Is KMO a significance test?

Its usual output is an index; it does not supply the conventional p-value associated with Bartlett's test.

Does a high result select the number of factors?

No. Factor retention requires additional analysis.

Can the overall result hide an item problem?

Yes. Inspect variable-level adequacy and substantive item behavior as well.

AI Glossary / Statistics and probability / Intermediate

Kaiser - Meyer - Olkin test

Definition

The Kaiser - Meyer - Olkin measure, commonly called the KMO test, is a diagnostic index used when assessing whether a correlation matrix is suitable for factor analysis. It compares squared correlations between variables with squared partial correlations after controlling for the other variables. Relatively small partial correlations lead to a higher index, suggesting a pattern that may be usefully summarized by common factors. Despite its common name, the usual KMO result is an adequacy measure rather than a conventional hypothesis test with a p-value.

What it means

Two questionnaire items can correlate partly because both relate to a broader shared construct. If their relationship becomes much smaller after accounting for the remaining items, that is consistent with a shared pattern of relationships across the set. KMO summarizes this contrast. It is often described as a measure of sampling adequacy, but it is not a direct calculation of the number of respondents required. It concerns the correlation structure observed for the selected variables. It cannot by itself establish the validity of a questionnaire, the number of factors, or whether a particular fitted factor model is good.

How it works

Let rᵢⱼ be a pairwise correlation and pᵢⱼ the partial correlation controlling for all other included variables. The overall index is the sum of rᵢⱼ² divided by that sum plus the sum of pᵢⱼ², excluding diagonal elements. Counting each pair once or twice gives the same ratio if done consistently. Individual measures use the corresponding sums for one variable. Partial correlations can be obtained from an invertible correlation matrix. Inspect the overall index and variable-level results together. Singular or unstable matrices, missing-data handling and the type of correlations used require attention before interpreting the calculation.

Three practical examples

Survey development: examine whether a set of service-experience items has a correlation structure worth investigating through factor analysis.
Measurement review: identify individual items whose variable-level adequacy differs from an acceptable-looking overall index.
Analysis preparation: compare diagnostics before and after correcting a coding error, while avoiding automatic item deletion merely to raise the score. Substantive coverage and measurement quality still matter.

Think of it this way

Think of a group of roads that may share an underlying transport network. Ordinary correlations show which places are connected; partial correlations ask how much direct connection remains after the other places are accounted for. KMO summarizes whether the observed connections appear largely shared or remain unusually pair-specific. The analogy suggests why the measure can help screen a structure, but it cannot tell you the correct map of hidden factors without further analysis.

Worked example

Consider three standardized variables with every pairwise correlation equal to 0.6. This correlation matrix is positive definite: its eigenvalues are 2.2, 0.4 and 0.4. For any pair, controlling for the third variable gives partial correlation (0.6 − 0.6 × 0.6)/√[(1 − 0.6²)(1 − 0.6²)] = 0.24/0.64 = 0.375. Counting each of the three pairs once, the squared-correlation sum is 3 × 0.6² = 1.08, and the squared-partial-correlation sum is 3 × 0.375² = 0.421875. KMO is therefore 1.08/(1.08 + 0.421875) = 64/89, approximately 0.7191. The symmetry makes each individual measure the same. This is a descriptive adequacy result, not a p-value of 0.7191 and not evidence that exactly one factor must be retained.

Illustrative case study

Fictional case study: Brookfield Research examines a customer-support questionnaire. Its overall KMO looks encouraging, but one item's individual measure is noticeably lower. Reviewing the item reveals that it asks about a different service channel used by only part of the sample. The team investigates missing responses and conceptual relevance before deciding how to model it. It then examines factor loadings, residual relationships and stability in additional data. The diagnostic helps direct investigation without replacing the substantive question of what the questionnaire is intended to measure.

Common misconceptions and mistakes

Common mistakes include treating a high KMO as proof of a valid factor model, confusing it with Bartlett's test of sphericity, and interpreting the name as a sample-size guarantee. Conventional verbal labels and cutoff values are heuristics, not universal acceptance rules. Removing items can raise an index while narrowing the construct being measured. The formula is undefined when both sums are zero, as with an exact identity correlation matrix; software may apply special conventions. A near-singular matrix can also make calculations unstable, so a displayed number should not bypass basic data checks.

Business use

Organizations use KMO when preparing employee, customer or product research for exploratory measurement analysis. Report what variables and correlation method were used, how missing data were handled, and whether the matrix was computationally well behaved. Combine the diagnostic with subject-matter reasoning, adequate sampling, factor retention methods and model assessment. If items are changed after inspecting results, document the process and seek validation rather than presenting the resulting score as independent confirmation. A useful measurement system requires more evidence than a favorable preliminary index.

Check your understanding

What does 0.7191 mean in the example?

Answer: It is the ratio of the summed squared correlations to that sum plus the summed squared partial correlations. It describes this matrix's adequacy pattern; it is neither the probability that factor analysis is valid nor the percentage of variance explained by a fitted factor.

Related terms

Learn next

compute individual adequacy measures and compare them with the overall result. Then study factor retention, residual correlations and validation on another sample so that the decision to use a factor model does not depend on one threshold.

Sources

Sources: EFAtools, KMO documentation, https://search.r-project.org/CRAN/refmans/EFAtools/html/KMO.html. REdaS, Kaiser-Meyer-Olkin Statistics, https://search.r-project.org/CRAN/refmans/REdaS/html/KMO.html. IBM, Kaiser-Meyer-Olkin measure for identity correlation matrix, https://www.ibm.com/support/pages/kaiser-meyer-olkin-measure-identity-correlation-matrix. Numerical example and fictional case are original.

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