ASSUMPTIONS IN FACTOR ANALYSIS

•  The critical assumptions underlying factor analysis are more conceptual than statistical.

Conceptual Issues

• The conceptual assumptions underlying factor analysis relate to the set of variables selected and the sample chosen. 

• A basic assumption of factor analysis is that some underlying structure does exist in the set of selected variables. The presence of correlated variables and the subsequent definition of factors do not guarantee relevance, even if they meet the statistical requirements. 

• It is the responsibility of the researcher to ensure that the observed patterns are conceptually valid and appropriate to study with factor analysis, because the technique has no means of determining appropriateness other than the correlations among variables. 
• For example, mixing dependent and independent variables in a single factor analysis and then using the derived factors to support dependence relationships is inappropriate.


• The researcher must also ensure that the sample is homogeneous with respect to the underlying factor structure. It is inappropriate to apply factor analysis to a sample of males and females for a set of items known to differ because of gender. 

• When the two subsamples (males and females) are combined, the resulting correlations and factor structure will be a poor representation of the unique structure of each group. Thus, whenever differing groups are expected in the sample, separate factor analyses should be performed, and the results should be compared to identify differences not reflected in the results of the combined sample.

Statistical Issues

• From a statistical standpoint, departures from normality, homoscedasticity, and linearity apply only to the extent that they diminish the observed correlations. 
• Only normality is necessary if a statistical test is applied to the significance of the factors, but these tests are rarely used. In fact, some degree of multicollinearity is desirable, because the objective is to identify interrelated sets of variables.
• Assuming the researcher has met the conceptual requirements for the variables included in the analysis, the next step is to ensure that the variables are sufficiently intercorrelated to produce representative factors. As we will see, we can assess this degree of interrelatedness from both overall and individual variable perspectives. 

The following are several empirical measures to aid in diagnosing the factorability of the correlation matrix.

• OVERALL MEASURES OF INTERCORRELATION
• In addition to the statistical bases for the correlations of the data matrix, the researcher must also ensure that the data matrix has sufficient correlations to justify the application of factor analysis. 




• If it is found that all of the correlations are low, or that all of the correlations are equal (denoting that no structure exists to group variables), then the researcher should question the application of factor analysis. 




• If visual inspection reveals no substantial number of correlations greater than .30, then factor analysis is probably inappropriate. The correlations among variables can also be analyzed by computing the partial correlations among variables. 

• If “true” factors exist in the data, the partial correlation should be small, because the variable can be explained by the variables loading on the factors. If the partial correlations are high, indicating no underlying factors, then factor analysis is inappropriate. 




• A high partial correlation is one with practical and statistical significance, and a rule of thumb would be to consider partial correlations above .7 as high. SPSS and SAS provide the anti-image correlation matrix, which is just the negative value of the partial . Larger partial or anti-image correlations are indicative of a data matrix perhaps not suited to factor analysis.






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• Another method of determining the appropriateness of factor analysis examines the entire correlation matrix. The Bartlett test of sphericity, a statistical test for the presence of correlations among the variables, is one such measure. It provides the statistical significance that the correlation matrix has significant correlations among at least some of the variables. The researcher should note, however, that increasing the sample size causes the Bartlett test to become more sensitive in detecting correlations among the variables

• A third measure to quantify the degree of intercorrelations among the variables and the appropriateness of factor analysis is the measure of sampling adequacy (MSA). This index ranges  from 0 to 1, reaching 1 when each variable is perfectly predicted without error by the other variables. The measure can be interpreted with the following guidelines: .80 or above, meritorious; .70 or above, middling; .60 or above, mediocre; .50 or above, miserable; and below .50, unacceptable. The MSA increases as (1) the sample size increases, (2) the average correlations increase, (3) the number of variables increases, or (4) the number of factors decreases. The researcher should always have an overall MSA value of above .50 before proceeding with the factor analysis. If the MSA value falls below .50, then the variablespecific MSA values (see the following discussion) can identify variables for deletion to achieve an overall value of .50

• VARIABLE-SPECIFIC MEASURES OF INTERCORRELATION

• In addition to a visual examination of a variable’s correlations with the other variables in the analysis, the MSA guidelines can be extended to individual variables. 

• The researcher should examine the MSA values for each variable and exclude those falling in the unacceptable range. In deleting variables, the researcher should first delete the variable with the lowest MSA and then recalculate the factor analysis.

• Continue this process of deleting the variable with the lowest MSA value under .50 until all variables have an acceptable MSA value. 

• Once the individual variables achieve an acceptable level, then the overall MSA can be evaluated and a decision made on continuance of the factor analysis.

Tài liệu gốc:

• Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2019). Multivariate data analysis (8th ed.). Boston: Cengage.