The analytical process is based on a matrix of correlations between the variables. Valuable insights can be gained from an examination of this matrix. For the factor analysis to be appropriate, the variables must be correlated. In practice, this is usually the case. If the correlations between all the variables are small, factor analysis may not be appropriate. We also expect that variables that are highly correlated with each other would also highly correlate with the same factor or factors
Formal statistics are available for testing the appropriateness of the factor model. Bartlett’s test of sphericity can be used to test the null hypothesis that variables are uncorrelated in the population; in other words, the population correlation matrix is an identity matrix. In an identity.
matrix, all the diagonal terms are I, and all off-diagonal terms are O.The test statistic for sphericity is based on a chi-square transformation of the determinant of the correlation matrix. A large value of the test statistic will favor the rejection of the null hypothesis. If this hypothesis cannot be rejected, then the appropriateness of factor analysis should be questioned. Another useful statistic is the Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy. This index compares the magnitudes of the observed correlation coefficients to the magnitudes of the partial correlation coefficients. Small values of the KMO statistic indicate that the correlations between pairs of variables cannot be explained by other variables and that factor analysis may not be appropriate. Generally, a value greater than 05 is desirable.
The correlation matrix, constructed from the data obtained to understand toothpaste benefits (Table 19.1), is shown in Table 192. There are relatively high correlations among VI (prevention of cavities), V3 (strong gums), and Vs (prevention of tooth decay). We would expect these variables to correlate with the same set of factors. Likewise, there are relatively high correlations among V2(shiny teeth), V4 (fresh breath). and V6(attractive teeth). These variables may also be expected to correlate with the same factors.6
The results of principal component, Elvis are given in Table 19.3. The null hypothesis, that the population correlation matrix j, .Entity matrix, is rejected by Bartlett’s test of sphericity. The approximate chi-square (J(NI<. I; 111.314 with 15 degrees of freedom, which is significant at the 0.05 level. The value of the·KMO statistic (0.660) is also large (> 0.5). Thus, factor analysis may be considered an appropriate technique for analyzing the correlation matrix ofTable 19.2.