Factor analysis is a general name denoting a class of procedures primarily used for data reduction and summarization. It is used in the following circumstances: •To identify underlying dimensions or factors, that explains the correlations among the set of variables. •To identify new, smaller set of uncorrelated variables to replace the original set of correlated variables in subsequent multivariate analysis. •To identify smaller set of salient variables from a larger set for use in subsequent multivariate analysis.
In our research, we are using factor analysis to find out a smaller set of uncorrelated factors to replace the original correlated variables like fear of side effects, packaging, brand image, quality, availability, ingredients, promotion, income group, frequency of purchase, color cosmetics, peer influence, price, fragrance, brand loyalty, age. . We have in all 15 independent variables. We are doing principal component analysis to determine the minimum number of factors that will account for maximum variance in the data for use in the subsequent multi variant analysis.KMO and Bartlett’s test of sphericity Bartlett’s test of sphericity can be used to test the null hypothesis that the variables are uncorrelated in the population or the population correlation matrix is an identity matrix .
The test statistics for sphericity is base in a Chi square transformation of the determinant of the correlation matrix. A large value of test statistic will favor the rejection of null hypothesis and hence will favor factor analysis. Its significance value should be less than 0. 005 to favor Factor analysis.The KMO statistic measures sampling adequacy . This index compares the magnitudes of the observed correlation coefficients to the magnitudes of partial correlation coefficients.
Small values of the KMO statistic indicate that the correlations between the pair of variable s cannot be explained by other variables and that factor analysis may not be appropriate. Generally, a value greater than 0. 5 is desirable. KMO and Bartlett’s Test Kaiser-Meyer-Olkin Measure of Sampling Adequacy.. 621 Bartlett’s Test of SphericityApprox.
Chi-Square562. 91 df105 Sig.. 000 Here in our test the value of KMO statistics is . 621 which sufficient enough to perform Factor analysis.
Also, the significance value in Bartlett’s test of sphericity is 0. 000 which indicates that correlation matrix is significantly different from identity matrix. This means that the variables are correlated highly enough to provide a reasonable basis for factor analysis. As per our analysis the approx chi square value is 562. 491 and degree of freedom is 105. This shows that data is fit for factor analysis.communalities Communalities is the squared multiple correlation for the variable as dependent using the factors as predictors. The communality measures the percent of variance in a given variable explained by all the factors jointly and may be interpreted as the reliability of the indicator. Communality for a variable is computed as the sum of squared factor loadings for that variable (row). Since factors are uncorrelated, the squared loadings may be added to get the total percent explained which is what communality is.For Principal Component Analysis the initial communality will be 1. 0 for all variables and all of the variance in the variables will be explained by all of the factors, which will be as many as there are variables.
The “extracted” communality is the percent of variance in a given variable explained by the factors which are extracted, which will usually be fewer than all the possible factors, resulting in coefficients less than 1. 0. When an indicator variable has a low communality (