KMO and Bartlet’s Test in Factor Analysis

Question:

How to use KMO and Bartlett's Test to Check Whether or not the Factor Analysis can be applied to my Data? 

 Answer:

Kaiser-Meyer-Olkin measure of sampling adequacy and Bartlett's test of sphericity are very important measures to conclude the worthiness of factor analysis. KMO takes values between 0 and 1. A value of 0 indicates that the sum of partial correlations is large relative to the sum of correlations, indicating diffusion in the pattern of correlations and the factor analysis is not appropriate to be conducted. A value close to 1 indicates that patterns of correlations are relatively compact and so factor analysis should yield distinct and reliable factors. 

In other words, KMO indicates the amount of variance shared among the items designed to measure a latent variable when compared to that shared with the error. Kaiser (1974) recommends accepting values greater than 0.5 as acceptable. More specifically, values between 0.5 and 0.7 are considered mediocre, values between 0.7 and 0.8 are considered good, values between 0.8 and 0.9 are deemed great and values above 0.9 are superb (Hutcheson and Sofroniou, 1999). A value more than 0.7 is the common threshold for confirmatory analysis (Hair et al., 2010).

Before being able to run the factor analysis, one should ensure that the data has an adequate level of multicolinearity, the multicolinearity issue is not desirable in regression analysis but it is a prerequisite here. Bartlett's measure tests the null hypothesis that the original correlation matrix is an identity matrix.

H₀:The Correlation Matrix= I(Identity Matrix)

H₁: The Correlation Matrix≠ I(Identity Matrix)

The identity matrix is the matrix in which all the diagonal elements are ones and the off diagonal elements are zeros. Meaning that there original data has no correlations among its variables.

Factor analysis cannot be performed on the data for which the correlation matrix is the identity matrix. Therefore, we want this test to be significant (i.e. has a significance value less than 0.05). If the P value is less than 0.05 we have to reject the null hypothesis thus there are some relationships between the variables we considered in the analysis.