Factor Analysis: Parallelism Test

In confirmatory factor analysis, one aspect of evaluating the adequacy of a scale involves examining the relationships with other items on other scales. The internal relationships between items of a scale represent internal consistency, whereas parallelism considers that relationship with items on other scales. When a measurement model is proposed, it should generate a set of relationships among the items of the scale consistent with the expectation of a single scale (or dimension) and designated as internal consistency. The underlying theoretical model also, if adequate, should be able to validate that underlying structure when comparing how that set of relationships is maintained with items on other scales; this set of relationships is considered parallelism.

The term structural equation modeling represents the idea of examining the relationships between constructs or items to test whether or not a particular set of ordered relations fits a predicted pattern. The test compares the predicted relationships to the actual relationships observed to determine whether the discrepancy observed is greater than expected due to random sampling error. If the discrepancy is large (indicated usually by a significant chi-square), the predicted model is considered not to fit. A nonsignificant chi-square result indicates that the predicted model is not inconsistent with the observed data and considered an adequate explanation for the set of relationships. If both the internal consistency for a scale and tests of parallelism are met, along with an adequate reliability, the scale may be considered as validated.

Internal consistency considers the relationship of items within a scale. The prediction of the relationship between items of the same scale is defined by multiplying the factor loadings of the factor on the same scale. Essentially, the argument becomes that the reason for the correlation between the two items is that the measurement or amount of the variable is “caused” by the underlying construct that is under consideration. What is argued in the case of a self-report scale is that the shared semantic space of the essential meaning of the trigger words (assuming a semantic differential) indicates a shared space. In the case of “tight” or closely shared meaning among items, the average correlation between each of the items is high. A scale in which there exists shared semantic space among items, but the interitem correlations are smaller, will reflect lower factor loadings.

Parallelism is defined with a bit more complication. It is argued that two items on separate scales should have a relationship to each other based on the relationship that the scales have to each other. Consider that an individual item is correlated (or loads) on the factor of interest. So, if there exists an item on Factor1 with a factor loading on that factor and then an item on Factor2 with a loading on that factor, there will exist a relationship between the two items. That relationship in a mathematical sense is the product of the following [Page 525]three terms: (a) factor loading of the item on Factor1, (b) factor loading of the item on Factor2, and (c) the correlation between the two

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