Factor Analysis: Internal Consistency

Test Method

Consider a scale measuring the credibility of communicator. The items might involve the assessment of the “expertise,” “competence,” “trust,” and “honesty” of the source of the message. The assumption of a reliable scale involving a single dimension implies a set of relationships between the individual items of the scale. The issue is that the relationships among the items are determined by the relationship with the scale. Consider the diagram set forth in Figure 1. The underlying “cause” becomes the reason that the items have any variability in common. The argument for correlation between any two items should reflect the underlying “cause” or the level of the construct that the respondent indicates in response to the item. The only reason that the items are related mathematically is that they share an underlying cause related to measuring a construct.

Figure 1 Construct to Measure

[Page 520]The implication of the underlying cause is that the factor loadings for Item A and Item B reflect the association of the item with the underlying construct. The particular relationship between Item A and Item B is the correlation between the two items, rAB. The correlation of the two items, rAB, should be equivalent to the multiplication of the two-factor loadings (factor loading of item A × factor loading of item B on the underlying construct).

The mathematical issues of confirmatory factor analysis require an examination of whether the underlying theoretical assumptions match the reported mathematical structure of relationships among the items. The internal consistency test provides an examination of the discrepancy between the actual relationships existing in the data versus the expected or predicted relationship based on the theoretical model proposed. Essentially, examining the difference between the predicted correlation generated by multiplying the factor loading of the two items of interest, and the observed correlation creates the basis for a chi-square statistical examination of the fit between the predicted and actual data generated in the test of the measurement model:

$${\mathrm{\chi }}^2=\frac{\mathrm{\Sigma }{d}^2}{2\cdot {\mathrm{\delta }}^2}.$$

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