on all other items, not a single one) and the final communality (after extraction: h² = sum of squared loadings across all retained factors — what the factor solution actually accounts for). The table in this tool always shows the final h². The uniqueness u² = 1 − h² contains measurement error and item-specific variance.
③ Eigenvalues & Scree Plot
A factor's eigenvalue shows how much variance (in units of standardized items) it explains. The scree plot shows the eigenvalues sorted by size. The Kaiser criterion (eigenvalue > 1) and the scree elbow are the most common criteria for determining the number of factors.
④ Rotation & Simple Structure
Varimax rotates the orthogonal axes so that each item loads as highly as possible on exactly one factor (Thurstone's simple structure). Oblique allows factor correlation (phi > 0) — more realistic, but harder to interpret. Communalities do not change under rotation.
⑤ Cronbach's Alpha
Cronbach's alpha estimates internal consistency as a function of the average inter-item correlation. A common misconception: alpha is not a measure of unidimensionality. A 2-factor structure can have identical alpha to a 1-factor structure — if the mean inter-item correlation stays the same.
⑥ Omega Total & Hierarchical
Omega total accounts for all factors and is a better lower bound on reliability than alpha. Omega hierarchical (Schmid-Leiman) measures only the share of a higher-order general factor — it rises with both phi and the cross-loading, and is 0 only when both are 0 (truly two separate factors with no shared variance at all).