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The core idea is that if a group of questionnaire items all measure the same latent construct — for example, anxiety — they will correlate with each other, and factor analysis will identify that shared variance as a distinct factor. Unlike observed variables, latent factors cannot be measured directly; they are inferred from the covariances of the items that load on them. 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For ordinal Likert items (≤ 4 categories), specify that polychoric correlations should be used instead of Pearson.",{"type":43,"tag":44,"props":380,"children":382},{"id":381},"interpreting-the-results",[383],{"type":49,"value":384},"Interpreting the Results",{"type":43,"tag":214,"props":386,"children":387},{},[388,404],{"type":43,"tag":218,"props":389,"children":390},{},[391],{"type":43,"tag":222,"props":392,"children":393},{},[394,399],{"type":43,"tag":226,"props":395,"children":396},{},[397],{"type":49,"value":398},"Output",{"type":43,"tag":226,"props":400,"children":401},{},[402],{"type":49,"value":403},"What it means",{"type":43,"tag":242,"props":405,"children":406},{},[407,423,439,455,471,487,503,519],{"type":43,"tag":222,"props":408,"children":409},{},[410,418],{"type":43,"tag":249,"props":411,"children":412},{},[413],{"type":43,"tag":56,"props":414,"children":415},{},[416],{"type":49,"value":417},"Factor loadings",{"type":43,"tag":249,"props":419,"children":420},{},[421],{"type":49,"value":422},"Correlation between each variable and each factor — values ≥ 0.40 define the factor",{"type":43,"tag":222,"props":424,"children":425},{},[426,434],{"type":43,"tag":249,"props":427,"children":428},{},[429],{"type":43,"tag":56,"props":430,"children":431},{},[432],{"type":49,"value":433},"Communality (h²)",{"type":43,"tag":249,"props":435,"children":436},{},[437],{"type":49,"value":438},"Proportion of each variable's variance explained by all factors — low h² (\u003C0.30) = item poorly explained",{"type":43,"tag":222,"props":440,"children":441},{},[442,450],{"type":43,"tag":249,"props":443,"children":444},{},[445],{"type":43,"tag":56,"props":446,"children":447},{},[448],{"type":49,"value":449},"Eigenvalue",{"type":43,"tag":249,"props":451,"children":452},{},[453],{"type":49,"value":454},"Variance extracted by each factor — Kaiser criterion: retain factors with eigenvalue > 1.0",{"type":43,"tag":222,"props":456,"children":457},{},[458,466],{"type":43,"tag":249,"props":459,"children":460},{},[461],{"type":43,"tag":56,"props":462,"children":463},{},[464],{"type":49,"value":465},"% Variance explained",{"type":43,"tag":249,"props":467,"children":468},{},[469],{"type":49,"value":470},"Fraction of total variance captured by each factor — report cumulative % for retained factors",{"type":43,"tag":222,"props":472,"children":473},{},[474,482],{"type":43,"tag":249,"props":475,"children":476},{},[477],{"type":43,"tag":56,"props":478,"children":479},{},[480],{"type":49,"value":481},"Scree plot",{"type":43,"tag":249,"props":483,"children":484},{},[485],{"type":49,"value":486},"Eigenvalue vs factor number — retain factors above the \"elbow\" (point of inflection)",{"type":43,"tag":222,"props":488,"children":489},{},[490,498],{"type":43,"tag":249,"props":491,"children":492},{},[493],{"type":43,"tag":56,"props":494,"children":495},{},[496],{"type":49,"value":497},"Rotated loading matrix",{"type":43,"tag":249,"props":499,"children":500},{},[501],{"type":49,"value":502},"Loadings after rotation — simpler structure, easier to interpret",{"type":43,"tag":222,"props":504,"children":505},{},[506,514],{"type":43,"tag":249,"props":507,"children":508},{},[509],{"type":43,"tag":56,"props":510,"children":511},{},[512],{"type":49,"value":513},"Factor correlation matrix",{"type":43,"tag":249,"props":515,"children":516},{},[517],{"type":49,"value":518},"For oblique rotation: correlations between factors — high r suggests factors are not truly distinct",{"type":43,"tag":222,"props":520,"children":521},{},[522,530],{"type":43,"tag":249,"props":523,"children":524},{},[525],{"type":43,"tag":56,"props":526,"children":527},{},[528],{"type":49,"value":529},"Factor scores",{"type":43,"tag":249,"props":531,"children":532},{},[533],{"type":49,"value":534},"Each observation's estimated position on each latent factor — for downstream analysis",{"type":43,"tag":44,"props":536,"children":538},{"id":537},"example-prompts",[539],{"type":49,"value":540},"Example Prompts",{"type":43,"tag":214,"props":542,"children":543},{},[544,560],{"type":43,"tag":218,"props":545,"children":546},{},[547],{"type":43,"tag":222,"props":548,"children":549},{},[550,555],{"type":43,"tag":226,"props":551,"children":552},{},[553],{"type":49,"value":554},"Scenario",{"type":43,"tag":226,"props":556,"children":557},{},[558],{"type":49,"value":559},"What to type",{"type":43,"tag":242,"props":561,"children":562},{},[563,580,597,614,631,648,664,681],{"type":43,"tag":222,"props":564,"children":565},{},[566,571],{"type":43,"tag":249,"props":567,"children":568},{},[569],{"type":49,"value":570},"Basic EFA",{"type":43,"tag":249,"props":572,"children":573},{},[574],{"type":43,"tag":253,"props":575,"children":577},{"className":576},[],[578],{"type":49,"value":579},"extract factors using principal axis factoring; 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report factor structure and reliability together",{"type":43,"tag":222,"props":682,"children":683},{},[684,689],{"type":43,"tag":249,"props":685,"children":686},{},[687],{"type":49,"value":688},"Polychoric EFA",{"type":43,"tag":249,"props":690,"children":691},{},[692],{"type":43,"tag":253,"props":693,"children":695},{"className":694},[],[696],{"type":49,"value":697},"items are 4-point Likert scales; use polychoric correlations for EFA instead of Pearson; compare loading patterns",{"type":43,"tag":44,"props":699,"children":701},{"id":700},"assumptions-to-check",[702],{"type":49,"value":703},"Assumptions to Check",{"type":43,"tag":705,"props":706,"children":707},"ul",{},[708,718,742,752,762],{"type":43,"tag":154,"props":709,"children":710},{},[711,716],{"type":43,"tag":56,"props":712,"children":713},{},[714],{"type":49,"value":715},"Sample size",{"type":49,"value":717}," — EFA requires at least n = 200 observations for stable results; n ≥ 300 is preferred; rules of thumb suggest 5–10 observations per variable; with small n, factor solutions are unstable and cross-validation on a holdout sample is essential",{"type":43,"tag":154,"props":719,"children":720},{},[721,726,728,733,735,740],{"type":43,"tag":56,"props":722,"children":723},{},[724],{"type":49,"value":725},"Factorability of the correlation matrix",{"type":49,"value":727}," — run ",{"type":43,"tag":56,"props":729,"children":730},{},[731],{"type":49,"value":732},"Bartlett's test of sphericity",{"type":49,"value":734}," (null: correlation matrix = identity; should be significant, p \u003C 0.05) and ",{"type":43,"tag":56,"props":736,"children":737},{},[738],{"type":49,"value":739},"Kaiser-Meyer-Olkin (KMO) measure",{"type":49,"value":741}," (> 0.60 acceptable, > 0.80 good, > 0.90 excellent) before EFA; low KMO indicates the variables do not share enough common variance for factor analysis",{"type":43,"tag":154,"props":743,"children":744},{},[745,750],{"type":43,"tag":56,"props":746,"children":747},{},[748],{"type":49,"value":749},"Linear relationships",{"type":49,"value":751}," — EFA assumes linear relationships among variables; non-linear relationships (e.g. U-shaped item responses) will not be captured; inspect scatterplots of item pairs before running EFA",{"type":43,"tag":154,"props":753,"children":754},{},[755,760],{"type":43,"tag":56,"props":756,"children":757},{},[758],{"type":49,"value":759},"No multicollinearity",{"type":49,"value":761}," — items with near-perfect correlations (r > 0.90) cause matrix inversion problems; check for and remove or combine redundant items; conversely, items with r \u003C 0.15 with all others are unlikely to load on any factor",{"type":43,"tag":154,"props":763,"children":764},{},[765,770,772,777,779,784],{"type":43,"tag":56,"props":766,"children":767},{},[768],{"type":49,"value":769},"Rotation choice",{"type":49,"value":771}," — use ",{"type":43,"tag":56,"props":773,"children":774},{},[775],{"type":49,"value":776},"orthogonal rotation (varimax)",{"type":49,"value":778}," when factors are expected to be uncorrelated (independent dimensions); use ",{"type":43,"tag":56,"props":780,"children":781},{},[782],{"type":49,"value":783},"oblique rotation (oblimin, promax)",{"type":49,"value":785}," when factors are expected to correlate (e.g., anxiety and depression subfactors of a general distress scale); always check the factor correlation matrix after oblique rotation — if all factor correlations are \u003C 0.30, orthogonal rotation is appropriate",{"type":43,"tag":44,"props":787,"children":789},{"id":788},"related-tools",[790],{"type":49,"value":791},"Related Tools",{"type":43,"tag":52,"props":793,"children":794},{},[795,797,803,805,811,813,819,821,827],{"type":49,"value":796},"Use the ",{"type":43,"tag":191,"props":798,"children":800},{"href":799},"/tools/pca",[801],{"type":49,"value":802},"PCA — Principal Component Analysis",{"type":49,"value":804}," when you want to reduce dimensionality for visualization or downstream modeling rather than identify latent psychological constructs — PCA maximizes explained variance while EFA models the shared (common) variance structure. Use the ",{"type":43,"tag":191,"props":806,"children":808},{"href":807},"/tools/cronbachs-alpha",[809],{"type":49,"value":810},"Cronbach's Alpha Calculator",{"type":49,"value":812}," to assess the internal consistency reliability of each subscale identified by factor analysis — EFA reveals the factor structure; Cronbach's alpha quantifies how reliably each factor is measured. Use the ",{"type":43,"tag":191,"props":814,"children":816},{"href":815},"/tools/correlation-matrix",[817],{"type":49,"value":818},"Correlation Matrix Calculator",{"type":49,"value":820}," to inspect the raw inter-item correlations before EFA and verify that the matrix is factorable. Use the ",{"type":43,"tag":191,"props":822,"children":824},{"href":823},"/tools/partial-correlation",[825],{"type":49,"value":826},"Partial Correlation Calculator",{"type":49,"value":828}," to examine relationships between variables after controlling for the influence of extracted factors.",{"type":43,"tag":44,"props":830,"children":832},{"id":831},"frequently-asked-questions",[833],{"type":49,"value":834},"Frequently Asked Questions",{"type":43,"tag":52,"props":836,"children":837},{},[838,843,847,849,854],{"type":43,"tag":56,"props":839,"children":840},{},[841],{"type":49,"value":842},"What is the difference between exploratory and confirmatory factor analysis?",{"type":43,"tag":56,"props":844,"children":845},{},[846],{"type":49,"value":60},{"type":49,"value":848}," makes no prior assumptions about which variables load on which factors — it discovers the factor structure from the data. It is used in scale development to identify dimensions before a theory is fully formed. ",{"type":43,"tag":56,"props":850,"children":851},{},[852],{"type":49,"value":853},"Confirmatory factor analysis (CFA)",{"type":49,"value":855}," tests a specific pre-specified model (e.g., \"items 1–4 load on Factor 1, items 5–8 on Factor 2\") and reports fit statistics (RMSEA, CFI, SRMR) indicating how well the model fits the data. CFA requires structural equation modeling software (e.g., R's lavaan). The standard workflow in psychometrics is EFA on a development sample → CFA on an independent validation sample. Running CFA on the same data used for EFA is circular and will produce inflated fit statistics.",{"type":43,"tag":52,"props":857,"children":858},{},[859,864,866,871,873,877,879,884,886,891],{"type":43,"tag":56,"props":860,"children":861},{},[862],{"type":49,"value":863},"How do I decide how many factors to retain?",{"type":49,"value":865},"\nUse multiple criteria and look for convergence: (1) ",{"type":43,"tag":56,"props":867,"children":868},{},[869],{"type":49,"value":870},"Kaiser criterion",{"type":49,"value":872}," — retain factors with eigenvalue > 1.0 (tends to over-extract); (2) ",{"type":43,"tag":56,"props":874,"children":875},{},[876],{"type":49,"value":481},{"type":49,"value":878}," — retain factors above the point where the curve flattens (the \"elbow\"); (3) ",{"type":43,"tag":56,"props":880,"children":881},{},[882],{"type":49,"value":883},"Parallel analysis",{"type":49,"value":885}," — compare eigenvalues from your data to those from random data of the same size; retain factors whose eigenvalues exceed the random-data 95th percentile (most accurate method); (4) ",{"type":43,"tag":56,"props":887,"children":888},{},[889],{"type":49,"value":890},"Interpretability",{"type":49,"value":892}," — does each factor have a clear, coherent meaning? Can you name it? Solutions that are mathematically defensible but conceptually opaque should be reconsidered. If parallel analysis says 3 factors but only 2 are interpretable, report the 2-factor solution with justification.",{"type":43,"tag":52,"props":894,"children":895},{},[896,901,906,908,913,915,920],{"type":43,"tag":56,"props":897,"children":898},{},[899],{"type":49,"value":900},"What is varimax rotation and when should I use a different rotation?",{"type":43,"tag":56,"props":902,"children":903},{},[904],{"type":49,"value":905},"Varimax",{"type":49,"value":907}," is an orthogonal rotation that maximizes the variance of squared loadings within each factor, pushing loadings toward 0 or ±1 and minimizing cross-loadings. It produces the simplest, most interpretable loading pattern when factors are truly independent. Use varimax for personality dimensions, cognitive abilities subtests, or any domain where independence between factors is theoretically expected. Use ",{"type":43,"tag":56,"props":909,"children":910},{},[911],{"type":49,"value":912},"oblimin",{"type":49,"value":914}," or ",{"type":43,"tag":56,"props":916,"children":917},{},[918],{"type":49,"value":919},"promax",{"type":49,"value":921}," (oblique rotations) when factors are expected to correlate — for example, anxiety and depression share substantial variance and their factors will be correlated even after rotation. The factor correlation matrix from an oblique rotation tells you whether you actually needed oblique rotation; if all factor correlations are small (\u003C 0.20–0.30), varimax would have given similar results.",{"type":43,"tag":52,"props":923,"children":924},{},[925,930,932,937],{"type":43,"tag":56,"props":926,"children":927},{},[928],{"type":49,"value":929},"Why do my factor loadings change when I add or remove variables?",{"type":49,"value":931},"\nFactor analysis is a ",{"type":43,"tag":56,"props":933,"children":934},{},[935],{"type":49,"value":936},"data-driven technique",{"type":49,"value":938}," — the extracted factors represent the common variance in the specific set of variables you include. Adding or removing variables changes the covariance structure and can alter both the number and interpretation of extracted factors. This is why EFA solutions from different studies rarely replicate perfectly: different item pools capture different aspects of a domain. This sensitivity to item selection is also why CFA on an independent sample is important — it tests whether the factor structure generalizes beyond the original development sample. Always report your exact item pool and sample characteristics so others can evaluate generalizability.",{"title":7,"searchDepth":940,"depth":940,"links":941},2,[942,943,944,945,946,947,948,949],{"id":46,"depth":940,"text":50},{"id":145,"depth":940,"text":148},{"id":209,"depth":940,"text":212},{"id":381,"depth":940,"text":384},{"id":537,"depth":940,"text":540},{"id":700,"depth":940,"text":703},{"id":788,"depth":940,"text":791},{"id":831,"depth":940,"text":834},"markdown","content:tools:073.factor-analysis.md","content","tools/073.factor-analysis.md","tools/073.factor-analysis","md",{"loc":4},1775502472518]