KMO and Bartlett’s test of sphericity calculator
Before a factor analysis (exploratory or confirmatory) you will be asked for two tests: KMO and Bartlett’s test of sphericity. Both answer the same question: are the items related enough to group into factors? Paste your data from Excel: one row per person and one column per item. If the first row has the question names, it is skipped automatically.
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How it’s calculated
The Pearson correlation matrix of all items is calculated and, from its inverse, the partial correlations (anti-image matrix).
KMO = sum of squared correlations / (sum of squared correlations + sum of squared partial correlations). It ranges from 0 to 1.
Each item’s MSA is the same calculation using only that item’s correlations.
Bartlett: χ² = −[n − 1 − (2p + 5) / 6] × ln|R|, with p(p − 1) / 2 degrees of freedom, where n is the number of people, p the number of items and |R| the determinant of the correlation matrix.
How to read the result
- KMO according to Kaiser (1974): 0.90 or more, marvelous; 0.80 to 0.89, meritorious; 0.70 to 0.79, middling; 0.60 to 0.69, mediocre; 0.50 to 0.59, miserable; below 0.50, unacceptable. Most authors ask for at least 0.60, and some accept 0.50.
- Bartlett’s test must be significant (p < .05): it means the items are not independent of each other and looking for factors makes sense.
- An item with an MSA below 0.50 relates little to the others. Consider removing it and recalculating.
- The determinant should be greater than 0.00001. If it is almost zero, some items are too similar (multicollinearity).
Frequently asked questions
- How many people do I need?
- At least more people than items to calculate it. For factor analysis, 5 to 10 people per item, or 200 in total, are common guidelines. With small samples the KMO comes out lower.
- Are KMO and Bartlett construct validity?
- No. They only tell you whether a factor analysis can be run. Construct validity comes from the factor analysis itself: how many factors appear, which items load on each and how much variance they explain.
- Do I calculate KMO for the whole scale or per dimension?
- With every item that goes into the factor analysis, usually the whole scale. Cronbach’s alpha and omega are calculated per dimension.
- Does it match SPSS?
- Yes. It uses the same formulas as SPSS (Analyze > Dimension Reduction > Factor > Descriptives > KMO and Bartlett’s test of sphericity). With the same data, the results match.
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