KMO and Bartlett’s test: what they are, how to read them and how to report them

If your study includes construct validity, at some point you will run a factor analysis. Before that, reviewers expect two tests: the KMO (Kaiser-Meyer-Olkin) measure and Bartlett’s test of sphericity. They say nothing about how many factors your scale has. They answer an earlier question: are your items related enough to group into factors?

What the KMO measures

The KMO compares two things: how much the items relate to each other (the correlations) and how much they relate once the effect of all the others is removed (the partial correlations). If the items share something, like a factor, the correlations are high and the partials low, and the KMO gets close to 1.

The formula is KMO = Σr² / (Σr² + Σa²), where r are the correlations between items and a the partial correlations. Besides the overall KMO, each item has its own measure of sampling adequacy (MSA), shown on the diagonal of the anti-image matrix in SPSS.

How to interpret the KMO

The most cited table is Kaiser’s (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 textbooks ask for at least 0.60, and some accept 0.50. Check what approved studies in your department use and cite the author of the cut-off.

What Bartlett’s test measures

Bartlett’s test of sphericity tests a very specific null hypothesis: that the correlation matrix is an identity matrix, meaning no item relates to any other. If that were true, there would be no factors to look for.

It is calculated from the determinant of the correlation matrix: χ² = −[n − 1 − (2p + 5) / 6] × ln|R|, with p(p − 1) / 2 degrees of freedom, where n is the number of people and p the number of items. You need it to be significant: p < .05.

Note: with large samples Bartlett’s test is almost always significant, so on its own it says little. That is why it is reported together with the KMO, which depends less on sample size.

An example

An 8-item scale answered by 40 people gives KMO = 0.72 and Bartlett χ²(28) = 120.04, p < .001. The KMO is middling, above 0.60, and Bartlett’s test is significant: the data are suitable for factor analysis. All 8 items have an MSA above 0.50, so none needs to go. What does stand out is the 40 people: 8 items usually call for at least 40 to 80, so the sample is tight.

What to do if the KMO is low

1. Check each item’s MSA. Items below 0.50 relate little to the rest. Remove the lowest one, recalculate and repeat if needed.
2. Check reverse-worded items. If an item is worded the other way and you didn’t recode it, it lowers the KMO and shows negative correlations.
3. Look at the sample size. With few people per item the KMO tends to be low. 5 to 10 people per item is the usual advice.
4. Check the determinant. If it is almost zero (below 0.00001), some items say almost the same thing or one column is the sum of others. Remove the duplicate.

How to report it

A typical paragraph in the results chapter or the instrument validity section:

The suitability of the data for factor analysis was assessed. The Kaiser-Meyer-Olkin measure was KMO = .72 (middling) and Bartlett’s test of sphericity was significant, χ²(28) = 120.04, p < .001, indicating that the data were suitable for factor analysis.

In APA style, the KMO and the p-value have no leading zero (.72, not 0.72) because they cannot be greater than 1.

The next step: factor analysis

Once KMO and Bartlett pass, the exploratory factor analysis comes next: how many factors appear, which items load on each and how much variance they explain. Ideally the factors match the dimensions you planned.

Then calculate the reliability of each dimension with Cronbach’s alpha or McDonald’s omega. In SurvIQ’s Thesis Mode, your survey responses are ready for those calculations. SurvIQ doesn’t write the thesis: it calculates, lays out the tables and shows what to review.