Exploratory Factor Analysis: Extraction, Rotation, and How Many Factors to Retain

Exploratory factor analysis (EFA) is a family of methods for recovering a small number of unobserved common factors from the correlations among a larger set of measured variables. Software will run it on almost any correlation matrix and return a clean-looking table. The table is not the analysis. Three decisions made before that table exists determine what it says, and all three are routinely made by accepting a default.

Those decisions are: which extraction method recovers the factors, how many factors are retained, and which rotation is applied to them. This guide covers each as a decision with a defensible answer, not as a menu.

The part most treatments soften: the retention rule that dominates published work — retain every factor with an eigenvalue greater than 1.00 — is the one the methodological literature has been rejecting since the 1960s. It is SPSS’s default, it over-extracts, and the alternative that outperforms it (Horn’s parallel analysis) has been available since 1965. If you read one section here, read the retention section; it is the decision that changes the number of factors you report.

Where EFA sits: two boundaries that are not stylistic

EFA is not principal component analysis

PCA and EFA appear in the same SPSS dialog box (Analyze → Dimension Reduction → Factor, with PCA as the default extraction), which is the single biggest reason they are conflated. They are different models answering different questions.

• PCA analyses total variance. It places 1.00 on the diagonal of the correlation matrix and produces components that are optimally weighted composites of the observed variables. A component is a function of the items. There is no latent variable and no error term. PCA is a data-reduction transform.
• Common factor analysis analyses common variance. It places communality estimates on the diagonal, and models each observed variable as a function of one or more common factors plus a unique factor that absorbs item-specific variance and measurement error. A factor is a hypothesised cause of the items.

The direction of the arrow is the whole distinction. If your research question is “do these twelve items reflect three underlying constructs?”, that is a common-factor question and PCA does not answer it. Fabrigar, Wegener, MacCallum and Strahan (1999, Psychological Methods 4(3), 272–299) made this the lead argument of their review of EFA practice in psychology, and it remains the standard reference for the point.

The practical consequence: because PCA attributes all of an item’s variance to the components, PCA loadings are systematically larger than the corresponding common-factor loadings. The discrepancy is small when communalities are high and the variable set is large, and grows as communalities fall and the number of variables shrinks — which is exactly the situation of a short, newly written scale, the case where EFA is most often used. A PCA solution on a 10-item pilot scale with modest communalities will look tidier than the truth.

EFA is not confirmatory factor analysis

In EFA every item is free to load on every factor; the factor structure is an output. In confirmatory factor analysis the pattern of which item loads on which factor is specified in advance as a hypothesis, most cross-loadings are fixed to zero, and the model is tested against the data. CFA can be wrong in a way EFA cannot: it can fail.

That ordering matters for what you can claim. EFA on a sample, followed by CFA on the same sample, is not confirmation — the structure being “confirmed” was estimated from those data. A real confirmatory test needs an independent sample or a held-out split. Downstream statistics inherit this: average variance extracted (AVE) and composite reliability are computed from CFA standardised loadings, and the SEM fit indices used to evaluate a measurement model apply to the confirmatory side. EFA sits upstream of both, and it produces no global fit test of the kind those indices provide.

Before you extract: is the matrix factorable?

Two diagnostics are conventionally reported, and both are weak gates rather than evidence of a good analysis.

• Bartlett’s test of sphericity tests whether the correlation matrix is an identity matrix. A significant result means only that some correlations differ from zero. With a realistic sample size it is significant almost regardless. A non-significant Bartlett’s test is informative (stop); a significant one is not.
• The Kaiser–Meyer–Olkin (KMO) measure of sampling adequacy compares the magnitude of observed correlations to that of partial correlations. Kaiser’s own descriptive labels are still the ones in use: values in the 0.90s “marvellous”, 0.80s “meritorious”, 0.70s “middling”, 0.60s “mediocre”, 0.50s “miserable”, below 0.50 “unacceptable”. Treat 0.60 as a floor, and read the item-level KMO values on the anti-image diagonal, not just the overall figure — a single item with an individual KMO below 0.50 is a candidate for removal, and the overall value can hide it.

On sample size, discard the fixed rules of thumb (N = 300, 10 subjects per item, and the rest). MacCallum, Widaman, Zhang and Hong (1999, Psychological Methods 4(1), 84–99) showed that the required N is not a constant or a fixed ratio to the number of items: it depends on communality level and on the number of indicators per factor (overdetermination). Where communalities are consistently high (around 0.60–0.70) and each factor is defined by several items, samples well under 200 can recover the structure adequately. Where communalities are low (around 0.30–0.40) and factors are defined by only three items, several hundred cases may still be insufficient. The honest reporting move is to state your observed communalities and factor overdetermination, not to cite a ratio.

Also check that your data can bear a Pearson correlation matrix at all. Ordinal items with few response categories — a five-point Likert scale, and worse for binary items — produce attenuated Pearson correlations, which can manufacture spurious “difficulty factors” that split items by response distribution rather than by content. The correct handling is a polychoric correlation matrix with an estimator suited to it, not Pearson correlations on integer codes.

Decision 1: Extraction method

Given that you want a common-factor model rather than components, two extraction methods cover nearly all real cases.

• Maximum likelihood (ML). Assumes multivariate normality of the observed variables. In exchange it provides a likelihood-ratio chi-square test of the m-factor model, standard errors for loadings, and a route into the wider latent-variable framework. If the normality assumption is tenable, ML is the stronger choice for exactly that reason: it produces inferential statistics the alternatives do not.
• Principal axis factoring (PAF), also called principal factors. Makes no distributional assumption; it iteratively substitutes communality estimates on the diagonal until they stabilise. It is more robust when the normality assumption is clearly violated, and it is less prone than ML to failing outright on a poorly conditioned matrix. It provides no significance test.

Fabrigar et al. (1999) frame the choice in exactly these terms: ML when the distributional assumption is defensible because of what it buys you, principal factors when it is not. This is the recommendation Costello and Osborne (2005) also carry forward in Best Practices in Exploratory Factor Analysis (Practical Assessment, Research & Evaluation, vol. 10, article 7), the most-cited practitioner treatment of these choices.

Two failure modes to recognise in the output:

• Heywood cases — a communality estimated at or above 1.00, or a negative unique variance. This is an improper solution, not a strong item. Usual causes: too many factors extracted, too few indicators per factor, or a sample too small for the model. Do not report it as if it were a result; re-examine the number of factors first.
• Non-convergence under ML on a matrix that PAF handles. This is usually the same underlying problem — over-extraction or an underdetermined factor — announcing itself earlier.

Decision 2: How many factors to retain

This is the decision that changes the answer. Extraction method typically shifts loadings at the second decimal; retaining four factors instead of two changes what the instrument is.

Why the eigenvalue-greater-than-one rule is the wrong default

The rule — retain every factor whose eigenvalue exceeds 1.00 — is usually credited to Kaiser (1960, Educational and Psychological Measurement 20(1), 141–151), building on Guttman’s (1954) result that the number of eigenvalues above unity is a lower bound for the number of components in the population correlation matrix. Three problems follow directly from that sentence:

1. It was derived for the population matrix, and you have a sample. Sample eigenvalues are inflated by sampling error. A criterion that is a lower bound in the population is not a decision rule in a sample.
2. It was derived for components, not common factors. The eigenvalues in question come from a correlation matrix with 1.00 on the diagonal. If you are running common factor analysis, the eigenvalues that matter come from the reduced matrix with communalities on the diagonal, and they are smaller. Applying a component-based threshold to a factor model is a category error — and it is what most software reports by default, because the eigenvalue table is generated before extraction.
3. The cut-point is a discontinuity with no substantive meaning. A factor with an eigenvalue of 1.01 is retained and one at 0.99 is discarded, though nothing distinguishes them. Because the number of eigenvalues above 1.00 tends to scale with the number of variables, the rule also systematically over-extracts on longer instruments, producing minor factors defined by two or three items that do not replicate.

The rule survives for reasons that have nothing to do with its merits: it is the default in SPSS, it is deterministic, and it requires no additional software. Its main competitor requires either R, or — in SPSS — an external syntax macro, of the kind O’Connor (2000, Behavior Research Methods, Instruments, & Computers 32(3), 396–402) published precisely because the procedures were not built in. A tooling gap, not a methodological argument, is why the weakest rule is the most-used one.

Parallel analysis (Horn, 1965)

Horn’s parallel analysis (Psychometrika 30(2), 179–185; doi:10.1007/BF02289447) fixes the specific defect above: it supplies the missing comparison against sampling error. The mechanism is simple enough to state in three steps.

1. Generate many random datasets — the standard implementations use several hundred to a few thousand — with the same number of cases and the same number of variables as your real data, but with no underlying structure (either random normal data or a random permutation of your own columns, which preserves the marginal distributions).
2. Extract eigenvalues from each random dataset and take, for each ordinal position, the mean or an upper percentile across replications. Glorfeld (1995) argued for the 95th percentile rather than the mean, on the grounds that the mean still retains roughly half the factors that arise from chance at that position; the 95th percentile is now the more defensible default and is what most implementations offer.
3. Retain a factor at each position where the observed eigenvalue exceeds the corresponding random-data eigenvalue, and stop at the first position where it does not.

The logic is a null comparison: a factor is worth retaining only if it explains more variance than a factor extracted from noise of the same shape would. That is what the eigenvalue-over-1.00 rule never asks.

One implementation detail that is frequently got wrong: run parallel analysis on the same kind of eigenvalues you intend to extract. Parallel analysis performed on PCA eigenvalues and then used to choose the number of common factors reintroduces the mismatch described above and tends to under-extract relative to a factor-based parallel analysis. In R, psych::fa.parallel() reports both the component and factor solutions in the same output — read the one that matches your extraction. JASP and jamovi both expose parallel analysis directly in their EFA dialogs (see our comparison of jamovi and JASP), which is the cheapest way out of the SPSS macro problem.

Velicer’s MAP test (1976)

Velicer’s minimum average partial test (Psychometrika 41(3), 321–327) approaches the question from the opposite direction. It partials out components one at a time and, after each step, computes the average of the squared partial correlations among the residuals. That average falls while systematic common variance is still being removed and rises once the procedure starts removing unique variance. The number of components retained is the step at which the average squared partial correlation reaches its minimum.

MAP and parallel analysis fail in different directions, which is what makes them useful together: MAP tends toward under-extraction, parallel analysis toward slight over-extraction, particularly where factors are strongly correlated. When they agree, the answer is well supported. When they disagree, the interval between them is the range worth examining substantively.

The scree plot (Cattell, 1966)

Cattell’s scree test plots eigenvalues against factor number and retains the factors above the “elbow” where the curve levels into scree. Its weakness is that it is a judgement call with no rule for ambiguous plots, and inter-rater agreement on real data is not high. Its strength is that it is the one method that shows you the shape of the whole eigenvalue sequence rather than a single number, which is how you notice a plot with two plausible elbows — a genuine finding about the data, not a nuisance.

The rule that actually holds

Run parallel analysis and MAP, look at the scree plot, and then examine the candidate solutions substantively. Retention is not a purely statistical decision, and no source that has looked at it carefully claims otherwise. A four-factor solution in which the fourth factor is defined by two items with loadings of 0.34 and 0.31 and no interpretable content is worse than the three-factor solution, whatever the eigenvalues said. Extract a range around the statistical indications, rotate each, and report the basis for the one you keep.

Illustrative comparison of the three rules

The following eigenvalues are an illustrative composite, constructed for this page to show how the rules diverge on a single dataset — they are not from a real study.

Factor	Observed eigenvalue	95th-percentile random eigenvalue	Kaiser (>1.00)	Parallel analysis
1	4.81	1.42	Retain	Retain
2	2.36	1.29	Retain	Retain
3	1.34	1.19	Retain	Retain
4	1.09	1.11	Retain	Stop
5	1.02	1.04	Retain	—
6	0.87	0.97	Stop	—

Kaiser’s rule returns five factors; parallel analysis returns three. Factors 4 and 5 clear 1.00 but do not clear the eigenvalues that random data of the same size produce at those positions — which is the definition of a factor you cannot distinguish from noise. This is the ordinary case, not a contrived one: the two rules disagree most often at exactly these marginal positions, and the marginal factors are the ones that fail to replicate.

Decision 3: Rotation

Rotation does not change model fit

This is the most important thing to understand about rotation, and the most commonly misstated. For a fixed number of factors, every rotation of that solution fits the data identically.

The reason is algebraic. The factor model reproduces the correlation matrix as ΛΦΛ′ + Ψ, where Λ is the loading matrix, Φ the factor correlation matrix and Ψ the unique variances. A rotation replaces Λ with ΛT for some invertible transformation matrix T, and adjusts Φ correspondingly. The product is unchanged. Consequently the rotated solution has the same communalities, the same unique variances, the same reproduced correlation matrix, the same residuals and the same total common variance explained as the unrotated one. Only the distribution of that variance across the factors changes — and with it, the interpretability of the pattern.

Two things follow. First, you cannot choose a rotation by comparing fit, because there is nothing to compare; a rotation is chosen for interpretability, and interpretability is the only criterion available. Second, and less comfortably, a rotated solution is not more correct than the unrotated one. It is one of infinitely many equivalent representations of the same m-dimensional factor space. This is rotational indeterminacy, and it is a real limit on the strength of any claim EFA supports. It is also precisely what CFA removes, by fixing cross-loadings in advance and thereby identifying a single solution that can be tested.

Orthogonal rotation constrains; oblique rotation does not

Orthogonal rotations — varimax (Kaiser, 1958), quartimax, equamax — keep the factor axes at right angles, which forces the correlation between every pair of factors to exactly zero. Oblique rotations — direct oblimin, promax, geomin — allow the axes to tilt, so factors may correlate.

Varimax is the most-used rotation in published EFA, and its popularity is again largely a default effect. But orthogonality is not a neutral setting. Fixing every inter-factor correlation at zero is a substantive constraint asserting that the underlying constructs are uncorrelated in the population. In most social-science and psychological applications that assertion is implausible on its face: the facets of a personality construct, the dimensions of job satisfaction, the sub-scales of a depression inventory and the components of organisational climate all correlate, sometimes strongly. An orthogonal rotation imposed on correlated factors distorts the loadings, spreading variance that belongs to a shared source across several factors and typically producing more cross-loadings, not fewer.

The argument for making oblique rotation the default is not that constructs always correlate. It is that oblique rotation is the choice that does not cost you anything if you are wrong:

• If the factors genuinely are uncorrelated, an oblique rotation will return a factor correlation matrix with near-zero off-diagonal entries, and its pattern matrix will closely reproduce the orthogonal solution. You lose nothing but a line of output.
• If the factors are correlated and you rotated orthogonally, you have imposed a false constraint, distorted the loading pattern, and thrown away the factor correlations — which are usually a substantively interesting result in themselves, and are exactly the quantities a later structural model will need.

The asymmetry is the whole argument: one choice is robust to being wrong and the other is not. Fabrigar et al. (1999) and Costello and Osborne (2005) both make oblique rotation the recommendation on essentially this reasoning. The practical rule that follows: rotate obliquely, inspect the factor correlation matrix, and report it. If the correlations are trivially small, say so — that is an empirical finding about your constructs, and it is a much better justification for a simple structure than having assumed it.

Pattern matrix versus structure matrix

Oblique rotation produces two loading matrices, and confusing them is a common reporting error.

• The pattern matrix holds the standardised regression weights of each item on each factor, controlling for the other factors — the unique contribution of that factor to that item.
• The structure matrix holds the simple correlations between items and factors, which include variance shared through the correlations between factors.

When factors correlate substantially, the structure matrix shows high loadings almost everywhere, because every item correlates with every factor partly through the factor correlations. It is not evidence of cross-loading. Interpret and report the pattern matrix, alongside the factor correlation matrix. Under orthogonal rotation the two matrices are identical, which is why the distinction never arises with varimax and catches people out the first time they rotate obliquely.

Reading the rotated solution

• Loading magnitude. Common practice treats 0.32 as a floor for interpretation (about 10% shared variance), 0.40 as a working minimum and 0.50 or above as strong. These are conventions, not tests. State the threshold you used before you look at the table, not after.
• Communalities. The proportion of an item’s variance explained by the retained factors. Items with communalities below about 0.20–0.30 contribute little and are candidates for removal — but remove them one at a time and re-run, because each removal changes the solution.
• Cross-loadings. An item loading above your threshold on two or more factors is not measuring one thing cleanly. The frequently used decision rule is to drop an item whose two highest loadings are within about 0.10–0.15 of each other. This is a judgement, and it should be reported as one.
• Factor overdetermination. A factor defined by fewer than three items with strong loadings is under-identified and unstable; treat it as provisional at best. Three is a minimum, four or five is a working target.
• Do not read reliability off the loadings. Internal consistency is a separate question — see Cronbach’s alpha and when to use omega instead — and a clean factor solution does not by itself establish either reliability or validity.

What to report

A methods section that lets a reader evaluate the analysis, rather than just observe its conclusion, states all of the following:

1. Sample size, and the item-level data type (continuous, ordinal) with the correlation matrix used (Pearson or polychoric).
2. KMO (overall, and any problematic item-level values) and Bartlett’s test.
3. Extraction method, named explicitly — “principal axis factoring”, not “factor analysis” — and the reason for it.
4. The retention criteria used, all of them, including where they disagreed. If parallel analysis and MAP returned different numbers, say what each returned and why you chose as you did.
5. The rotation, named, with its family (oblique or orthogonal) and the justification. If oblique, the factor correlation matrix.
6. The pattern matrix, with all loadings shown (not only those above a threshold), the interpretation threshold used, and the communalities.
7. The variance explained, and any items removed with the criterion that removed them.

The reason to report retention criteria that disagreed is that this is the single most consequential and most arbitrary point in the procedure. A methods section that reports only the final number has removed the reader’s ability to assess it.

The errors that recur

• Running PCA and calling it factor analysis. The most common single error, and it starts with an unchanged default dialog.
• Accepting eigenvalue > 1.00. Over-extracts; produces minor factors that do not replicate.
• Varimax by default on correlated constructs. Imposes a false constraint and discards the factor correlations.
• Interpreting the structure matrix as though it were the pattern matrix. Manufactures apparent cross-loadings.
• Treating a rotated solution as more true than an unrotated one. Rotation changes interpretability, not fit.
• Calling EFA followed by CFA on the same sample “confirmation”. It is not; the structure was estimated from those data.
• Naming a factor and then reasoning from the name. The label is a summary of the loading pattern, not evidence about it.

Frequently asked questions

What is the difference between EFA and PCA?

EFA models common variance and treats factors as latent causes of the observed items, with a unique-variance term for each item. PCA analyses total variance and produces components that are weighted composites of the items, with no latent variable and no error term. PCA is a data-reduction technique, not factor analysis. Use EFA when the question is about underlying constructs.

How many factors should I retain?

Use parallel analysis (Horn, 1965) as the primary criterion, corroborate it with Velicer’s MAP test and the scree plot, and check the candidate solutions for interpretability. Do not use the eigenvalue-greater-than-one rule as a decision rule — it was derived as a lower bound for the population correlation matrix and over-extracts in samples.

Should I use varimax or oblimin?

Use an oblique rotation such as direct oblimin or promax unless you have a specific theoretical reason to require uncorrelated factors. If the factors turn out to be uncorrelated, the oblique solution reproduces the orthogonal one, so oblique rotation costs nothing when it is unnecessary and prevents a real distortion when orthogonality is false.

Does rotation change how well the model fits?

No. For a fixed number of factors, all rotations fit the data identically: communalities, unique variances, the reproduced correlation matrix and the total common variance explained are all unchanged. Rotation redistributes variance across factors to make the pattern interpretable. It cannot be chosen or defended on fit grounds.

What is a good factor loading?

Conventionally 0.32 is a floor for interpretation, 0.40 a working minimum and 0.50 or higher strong. These are conventions rather than statistical tests, and the threshold should be stated in advance and applied consistently.

What sample size do I need for EFA?

There is no valid fixed number or subject-to-item ratio. MacCallum et al. (1999) showed that adequate N depends on communality level and on how many indicators define each factor: with high communalities and well-determined factors, samples below 200 can suffice; with low communalities and three-indicator factors, several hundred cases may not. Report your communalities rather than a ratio.

Can I run EFA on Likert items?

Yes, but not on a Pearson correlation matrix of the integer codes if the items have few categories. Pearson correlations between coarse ordinal items are attenuated and can produce spurious factors that group items by response distribution. Use a polychoric correlation matrix with an appropriate estimator.

Sources and a note on one of them

• Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30(2), 179–185. doi:10.1007/BF02289447
• Velicer, W. F. (1976). Determining the number of components from the matrix of partial correlations. Psychometrika, 41(3), 321–327.
• Fabrigar, L. R., Wegener, D. T., MacCallum, R. C., & Strahan, E. J. (1999). Evaluating the use of exploratory factor analysis in psychological research. Psychological Methods, 4(3), 272–299.
• MacCallum, R. C., Widaman, K. F., Zhang, S., & Hong, S. (1999). Sample size in factor analysis. Psychological Methods, 4(1), 84–99.
• Kaiser, H. F. (1960). The application of electronic computers to factor analysis. Educational and Psychological Measurement, 20(1), 141–151.
• Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, 1(2), 245–276.
• O’Connor, B. P. (2000). SPSS and SAS programs for determining the number of components using parallel analysis and Velicer’s MAP test. Behavior Research Methods, Instruments, & Computers, 32(3), 396–402.
• Costello, A. B., & Osborne, J. W. (2005). Best practices in exploratory factor analysis: four recommendations for getting the most from your analysis. Practical Assessment, Research & Evaluation, 10, article 7.

A disclosure about the Costello and Osborne citation. That article reports a survey of EFA practice in the published literature, and its finding — that the dominant combination in use was principal components extraction with varimax rotation and eigenvalue-greater-than-one retention — is the empirical basis for the claim on this page that the weakest options are the most common ones. At the time of writing we were unable to retrieve the article’s full text directly: the publisher’s site (Practical Assessment, Research & Evaluation, University of Massachusetts Amherst) returned a bot-protection interstitial and the legacy repository copy returned HTTP 403. We have therefore stated that finding qualitatively and have not reproduced any specific percentages from it. If you need the exact figures, take them from the article itself rather than from a secondary source; paraphrases of those numbers circulate widely and are not consistent with one another.

For related quantitative-methods material on this site, see the research methods pillar, and in particular confirmatory factor analysis, average variance extracted, SEM fit indices and multicollinearity and VIF.